A Glimpse at Set Theory. The Real Numbers. The Topology of Cartesian Spaces. Convergence. Continuous Functions. Functions of One Variable. Infinite Series. Differentiation in RP Integration in RP.

The elements of real analysis (2nd edition), by Robert G. Bartle. Pp xv, 480. £10. 1976. SBM 0 471 05464 X (Wiley)

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Theory of Linear Active Networks

Higher-order elements are introduced to provide a logically complete formulation for nonlinear circuit theory. A distinctive feature of higher-order elements is that they possess internal dynamics that are more complicated than those of conventional circuit elements (namely, the resistor, inductor, capacitor and memristor). In this paper, we provide a state space formulation for studying two-terminal higher-order elements. State-space properties such as local controllability, input-observability, passivity and losslessness are investigated in detail.

State space theory of nonlinear two- terminal higher-order elements

The above paper[1] proposes basic definitions and conventi which are not appropriate for nonlinear networks in general, although I may be appropriate for the class of networks to which the paper addressed. Since the paper[1] does not make this distinction clear, this l is written in the hope that these concepts will be subjected to crit scrutiny before adoption elsewhere in the literature. The new res appearing in the paper[1] are not affected by this discussion of fundamen

Comments on "Analysis and Synthesis of Nonlinear Reciprocal Networks containing two Element Types and Transformers

This paper is the second in a two part series [1] which aims to provide a rigorous foundation in the nonlinear domain for the two energy-based concepts which are fundamental to network theory: passivity and losslessness. We hope to clarify the way they enter into both the state-space and the inputoutput viewpoints. Our definition of losslessness is inspired by that of a "conservative system" in classical mechanics, and we use several examples to compareit with other concepts of losslessness found in the literature. We show in detail how our definition avoids the anomalies and contradictions which many current definitions produce. This concept of losslessness has the desirable property of being preserved under interconnections, and we extend it to one which is representation independent as well. Applied to five common classes of n-ports, it allows us to define explicit criteria for losslessness in terms of the state and output equations. In particular we give a rigorous justification for the various equivalent criteria in the linear case. And we give a canonical network realization for a large class of lossless systems.

John Lanier. Wyatt

Papers

Foundations of nonlinear network theory : part II: losslessness