Software Engineering GLOSSARY
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Analysis of Functions of a Single Variable a Detailed Development
ANALYSIS OF FUNCTIONS OF A SINGLE VARIABLE A DETAILED DEVELOPMENT LAWRENCE W. BAGGETT University of Colorado OCTOBER 29, 2006 2 For Christy My Light i PREFACE I have written this book primarily for serious and talented mathematics scholars , seniors or first-year graduate students, who by the time they finish their schooling should have had the opportunity to study in some detail the great discoveries of our subject. What did we know and how and when did we know it? I hope this book is useful toward that goal, especially when it comes to the great achievements of that part of mathematics known as analysis. I have tried to write a complete and thorough account of the elementary theories of functions of a single real variable and functions of a single complex variable. Separating these two subjects does not at all jive with their development historically, and to me it seems unnecessary and potentially confusing to do so. On the other hand, functions of several variables seems to me to be a very different kettle of fish, so I have decided to limit this book by concentrating on one variable at a time. Everyone is taught (told) in school that the area of a circle is given by the formula A = πr2: We are also told that the product of two negatives is a positive, that you cant trisect an angle, and that the square root of 2 is irrational. Students of natural sciences learn that eiπ = 1 and that sin2 + cos2 = 1: More sophisticated students are taught the Fundamental− Theorem of calculus and the Fundamental Theorem of Algebra. -
Variables in Mathematics Education
Variables in Mathematics Education Susanna S. Epp DePaul University, Department of Mathematical Sciences, Chicago, IL 60614, USA http://www.springer.com/lncs Abstract. This paper suggests that consistently referring to variables as placeholders is an effective countermeasure for addressing a number of the difficulties students’ encounter in learning mathematics. The sug- gestion is supported by examples discussing ways in which variables are used to express unknown quantities, define functions and express other universal statements, and serve as generic elements in mathematical dis- course. In addition, making greater use of the term “dummy variable” and phrasing statements both with and without variables may help stu- dents avoid mistakes that result from misinterpreting the scope of a bound variable. Keywords: variable, bound variable, mathematics education, placeholder. 1 Introduction Variables are of critical importance in mathematics. For instance, Felix Klein wrote in 1908 that “one may well declare that real mathematics begins with operations with letters,”[3] and Alfred Tarski wrote in 1941 that “the invention of variables constitutes a turning point in the history of mathematics.”[5] In 1911, A. N. Whitehead expressly linked the concepts of variables and quantification to their expressions in informal English when he wrote: “The ideas of ‘any’ and ‘some’ are introduced to algebra by the use of letters. it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics.”[6] There is a question, however, about how to describe the use of variables in mathematics instruction and even what word to use for them. -
A Quick Algebra Review
A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation. -
Leibniz and the Infinite
Quaderns d’Història de l’Enginyeria volum xvi 2018 LEIBNIZ AND THE INFINITE Eberhard Knobloch [email protected] 1.-Introduction. On the 5th (15th) of September, 1695 Leibniz wrote to Vincentius Placcius: “But I have so many new insights in mathematics, so many thoughts in phi- losophy, so many other literary observations that I am often irresolutely at a loss which as I wish should not perish1”. Leibniz’s extraordinary creativity especially concerned his handling of the infinite in mathematics. He was not always consistent in this respect. This paper will try to shed new light on some difficulties of this subject mainly analysing his treatise On the arithmetical quadrature of the circle, the ellipse, and the hyperbola elaborated at the end of his Parisian sojourn. 2.- Infinitely small and infinite quantities. In the Parisian treatise Leibniz introduces the notion of infinitely small rather late. First of all he uses descriptions like: ad differentiam assignata quavis minorem sibi appropinquare (to approach each other up to a difference that is smaller than any assigned difference)2, differat quantitate minore quavis data (it differs by a quantity that is smaller than any given quantity)3, differentia data quantitate minor reddi potest (the difference can be made smaller than a 1 “Habeo vero tam multa nova in Mathematicis, tot cogitationes in Philosophicis, tot alias litterarias observationes, quas vellem non perire, ut saepe inter agenda anceps haeream.” (LEIBNIZ, since 1923: II, 3, 80). 2 LEIBNIZ (2016), 18. 3 Ibid., 20. 11 Eberhard Knobloch volum xvi 2018 given quantity)4. Such a difference or such a quantity necessarily is a variable quantity. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Reverse Engineering Digital Forensics Rodrigo Lopes October 22, 2006
