Introduction to Systems of Equations and Inequalities

1 IDIC – SANS GIAC LevelTwo ©2000, 2001 1 Network Based Intrusion Detection Tutorial 1 Introduction to the basic approaches and issues of Intrusion Detection Hello! Welcome to the first half of our network based intrusion detection tutorial, where we will introduce you to the basic approaches of intrusion detection. In this section, we will discuss a rule- based analysis process by going through the topics listed on your next slide. At the end of the section we will talk about some of the methods currently used to perform intrusion detection. 2 IDIC - SANS GIAC LevelTwo ©2000, 2001 2 • False positives, False negatives • EOI, dictionary signatures, profile changes • Severity = (criticality + lethality) – countermeasures (system + network) • Long term conditions Before We Begin We will begin our discussion by talking about false positives and false negatives, which are ever present factors in the life of an intrusion analyst. We will then discuss the notion of Events of Interest (EOI), and their relevance to the event analysis process. We will also go over techniques for judging the severity of a particular event. Additionally, we will propose a way to handle long term conditions that might result from a prolonged exposure to attacks. 3 IDIC - SANS GIAC LevelTwo ©2000, 2001 3 Sources of Data All data: observable or not Collectable Events of Interest There are very few situations in which we are able to collect all the data. We need to develop techniques that allow us to routinely locate Events of Interest (EOI) in the data we are able to collect, so that we know where to focus our attention. 4 IDIC - SANS GIAC LevelTwo ©2000, 2001 4 False Positives and Negatives False positives False negatives All Data Real EOI False positives are “false alarms.” The detects match only some of the criteria for indicators of possible intrusion. False positives tend to wear down incident handling resources and make us slower to react in the future. False negatives are the actual intrusions and intrusion attempts that we do not detect. These can allow an adversary to establish a significant presence in our information systems before we begin to react. 5 IDIC - SANS GIAC LevelTwo ©2000, 2001 5 What Are Events Of Interest • Since we can’t collect, store, or analyze all possible events, we focus our collection efforts on stuff that might prove useful, EOI. – Dictionary: known attack signatures, known attackers – Short term significant changes in system or user profile The reality of limited computing and personnel resources is such that we cannot collect, store, and analyze all possible events. Therefore, analysts tend to focus their collection efforts on events that might prove useful – Events of Interest (EOI). Unfortunately, focusing helps reduce the false alarms or false positives, but increases the chance of missing an EOI. One of the ways to help ensure that an EOI is not missed is to compare suspicious events against a dictionary of known attacks or attackers. You can’t afford not to test against a dictionary! Another way to widen our field of vision is to monitor for changes in system or user profiles. Consider the following example: You have noticed an increase in the number of probes and intrusion attempts at your site. DNS and mail relay systems are always high profile targets, so you are Introduction to Systems of Equations and Inequalities Introduction to Systems of Equations and Inequalities By: OpenStaxCollege Enigma machines like this one, once owned by Italian dictator Benito Mussolini, were used by government and military officials for enciphering and deciphering top-secret communications during World War II (credit: Dave Addey, Flickr) By 1943, it was obvious to the Nazi regime that defeat was imminent unless it could build a weapon with unlimited destructive power, one that had never been seen before in the history of the world In September, Adolf Hitler ordered German scientists to begin building an atomic bomb Rumors and whispers began to spread from across the ocean Refugees and diplomats told of the experiments happening in Norway However, Franklin D Roosevelt wasn’t sold, and even doubted British Prime Minister Winston Churchill’s warning Roosevelt wanted undeniable proof Fortunately, he soon received the proof he wanted when a group of mathematicians cracked the “Enigma” code, proving beyond a doubt that Hitler was building an atomic bomb The next day, Roosevelt gave the order that the United States begin work on the same 1/2 Introduction to Systems of Equations and Inequalities The Enigma is perhaps the most famous cryptographic device ever known It stands as an example of the pivotal role cryptography has played in society Now, technology has moved cryptanalysis to the digital world Many ciphers are designed using invertible matrices as the method of message transference, as finding the inverse of a matrix is generally part of the process of decoding In addition to knowing the matrix and its inverse, the receiver must also know the key that, when used with the matrix inverse, will allow the message to be read In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions We will not be breaking any secret codes here, but we will lay the foundation for future courses 2/2 I.D.Chueshov D issipative S ystems Infinite-Dimensional Introduction Theory I. D. Chueshov Introduction to the Theory of InfiniteDimensional Dissipative Systems 966–7021–64–5 ORDER www.acta.com.ua I . D . Chueshov Chueshov Universitylecturesincontemporarymathematics D issipative issipative S ystems ystems of Infinite-Dimensional Infinite-Dimensional Introduction ntroduction Theory Theory to the This book provides an exhau - stive introduction to the scope of main ideas and methods of the theory of infinite-dimensional dis - sipative dynamical systems which has been rapidly developing in re - cent years. In the examples sys tems generated by nonlinear partial differential equations arising in the different problems of modern mechanics of continua are considered. The main goal of the book is to help the reader to master the basic