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Author: Angelo J. Nasca Publisher: ISBN: Category : Languages : en Pages :
Book Description
Linear dynamics is the study of orbits of linear operators on Hilbert spaces (or more general topological vector spaces). If a linear operator has a dense orbit, we say the operator is hypercyclic. One of the fundamental results in linear dynamics is Ansari's Theorem: If a bounded linear operator T is hypercyclic, then for any fixed n natural number, the operator T^n is also hypercyclic. The phenomenon exhibited by Ansari's Theorem is a driving force behind much of the work in this document. This thesis has an introductory chapter and two main chapters. The first main chapter has the goal of extending Ansari's Theorem to actions generated by several commuting linear operators. This goal is realized under the assumption that the action in question is weakly mixing. This chapter also contains several results about semigroups of linear operators on finite dimensional spaces, including a multi-parameter analogue of Ansari's Theorem, and ends with a treatment of affine hypercyclic operators. The second main chapter extends the construction of a probability measure of full support on a Hilbert space which is preserved by a linear operator, to a situation involving the action generated by several commuting operators. Conditions are given that ensure mixing properties of such an action. Following this, we use joint ergodicity to establish some new theorems about joint hypercyclicity, including a theorem which is a variation on both Ansari's Theorem and the related Leon-Muller Theorem. We end the chapter with an example of an operator which is jointly hypercyclic along the polynomial sequences n,n^2,...,n^{d-1}, and yet fails to be hypercyclic along the sequence n^d.
Author: Angelo J. Nasca Publisher: ISBN: Category : Languages : en Pages :
Book Description
Linear dynamics is the study of orbits of linear operators on Hilbert spaces (or more general topological vector spaces). If a linear operator has a dense orbit, we say the operator is hypercyclic. One of the fundamental results in linear dynamics is Ansari's Theorem: If a bounded linear operator T is hypercyclic, then for any fixed n natural number, the operator T^n is also hypercyclic. The phenomenon exhibited by Ansari's Theorem is a driving force behind much of the work in this document. This thesis has an introductory chapter and two main chapters. The first main chapter has the goal of extending Ansari's Theorem to actions generated by several commuting linear operators. This goal is realized under the assumption that the action in question is weakly mixing. This chapter also contains several results about semigroups of linear operators on finite dimensional spaces, including a multi-parameter analogue of Ansari's Theorem, and ends with a treatment of affine hypercyclic operators. The second main chapter extends the construction of a probability measure of full support on a Hilbert space which is preserved by a linear operator, to a situation involving the action generated by several commuting operators. Conditions are given that ensure mixing properties of such an action. Following this, we use joint ergodicity to establish some new theorems about joint hypercyclicity, including a theorem which is a variation on both Ansari's Theorem and the related Leon-Muller Theorem. We end the chapter with an example of an operator which is jointly hypercyclic along the polynomial sequences n,n^2,...,n^{d-1}, and yet fails to be hypercyclic along the sequence n^d.
Author: M.S. Livsic Publisher: Springer Science & Business Media ISBN: 940158561X Category : Mathematics Languages : en Pages : 329
Book Description
Considering integral transformations of Volterra type, F. Riesz and B. Sz.-Nagy no ticed in 1952 that [49]: "The existence of such a variety of linear transformations, having the same spectrum concentrated at a single point, brings out the difficulties of characterization of linear transformations of general type by means of their spectra." Subsequently, spectral analysis has been developed for different classes of non selfadjoint operators [6,7,14,20,21,36,44,46,54]. It was then realized that this analysis forms a natural basis for the theory of systems interacting with the environment. The success of this theory in the single operator case inspired attempts to create a general theory in the much more complicated case of several commuting operators with finite-dimensional imaginary parts. During the past 10-15 years such a theory has been developed, yielding fruitful connections with algebraic geometry and sys tem theory. Our purpose in this book is to formulate the basic problems appearing in this theory and to present its main results. It is worth noting that, in addition to the joint spectrum, the corresponding algebraic variety and its global topological characteristics play an important role in the classification of commuting operators. For the case of a pair of operators these are: 1. The corresponding algebraic curve, and especially its genus. 2. Certain classes of divisors - or certain line bundles - on this curve.
Author: Frédéric Bayart Publisher: Cambridge University Press ISBN: 0521514967 Category : Mathematics Languages : en Pages : 352
Book Description
The first book to assemble the wide body of theory which has rapidly developed on the dynamics of linear operators. Written for researchers in operator theory, but also accessible to anyone with a reasonable background in functional analysis at the graduate level.
Author: Thomas F. Jordan Publisher: Courier Corporation ISBN: 0486140547 Category : Science Languages : en Pages : 162
Book Description
Suitable for advanced undergraduates and graduate students, this compact treatment examines linear space, functionals, and operators; diagonalizing operators; operator algebras; and equations of motion. 1969 edition.
Author: Moshe S. Livšic Publisher: ISBN: Category : Nonselfadjoint operators Languages : en Pages : 318
Book Description
Considering integral transformations of Volterra type, F. Riesz and B. Sz.-Nagy no ticed in 1952 that [49]: "The existence of such a variety of linear transformations, having the same spectrum concentrated at a single point, brings out the difficulties of characterization of linear transformations of general type by means of their spectra." Subsequently, spectral analysis has been developed for different classes of non selfadjoint operators [6,7,14,20,21,36,44,46,54]. It was then realized that this analysis forms a natural basis for the theory of systems interacting with the environment. The success of this theory in the single operator case inspired attempts to create a general theory in the much more complicated case of several commuting operators with finite-dimensional imaginary parts. During the past 10-15 years such a theory has been developed, yielding fruitful connections with algebraic geometry and sys tem theory. Our purpose in this book is to formulate the basic problems appearing in this theory and to present its main results. It is worth noting that, in addition to the joint spectrum, the corresponding algebraic variety and its global topological characteristics play an important role in the classification of commuting operators. For the case of a pair of operators these are: 1. The corresponding algebraic curve, and especially its genus. 2. Certain classes of divisors - or certain line bundles - on this curve.
