A year after Kazuto and Asuna escaped from Aincrad, the deadly setting of Sword Art Online, they're still coping with the aftermath as they try to recover from the harrowing experience. Meanwhile, there's a new VRMMO game sweeping the world's gamers--Gun Gale Online. When a mysterious player who calls himself "Death Gun" claims to be responsible for real-world deaths, Kazuto (aka Kirito) is recruited to enter the game to investigate, bringing an end to his hard-won peace!

This book provides a unique and highly accessible approach to singularity theory from the perspective of differential geometry of curves and surfaces. It is written by three leading experts on the interplay between two important fields — singularity theory and differential geometry. The book introduces singularities and their recognition theorems, and describes their applications to geometry and topology, restricting the objects of attention to singularities of plane curves and surfaces in the Euclidean 3-space. In particular, by presenting the singular curvature, which originated through research by the authors, the Gauss–Bonnet theorem for surfaces is generalized to those with singularities. The Gauss–Bonnet theorem is intrinsic in nature, that is, it is a theorem not only for surfaces but also for 2-dimensional Riemannian manifolds. The book also elucidates the notion of Riemannian manifolds with singularities. These topics, as well as elementary descriptions of proofs of the recognition theorems, cannot be found in other books. Explicit examples and models are provided in abundance, along with insightful explanations of the underlying theory as well. Numerous figures and exercise problems are given, becoming strong aids in developing an understanding of the material. Readers will gain from this text a unique introduction to the singularities of curves and surfaces from the viewpoint of differential geometry, and it will be a useful guide for students and researchers interested in this subject.

'In a class populated by students who already have some exposure to the concept of a manifold, the presence of chapter 3 in this text may make for an unusual and interesting course. The primary function of this book will be as a text for a more conventional course in the classical theory of curves and surfaces.'MAA ReviewsThis engrossing volume on curve and surface theories is the result of many years of experience the authors have had with teaching the most essential aspects of this subject. The first half of the text is suitable for a university-level course, without the need for referencing other texts, as it is completely self-contained. More advanced material in the second half of the book, including appendices, also serves more experienced students well.Furthermore, this text is also suitable for a seminar for graduate students, and for self-study. It is written in a robust style that gives the student the opportunity to continue his study at a higher level beyond what a course would usually offer. Further material is included, for example, closed curves, enveloping curves, curves of constant width, the fundamental theorem of surface theory, constant mean curvature surfaces, and existence of curvature line coordinates.Surface theory from the viewpoint of manifolds theory is explained, and encompasses higher level material that is useful for the more advanced student. This includes, but is not limited to, indices of umbilics, properties of cycloids, existence of conformal coordinates, and characterizing conditions for singularities.In summary, this textbook succeeds in elucidating detailed explanations of fundamental material, where the most essential basic notions stand out clearly, but does not shy away from the more advanced topics needed for research in this field. It provides a large collection of mathematically rich supporting topics. Thus, it is an ideal first textbook in this field.

Deep in an unfamiliar forest, Kirito awakens with sensations far too convincing for a virtual world... With memories of a life he doesn't remember living. A boy and a girl...and a name that sticks in his mind: Alice.

After turning the tables on the Yamato dominion's overwhelmingly more powerful forces, the high school prodigies press their advantage all the way to the inner defenses of their enemy's stronghold, Azuchi Castle. In a desperate, last-ditch effort to win a battle that appears all but already lost, the cornered Princess Mayoi unleashes her brainwashing magic once more. As the fighting spirals out of anyone's control, Mayoi's tragic past and the great, terrible secret of Yamato's history will be revealed...

In order to uncover Death Gun's true identity, Kirito prepares to enter the Bullet of Bullets tournament! As Death Gun's newest target, Sinon stands side-by-side with Kirito, ready to overcome her past trauma by defeating their common foe. At long last, it's time for Kirito's showdown against Death Gun! Why and how did all those players die, and who really killed them? Seeking answers, Kirito steels himself and prepares for battle in this thrilling conclusion to the series!

After Ronie and Tiese try to save their friend from the clutches of noble knights Raios and Humbert, they end up getting captured themselves! With the two girls bound and helpless, Kirito's going to need Eugeo's help to save them...but Eugeo finds himself paralyzed by the thought of breaking the Taboo Index! With the fate of his page hanging in the balance, can he violate his world's most sacred code to rescue her?

After Ronie and Tiese try to save their friend from the clutches of noble knights Raios and Humbert, they end up getting captured themselves! With the two girls bound and helpless, Kirito's going to need Eugeo's help to save them...but Eugeo finds himself paralyzed by the thought of breaking the Taboo Index! With the fate of his page hanging in the balance, can he violate his world's most sacred code to rescue her?

The curtain closes on the story of the high school prodigies' revolution in another world! Having put an end to the civil war in Yamato, Tsukasa and the others can finally head for the elf village and put what they learned from Adel's memoir to use. Just who or what exactly is the "wicked dragon" Yggrda, and why did she summon them to this world? As their long journey comes to an end, it's finally time for them to learn the truth...

