Auxiliary function

Abstract: I. Elements of Stability Theory.- 1. A First Glance at Stability Concepts.- 2. Various Definitions of Stability and Attractivity.- 3. Auxiliary Functions.- 4. Stability and Partial Stability.- 5. Instability.- 6. Asymptotic Stability.- 7. Converse Theorems.- 8. Bibliographical Note.- II. Simple Topics in Stability Theory.- 1. Theorems of E.A. Barbashin and N.N. Krasovski for Autonomous and Periodic Systems.- 2. A Theorem of V.M. Matrosov on Asymptotic Stability.- 3. Introduction to the Comparison Method.- 4. Total Stability.- 5. The Frequency Method for Stability of Control Systems.- 6. Non-Differentiable Liapunov Functions.- 7. Bibliographical Note.- III. Stability of a Mechanical Equilibrium.- 1. Introduction.- 2. The Lagrange-Dirichlet Theorem and Its Variants.- 3. Inversion of the Lagrange-Dirichlet Theorem Using Auxiliary Functions.- 4. Inversion of the Lagrange-Dirichlet Theorem Using the First Approximation.- 5. Mechanical Equilibrium in the Presence of Dissipative Forces.- 6. Mechanical Equilibrium in the Presence of Gyroscopic Forces.- 7. Bibliographical Note.- IV. Stability in the Presence of First Integrals.- 1. Introduction.- 2. General Hypotheses.- 3. How to Construct Liapunov Functions.- 4. Eliminating Part of the Variables.- 5. Stability of Stationary Motions.- 6. Stability of a Betatron.- 7. Construction of Positive Definite Functions.- 8. Bibliographical Note.- V. Instability.- 1. Introduction.- 2. Definitions and General Hypotheses.- 3. Fundamental Proposition.- 4. Sectors.- 5. Expellers.- 6. Example of an Equation of N Order.- 7. Instability of the Betatron.- 8. Example of an Equation of Third Order.- 9. Exercises.- 10. Bibliographical Notes.- VI. A Survey of Qualitative Concepts.- 1. Introduction.- 2. A View of Stability and Attractivity Concepts.- 3. Qualitative Concepts in General.- 4. Equivalence Theorems for Qualitative Concepts.- 5. A Tentative Classification of Concepts.- 6. Weak Attractivity, Boundedness, Ultimate Boundedness.- 7. Asymptotic Stability.- 8. Bibliographical Note.- VII. Attractivity for Autonomous Equations.- 1. Introduction.- 2. General Hypotheses.- 3. The Invariance Principle.- 4. An Attractivity and a Weak Attractivity Theorem.- 5. Attraction of a Particle by a Fixed Center.- 6. A Class of Nonlinear Electrical Networks.- 7. The Ecological Problem of Interacting Populations.- 8. Bibliographical Note.- VIII. Attractivity for Non Autonomous Equations.- 1. Introduction, General Hypotheses.- 2. The Families of Auxiliary Functions.- 3. Another Asymptotic Stability Theorem.- 4. Extensions of the Invariance Principle and Related Questions.- 5. The Invariance Principle for Asymptotically Autonomous and Related Equations.- 6. Dissipative Periodic Systems.- 7. Bibliographical Note.- IX. The Comparison Method.- 1. Introduction.- 2. Differential Inequalities.- 3. A Vectorial Comparison Equation in Stability Theory.- 4. Stability of Composite Systems.- 5. An Example from Economics.- 6. A General Comparison Principle.- 7. Bibliographical Note.- Appendix I. DINI Derivatives and Monotonic Functions.- 1. The Dini Derivatives.- 2. Continuous Monotonic Functions.- 3. The Derivative of a Monotonic Function.- 4. Dini Derivative of a Function along the Solutions of a Differential Equation.- Appendix II. The Equations of Mechanical Systems.- Appendix III. Limit Sets.- List of Examples.- Author Index.

Abstract: A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.

Abstract: This letter describes algorithms for nonnegative matrix factorization (NMF) with the β-divergence (β-NMF). The β-divergence is a family of cost functions parameterized by a single shape parameter β that takes the Euclidean distance, the Kullback-Leibler divergence, and the Itakura-Saito divergence as special cases (β = 2, 1, 0 respectively). The proposed algorithms are based on a surrogate auxiliary function (a local majorization of the criterion function). We first describe a majorization-minimization algorithm that leads to multiplicative updates, which differ from standard heuristic multiplicative updates by a β-dependent power exponent. The monotonicity of the heuristic algorithm can, however, be proven for β ∈ (0, 1) using the proposed auxiliary function. Then we introduce the concept of the majorization-equalization (ME) algorithm, which produces updates that move along constant level sets of the auxiliary function and lead to larger steps than MM. Simulations on synthetic and real data illustrate the faster convergence of the ME approach. The letter also describes how the proposed algorithms can be adapted to two common variants of NMF: penalized NMF (when a penalty function of the factors is added to the criterion function) and convex NMF (when the dictionary is assumed to belong to a known subspace).

Abstract: Finding integral inequalities for quadratic functions plays a key role in the field of stability analysis. In such circumstances, the Jensen inequality has become a powerful mathematical tool for stability analysis of time-delay systems. This paper suggests a new class of integral inequalities for quadratic functions via intermediate terms called auxiliary functions, which produce more tighter bounds than what the Jensen inequality produces. To show the strength of the new inequalities, their applications to stability analysis for time-delay systems are given with numerical examples.