Operators Books

This monograph contains an exposition of the foundations of the spectral theory of polynomial operator pencils acting in a Hilbert space. Spectral problems for polynomial pencils have attracted a steady interest in the last 35 years, mainly because they arise naturally in such diverse areas of mathematical physics as differential equations and boundary value problems, controllable systems, the theory of oscillations and waves, elasticity theory, and hydromechanics.

In this book, the author devotes most of his attention to the fundamental results of Keldysh on multiple completeness of the eigenvectors and associate vectors of a pencil, and on the asymptotic behavior of its eigenvalues and generalizations of these results. The author also presents various theorems on spectral factorization of pencils which grew out of known results of M. G. Krein and Heinz Langer. A large portion of the book involves the theory of selfadjoint pencils, an area having numerous applications. Intended for mathematicians, researchers in mechanics, and theoretical physicists interested in spectral theory and its applications, the book assumes a familiarity with the fundamentals of spectral theory of operators acting in a Hilbert space.

This book by Prof Yang is an excellent book for people want to know more about piezoelectricity and piezoelectric devices.

It starts with a complete theory of piezoelectricity with the intention to get readers familirize with the fundamentals. This part is complicated from its appearance, but there is a purpose. Such formulation is essential in dealing with complicated issues which discussed in detail later, ie the effect of various bias fields like thermo, stress, acceleration and so on.

After descriptions of the basic nonlinear theory, the chapters followed restore to the linear theory for various strctural elements and applications as devices. It touched almost all applications of acosutic devices with simple analytical solutions.

The methods and techniques are not directly related to the engineering solutions of actual problems, but it is the first step you have to take towards the direction.

I highly recommend this book to graduate students and engineers who are interested in advanced knowledge on this subject.

I never thought that the mystery of differential operators could be revealed in such an amusing and eloquent fashion. A book with real style - no doubt due to the wonderfully idiosyncratic translation by the sultry Helen Zahavi.

Just Done Publishing has done it again with another outstanding title! This book is about Andre Diedericks exploits while severing with the South African elite 5 Recce and in particular with Small Teams. This book does a great job of covering the evolution of Small Team operations within the South African Special Forces, this is topic not covered in many other books, like the detail covered in "Journey Without Boundaries." But with that I also recommend reading the Rhodesian side of Small Team Ops in the book "Bush War in Rhodesia - The Extraordinary Combat Memoir of a Rhodesian Reconnaissance Specialist.

Diedericks was a true warrior elite with such a colorful background, this book does him justice and is a fine tribute to his life and legacy of Small Team Ops. If you like to read detailed information on tactics, techniques and procedures you will truly enjoy this book, there is nuggets of hard learned lessons throughout the book that most soldiers/operators can take away from the book and apply to their soldiering/operations. Do not to get confused from that statement there is a story here and it is hair raising at times, but I will not spoil it in this review, you will have to read about the exploits yourself.

The book itself comes in two versions. One is hardcover with black and white pictures and the second is soft cover with color pictures. I have the hardcover and the quality is outstanding the pictures are clear and crisp. I may order another copy in soft cover to see the pictures in color and if you prefer color you may want to order the soft cover. The last third (or so) of the book is all pictures which alone is worth the money of the book. The book also has an extensive glossary, index and is footnoted. This is fast read of a book and I can not recommend this book more to the military professional.

This book is not a manual in the usual sense, but a compilation of facts concerning Lie algebras that continuously appear in physical problems. The material covered is the result of various seminars given by the author during many years, and synthetize the main facts that should be known to any physicist.

The material is divided into 12 chapters of variable length. The first two present the main theory of semisimple Lie algebras, enumerating the key results from root theory and Dynkin-Coxeter diagrams to classify the complex simple algebras. The real forms for the classical algebras are given in table form, without going into its detailed obtainment. It should also be taken into account that for Dynkin diagrams, the author does not distinguish between positively and negatively oriented angles, thus the angles between roots in equation (2.8) are reduced to five (unoriented) angles instead of the usual eight (oriented) angles.

