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Beyond Perturbation: Introduction to the Homotopy Analysis Method (Modern Mathematics and Mechanics)
Shijun Liao
Manufacturer: Chapman & Hall/CRC
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ASIN: 158488407X |
Book Description
Solving nonlinear problems is inherently difficult, and the stronger the nonlinearity, the more intractable solutions become. Analytic approximations often break down as nonlinearity becomes strong, and even perturbation approximations are valid only for problems with weak nonlinearity. This book introduces a powerful new analytic method for nonlinear problems-homotopy analysis-that remains valid even with strong nonlinearity. In Part I, the author starts with a very simple example, then presents the basic ideas, detailed procedures, and the advantages (and limitations) of homotopy analysis. Part II illustrates the application of homotopy analysis to many interesting nonlinear problems. These range from simple bifurcations of a nonlinear boundary-value problem to the Thomas-Fermi atom model, Volterra's population model, Von Kármán swirling viscous flow, and nonlinear progressive waves in deep water. Although the homotopy analysis method has been verified in a number of prestigious journals, it has yet to be fully detailed in book form. Written by a pioneer in its development, Beyond Pertubation: Introduction to the Homotopy Analysis Method is your first opportunity to explore the details of this valuable new approach, add it to your analytic toolbox, and perhaps make contributions to some of the questions that remain open.
Book Description
This text is based on a one-semester graduate course taught by the author at The Fields Institute in fall 1995 as part of the homotopy theory program which constituted the Institute's major program that year. The intent of the course was to bring graduate students who had completed a first course in algebraic topology to the point where they could understand research lectures in homotopy theory and to prepare them for the other, more specialized graduate courses being held in conjunction with the program. The notes are divided into two parts: prerequisites and the course proper.
Part I, the prerequisites, contains a review of material often taught in a first course in algebraic topology. It should provide a useful summary for students and non-specialists who are interested in learning the basics of algebraic topology. Included are some basic category theory, point set topology, the fundamental group, homological algebra, singular and cellular homology, and Poincaré duality.
Part II covers fibrations and cofibrations, Hurewicz and cellular approximation theorems, topics in classical homotopy theory, simplicial sets, fiber bundles, Hopf algebras, spectral sequences, localization, generalized homology, and cohomology operations.
Book Description
This book is primarily directed to graduate students interested in the field and to algebraic topologists who wish to learn something about BP. Beginning with the geometric background of complex bordism, the author goes on to a discussion of formal groups and an introduction to BP-homology. He then presents his view of the major developments in the field in the last decade (the calculation of the homology of Eilenberg-MacLane spaces in this section may be useful in teaching advanced algebraic topology courses). The book concludes with a section on unstable operations with comments on where applications may come from in the future.
Customer Reviews:
A fine introduction.......2002-10-13
Homotopy theory is one of the hardcore topics in algebraic topology that usually takes a formidable amount of technical machinery in order to progress in its development. This is somewhat paradoxical considering that defining homotopy groups is very straightforward. The author has given the reader a fine introduction to homotopy theory in this book, and one that still could be read even now, in spite of the developments in homotopy theory that have taken place since the book was published (1975). The book emphasizes how homotopy theory fits in with the rest of algebraic topology, and so less emphasis is placed on the actual calculation of homotopy groups, although there is enough of the latter to satisfy the reader's curiosity in this regard. In the book the author states that "the deeper one gets into mathematics, the closer one sees the connections". This is readily apparent in his coverage, as he gives a good general view of how algebra and topology are intertwined in the study of homotopy theory.
The calculation of the fundamental group in homotopy theory is done by first considering covering spaces. Noting that this approach is useless in proving that a space is simply connected, the author moves on to the van Kampen theorem, and he uses it to show that the n-dimensional sphere is simply-connected. The calculation of the nth homotopy group for n > 1 is done using locally trivial bundles, which are the simplest generalizations of covering spaces. These bundles have the homotopy lifting property, and one can use this to relate the homotopy on the fibers to that of the base of the bundle. The author also shows how to get homotopy information from projective space fiberings.