Reverse Engineering Digital Forensics Rodrigo Lopes October 22, 2006 Introduction Engineering is many times described as making practical application of the knowledge of pure sciences in the solution of a problem or the application of scientific and mathematical principles to develop economical solutions to technical problems, creating products, facilities, and structures that are useful to people. What if the opposite occurs? There is some product that may be a solution to some problem but the inner workings of the solution or even the problem it addresses may be unknown. Reverse engineering is the process of analyzing and understanding a product which functioning and purpose are unknown. In Computer Science in particular, reverse engineering may be defined as the process of analyzing a system's code, documentation, and behavior to identify its current components and their dependencies to extract and create system abstractions and design information. The subject system is not altered; however, additional knowledge about the system is produced. The definition of Reverse Engineering is not peaceful though, especially when it concerns to court and lawsuits. The Reverse Engineering of products protected by copyrighting may be a crime, even if no code is copied. From the software companies’ point of view, Reverse Engineering is many times defined as “Analyzing a product or other output of a process in order to determine how to duplicate the know-how which has been used to create a product or process”. Scope and Goals In the Digital Forensics’ scope, reverse engineering can directly be applied to analyzing unknown and suspicious code in the system, to understand both its goal and inner functioning. -
AIX Globalization
AIX Version 7.1 AIX globalization IBM Note Before using this information and the product it supports, read the information in “Notices” on page 233 . This edition applies to AIX Version 7.1 and to all subsequent releases and modifications until otherwise indicated in new editions. © Copyright International Business Machines Corporation 2010, 2018. US Government Users Restricted Rights – Use, duplication or disclosure restricted by GSA ADP Schedule Contract with IBM Corp. Contents About this document............................................................................................vii Highlighting.................................................................................................................................................vii Case-sensitivity in AIX................................................................................................................................vii ISO 9000.....................................................................................................................................................vii AIX globalization...................................................................................................1 What's new...................................................................................................................................................1 Separation of messages from programs..................................................................................................... 1 Conversion between code sets............................................................................................................. -
245533753-MIT.Pdf
THE VULNERABILITY OF TECHNICAL SECRETS TO REVERSE ENGINEERING: IMPLICATIONS FOR COMPANY POLICY By Cenkhan Kodak M.S. in Electrical and Computer Systems Engineering (2001) University of Massachusetts at Amherst Submitted to the Systems Design and Management Program In partial fulfillment of the requirements for the degree of Master of Science in Engineering and Management At the MASSACHUSETTS INSTITUTE OF TECHNOLOGY FEBRUARY 2008 © 2008 Cenkhan Kodak. All rights reserved. The author hereby grants to MIT permission to reproduce and Distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created Signature of the Author: m- /7 Systems Desigq and Management Program r\ Ja iry 2008 Certified by: 7 Professoi ,ric von Hippel Thesis Supervisor, MIT mSchgQl o•.Ma genfer t Certified by: MASSACHUSES INSTITUTE= Pat Hale OF TEOHiNOLOGY Director, Systems Design and Management Program MAY 0 6 2008 I-I .a,:IARCHIVES -2- THE VULNERABILITY OF TECHNICAL SECRETS TO REVERSE ENGINEERING: IMPLICATIONS FOR COMPANY POLICY By Cenkhan Kodak Submitted to the Systems Design and Engineering Program On February 04 2008, in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering and Management Abstract In this thesis I will explore the controversial topic of reverse engineering, illustrating with case examples drawn from the data storage industry. I will explore intellectual property rights issues including, users' fair-use provisions that permit reverse engineering. I will also explore the nature of the practice via several types of analyses including: the costs and benefits of reverse engineering; the practical limitations of reverse engineering; and a layered approach to reverse engineering as it applies to complex systems. -
Lecture Notes
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 1 18.01 Fall 2006 Unit 1: Derivatives A. What is a derivative? • Geometric interpretation • Physical interpretation • Important for any measurement (economics, political science, finance, physics, etc.) B. How to differentiate any function you know. d • For example: �e x arctan x �. We will discuss what a derivative is today. Figuring out how to dx differentiate any function is the subject of the first two weeks of this course. Lecture 1: Derivatives, Slope, Velocity, and Rate of Change Geometric Viewpoint on Derivatives y Q Secant line Tangent line P f(x) x0 x0+∆x Figure 1: A function with secant and tangent lines The derivative is the slope of the line tangent to the graph of f(x). But what is a tangent line, exactly? 1 Lecture 1 18.01 Fall 2006 • It is NOT just a line that meets the graph at one point. • It is the limit of the secant line (a line drawn between two points on the graph) as the distance between the two points goes to zero. Geometric definition of the derivative: Limit of slopes of secant lines P Q as Q ! P (P fixed). The slope of P Q: Q (x0+∆x, f(x0+∆x)) Secant Line ∆f (x0, f(x0)) P ∆x Figure 2: Geometric definition of the derivative Δf f(x0 + Δx) − f(x0) 0 lim = lim = f (x0) Δx!0 Δx Δx!0 Δx | {z } | {z } \derivative of f at x0 " “difference quotient" 1 Example 1. -