Someone is manipulating the Republic of Elm's first national elections from the shadows to fulfill their own selfish desires, and it's up to God Akatsuki to stop them! Meanwhile, Tsukasa and the Resistance carry out a shocking plan to overturn the immense difference in numbers between them and the Yamato Dominion and take down this inhumane government...

The high school prodigies have secured the Republic of Elm's place in the world by issuing their new currency, but before they can even catch their breath from dealing with the last international crisis, a new one arises in the form of the conquered Yamato Empire's Princess Kaguya. When she pleads for the salvation of her country from the Freyjagard Empire under the pretense of the Seven Luminaries' hallowed ideal of "equality for all," the prodigies may have no choice but to renew tensions with their former foe...

After discovering what Princess Mayoi has done to the minds of the people of Yamato, Tsukasa decides to side with the Resistance to overthrow the Yamato government. Meanwhile, the Republic of Elm's national elections are coming to a close, but will a horrific incident at the last minute end up swaying its results...?!

Taking down the local lord was child's play for the seven high school prodigies, and though they're well on their way to all out war with the empire, they're first going...to church?! The geniuses set their sights on religion in the alternate realm of Freyjagard, where abuse of power runs rampant in all aspects of society. Intent on wielding their special skillsets for the good of the little guy, the brilliant high schoolers set about changing the world and making history!

After Kirito emerges the victor of his battle with Shinon, all that remains is the final round! Who will triumph in the Bullet of Bullets tournament?

In Japan, there are seven superhuman highschoolers who are world-class geniuses in various fields ranging from governance to economics and beyond. On one fateful day, these seven wind up in a plane accident and wake up only to find themselves in another world. Finding themselves in a foreign place where magic and beast-people are real, they immediately proceed to panic-or at least that's what would've happened if they were some run of the mill students!! If anything, these seven are actually using their talents to do absolutely ludicrous things so they can go home... "Let's take it easy, because if we really let loose, I'm pretty sure we'll end up destroying this world."

The prodigies, continuing their revolution against the empire, are in for a huge shock when the truth of the Blue Brigade's loyalties comes to light! And it seems like surprises aren't in short supply. Despite the victory the prodigies achieved in the battle against Gustav, it may be that the fight against him isn't quite over...

As the day of the Goss Currency Exchange draws near, it's finally time for Masato Sanada, Earth's greatest businessman, to show this new world what he can really do. In order to ensure a future for the common people, the High School Prodigies are ready and able to overturn the entirety of this world's status quo.

'In a class populated by students who already have some exposure to the concept of a manifold, the presence of chapter 3 in this text may make for an unusual and interesting course. The primary function of this book will be as a text for a more conventional course in the classical theory of curves and surfaces.'MAA ReviewsThis engrossing volume on curve and surface theories is the result of many years of experience the authors have had with teaching the most essential aspects of this subject. The first half of the text is suitable for a university-level course, without the need for referencing other texts, as it is completely self-contained. More advanced material in the second half of the book, including appendices, also serves more experienced students well.Furthermore, this text is also suitable for a seminar for graduate students, and for self-study. It is written in a robust style that gives the student the opportunity to continue his study at a higher level beyond what a course would usually offer. Further material is included, for example, closed curves, enveloping curves, curves of constant width, the fundamental theorem of surface theory, constant mean curvature surfaces, and existence of curvature line coordinates.Surface theory from the viewpoint of manifolds theory is explained, and encompasses higher level material that is useful for the more advanced student. This includes, but is not limited to, indices of umbilics, properties of cycloids, existence of conformal coordinates, and characterizing conditions for singularities.In summary, this textbook succeeds in elucidating detailed explanations of fundamental material, where the most essential basic notions stand out clearly, but does not shy away from the more advanced topics needed for research in this field. It provides a large collection of mathematically rich supporting topics. Thus, it is an ideal first textbook in this field.

With the successful establishment of the Elm Republic, the Freyjagard Empire is engulfed in civil war, and the Yamato Empire finds itself embroiled in the Resistance's battle. Now it's up to Tsukasa, Aoi, and Ringo to lead the charge using their resourcefulness, prowess in combat, and technological mastery!

After starting an economic revolution in the city of Dormundt, the high school prodigies are up to their eyeballs in money and resources. They've also managed to save little Roo from a life of slavery, so they've earned a well-deserved rest. That is, until "Silver Knight" Inzaghi decides to attack Elm Village, pillage its newfound wealth, and abduct poor Lyrule as a concubine for the feudal lord. With the greatest minds and talents known on Earth, this dream team is now set to revolutionize the other world!!

In Underworld, a virtual land from which there seems no escape, Kirito meets Eugeo, an NPC destined to follow his Calling-his life's goal that must be achieved before he can follow his own path. What awaits Kirito in this new land?

After successfully garnering support for their religion, "the Seven Luminaries," the high school prodigies prepare to head into an all-out war with the empire. But when Gustav uses magic power to suddenly set everything ablaze in a fit of fury, its up to this dream team to use their superior abilities and technology to tilt the scales back in their favor!