That the n-th homotopy group can be given a group structure is done in the context of compactly generated Hausdorff spaces by first using the reduced suspension as the domain. The group structure is alternately defined using an H-space structure on the range. The duality between these points of view is then proved by the author.
In the simplicial category, the author proves the Blakers-Massey Theorem. The homotopy groups of spheres in certain selections of dimensions are then calculated. The homotopy theory of spaces more general than simplicial complexes, the CW complexes, is treated in detail by the author. The notion of weak homotopy equivalence is introduced, and a proof of the Whitehead theorem, showing that weak homotopy equivalence between CW complexes is the same as homotopy equivalence, is proven.
The author does a fine job of discussing K(pi,n)'s and Postnikov systems, which are introduced as tools to find a space that will realize a sequence of homotopy groups. Geometric intuition takes its leave here, the reader now being properly embedded in the true abstraction of algebraic topology. Obstruction theory makes its first appearance here.
Spectra, one of the most esoteric of topics in homotopy theory, also makes its appearance in this book. Its relation to homology and cohomology is brought about via the suspension functor. The homology of CW complexes is discussed, along with the generalization to more general spaces, using singular homology, which is defined in terms of spectra. This approach is different than what is usually done in books on algebraic topology. Homotopy theory is related to ordinary homology in 0 and higher dimensions and the Whitehead theorem, giving a homotopy equivalence if the homology of simply connected CW complexes is an isomorphism, is proven.
The multiplicative properties of cohomology is discussed in detail, and the author brings in the heavy guns from homological algebra. These tools are all used to analyze orientation and duality issues in paracompact topological manifolds. The author introduces duality as a generalization of that in Euclidean n-space, wherein one can find an (n-k)-dimensional subspace for each k-dimensional subspace.
Cohomology operations, which are the modern tour-de-force of algebraic topology, are discussed first as coefficient transformations, and then as natural transformations between spectra. The cup and cap products, and their generalizations in the Steenrod squaring operations , are discussed in fair detail. Spectral sequences are not used in the book, and so they are only assumed in order to study the algebra of stable operations over the integers modulo 2. This is done with the assistance also of Adem relations, which are relations among the Steenrod squares.
K-theories, which are introduced as examples of 'extraordinary' cohomology theories, are discussed briefly, in the context of vector bundles, but the Bott periodicity theorem is not proven. Instead, the author uses it to solve the Hopf invariant and vector field problems. The Gauss map is defined and then used to give the classification theorem for vector bundles. The Whitney sum of vector bundles, along with the Grothendieck construction, give the K-theory functor. Applications of K-theory to Lie groups are delegated to the exercises.
The author also includes a brief discussion of cobordism, which is done with the assistance of some notions from differential topology, such as the normal bundle and the concept of a tubulur neighborhood. The cobordism ring is shown to be graded, and assuming the Whitney embedding theorem, the Thom isomorphism between the cobordism ring and the homotopy of MO, where MO(k) is the tangent bundle over the universal k-plane bundle over BO(k). The homotopy of MO is calculated by first calculating the cohomology of BO and MO over the integers modulo 2. The Stiefel-Whitney classes are introduced here, and used to show that real projective 2n-space can be viewed as a ring generator of the cobordism ring. A most interesting discussion, as it shows to what extent the homology and cohomology derived from unoriented cobordism is different from ordinary homology and cohomology over the integers modulo 2. As is shown, every homology class over the integers modulo 2 is represented by a map from a manifold.
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An Introduction to Homotopy Theory
P. J. Hilton
Manufacturer: Cambridge University Press
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ASIN: 0521052653 |
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An Introduction to Topology and Homotopy
Allan J. Sieradski
Manufacturer: P W S Publishers
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ASIN: 0534929605 |
Book Description
The treatment of the subject of this text is not encyclopedic, nor was it designed to be suitable as a reference manual for experts. Rather, it introduces the topics slowly in their historic manner, so that students are not overwhelmed by the ultimate achievements of several generations of mathematicians. Careful readers will see how topologists have gradually refined and extended the work of their predecessors and how most good ideas reach beyond what their originators envisioned. To encourage the development of topological intuition, the text is amply illustrated. Examples, too numerous to be completely covered in two semesters of lectures, make this text suitable for independent study and allow instructors the freedom to select what they will emphasize. The first eight chapters are suitable for a one-semester course in general topology. The entire text is suitable for a year-long undergraduate or graduate level curse, and provides a strong foundation for a subsequent algebraic topology course devoted to the higher homotopy groups, homology, and cohomology.