strategies used in the study of infinite-dimensional dissipative systems and to qualify him/her for an independent scien - tific research in the given branch. Experts in nonlinear dynamics will find many fundamental facts in the convenient and practical form in this book. The core of the book is com - posed of the courses given by the author at the Department of Me chanics and Mathematics at Kharkov University during a number of years. This book con - tains a large number of exercises which make the main text more complete. It is sufficient to know the fundamentals of functional analysis and ordinary differential equations to read the book. Translated by Constantin I. Chueshov from the Russian edition («ACTA», 1999) Translation edited by Maryna B. Khorolska Author: I. D. Chueshov I. D. Chueshov Title: Introduction to the Theory Introduction to the Theory of InfiniteDimensional of InfiniteDimensional Dissipative Systems Dissipative Systems ISBN: 966–7021–64–5 You can ORDER ORDER this book while visiting the website of «A CTA» Scientific Publishing House http://www.acta.com.ua www.acta.com.ua /en/ « A CTA » 2002 I. D. Chueshov Introduction IntroductionIntroduction Introduction to the Theory of Infinite-Dimensional to the Theory of Infinite-Dimensionalto the Theory of Infinite-Dimensional to the Theory of Infinite-Dimensional Dissipative Systems Dissipative SystemsDissipative Systems Dissipative Systems A CTA 2002 UDC 517 2000 Mathematics Subject Classification: primary 37L05; secondary 37L30, 37L25. This book provides an exhaustive introduction to the scope of main ideas and methods of the theory of infinite-dimen- sional dissipative dynamical systems which has been rapidly developing in recent years. In the examples systems genera- ted by nonlinear partial differential equations arising in the different problems of modern mechanics of continua are con- sidered. The main goal of the book is to help the reader to master the basic strategies used in the study of infinite-di- mensional dissipative systems and to qualify him/her for an independent scientific research in the given branch. Experts in nonlinear dynamics will find many fundamental facts in the convenient and practical form in this book. The core of the book is composed of the courses given by the author at the Department of Mechanics and Mathematics at Kharkov University during a number of years. This book contains a large number of exercises which make the main text more complete. It is sufficient to know the fundamentals of functional analysis [...]... objected whether an exposition devoted entirely to a single equation (Cauchy’s Functional Equation) and a single inequality (Jensen’s Inequality) deserves the name An introduction to the Theory of Functional Equations and Inequalities However, the Cauchy equation plays such a prominent role in the theory of functional equations that the title seemed appropriate Every adept of the theory of functional equations. .. be different from A, and hence of the form P (a) with an a ∈ A If a ∈ P2 , then P1 is an initial segment of (P2 , ) Indeed, if x ∈ P1 and y < x, then y < a and hence y ∈ P (a) = P1 And if a ∈ P2 , then P2 is an initial segment of (P1 , ) Indeed, then P2 = A and hence of / the form P (b) with a b ∈ A If x ∈ P2 and y < x, then y < b and y ∈ P2 Thus one of the sets P1 , P2 is similar to its initial segment... appropriate to speak about the two classes of functions together Even in such a large book it was impossible to cover the whole material pertinent to the theory of the Cauchy equation and Jensen’s inequality The exercises at the end of each chapter and various bibliographical hints will help the reader to pursue further his studies of the subject if he feels interested in further developments of the theory. .. A and int A Some special letters are used to denote particular sets of numbers And so N denotes the set of positive integers, whereas Z denotes the set of all integers Q stands for the set of all rational numbers, R for the set of all real numbers, and C for the set of all complex numbers The letter N is reserved to denote the dimension of the underlying space The end of every proof is marked by the. .. use of the trichotomy law for the cardinals, which, however, cannot be proved at this stage What we actually prove here is that the m is impossible This is sufficient to prove the remaining theorems inequality α of the present chapter, and then the trichotomy law for the cardinals follows from Theorem 1.4.1 and 1.7.1, and hence also condition (1.5.3) We define Ω to be the order type of the set M (ℵ0 ), and. .. consider some functional equations on groups or related algebraic structures We assume that the reader has a basic knowledge of the calculus, theory of Lebesgue’s measure and integral, algebra, topology and set theory However, for the convenience of the reader, in the first part of the book we present such fragments of those theories which are often left out from the university courses devoted to them Also,... Thus f is of type β and its range γ∈Γ(β) is X In this proof of Zermelo theorem we have used the Axiom of Choice This could not be avoided, the Axiom of Choice had to be used in the proof Actually the Zermelo theorem is equivalent to the Axiom of Choice, as results from the following Theorem 1.7.2 If every set can be well ordered, then for every set there exists a choice function Proof Let X be an arbitrary... the sign Other symbols are introduced in the text, and for Chapter 1: Introduction to Systems Analysis and Design PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved Objectives • Understand the fundamental systems development life cycle and its four phases • Understand the evolution of systems development methodologies • Be familiar with the Unified Process and its extensions • Be familiar with the different roles on the project team PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved Why Should We Care? Would you buy a car that only had a 28% chance of driving off the lot with no problems? PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved Recent Significant IT Failures PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved SYSTEMS DEVELOPMENT LIFE CYCLE PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved Systems Development Life Cycle PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved SDLC: Planning Project Initiation – Develop a system request – Conduct a feasibility analysis Project Management – Develop work plan – Staff the project – Control and direct the project Why should we build this system? PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved SDLC: Analysis Develop analysis strategy Gather requirements Develop a system proposal What should the system for us? Where and when will it be used? PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved SDLC: Design Develop a design strategy Design architecture and interfaces Develop databases and file specifications Develop the program design How will we build the system? PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved SDLC: Implementation Construct system Install system – Implement a training plan for the users Establish a support plan Build the system! PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved THE UNIFIED PROCESS PowerPoint Presentation for Dennis, Wixom, & Tegarden Systems Analysis and Design with UML, 3rd Edition Copyright © 2009 John Wiley & Sons, Inc All rights reserved The Unified Process • A specific methodology that maps out when and how to use the various UML techniques for object-oriented analysis and design • A two-dimensional process consisting of phases and flows – Phases describe how the system evolves over time – Chapter Chapter 10 10 Feasibility Feasibility Analysis Analysis and and the the System System Proposal Proposal McGraw-Hill/Irwin © 2008 The McGraw-Hill Companies, All Rights Reserved 10-2 Objectives • Identify feasibility checkpoints in the systems life cycle • Identify alternative system solutions • Define and describe four types of feasibility and their respective criteria • Perform various cost-benefit analyses using time-adjusted costs and benefits • Write suitable system proposal reports for different audiences • Plan for a formal presentation to system owners and users 10-3 Feasibility Analysis Feasibility – the measure of how beneficial or practical an information system will be to an organization Feasibility analysis – the process by which feasibility is measured Creeping Commitment – an approach to feasibility that proposes that feasibility should be measured throughout the life cycle 10-4 Feasibility Checkpoints During Systems Analysis 10-5 Four Tests For Feasibility Operational feasibility – a measure of how well a solution meets the system requirements Technical feasibility – a measure of the practicality of a technical solution and the availability of technical resources and expertise Schedule feasibility – a measure of how reasonable the project timetable is Economic feasibility - a measure of the costeffectiveness of a project or solution 10-6 Operational Feasibility • How well proposed system solves the problems and takes advantage of opportunities identified during the scope definition and problem analysis phases • How well proposed system satisfies system requirements identified in the requirements analysis phase • Is the problem still worth solving? 10-7 Technical feasibility • Is the proposed technology or solution practical? • Do we currently possess the necessary technology? • Do we possess the necessary technical expertise? 10-8 Schedule feasibility • Are specified deadlines mandatory or desirable? • Are mandatory deadlines realistic for proposed solution? 10-9 Economic feasibility • During Scope Definition – Do the problems or opportunities warrant the cost of a detailed study and analysis of the current system? • During Problem Analysis – After a detailed study of the current system – Better estimates of development costs and benefits • During Decision Analysis – Requirements now defined – Development costs can be better estimated 10-10 Information System Costs • Development costs - one time costs that will not recur after the project has been completed – – – – – Personnel Computer usage Training Supply, duplication, and equipment Computer equipment and software • Operating costs - costs that recur throughout the lifetime of the system – Fixed costs — occur at regular intervals but at relatively fixed rates – Variable costs — occur in proportion to usage 10-25 Sample Feasibility Analysis Matrix Wt Candidate Purchase commercial off-the-shelf package for member services Candidate Write new application inhouse using new company standard VB.NET and SQL Server database Candidate Rewrite current in-house application using Powerbuilder 15% Supports only Member Services requirements Current business process would have to be modified to take advantage of software functionality Also there is concern about security in the system Fully supports userrequired functionality Fully supports userrequired functionality Description Operational feasibility Score: 60 Score: 100 Score: 100 10-26 Sample Feasibility Analysis Matrix (cont.) Technical feasibility Wt 20% Candidate Current production release of Platinum Plus package is version 1.0 and has been on the market for only weeks Maturity of product is a risk, and company charges and additional monthly fee for technical support Candidate Solution requires writing application in VB NET Although current technical staff has only Powerbuilder experience, it should be relatively easy to find programmers with VB NET experience Required to hire or train Java J2EE expertise ... investigate matrices and their inverses, and various ways to use matrices to solve systems of equations First, however, we will study systems of equations on their own: linear and nonlinear, and then partial.. .Introduction to Systems of Equations and Inequalities The Enigma is perhaps the most famous cryptographic device ever known It stands as an example of the pivotal role cryptography has... cryptanalysis to the digital world Many ciphers are designed using invertible matrices as the method of message transference, as finding the inverse of a matrix is generally part of the process of decoding