Algebraic Analysis

ISBN-13: 9789401071390 / Angielski / Miękka / 2011 / 620 str.

D. Przeworska-Rolewicz

Opis

Approach your problems from the right It isn't that they can't see the solution end and begin with the answers. Then, It is that they can't see the problem. one day, perhaps. you will find the final G. K. Chesterton. The Scandal of Fa question. ther Brown 'The point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of mon 0 graphs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact. that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics algebraic geometry interacts with physics; the Minkowsky lemma."

Spis treści

1. Fundamental properties of linear operators in lineal spaces.- 1.1. Groups and rings.- 1.2. Linear spaces.- 1.3. Linear operators and linear functionals.- 1.4. Eigenspaces and principal spaces. Algebraic operators. Volterra operators.- 1.5. Conjugate spaces and conjugate operators.- 2. Calculus of right invertible operators.- 2.1. Properties of right inverses. Indefinite integrals.- 2.2. Initial operators. Taylor—Gontcharov formula. Definite integrals.- 2.3. Exponentials, sine and cosine operators and Volterra right inverses.- 2.4. D-polynomials.- 2.5. Remarks on left invertible operators.- 3. General solution of equations with right invertible operators.- 3.1. Equations of order one with non-commutative coefficients.- 3.2. Equations of higher order with non-commutative coefficients.- 3.3. Equations with stationary coefficients.- 3.4. Equations with scalar coefficients. Operational calculus of right invertible operators. D-hull.- 3.5. Systems of equations with scalar coefficients.- 3.6. General solution of equations with left invertible operators.- 4. Initial and boundary value problems.- 4.1. Well-posed and ill-posed initial value problems.- 4.2. Hyperbolic equations.- 4.3. Elliptic and polyharmonic equations.- 4.4. Differential equations with delayed and advanced argument.- 4.5. Equations with involutions of order n.- 4.6. Well-posed and ill-posed boundary value problems.- 5. Periodic operators and elements. Shift operators. Shift invariant spaces.- 5.1. Periodic operators and elements.- 5.2. R-shifts and D-shifts.- 5.3. Existence of periodic solutions.- 5.4. Canonical mapping.- 5.5. Boundary value problems for stationary linear systems with shifts.- 6. D-algebras.- 6.1. Classification and examples of D-algebras.- 6.2. Integration. Exponentials. Trigonometric identity.- 6.3. Constants variation method. Wro?ski theorems.- 6.4. Fourier method.- 6.5. Green formulae. Picone identity. Euler—Lagrange equation.- 7. Perturbations and nonlinear problems.- 7.1. Finite dimensional perturbations.- 7.2. Perturbations by means of right inverses.- 7.3. Quasi-linear and nonlinear problems.- 7.4. Method of variables separable.- 8. Why using metric properties in algebraic analysis?.- 8.1. Right invertible operators in linear metric spaces.- 8.2. Canonical mapping and semigroups.- 8.3, Perturbations and periodic problems.- 8.4. Bielecki method and its applications.- 9. Miscellanea.- 9.1. Accelerating convergence of orthonormal series.- 9.2. Von Trotha principle of contractive mappings.- 9.3. Linear systems. F1-controllability.- 9.4. Dualities and conjugate systems.- Authors index.- List of symbols.