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Shape Theory: An Introduction (Lecture Notes in Mathematics; 687)
Jerzy Dydak
Manufacturer: Springer
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ASIN: 0387089551 |
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- Not Free SF Reader
- weakest of the series
- THERE ARE OLD NARNIANS, AND BOLD NARNIANS, BUT NO OLD BOLD NARNIANS
- Faith and Rebuilding
- Prince Caspian by Hayden P.
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Prince Caspian (Narnia)
C. S. Lewis
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The Voyage of the Dawn Treader (The Chronicles of Narnia, Book 5, Full-Color Collector's Edition)
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The Magician's Nephew (The Chronicles of Narnia)
ASIN: 0064471055 |
Book Description
The four Pevensies help Caspian battle Miraz and ascend his rightful throne.
Customer Reviews:
Not Free SF Reader.......2007-09-03
The children from the first Narnia book return there to find that many centuries have passed in Narnia time.
Those who were once boss types in Narnia have to solve a bit of a usurpation problem, with the aid of a couple of characters, including the old fleabag himself, Aslan.
weakest of the series.......2007-07-17
'Prince Caspian' finds Peter, Susan, Edmund and Lucy getting ready to return to school at the end of summer holidays, when they are suddenly pulled by magic into another world. Of course, at once they suspect they are back in Narnia, but nothing seems familiar to them. It turns out while only a year has passed in our wold, centuries have passed in Narnia.
This is the second published book in the series (4th in the internal chronolgy). The first 2/3rds of the book are quite good, as the kids realise where (and when) they are, encounter Trumpkin the dwarf and learn of the recent history of Narnia, and the story of young Prince Caspian.
However once they reach Caspian the story really sags, as it just seems the characters wait around until Aslan fixes everything.
The Christian references are toned down a little in this one. All in all the weakest of the series -probably closer to 3 1/2 stars.
THERE ARE OLD NARNIANS, AND BOLD NARNIANS, BUT NO OLD BOLD NARNIANS.......2007-05-08
It is a bleak time for the Narnians of old. The talking animals are all in hiding and men who call themselves Telmarines are ruling the land. The men of Telmar are afraid of and hostile to talking animals, the dryads and hamadryads, the naiads, centaurs, dwarves, and satyrs. They fear them and have tried to destroy them. The woods are silent and the dryads sleep, dreaming of a free Narnia and better times. But the memories of old, free Narnia are alive and are passed on in secret. The nurse of Prince Caspian is just one who knows the exciting secrets of old, and there are many others. And so it happens that the young prince comes to love the old that is hidden more than the new that he will rule. But although the stories of old may feed the soul, they are dangerous to know. And that is the start of the prince's dangers and adventures. He may call on those free creatures who are in hiding, they may rally to his call, but will they be strong enough to overthrow their oppressors? He has one more magical link with the past, and he will use it at the moment of greatest need - the magical horn of Queen Susan bringing unknown help to those who use it, which has been preserved as a relic by the faithful.
The Chronicles of Narnia begin, as everyone knows, with `The Lion, the Witch, and the Wardrobe'. This story, `Prince Caspian', is probably best read second in the sequence as it is a continuation of the original four's adventures. The High King Peter, King Edmund, Queen Susan, and Queen Lucy are summoned by magic back to Narnia to aid Prince Caspian in time of crisis. The story of `The Magician's Nephew' goes back to the beginning of Narnian time and a little earlier in our world's time to tell how Narnia was created in the first place, and it is probably best read about fifth or sixth in the sequence, but at any rate before `The Last Battle' which tells how Narnia ends and is more frightening than the rest. The best loved of all the stories is probably `The Voyage of the Dawn Treader', partly because it is the longest and richest story in the series and is supremely well written. It is very easy to read but full of interweaving plots, and thrills on land and sea, and full of hard realities like how people behave while thirsty on short water rations and no land in sight. It is the jewel of the set, and fits perfectly in the middle. Reading about prince Caspian will help set the jewel in your mind.
