# Survey of the Mathematics of Big Data

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Survey of the Mathematics of Big Data Philippe B. Laval KSU September 12, 2014 Philippe B. Laval (KSU) Math & Big Data September 12, / 23

2 Introduction We survey some mathematical techniques used with Big Data. The goal here is to make you aware of these techniques rather than giving you detail about them. That task would take several semesters for each technique. Philippe B. Laval (KSU) Math & Big Data September 12, / 23

3 Outline 1 How Big is Big Data? 2 How Can Mathematics Help Types of Data Mathematics to the Rescue! 3 Acquisition of Data Information Content Compressed Sensing 4 Data Analysis 5 Conclusion High Dimensional Data Imaging Data Philippe B. Laval (KSU) Math & Big Data September 12, / 23

4 How Big is Big Data? We live in a digital world, which generates a lot of data. New technologies produce enormous amount of data. We are gathering more data than ever, even from old technologies. Problem: Acquisition/storage, analysis and transmission of data. Total data generated > total storage. Increase in generation rate >> increase in communication rate. Analysis can be very complex. Problem: Data is noisy, unstructured and dynamic. Philippe B. Laval (KSU) Math & Big Data September 12, / 23

5 How Can Mathematics Help? One answer is given in a video from a previous lecture, remembering that many of the skills mathematicians have are needed in programming. But also, mathematics......allows us to formalize both the data and the problem....provides a big "chest" of tools or methodologies....allows validation of the methodologies (proof of functionality). Philippe B. Laval (KSU) Math & Big Data September 12, / 23

6 How Can Mathematics Help? - Types of Data Signals. Can be represented by a function f : R R Philippe B. Laval (KSU) Math & Big Data September 12, / 23

7 How Can Mathematics Help? - Types of Data Images. Can be represented by a function f : R 2 R Philippe B. Laval (KSU) Math & Big Data September 12, / 23

8 How Can Mathematics Help? - Types of Data Data on manifolds. Can be represented by a function f : S 2 R Philippe B. Laval (KSU) Math & Big Data September 12, / 23

9 Acquisition of Data As noted above, the main problem is the size of the data. However, this cannot be solved by increasing storage capacity. Possible solutions: Search for better ways to compress. Acquire less data. Philippe B. Laval (KSU) Math & Big Data September 12, / 23

10 Acquisition of Data - Better Ways to Compress Data Compression saves storage but also decreases bandwidth. General ideas behind compression: Take advantage of redundant information. Only keep the most relevant information. Einstein: "Not everything that can be counted counts, and not everything hat counts can be counted." Philippe B. Laval (KSU) Math & Big Data September 12, / 23

11 Acquisition of Data - Better Ways to Compress The standard for data compression was JPEG. It is based on the discrete cosine transform (DCT) (see Fourier analysis, Fourier transform, fast Fourier transform (FFT) and the discrete fast Fourier transform (DFFT). It expresses the image in terms of building blocks, the functions cos [ π n ( i ) k ]. Definition The DCT is a linear, invertible function f : R n R n. It transforms the numbers (x 0, x 1,..., x n 1 ) into (X 0, X 1,..., X n 1 ) by the formula n 1 [ ( π X k = x i cos i + 1 ) ] k n 2 i=0 for k = 0, 1,..., n 1 Philippe B. Laval (KSU) Math & Big Data September 12, / 23

12 Acquisition of Data - Better Ways to Compress The latest standard for data compression is JPEG2000. It is similar to JPEG, but it uses wavelets instead of the discrete cosine transform. The branch of mathematics which studies wavelets is called Harmonic Analysis. Like the DCT, wavelets decompose the image in terms of building blocks. But the building blocks are not limited to the cosine function. Both the DCT and wavelets use the divide and conquer method. Decompose an image in terms of its building blocks. Only keep the most relevant building blocks. Philippe B. Laval (KSU) Math & Big Data September 12, / 23

13 Acquisition of Data - Better Ways to Compress Problem: The raw data still has to be stored first Solution: Compressed sensing (Donoho, Candès, Romberg, Tao (2006) Philippe B. Laval (KSU) Math & Big Data September 12, / 23