Chapter three compiles the most important facts about Lie algebras of Lie groups, mainly focused on matrix groups. Important techniques like the exponential map and the covering of groups are nicely illustrated with the classical unitary algebra su(2) and the Lorentz group (in one dimension). I personally miss some comment on the left invariant vector fields or 1-forms (Maurer-Cartan equations), of importance in many applications to cosmology.

The fourth chapter is devoted to representation theory. Although the Weyl decomposition theorem is not included, it is assumed that any representation decomposes as a direct sum of irreducible modules (valid for semisimple Lie algebras). The fundamental representations are discussed for the classical algebras (symplectic, unitary and orthogonal), and for the latter, the spinor representations are also given. The dimension formulae are given, and the tensor products (Clebsch-Gordan problem) is developed by means of Young tableaux. This is applied to the branching rules of representations with respect to some chain and the missing label problem, illustrated by examples that are typical in the interacting boson model.

In chapter five, Casimir operators of Lie algebras are defined and obtained for the classical Lie algebras. Here the author uses the Perelomov-Popov approach of operators that can be identified with symmetric elements in the universal enveloping algebra. At the beginning of the chapter it is said that the number of Casimir operators equals the rank of the algebra. Again, this is only valid for semisimple Lie algebras, and generally false for arbitrary Lie algebras. The eigenvalue problem is presented using important examples, and the results resumed in a table at the end of the chapter.

The previous chapter is a nice motivation for tensor operators in general, which comprise essential techniques like the coupling and recoupling coefficients, how to determine them and their symmetries (much of this material was originally developed by Racah in his Princeton lectures of 1951). This chapter is of great importance for applications.

Chapters 8 and 9 are devoted to another technique of great relevance, the realizations of Lie algebras by means of creation and annihilation operators, divided into boson and fermion operators, according to commutation or anticommutation relations. Here the unitary case is exploited, and many subalgebra chains are analyzed with respect to these realizations. Of special interest are the sections concerning the L-S and j-j couplings used in spectroscopy of light nuclei and shell models, and where original examples have been used.

Chapter 9 presents another possibility for realizing Lie algebras, namely by differential operators. Although a short chapter, important topics like the Casimir operators as differential operators or the Laplace-Beltrami form is presented. In chapter 10, the classical matrix realizations (in fact representations by linear operators) are briefly recalled, and the classical interpretation of the Casimir operators is recovered (without using the Schur lemma).

The two last chapters deal with quite more specific topics, like dynamic symmetries, studied in both fermionic and bosonic systems, in the unitary algebras u(6) and u(4), in order to obtain mass and energy level diagrams. For the part of degeneracy algebras, the problems illustrated are the isotropic harmonic oscillator, the Coulomb problem and the Teller-Pöschl and Morse potentials. In all these problems the reader is referred to original articles to complete the information presented.

The chapters of the book do not develop the theory systematically, but rather focus on a type of problem or technique which is developed using the main Lie algebras appearing (mainly) in spectroscopy, atomic, nuclear and molecular physics, as well as quantum mechanics. No proofs are given, which prevents the reader from being distracted from the main objective of the lectures. To fill the gaps, the reader is led, at many places, to consult either original references or more formal books.

The book is written in an informal style, which simplifies its reading and makes it a suitable consultation work. The profusion of examples (many of them actually coming from original references) explains quite well the topics studied, and gives a concrete idea how to apply the techniques. It is a very welcomed addition to the literature that contains much topics treated for the first time in textbook form.

There are few misprints and mistakes in the text, which can however confuse the reader having no previous knowledge on Lie algebras. For example, on page 7, the definition of semidirect sum is confusing and wrong. It is actually not necessary that one of the algebras is an ideal in the other, it suffices that one of them acts by derivations on the other. The definition given in the book is incompatible with example 13 on the same page. Another confusing point is subsection 1.13. Here by derivations the author means the derived series of an algebra, determining whether it is solvable or not. Derivations are linear maps satisfying the Leibniz rule, and are completely independent on the solvable character. The notation is very confusing, since the derived subalgebra (commutator ideal) is denoted in the same manner as the Lie algebra of derivations (which is actually a linear Lie algebra).