`Prince Caspian' is also an interesting story because it explains so much of the magic of Narnia, and gives those who wish to see an insight into politics, history (ours and Narnia's), battles, and human psychology. It is particularly revealing to see what a prince's education involves: some literature, some mathematics, some social graces, some skills in entertainment and music, some politics. Some people do not like this story because it is about a war, but it really is about what leads up to war, what happens after, and how the individuals involved all react and cope. The actual fighting is a small part of the whole, unlike a modern action film which is heavy on the fighting and light on the people. Having said that, the storyline is one of the simplest in the set as we stay almost all the time with the four children together, who quickly resume their adult roles once in Narnia. When things threaten to overwhelm the brave few, Aslan is at hand but to their surprise he is not always easy to see.
Faith and Rebuilding.......2007-04-16
Peter, Lucy, Susan, and Edmund are once again whisked into the land of Narnia when they least expect it: while in a subway together several years after "The Lion, the Witch, and the Wardrobe." Narnia has been conquered by a plotting king, and the "old Narnians" are now scattered in hiding from the wicked humans. The children must aid Prince Caspian, the exiled rightful owner of Narnia, and win their kingdom back for the good of all.
This is an epic tale with many new and fascinating creatures introduced, with a central theme of faith. The fact that Lucy is the only one able to see the glimpses of Aslan when the struggle becomes bitter speaks volumes about the virtue of faith, and we can already see in this installment how Susan began to lose hers. The children are not only called to Narnia to save it once again, but to teach the new Prince the value of the old and magical ways of Narnia, and to learn a spiritual lesson themselves.
J. Lyon Layden
The Other Side of Yore
Prince Caspian by Hayden P........2007-03-15
I enjoyed Prince Caspian, but not as much as The Lion, The Witch, and The Wardrobe. They were similar in terms of the characters (with new ones too), war, celebrations, and the search for Aslan. They were different in the timing - several hundred years had passed between the first book and Prince Caspian and Cair Paravel was now in ruins. All four children are back in this story, but they are aging faster than in the 'real' life. In fact, by the end of the story you learn that two will not be back to Narnia because they are 'too old'. You meet the new and next King of Narnia, Caspian the Ninth, learn his history and about his evil uncle. The plot involves a conflict between old Narnia and the Telmarines, which leads to a battle between good and evil, and of course, good wins. Aslan becomes apparent and visible to the children when they finally believe in him again. Then, Aslan helps them and all the creatures of old Narnia win the battle. In the end, and before he leaves Narnia, Aslan banishes the Telmarines but gives them another place to live, heals the sick, and rewards those who diplay loyalty.
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Prince Caspian
C.S. Lewis
Manufacturer: The Macmillan Company
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ASIN: B000GZJLCM |
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The second book in the Chronicles of Narnia Series. "This sequel to an earlier adventure in Fairyland ... has the same down-to-earth qualities." ---The New York Times
Customer Reviews:
Magic Wonderland.......2006-02-10
This book is about a boy named Digory and his friend Polly. Digory lived with his uncle Andrew, his aunt, and his mother who was in bed with an illness. Polly was Digory's neighbor and she lived with her parents. One day when Digory was over at Polly's, they went through this tunnel that they believed went up through the attic of every house in London. As they were walking through it they discovered a door. As they opened the door they found themselves inside of a room, and Digory recognized it as his house. They also found that Uncle Andrew was in there also. Polly wasn't paying much attention to Uncle Andrew because there were these beautiful rings right next to him on the side table. When Uncle Andrew noticed she was looking at them he told her she could have one. When Polly touched the ring she disappeared and Digory demanded to know what happened to her. Finally Uncle Andrew told Digory that the ring was a magic one and that he transported Polly into another world and that the only way she could come back is if he, digory, would go in and save her. So digory was also transported, but once he got to the new world he saw a big forest full of trees and on the ground was small ponds as far he could see. Digory found Polly and they jumped into one of the pools to see what would happen. They again found themselves into another world. From there they awaken a white witch, and discover Narnia.
