William F. Trench

Bio: William F. Trench is an academic researcher. The author has contributed to research in topics: Least-upper-bound property & Absolute continuity. The author has an hindex of 1, co-authored 1 publications receiving 599 citations.

Introduction to Real Analysis

Abstract: CHAPTER 1 PRELIMINARIES. 1.1 Sets and Functions. 1.2 Mathematical Induction. 1.3 Finite and Infinite Sets. CHAPTER 2 THE REAL NUMBERS. 2.1 The Algebraic and Order Properties of R. 2.2 Absolute Value and the Real Line. 2.3 The Completeness Property of R. 2.4 Applications of the Supremum Property. 2.5 Intervals. CHAPTER 3 SEQUENCES AND SERIES. 3.1 Sequences and Their Limits. 3.2 Limit Theorems. 3.3 Monotone Sequences. 3.4 Subsequences and the Bolzano-Weierstrass Theorem. 3.5 The Cauchy Criterion. 3.6 Properly Divergent Sequences. 3.7 Introduction to Infinite Series. CHAPTER 4 LIMITS. 4.1 Limits of Functions. 4.2 Limit Theorems. 4.3 Some Extensions of the Limit Concept. CHAPTER 5 CONTINUOUS FUNCTIONS. 5.1 Continuous Functions. 5.2 Combinations of Continuous Functions. 5.3 Continuous Functions on Intervals. 5.4 Uniform Continuity. 5.5 Continuity and Gauges. 5.6 Monotone and Inverse Functions. CHAPTER 6 DIFFERENTIATION. 6.1 The Derivative. 6.2 The Mean Value Theorem. 6.3 L'Hospital's Rules. 6.4 Taylor's Theorem. CHAPTER 7 THE RIEMANN INTEGRAL. 7.1 Riemann Integral. 7.2 Riemann Integrable Functions. 7.3 The Fundamental Theorem. 7.4 The Darboux Integral. 7.5 Approximate Integration. CHAPTER 8 SEQUENCES OF FUNCTIONS. 8.1 Pointwise and Uniform Convergence. 8.2 Interchange of Limits. 8.3 The Exponential and Logarithmic Functions. 8.4 The Trigonometric Functions. CHAPTER 9 INFINITE SERIES. 9.1 Absolute Convergence. 9.2 Tests for Absolute Convergence. 9.3 Tests for Nonabsolute Convergence. 9.4 Series of Functions. CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL. 10.1 Definition and Main Properties. 10.2 Improper and Lebesgue Integrals. 10.3 Infinite Intervals. 10.4 Convergence Theorems. CHAPTER 11 A GLIMPSE INTO TOPOLOGY. 11.1 Open and Closed Sets in R. 11.2 Compact Sets. 11.3 Continuous Functions. 11.4 Metric Spaces. APPENDIX A LOGIC AND PROOFS. APPENDIX B FINITE AND COUNTABLE SETS. APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA. APPENDIX D APPROXIMATE INTEGRATION. APPENDIX E TWO EXAMPLES. REFERENCES. PHOTO CREDITS. HINTS FOR SELECTED EXERCISES. INDEX.

A data mining framework for building intrusion detection models

Abstract: There is often the need to update an installed intrusion detection system (IDS) due to new attack methods or upgraded computing environments. Since many current IDSs are constructed by manual encoding of expert knowledge, changes to IDSs are expensive and slow. We describe a data mining framework for adaptively building Intrusion Detection (ID) models. The central idea is to utilize auditing programs to extract an extensive set of features that describe each network connection or host session, and apply data mining programs to learn rules that accurately capture the behavior of intrusions and normal activities. These rules can then be used for misuse detection and anomaly detection. New detection models are incorporated into an existing IDS through a meta-learning (or co-operative learning) process, which produces a meta detection model that combines evidence from multiple models. We discuss the strengths of our data mining programs, namely, classification, meta-learning, association rules, and frequent episodes. We report on the results of applying these programs to the extensively gathered network audit data for the 1998 DARPA Intrusion Detection Evaluation Program.

Oscillations and concentrations in weak solutions of the incompressible fluid equations

Abstract: The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.

Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems

Abstract: Motivated by control Lyapunov functions and Razumikhin theorems on stability of time delay systems, we introduce the concept of control Lyapunov-Razumikhin functions (CLRF). The main reason for considering CLRFs is construction of robust stabilizing control laws for time delay systems. Most existing universal formulas that apply to CLFs, are not applicable to CLRFs. It turns out that the domination redesign control law applies, achieving global practical stability and, under an additional assumption, global asymptotic stability. This additional assumption is satisfied in the practically important case when the quadratic part of a CLRF is itself a CLRF for the Jacobian linearization of the system. The CLRF based domination redesign possesses robustness to input unmodeled dynamics including an infinite gain margin. While, in general, construction of CLRFs is an open problem, we show that for several classes of time delay systems a CLRF can be constructed in a systematic way.

Uniform estimates and limiting arguments for nonlocal minimal surfaces

Abstract: We consider nonlocal minimal surfaces obtained by a fractional type energy functional, parameterized by \({s\in(0,1)}\). We show that the s-energy approaches the perimeter as s → 1−. We also provide density properties and clean ball conditions, which are uniform as s → 1−, and optimal lower bounds obtained by a rearrangement result. Then, we show that s-minimal sets approach sets of minimal perimeter as s → 1−.

Characteristic wave detection in ECG signal using morphological transform

Abstract: Background Detection of characteristic waves, such as QRS complex, P wave and T wave, is one of the essential tasks in the cardiovascular arrhythmia recognition from Electrocardiogram (ECG).