Author: Julian Schwinger Publisher: Courier Corporation ISBN: 0486604446 Category : Science Languages : en Pages : 450
Book Description
This monumental collection of 34 historical papers on quantum electrodynamics features contributions by the 20th century's leading physicists: Dyson, Fermi, Feynman, Foley, Oppenheimer, Pauli, Weisskopf, and others. Twenty-nine are in English, three in German, and one each in French and Italian. Editor Julian Schwinger won a Nobel Prize for his pioneering work in quantum electrodynamics.
Author: Kimball A Milton Publisher: World Scientific ISBN: 981449450X Category : Science Languages : en Pages : 809
Book Description
Julian Schwinger (1918-1994) was one of the giants of 20th Century science. He contributed to a broad range of topics in theoretical physics, ranging from classical electrodynamics to quantum mechanics, from nuclear physics through quantum electrodynamics to the general theory of quantum fields. Although his mathematical prowess was legendary, he was fundamentally a phenomenologist. He received many awards, including the first Einstein Prize in 1951, and the Nobel Prize in 1965, which he shared with Richard Feynman and Sin-itiro Tomonaga for the self-consistent formulation of quantum electrodynamics into a practical theory. His more than 70 doctoral students have played a decisive role in the development of science in the second half of this century.This important volume includes many of Schwinger's most important papers, on the above and other topics, such as the theory of angular momentum and the theory of many-body systems. The papers collected here continue to underlie much of the work done by theoretical physicists today.
Author: J Hogan Publisher: CRC Press ISBN: 1420033832 Category : Mathematics Languages : en Pages : 370
Book Description
Nonlinear dynamics has been successful in explaining complicated phenomena in well-defined low-dimensional systems. Now it is time to focus on real-life problems that are high-dimensional or ill-defined, for example, due to delay, spatial extent, stochasticity, or the limited nature of available data. How can one understand the dynamics of such sys
Author: Anna Marmodoro Publisher: Oxford University Press ISBN: 0198735871 Category : Philosophy Languages : en Pages : 299
Book Description
This volume presents thirteen original essays which explore both traditional and contemporary aspects of the metaphysics of relations. It is uncontroversial that there are true relational predications-'Abelard loves Eloise', 'Simmias is taller than Socrates', 'smoking causes cancer', and so forth. More controversial is whether any true relational predications have irreducibly relational truthmakers. Do any of the statements above involve their subjects jointly instantiating polyadic properties, or can we explain their truths solely in terms of monadic, non-relational properties of the relata? According to a tradition dating back to Plato and Aristotle, and continued by medieval philosophers, polyadic properties are metaphysically dubious. In non-symmetric relations such as the amatory relation, a property would have to inhere in two things at once-lover and beloved-but characterise each differently, and this puzzled the ancients. More recent work on non-symmetric relations highlights difficulties with their directionality. Such problems offer clear motivation for attempting to reduce relations to monadic properties. By contrast, ontic structural realists hold that the nature of physical reality is exhausted by the relational structure expressed in the equations of fundamental physics. On this view, there must be some irreducible relations, for its fundamental ontology is purely relational. The Metaphysics of Relations draws together the work of a team of leading metaphysicians, to address topics as diverse as ancient and medieval reasons for scepticism about polyadic properties; recent attempts to reduce causal and spatiotemporal relations; recent work on the directionality of relational properties; powers ontologies and their associated problems; whether the most promising interpretations of quantum mechanics posit a fundamentally relational world; and whether the very idea of such a world is coherent. From those who question whether there are relational properties at all, to those who hold they are a fundamental part of reality, this book covers a broad spectrum of positions on the nature and ontological status of relations, from antiquity to the present day.
Author: R. F. Streater Publisher: Imperial College Press ISBN: 1848162448 Category : Science Languages : en Pages : 393
Book Description
How can one construct dynamical systems obeying the first and second laws of thermodynamics: mean energy is conserved and entropy increases with time? This book answers the question for classical probability (Part I) and quantum probability (Part II). A novel feature is the introduction of heat particles which supply thermal noise and represent the kinetic energy of the molecules. When applied to chemical reactions, the theory leads to the usual nonlinear reaction-diffusion equations as well as modifications of them. These can exhibit oscillations, or can converge to equilibrium.In this second edition, the text is simplified in parts and the bibliography has been expanded. The main difference is the addition of two new chapters; in the first, classical fluid dynamics is introduced. A lattice model is developed, which in the continuum limit gives us the Euler equations. The five Navier-Stokes equations are also presented, modified by a diffusion term in the continuity equation. The second addition is in the last chapter, which now includes estimation theory, both classical and quantum, using information geometry.
Author: Angelo J. Nasca
Author: Angelo J. Nasca
Author: M.S. Livsic
Author: Frédéric Bayart
Author: Thomas F. Jordan
Author: Moshe S. Livšic
Author: Julian Schwinger
Author: Kimball A Milton
Author: J Hogan
Author: Anna Marmodoro
Author: R. F. Streater