14 Acquisition of Data - Compressed Sensing Main idea: Goal: Assumption: Only a small percentage of the data to be captured is really relevant (sparse data). Problem: We don t know which. Solution: Use random building blocks. This is pushing the sampling theorem beyond its limits! Goal: To capture a signal with as few points as possible. Thus the raw data will be already compressed. Requires a lot of linear algebra, solving sparse systems. Philippe B. Laval (KSU) Math & Big Data September 12, / 23

15 Applications of Compressed Sensing Business Compression Astronomy Radar Biology Communications Imaging Science Geology Medicine Information theory Remote sensing Optics Philippe B. Laval (KSU) Math & Big Data September 12, / 23

16 Data Analysis There are many techniques, statistical and mathematical which already exist. Problem: The analysis maybe be too complex for current computing power. Data can be noisy, unstructured and dynamic. However, these problems cannot be simply solved by more computing power. Possible solutions include: Graph Theory. TDA (Topological Data Analysis). Philippe B. Laval (KSU) Math & Big Data September 12, / 23

17 Data Analysis - High-Dimensional Data Strategy: Detection of inner structure. Reduction of dimension Examples: A circle can be approximated by lines. A sphere can be approximated by flat surfaces. Philippe B. Laval (KSU) Math & Big Data September 12, / 23

18 Data Analysis - High-Dimensional Data Philippe B. Laval (KSU) Math & Big Data September 12, / 23

19 Data Analysis - High-Dimensional Data Question: How do we identify inner structure? By using topology. The link will open a website. Scroll down to play the video. Philippe B. Laval (KSU) Math & Big Data September 12, / 23

20 Data Analysis - Imaging Data Tasks: Removing Noise. Pattern recognition. Reconstructing missing data.... Tools: Statistics. Fourier and harmonic analysis. Partial differential equations. Compressed sensing. Linear Algebra. Philippe B. Laval (KSU) Math & Big Data September 12, / 23

21 Conclusion Big Data is here to stay. We need to tame it! The techniques presented are fairly diffi cult and require a lot knowledge in various branches of mathematics, statistics and science. Not everybody needs to be a mathematician. But chances are you will work with one. Knowing at least some mathematics will make the communication and the teamwork easier. Questions? Philippe B. Laval (KSU) Math & Big Data September 12, / 23

22 Some Questions 1 Name some new technologies which generate data which were not mentioned during the talk. 2 Give examples of unstructured data not mentioned during the talk. 3 Why is removing noise from data (images as well as other data) important? 4 Give applications of pattern recognition in images. Philippe B. Laval (KSU) Math & Big Data September 12, / 23

23 Online Articles to Read 1 The Mathematical Shape of Things to Come from Quanta Magazine. 2 The Real Secret to Unlocking Big Data? Math. 3 The New Shape of Big Data. 4 New Mathematical Method May Help Tame Big Data from Communications of the ACM. 5 Better Way to Make Sense of Big Data from Science Daily. Philippe B. Laval (KSU) Math & Big Data September 12, / 23

### Math Courses Available After College Algebra. nsm.uh.edu

Math Courses Available After College Algebra This video is designed for: Students transferring into the University of Houston from a community college UH students with college algebra credit I have College

### Image Processing Introduction and Overview

Image Processing Introduction and Overview Prof. Eric Miller elmiller@ece.tufts.edu Fall 2007 EN 74-ECE Image Processing Lecture 1-1 Today s Lecture All the dull administrative stuff first Introduction

### MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

### Consolidation,Work,Group, Final,Report!

Page1of9 ConsolidationWorkGroup FinalReport WorkGroupDetails Date: WorkGroupName: MathOWG8 WorkGroupCo8Chairs: MarkR.AndersonandAndrewMcMorran FunctionalArea: AcademicDegreesandPrograms FunctionalAreaCoordinator:

### If n is odd, then 3n + 7 is even.

Proof: Proof: We suppose... that 3n + 7 is even. that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. that 3n + 7 is even. Since n is odd, there exists an integer k so that