If you read this book you can find out what Narnia is and what they will do when the existences of Narnia is in worldly hands.
Things I liked about this book were how much detail there was in the writing. Some things I didn't like were how it lacked in adventure and excitement, but other than that it was really good.
I would recommend boys and girls to read this book. Also if the person likes fantasies and long series.
Average customer rating:
- Timeless Set, Great Presentation
- Wonderful book , very absorbing
- Meets expectations
- Narnia boxed set
- The Real Fantasy
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The Chronicles of Narnia: The Magician's Nephew/The Lion, the Witch and the Wardrobe/The Horse and His Boy/Prince Caspian/Voyage of the Dawn Treader/The Silver Chair/The Last Battle
C. S. Lewis , and
Pauline Baynes
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The Lion, the Witch and the Wardrobe Color Gift Edition (Narnia)
ASIN: 0064405370 |
Amazon.com
The Chronicles of Narnia, by C.S. Lewis, is one of the very few sets of books that should be read three times: in childhood, early adulthood, and late in life. In brief, four children travel repeatedly to a world in which they are far more than mere children and everything is far more than it seems. Richly told, populated with fascinating characters, perfectly realized in detail of world and pacing of plot, and profoundly allegorical, the story is infused throughout with the timeless issues of good and evil, faith and hope. This boxed set edition includes all seven volumes.
Book Description
Collection includes all seven novels in the series.
Customer Reviews:
Timeless Set, Great Presentation.......2007-10-05
This set is a great gift whether you were a kid 40 years ago or you're a kid now. For years this set of books has taken millions of readers an adventure after adventure. If you enjoyed them as a child or even as an adult, get this set and share the adventure with your loved ones.
Wonderful book , very absorbing.......2007-09-28
I am almost thru with the Chronicles of Narnia and have thoroughly enjoyed it. Great read and so absorbing.
Meets expectations.......2007-09-26
Boxed book set was in like-new condition; on line description was precise. Shipment was prompt. Would not hesitate to order from here again.
Narnia boxed set.......2007-09-17
Nice box. Paperbacks are cheap productions but it is an ok set for the price.
The Real Fantasy.......2007-08-25
Well, usually I despise the kind of fantasy with talking animals and little kids hopping around, as the talking animals are a consequence of the author's inability to paint realistic humans, and the little kids a clear admission that the story is too simple to be appreciated by anyone with more critical sense of a kid.I also hate book "with a moral", they are stiff and feel "constructed" and artificial. This book is an exception in all senses. It's masterfully written, with a class and style the only the greatest writers possess. Characters are alive and breathing, both the humans and the animals, and merge beautifully, and the Christian inspiration flows silently, discreetly and subterraneanly almost all the time, with a class and reserve that reminds of the only Master. A precious , captivating , charming book that would be a real shame to miss if you have not already read it! And don't be discouraged by the horrible movie they made out of it. It had its usefulness for me, as I didn't even know this book existed until I heard about the movie, but luckily I expected the book to outclass that wretched excuse for a movie they did, and, man was I right!
So if you are up for a book that will bring you in a wonderful, colorful, exciting and enriching adventure hurry and get this book, and , as others have said, take care to read it in the original order on which it was written, not the "chronological Narnian order". Some publishers have been stupid enough to publish the various tales in this latter order which spoils most of the surprise and suspense.
The order in which you should read it is:
1) The Lion, The Witch, and The Wardrobe, 2) Prince Caspian, 3)The Voyage of the Dawn Treader, 4) The Silver Chair, 5) The Horse and His Boy, 6) The Magician's Nephew, and 7) The Last Battle
The only shadow on this masterpiece is the superficial and ill-chosen choice of the Narnians villains and enemies as people who are way too reminiscent (you could say copied) from our world stereotypical muslim people. besides being superficial and offensive, it also spoils the beauty of the tale with his gross oversimplification and stink of prejudice, ignorance and lack of fantasy. It's sad to see how the dark sides of religion , prejudice, closure, and lack of tolerance pollute not only our everyday life but even a literature masterpiece. Also, I would consider this book as made by the first 6 tales, the last one "The Last Battle" is hugely inferior to the other six, just a quick closure that lacks structure, coherence and , well, "magic".