Inspite of these minor details, the book will certainly be of great use for students or specialists that want to refresh their knowledge on Lie algebras applied to physics. The list of references is quite complete and provides a deeper insight into the problems where these structures appear. However, there are also some surprising absences in the references, such as the books of J. F. Cornwell or H. Lipkin, in my opinion two classicals on group theory in physics. Among the original articles, I miss for example the relevant review article by R. Slansky [Phys. Rep. 79 (1981), 1-128], although it is clear that giving a complete reference list is impossible.

Resuming, the book by Iachello constitutes an excellent reference for those interested in the practical application and techniques of Lie algebras to physics, and that try to avoid the often embarrassing theoretical works. It should also be mentioned that much of the material is divided into hundreds of original articles, and therefore a unified presentation will be of great use for the physical community.

John Wiley should be ashamed of your self This unbelivable price for a miserable paperbak print

As an early elementary school teacher, I read a lot of children's books. Mary Alice is one of my all-time favorites. She is a very efficient telephone operator who tells people the time. "At the tone, the time will be 1 o'clock exactly. Quack". (Did I mention, she's a duck?) Mary Alice catches a cold so Boss Chicken insists that she go home to take care of herself. Her boss tries to assure her the position could easily be filled by someone, but Mary Alice is hurt by the remarks and frets over someone permanently replacing her. Boss Chicken interviews and hires a series of different operators, who, for various reasons, don't fit the bill (no pun intended). The snake hisses and scares the customers, the dog can't tell time, an armandillo has trouble seeing the clock, and the beaver is so eager that she can't wait for people to call her so SHE calls them. In the end, Boss Chicken fully understands the difficulty of the job and appreciates Mary Alice.
The story and the pictures are wonderfully funny. I especially love the picture of Mary Alice wearing her pink bunny slippers.
What makes this story even better is that it lends itself to dramatic storytelling. I use different voices for the various characters in the story.
My kids and I thoroughly enjoy Mary Alice Operator Number 9 and I know you will, too.

I have hardcover copy of this book published by Dutton in 1977. To this day I take THIS book off my shelf when I feel like good selfhelp advice to just pull myself together again a little bit. I believe I am not the only woman who sometimes takes one look in the mirror and thinks : yup, it's time for a bit of spitting and polishing MYSELF, too! I found the book on a sale. Find a copy - it's like listening to a good friend yakking away while he is doing wondeful things for you. Is George Masters still around?

The book by Prof Gustafson and Prof Sigal has a good balance between mathematical rigor and not too much technicalities. Some results of more mathematical sofisticated nature are adressed to "Int to Spectral Theory. With Applications to Schrodinger Operators", by P. Hilsop
and I. Sigal.
In this way some of the main ideas of Quantum Mechanics are explained in this text in a clear way for someone with a mathematical
mind. It is apropriate for a person that does not know
well
the physics of Quantum Mechanics.
I believe any mathematician with some background in Functional Analysis and some basic knowledge of Partial Differential Equations and Fourier Analysis will enjoy this book.
There is some other books in the subject which contains all proofs,
all dificult technicalities, all mathematical details, but, in my opinion,
for a mathematician first lecture on the subject this the the one.

A. O. Lopes

Blanchard's text is an excelent journey through a variety of topics in mathematical physics. The language is clearly more closely related to an advanced physics text than a math text, and Blanchard strives to give explanations as well as just rigorous proofs to accompany his results. It makes for a truly interesting read where one can gain essential details without having to necessarily be weighed down by long proofs (which are usually set in a smaller font so as to allow one to skip them at will).

The choice of exercises is likewise superb, ranging from the trivial to the truly difficult. It is not, however, too uncommon to find that one needs more knoweldge than can be found in the text to really complete a problem. Students of this text would be better suited adding at least two courses in real analysis to Blanchard's suggested list.

Still, the selection of important topics to cover is great, the examples chosen are lucid, and the exposition is memorable. I highly recommend this book to anyone looking for an interesting read, who is unfamiliar with distributions or hilbert space operators in particular.