### Allocation of Mathematics Modules at Maynooth to Areas of Study for PME (Professional Masters in Education)

Allocation of Mathematics Modules at Maynooth to Areas of Study for PME (Professional Masters in Education) Module Code Module Title Credits Area 1 Area 2 MT101 Differential Calculus In One Real Variable

### HANDBOOK FOR THE APPLIED AND COMPUTATIONAL MATHEMATICS OPTION. Department of Mathematics Virginia Polytechnic Institute & State University

HANDBOOK FOR THE APPLIED AND COMPUTATIONAL MATHEMATICS OPTION Department of Mathematics Virginia Polytechnic Institute & State University Revised June 2013 2 THE APPLIED AND COMPUTATIONAL MATHEMATICS OPTION

### A Negative Result Concerning Explicit Matrices With The Restricted Isometry Property

A Negative Result Concerning Explicit Matrices With The Restricted Isometry Property Venkat Chandar March 1, 2008 Abstract In this note, we prove that matrices whose entries are all 0 or 1 cannot achieve

### Applied Mathematics and Mathematical Modeling

Applied Mathematics and Mathematical Modeling Joseph Malkevitch, York College (CUNY), Chair Ricardo Cortez, Tulane University Michael A. Jones, Mathematical Reviews Michael O Neill, Claremont McKenna College

### G C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.

### Credit Number Lecture Lab / Shop Clinic / Co-op Hours. MAC 224 Advanced CNC Milling 1 3 0 2. MAC 229 CNC Programming 2 0 0 2

MAC 224 Advanced CNC Milling 1 3 0 2 This course covers advanced methods in setup and operation of CNC machining centers. Emphasis is placed on programming and production of complex parts. Upon completion,

### Image Compression through DCT and Huffman Coding Technique

International Journal of Current Engineering and Technology E-ISSN 2277 4106, P-ISSN 2347 5161 2015 INPRESSCO, All Rights Reserved Available at http://inpressco.com/category/ijcet Research Article Rahul

### Tim Hsu. Updated Fall 2012

Updated Fall 2012 You ve heard about teaching math... A bachelor s degree in math, plus additional training (single-subject credential) qualifies you to teach high school math. A master s degree (2 extra

### ISU Department of Mathematics. Graduate Examination Policies and Procedures

ISU Department of Mathematics Graduate Examination Policies and Procedures There are four primary criteria to be used in evaluating competence on written or oral exams. 1. Knowledge Has the student demonstrated

### Applications to Data Smoothing and Image Processing I

Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is

### CS 591.03 Introduction to Data Mining Instructor: Abdullah Mueen

CS 591.03 Introduction to Data Mining Instructor: Abdullah Mueen LECTURE 3: DATA TRANSFORMATION AND DIMENSIONALITY REDUCTION Chapter 3: Data Preprocessing Data Preprocessing: An Overview Data Quality Major

### Chapter 14. MPEG Audio Compression

Chapter 14 MPEG Audio Compression 14.1 Psychoacoustics 14.2 MPEG Audio 14.3 Other Commercial Audio Codecs 14.4 The Future: MPEG-7 and MPEG-21 14.5 Further Exploration 1 Li & Drew c Prentice Hall 2003 14.1

### 4.1. Title: data analysis (systems analysis). 4.2. Annotation of educational discipline: educational discipline includes in itself the mastery of the

4.1. Title: data analysis (systems analysis). 4.4. Term of study: 7th semester. 4.1. Title: data analysis (applied mathematics). 4.4. Term of study: 6th semester. 4.1. Title: data analysis (computer science).

