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Finite fields (Algebra). --- Polynomial rings. --- Probabilities.

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"One of the first textbooks ever to explain polynomial rings and affine spaces Description of contents is simple, concrete and easier to understand so that readers have as less obstacles as possible Almost all results are proved when they are used or referred to, except for the knowledge of algebra or topology to be acquired in the undergraduate study At the end of each chapter, there are many problems attached, which the author hope to guide the readers to advance into this area of research"--

Polynomial rings --- Geometry, Affine --- Geometry, Algebraic

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Anneaux de polynômes. --- Polynomial rings. --- Polynômes. --- Polynomials.

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PI-algebras --- Algebras with polynomial identities --- Polynomial identities, Algebras with --- Associative algebras --- Anells associatius --- Anells (Àlgebra) --- Anells artinians --- Anells noetherians --- Cossos no commutatius

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The theoretical part of this monograph examines the distribution of the spectrum of operator polynomials, focusing on quadratic operator polynomials with discrete spectra. The second part is devoted to applications. Standard spectral problems in Hilbert spaces are of the form A-λI for an operator A, and self-adjoint operators are of particular interest and importance, both theoretically and in terms of applications. A characteristic feature of self-adjoint operators is that their spectra are real, and many spectral problems in theoretical physics and engineering can be described by using them. However, a large class of problems, in particular vibration problems with boundary conditions depending on the spectral parameter, are represented by operator polynomials that are quadratic in the eigenvalue parameter and whose coefficients are self-adjoint operators. The spectra of such operator polynomials are in general no more real, but still exhibit certain patterns. The distribution of these spectra is the main focus of the present volume. For some classes of quadratic operator polynomials, inverse problems are also considered. The connection between the spectra of such quadratic operator polynomials and generalized Hermite-Biehler functions is discussed in detail. Many applications are thoroughly investigated, such as the Regge problem and damped vibrations of smooth strings, Stieltjes strings, beams, star graphs of strings and quantum graphs. Some chapters summarize advanced background material, which is supplemented with detailed proofs. With regard to the reader’s background knowledge, only the basic properties of operators in Hilbert spaces and well-known results from complex analysis are assumed.

Mathematics. --- Operator Theory. --- Ordinary Differential Equations. --- Mathematical Physics. --- Operator theory. --- Differential Equations. --- Mathématiques --- Théorie des opérateurs --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Spectral theory (Mathematics) --- Polynomial operator pencils. --- Operator pencils, Polynomial --- Operator polynomials --- Pencils, Polynomial operator --- Polynomial pencils --- Polynomials, Operator --- Differential equations. --- Mathematical physics. --- Linear operators --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- 517.91 Differential equations --- Differential equations --- Physical mathematics --- Physics