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Full of Depth and Insight as to this position in the Church

Simply, the best I have read...It puts you in the action from the fist chapter, from the begining, the concepts related to the basic analitical tools that you'll need to understand how a simulation software is capable of caracterize an electric circuit are presented ...It asumes you to know electric circuit analisis, so it don't enphatize the teoretical stuff, instead of this, it tell you how to use it to develop a simulation program. The Numerical stuff necessary for the software is treated as well in the book...
You'll be able to write a basic simulation program using just the concepts presented in the first chapter!!!...


Interesting Treatise on Drawing from a 19th Century MasterAnother fascinating aspect of this book is the snapshot it gives into the mind of a prominant 19th century art critic. Rushkin not only was a master draughtsman and painter but a widely respected art critic in his day. Monet was quoted by a British journalist to have said, "90% of the theory of Impressionist painting is in Rushkin's Elements of Drawing." A young George Seurat obtained a copy and admitted to having read it carefully. Now I'm no Monet or Seurat but I figure if these guys valued Rushkin's instruction I should certainly pay attention to what he had to say.
Rushkin explains exactly what the goal of each exercise is. He also recommends specific paintings or drawings to examine along with critiques of why this or that area in the drawing/painting is superior or lacking. He strongly believed it more profitable to study in-depth a few highly superior drawings/paintings to a wider assortment of middling/average execution. And he believed this even of famous artist's work - famous or not he advises to ignore for the moment their less masterful work and focus on the truly great ones. Rushkin pulled no punches. The entire treatise is full of his opinions right along side the exercises - yet I would say they are not opinions without merit. He gives you something to think about when looking at works of the art masters and something to strive for in your drawings and paintings so that you can become more than just technically competent. He addresses the heart and soul of drawing and painting. It made me think of why this or that particular line, shading or painting technique in an art master's drawing/painting touches me the way it does.
This is the best marriage between technical competence and artistry. And you grow in understanding that all the exercises he gives are only in service to the spirit of art. It is an emphasis that most modern how-to books don't touch. Analysis this deep in modern art books are left for books that are advertised as art critiques. Since almost all my art books fall under the "how-to" category (as anyone who's read my other book reviews will see) I found this critique aspect rather refreshing and wanting to read more such types of books.
I strongly recommend this book. Despite the lack of photos or modern step-by-step illustrations (the illustrations are line art - the most up-to-date technology for book illustration then available in an affordably priced book) I think it is very worth getting and reading. Perhaps artists who have been formally trained in universities or art academies will find this kind of instruction typical. But for someone like me who is entirely self-taught from the books he/she buys it is a great investment into expanding boundaries and knowledge of art in general.


Helpful and Constructive Debate Text!

ExcellentThe author begins the book with two neat problems that motivate well the subject of elliptic curves: the pyramid of cannonballs and the right triangle problem, i.e. which integers can occur as areas of right triangles with integer sides? He then immediately begins the elementary theory of ECs in chapter 2. The treatment is pretty standard, although he proves Pascal's and Pappus's theorems using the associativity of the group operation on ECs, which is not usually done in books on ECs. Also somewhat non-standard this early in the game is the discussion of reduction of ECs modulo various primes, and the subsequent definitions of additive, split multiplicative, and non-split multiplicative reduction.
The study of torsion points is done in chapter 3 with the Weil pairing on the n-torsion of an EC taking center stage. A fairly short chapter, the author delays the proof of the properties of the Weil pairing until chapter 11, where it is done with divisors.
Chapter 4 deals with elliptic curves over finite fields, and is one of the most important in the book from the standpoint of cryptographic applications of ECs. Hasse's theorem, giving the bounds for the group of points on an EC over a finite field, is proven in detail. The Frobenius endomorphism is introduced, and a proof of Schoof's algorithm for computing the number of points on ECs over a finite field is given a detailed treatment. There are many symbolic computational software packages in both the open and commerical realm which will do the counting straightforwardly, and anyone interested in cryptography will need to be familiar with some of these. Supersingular curves in characteristic p are introduced, and the author gives a good discussion of the reason why they are named as such.
The discrete logarithm problem, a topic also very important for cryptographic applications, is discussed in chapter 5. The chapter beings with the index calculus, and, recognizing that it does not apply to general groups, the Pohlig-Hellman, baby step-giant step method, and Pollards rho and lambda methods are discussed in details. The author then shows that for supersingular and "anomalous" curves, that the discrete logarithm problem can be reduced to an easier discrete logarithm problem. Along the way, two important concepts are introduced: the p-adic valuation, and the Tate-Lichtenbaum pairing, the latter of which is related to the Weil pairing, but applies to situations where the Weil pairing does not.
Elliptic curve cryptography is then discussed in chapter 6, and the treatment is fairly thorough. The author shows to what extent the Decision Diffie-Hellman problem can be solved using the Weil pairing. He also shows how to represent a message on an elliptic curve, satisfying early on any reader's curiosity on just how this is done. The El Gamal and ECDSA are compared in terms of their computational efficiency. An EC generalization of RSA is also discussed in some detail, along with a cryptosystem based on the Weil pairing. Chapter 7 then gives other applications of ECs, such as factoring and primality testing.
Chapter 8 marks the beginning of the "heavy artillery" in the theory of ECs, for here the author begins the discussion of elliptic curves over the rational numbers, which can be viewed as an example of Diophantine geometry. The famous Mordell-Weil theorem is proved, and as a sign that one is definitely in the arena of modern mathematics, the proof is given in terms of Galois cohomology, which is an abstraction of the Fermat method of descent. The reader gets a taste of height functions, and via some good examples, gets insight into why the rank of the EC is so difficult to compute. A neat example is given of a nontrivial Shafarevich-Tate group.
I did not read the chapters 9, 10, or 11 on ECs over the complex numbers, complex multiplication, and divisors, so I will omit their review. Chapter 12 introduces the famous zeta functions, and their use in obtaining arithmetic information about an EC. Zeta functions motivate the definition of an L-function of an EC, these being tremendously important in modern developments in the theory of ECs, such as the Swinnerton-Dyer and Birch conjecture, the latter of which is motivated rather nicely in this chapter.
The last chapter of the book is an excellent introduction to the proof of Fermat's Last Theorem. Considering the level of the book, the author captures very well the essential ideas. Readers will be well prepared, after studying more algebraic number theory and the theory of Galois representations (which the author only skims in the book), to tackle the full proof if so desired.


The invention of the modern university

A masterpiece!

A Simpler TimeEmily is a little girl who lives with her parents on a property in rural England. One day she and her mother visit the zoo in London to see the different animals. Just as they are about to leave the Children's zoo, Emily learns that the zookeeper is in a terrible predicament with Jumbo, the baby elephant. A concerned and caring Emily devises a plan and, with her mother's support, sets out to help Jumbo and the zookeeper, and in the end helps her father too.
"Emily's Own Elephant" is a lovely book for children aged 3 to 8 years, and of course their parents! But parents be warned - as soon as the story is finished your child will want you to read it to them over and over. And I'm guessing that with such a lovely story, you will want to read it again too.


Major historical events and their repercussions

The title says it all