### How mathematicians deal with new trends like big data, etc.

How mathematicians deal with new trends like big data, etc. Peter Bühlmann Seminar für Statistik, ETH Zürich New trends: some history Felix Klein (1872): Erlangen program (Vergleichende Betrachtungen über

### 3. Interpolation. Closing the Gaps of Discretization... Beyond Polynomials

3. Interpolation Closing the Gaps of Discretization... Beyond Polynomials Closing the Gaps of Discretization... Beyond Polynomials, December 19, 2012 1 3.3. Polynomial Splines Idea of Polynomial Splines

### Prentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009

Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level

### So which is the best?

Manifold Learning Techniques: So which is the best? Todd Wittman Math 8600: Geometric Data Analysis Instructor: Gilad Lerman Spring 2005 Note: This presentation does not contain information on LTSA, which

### Figure 1: Relation between codec, data containers and compression algorithms.

Video Compression Djordje Mitrovic University of Edinburgh This document deals with the issues of video compression. The algorithm, which is used by the MPEG standards, will be elucidated upon in order

### Mathematics. Mathematics MATHEMATICS. 298 2015-16 Sacramento City College Catalog. Degree: A.S. Mathematics AS-T Mathematics for Transfer

MATH Degree: A.S. AS-T for Transfer Division of /Statistics & Engineering Anne E. Licciardi, Dean South Gym 220 916-558-2202 Associate in Science Degree Program Information The mathematics program provides

### Diablo Valley College Catalog 2014-2015

Mathematics MATH Michael Norris, Interim Dean Math and Computer Science Division Math Building, Room 267 Possible career opportunities Mathematicians work in a variety of fields, among them statistics,

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### Image Compression Effects on Face Recognition for Images with Reduction in Size

Image Compression Effects on Face Recognition for Images with Reduction in Size Padmaja.V.K Jawaharlal Nehru Technological University, Anantapur Giri Prasad, PhD. B. Chandrasekhar, PhD. ABSTRACT In this

### Introduction to the Mathematics of Big Data. Philippe B. Laval

Introduction to the Mathematics of Big Data Philippe B. Laval Fall 2015 Introduction In recent years, Big Data has become more than just a buzz word. Every major field of science, engineering, business,

### The Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better!

The Fourth International DERIVE-TI9/89 Conference Liverpool, U.K., -5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue Notre-Dame Ouest Montréal

### ANALYSIS OF THE EFFECTIVENESS IN IMAGE COMPRESSION FOR CLOUD STORAGE FOR VARIOUS IMAGE FORMATS

ANALYSIS OF THE EFFECTIVENESS IN IMAGE COMPRESSION FOR CLOUD STORAGE FOR VARIOUS IMAGE FORMATS Dasaradha Ramaiah K. 1 and T. Venugopal 2 1 IT Department, BVRIT, Hyderabad, India 2 CSE Department, JNTUH,

### Multimedia Databases. Wolf-Tilo Balke Philipp Wille Institut für Informationssysteme Technische Universität Braunschweig http://www.ifis.cs.tu-bs.

Multimedia Databases Wolf-Tilo Balke Philipp Wille Institut für Informationssysteme Technische Universität Braunschweig http://www.ifis.cs.tu-bs.de 14 Previous Lecture 13 Indexes for Multimedia Data 13.1

### Prerequisite: High School Chemistry.

ACT 101 Financial Accounting The course will provide the student with a fundamental understanding of accounting as a means for decision making by integrating preparation of financial information and written

### Paper 2 Revision. (compiled in light of the contents of paper1) Higher Tier Edexcel

Paper 2 Revision (compiled in light of the contents of paper1) Higher Tier Edexcel 1 Topic Areas 1. Data Handling 2. Number 3. Shape, Space and Measure 4. Algebra 2 Data Handling Averages Two-way table

### Sparsity-promoting recovery from simultaneous data: a compressive sensing approach

SEG 2011 San Antonio Sparsity-promoting recovery from simultaneous data: a compressive sensing approach Haneet Wason*, Tim T. Y. Lin, and Felix J. Herrmann September 19, 2011 SLIM University of British

### Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours

MAT 051 Pre-Algebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT

### The Applied and Computational Mathematics (ACM) Program at The Johns Hopkins University (JHU) is

The Applied and Computational Mathematics Program at The Johns Hopkins University James C. Spall The Applied and Computational Mathematics Program emphasizes mathematical and computational techniques of

### When is missing data recoverable?

When is missing data recoverable? Yin Zhang CAAM Technical Report TR06-15 Department of Computational and Applied Mathematics Rice University, Houston, TX 77005 October, 2006 Abstract Suppose a non-random

### Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price

Math 1314 Lesson 8: Business Applications: Break Even Analysis, Equilibrium Quantity/Price Cost functions model the cost of producing goods or providing services. Examples: rent, utilities, insurance,

### DEPARTMENT of CHEMISTRY AND BIOCHEMISTRY

DEPARTMENT of CHEMISTRY AND BIOCHEMISTRY ACADEMIC GUIDANCE 2013-2014 PROGRAMS B.S. in Chemistry B.A. in Chemistry B.S. in Biochemistry B.S. in Physical Sciences with specialization in Chemistry or Physics

### CS 5614: (Big) Data Management Systems. B. Aditya Prakash Lecture #18: Dimensionality Reduc7on

CS 5614: (Big) Data Management Systems B. Aditya Prakash Lecture #18: Dimensionality Reduc7on Dimensionality Reduc=on Assump=on: Data lies on or near a low d- dimensional subspace Axes of this subspace

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### Computer and Systems Engineering (CSE) Master of Science Programs

Computer and Systems Engineering (CSE) Master of Science Programs The Computer and Systems Engineering (CSE) degree offered by the University of Houston (UH) is a graduate level interdisciplinary program

### SEMESTER PLANS FOR MATH COURSES, FOR MAJORS OUTSIDE MATH.

SEMESTER PLANS FOR MATH COURSES, FOR MAJORS OUTSIDE MATH. CONTENTS: AP calculus credit and Math Placement levels. List of semester math courses. Student pathways through the semester math courses Transition

### Introduction to Medical Imaging. Lecture 11: Cone-Beam CT Theory. Introduction. Available cone-beam reconstruction methods: Our discussion:

Introduction Introduction to Medical Imaging Lecture 11: Cone-Beam CT Theory Klaus Mueller Available cone-beam reconstruction methods: exact approximate algebraic Our discussion: exact (now) approximate

### Mathematics. Pre-Calculus ( -1 Course Sequence)

Mathematics Courses in Mathematics are offered with instruction in English, French (F) and Spanish (S) where enrolment warrants. Note: Five credits at the 20 level are required to obtain an Alberta High

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### Computational Optical Imaging - Optique Numerique. -- Deconvolution --

Computational Optical Imaging - Optique Numerique -- Deconvolution -- Winter 2014 Ivo Ihrke Deconvolution Ivo Ihrke Outline Deconvolution Theory example 1D deconvolution Fourier method Algebraic method

### School of Mathematics, Computer Science and Engineering. Mathematics* Associate in Arts Degree COURSES, PROGRAMS AND MAJORS

Mathematics School of Mathematics, Computer Science and Engineering Dean: Lianna Zhao, MD Academic Chair: Miriam Castroconde Faculty: Miriam Castroconde; Terry Cheng; Howard Dachslager, PhD; Ilknur Erbas

### Geometry Course Summary Department: Math. Semester 1

Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give

### Implementation of the 5/3 Lifting 2D Discrete Wavelet Transform

Implementation of the 5/3 Lifting 2D Discrete Wavelet Transform 1 Jinal Patel, 2 Ketki Pathak 1 Post Graduate Student, 2 Assistant Professor Electronics and Communication Engineering Department, Sarvajanik

### Recommended Courses by ECE Topic Area

Recommended Courses by ECE Topic Area Undergraduate Students: Verify a course is an approved Science/Math/Engineering Elective or Technical Elective for your major. Graduate Students: A maximum of 6 credits

### Sequence of Mathematics Courses

Sequence of ematics Courses Where do I begin? Associates Degree and Non-transferable Courses (For math course below pre-algebra, see the Learning Skills section of the catalog) MATH M09 PRE-ALGEBRA 3 UNITS

### Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks

Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Wavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)

Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2π-periodic square-integrable function is generated by a superposition

### MPEG, the MP3 Standard, and Audio Compression

MPEG, the MP3 Standard, and Audio Compression Mark ilgore and Jamie Wu Mathematics of the Information Age September 16, 23 Audio Compression Basic Audio Coding. Why beneficial to compress? Lossless versus