From Laptop to Lambda: Outsourcing Everyday Jobs to Thousands of Transient Functional Containers
From Laptop to Lambda: Outsourcing Everyday Jobs to Thousands of Transient Functional Containers Sadjad Fouladi Francisco Romero Dan Iter Qian Li Shuvo Chatterjee+ Christos Kozyrakis Matei Zaharia Keith Winstein Stanford University, +Unaffiliated Abstract build system ExCamera Google Test Scanner (make, ninja) (video encoder) (unit tests) (video analysis) We present gg, a framework and a set of command-line front-ends gg tools that helps people execute everyday applications—e.g., model substitution substitution scriptingscripting SDK SDK C++C++ SDK SDK Python SDKSDK software compilation, unit tests, video encoding, or object recognition—using thousands of parallel threads on a cloud- functions service to achieve near-interactive completion times. IR In the future, instead of running these tasks on a laptop, or gg back-ends keeping a warm cluster running in the cloud, users might gg push a button that spawns 10,000 parallel cloud functions to Lambda/S3 Lambda/RediscomputeOpenWhisk and storageGoogle Cloudengines Functions EC2 Local execute a large job in a few seconds from start. gg is designed to make this practical and easy. Lambda/S3 Lambda/Redis OpenWhisk Google Cloud Functions EC2 Local With gg, applications express a job as a composition of lightweight OS containers that are individually transient (life- Figure 1: gg helps applications express their jobs as a composition times of 1–60 seconds) and functional (each container is her- of interdependent Linux containers, and provides back-end engines metically sealed and deterministic). gg takes care of instantiat- to execute the job on different cloud-computing platforms. ing these containers on cloud functions, loading dependencies, minimizing data movement, moving data between containers, and dealing with failure and stragglers. -
Software Security and Reverse Engineering
Software Security and Reverse Engineering What is reverse engineering? Today the market of software is covered by an incredible number of protected applications, which don't allow you to use all features of programs if you aren't a registered user of these. Reverse engineering is simply the art of removing protection from programs also known as “cracking”. In Some other words cracking is described as follows: - “When you create a program you engineer it, in fact you build the executable from the source-code. The reverse engineering is simply the art of generate a source-code from an executable. Reverse engineering is used to understand how a program does an action, to bypass protection etc. Usually it's not necessary to disassemble all code of the application not only the part of the application that we are interested must be reversed. Reverse engineering used by a cracker to understand the protection scheme and to break it, so it's a very important thing in the whole world of the crack.” In short: - "Reverse Engineering referred to a way to modify a program such that it behaves as the way a reverse engineer wish." “Cracking is a method of making a software program function other than it was Originally intended by means of investigating the code, and, if necessary, patching It.” A Little bit of history Reveres egg. Most probably start with the DOS based computer games. The aim is that a player has full life and armed in the final stage of the game. So what a reverse egg. -
Using Lame's Petrophysical Parameters for Fluid Detection and Lithology Determination in Parts of Niger Delta
DOI: http://dx.doi.org/10.4314/gjgs.v13i1.4 GLOBAL JOURNAL OF GEOLOGICAL SCIENCES VOL. 13, 2015: 23-33 23 COPYRIGHT© BACHUDO SCIENCE CO. LTD PRINTED IN NIGERIA ISSN 1118-0579 www.globaljournalseries.com , Email: [email protected] USING LAME’S PETROPHYSICAL PARAMETERS FOR FLUID DETECTION AND LITHOLOGY DETERMINATION IN PARTS OF NIGER DELTA C. C. Ezeh (Received 3 March 2014; Revision Accepted 8 July 2014) ABSTRACT This study analyzed seismic and well log data in an attempt to interpret relationship between rock properties and acoustic impedance and to construct amplitude versus offset (AVO) models. The objective was to study sand properties and predict potential AVO classification. Logs of Briga 84 well provided compressional and shear velocities. AVO synthetic gather models were created using various gas/brine/oil substitutions. The Fluid Replacement Modeling predicted Class 3 AVO responses from gas sands. Log calibrated Lambda Mu Rho (LMR) inversion provided a quantitative extraction of rock properties to clearly determine lithology and fluids. Beyond the standard LMR cross-plotting that isolated gas sand clusters in the log, an improved separation of high porosity sands from shale was also achieved using λρ - µρ vs. AI. When applied to the calibrated AVO/LMR attributes inverted from 3D seismic, the λρ analysis permitted a better isolation of prospective hydrocarbon zones than standard AI inversion Thus, as an exploration tool for hydrocarbon accumulation, the Lambda Mu Rho (λµρ) technique is a good indicator of the presence of low impedance sands encased in shales which are good gas reservoirs. KEYWORDS: Impedance, synthetic gather, cross-plotting, porosity, inversion INTRODUCTION regional field structure comprises of large collapsed crest rollover anticline trending east – west and bounded The study area is situated in the eastern coastal to the north by the central swamp II depobelt.