Especially painful since it attempts to deal with the core of religion.. want to know what I mean by this? Go read the book!
Average customer rating:
- Armstrong puts Lewis to life.
- Lovely artwork, but extremely abridged!
- Matthew Armstrong - Illustrator Extraordinaire!
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The Return to Narnia: The Rescue of Prince Caspian (Narnia)
C. S. Lewis
Manufacturer: HarperCollins
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Similar Items:
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The Lion, the Witch and the Wardrobe (Narnia)
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Prince Caspian (Narnia)
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Peter Pan in Scarlet
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Flotsam (Caldecott Medal Book)
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The Chronicles of Narnia Boxed Set
ASIN: 0061131105
Release Date: 2006-09-26 |
Book Description
Peter, Susan, Edmund and Lucy return to Narnia for the first time since The Lion, the Witch and the Wardrobe to help Prince Caspian in his fight for the throne.
Customer Reviews:
Armstrong puts Lewis to life........2007-03-05
The art work in this book is incredible. Armstrong took the images from my childhood dreams and put them on paper. The colors are rhythmic and images are fierce. I recommend this book, for the story everyone grew up with, and no one should grow up without. The art work brings the story to life. I love this book.
Lovely artwork, but extremely abridged!.......2006-12-28
I bought this book for my 5 year old son, who loves all things Narnia. It was an impulse buy, and I wish that I had examined it more closely.
Matthew Armstrong is a lovely illustrator. I particularly like his depiction of Edmund- hair slightly messy, head held at a rakish angle. It gives Edmund an impish look. The lively battle scenes also stand out, full of movement and gleaming swords. But the drawing of Aslan is a little disappointing. I can't quite put my finger on it...he's just a little stiff or flat.
While the illustrations are charming, the writing is not. Too many of the sentences start with a "th" word: then, they, there, that, the... As I read it aloud, the wording felt awkward. Perhaps this book is overly simplified for very young audiences? (Although I would argue that even small children can appreciate good writing along with good illustrations)
I won't voluntarily pull this book off the shelf often, as it is not a pleasure to read. But my son will probably enjoy looking at the pictures as he listens to the books on tape.
For illustrations- 4 stars
For writing- 2 stars
Matthew Armstrong - Illustrator Extraordinaire!.......2006-11-16
Of course this book is great because of C.S. Lewis. But Matthew Armstrong's illustrations make this book shine. Matthew worked in Utah for Sony video games, but we met and he illustrated "The Blacksmith's Gift" for me - and made that book shine as well. After "The Blacksmith's Gift" won a Benjamin Franklin Award, he had the opportunity to move up to Narnia! I'd recommend getting any book that Matthew does - he brings his unique abilities to each and every work!
Customer Reviews:
Great 3rd Book.......2007-04-16
Peter, Lucy, Susan, and Edmund are once again whisked into the land of Narnia when they least expect it: while in a subway together several years after "The Lion, the Witch, and the Wardrobe." Narnia has been conquered by a plotting king, and the "old Narnians" are now scattered in hiding from the wicked humans. The children must aid Prince Caspian, the exiled rightful owner of Narnia, and win their kingdom back for the good of all.
This is an epic tale with many new and fascinating creatures introduced, with a central theme of faith. The fact that Lucy is the only one able to see the glimpses of Aslan when the struggle becomes bitter speaks volumes about the virtue of faith, and we can already see in this installment how Susan began to lose hers. The children are not only called to Narnia to save it once again, but to teach the new Prince the value of the old and magical ways of Narnia, and to learn a spiritual lesson themselves.
J. Lyon Layden
The Other Side of Yore
Book Description
Easy-to-use, reproducible lessons on literary terms, comprehension and analysis, critical thinking, related scriptural principles, vocabulary, activities, plus a complete answer key.
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