### International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. XXXIV-5/W10

Accurate 3D information extraction from large-scale data compressed image and the study of the optimum stereo imaging method Riichi NAGURA *, * Kanagawa Institute of Technology nagura@ele.kanagawa-it.ac.jp

### Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary

Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:

### Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

### Compression and Image Formats

Compression Compression and Image Formats Reduce amount of data used to represent an image/video Bit rate and quality requirements Necessary to facilitate transmission and storage Required quality is application

### ANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 2-1

WHAT IS AN FFT SPECTRUM ANALYZER? ANALYZER BASICS The SR760 FFT Spectrum Analyzer takes a time varying input signal, like you would see on an oscilloscope trace, and computes its frequency spectrum. Fourier's

### (2) (3) (4) (5) 3 J. M. Whittaker, Interpolatory Function Theory, Cambridge Tracts

Communication in the Presence of Noise CLAUDE E. SHANNON, MEMBER, IRE Classic Paper A method is developed for representing any communication system geometrically. Messages and the corresponding signals

### Combined BS/MS Degree Program in the Department of Electrical Engineering at Wright State University

Combined BS/MS Degree Program in the Department of Electrical Engineering at Wright State University Approved by: Dept Graduate Studies Committee January 20, 2011 Approved by: EE Dept Faculty February

### Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals. Introduction

Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals Modified from the lecture slides of Lami Kaya (LKaya@ieee.org) for use CECS 474, Fall 2008. 2009 Pearson Education Inc., Upper

### On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information

Finance 400 A. Penati - G. Pennacchi Notes on On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information by Sanford Grossman This model shows how the heterogeneous information

### Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

### Big Ideas in Mathematics

Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards

### This material is intended for use by undergraduate students who are contemplating taking a course with me.

This material is intended for use by undergraduate students who are contemplating taking a course with me. I have pasted, without edits, all Section C comments from my recent online evaluations from SIRS

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### FIELDS-MITACS Conference. on the Mathematics of Medical Imaging. Photoacoustic and Thermoacoustic Tomography with a variable sound speed

FIELDS-MITACS Conference on the Mathematics of Medical Imaging Photoacoustic and Thermoacoustic Tomography with a variable sound speed Gunther Uhlmann UC Irvine & University of Washington Toronto, Canada,

### Graph Based Processing of Big Images

Graph Based Processing of Big Images HUI HAN CHIN DSO NATIONAL LABORATORIES, SINGAPORE 31 OCT 2014 Introduction Member of Technical Staff at Signal Processing Laboratory, Sensors Division, DSO National

### Conceptual similarity to linear algebra

Modern approach to packing more carrier frequencies within agivenfrequencyband orthogonal FDM Conceptual similarity to linear algebra 3-D space: Given two vectors x =(x 1,x 2,x 3 )andy = (y 1,y 2,y 3 ),

### Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula

Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula An equation is a mathematical statement that two mathematical expressions are equal For example the statement 1 + 2 = 3 is read as

### Probability Concepts Probability Distributions. Margaret Priest (margaret.priest@wgc.school.nz) Nokuthaba Sibanda (nokuthaba.sibanda@vuw.ac.

Probability Concepts Probability Distributions Margaret Priest (margaret.priest@wgc.school.nz) Nokuthaba Sibanda (nokuthaba.sibanda@vuw.ac.nz) Our Year 13 students who are most likely to want to continue

### Moving Average Filters

CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average

### EASTERN ARIZONA COLLEGE Differential Equations

EASTERN ARIZONA COLLEGE Differential Equations Course Design 2015-2016 Course Information Division Mathematics Course Number MAT 260 (SUN# MAT 2262) Title Differential Equations Credits 3 Developed by

### PCHS ALGEBRA PLACEMENT TEST

MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If

### MATH 1314 College Algebra Departmental Syllabus

MATH 1314 College Algebra Departmental Syllabus Fall 2010 Catalog Description MATH 1314. College Algebra (3-3-0) This course is a study of solutions to equations and inequalities, algebraic functions and

### The following is a reprint of an interview with

An Interview with Sun-Yung Alice Chang Y K Leong Asia Pacific Mathematics Newsletter Sun-Yung Alice Chang The following is a reprint of an interview with Professor Sun-Yung Alice Chang of Princeton University

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected

### MATHEMATICAL SCIENCES, BACHELOR OF SCIENCE (B.S.) WITH A CONCENTRATION IN APPLIED MATHEMATICS

VCU MATHEMATICAL SCIENCES, BACHELOR OF SCIENCE (B.S.) WITH A CONCENTRATION IN APPLIED MATHEMATICS The curriculum in mathematical sciences promotes understanding of the mathematical sciences and their structures,

### Prerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.

Course Syllabus Math 1314 College Algebra Revision Date: 8-21-15 Catalog Description: In-depth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems

### Welcome to Math 7 Accelerated Courses (Preparation for Algebra in 8 th grade)

Welcome to Math 7 Accelerated Courses (Preparation for Algebra in 8 th grade) Teacher: School Phone: Email: Kim Schnakenberg 402-443- 3101 kschnakenberg@esu2.org Course Descriptions: Both Concept and Application

### Compressive sensing in wireless mobile communication system at high data rate transmission

International Journal of Engineering and Technical Research (IJETR) Compressive sensing in wireless mobile communication system at high data rate transmission Narendra Shukla, Anil Kumar, A. K. Jaiswal,

### Spectrum Level and Band Level

Spectrum Level and Band Level ntensity, ntensity Level, and ntensity Spectrum Level As a review, earlier we talked about the intensity of a sound wave. We related the intensity of a sound wave to the acoustic

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### Department of Mathematics

Department of Mathematics 220 Yost Hall http://www.case.edu/artsci/math Daniela Calvetti, Department Chair daniela.calvetti@case.edu The Department of Mathematics at Case Western Reserve University is

### Derive 5: The Easiest... Just Got Better!

Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering

### Learning mathematics Some hints from the psychologists Examples:

Learning mathematics Some hints from the psychologists In your degree course you will learn, and be examined on: 1. Facts (= knowing that ) 2. Skills (= knowing how ). Examples: An example of a fact to

### How to write a technique essay? A student version. Presenter: Wei-Lun Chao Date: May 17, 2012

How to write a technique essay? A student version Presenter: Wei-Lun Chao Date: May 17, 2012 1 Why this topic? Don t expect others to revise / modify your paper! Everyone has his own writing / thinkingstyle.

### Research Article A Method of Data Recovery Based on Compressive Sensing in Wireless Structural Health Monitoring

Mathematical Problems in Engineering Volume 214, Article ID 46478, 9 pages http://dx.doi.org/1.11/214/46478 Research Article A Method of Data Recovery Based on Compressive Sensing in Wireless Structural

### You know from calculus that functions play a fundamental role in mathematics.

CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.

### Conceptual Framework Strategies for Image Compression: A Review

International Journal of Computer Sciences and Engineering Open Access Review Paper Volume-4, Special Issue-1 E-ISSN: 2347-2693 Conceptual Framework Strategies for Image Compression: A Review Sumanta Lal

### Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture - 2 Simple Linear Regression

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 2 Simple Linear Regression Hi, this is my second lecture in module one and on simple

### NMR Measurement of T1-T2 Spectra with Partial Measurements using Compressive Sensing

NMR Measurement of T1-T2 Spectra with Partial Measurements using Compressive Sensing Alex Cloninger Norbert Wiener Center Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu

### MATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS

* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSA-MAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices

### Signal and Information Processing

The Fu Foundation School of Engineering and Applied Science Department of Electrical Engineering COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK Signal and Information Processing Prof. John Wright SIGNAL AND

### High School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable.

Performance Assessment Task Quadratic (2009) Grade 9 The task challenges a student to demonstrate an understanding of quadratic functions in various forms. A student must make sense of the meaning of relations

### One LAR Course Credits: 3. Page 4

Course Descriptions Year 1 30 credits Course Title: Calculus I Course Code: COS 101 This course introduces higher mathematics by examining the fundamental principles of calculus-- functions, graphs, limits,