The Cortex and the Critical Point

Author: John M. Beggs
Publisher: MIT Press
ISBN: 0262544032
Category : Science
Languages : en
Pages : 217

Book Description
How the cerebral cortex operates near a critical phase transition point for optimum performance. Individual neurons have limited computational powers, but when they work together, it is almost like magic. Firing synchronously and then breaking off to improvise by themselves, they can be paradoxically both independent and interdependent. This happens near the critical point: when neurons are poised between a phase where activity is damped and a phase where it is amplified, where information processing is optimized, and complex emergent activity patterns arise. The claim that neurons in the cortex work best when they operate near the critical point is known as the criticality hypothesis. In this book John Beggs—one of the pioneers of this hypothesis—offers an introduction to the critical point and its relevance to the brain. Drawing on recent experimental evidence, Beggs first explains the main ideas underlying the criticality hypotheses and emergent phenomena. He then discusses the critical point and its two main consequences—first, scale-free properties that confer optimum information processing; and second, universality, or the idea that complex emergent phenomena, like that seen near the critical point, can be explained by relatively simple models that are applicable across species and scale. Finally, Beggs considers future directions for the field, including research on homeostatic regulation, quasicriticality, and the expansion of the cortex and intelligence. An appendix provides technical material; many chapters include exercises that use freely available code and data sets.

Critical Point

Author: S. L. Huang
Publisher: Tor Books
ISBN: 9781250180360
Category : Fiction
Languages : en
Pages : 368

Book Description
S. L. Huang's Critical Point is a breakout SF thriller for fans of John Scalzi and Greg Rucka. Math-genius mercenary Cas Russell has stopped a shadow organization from brainwashing the world and discovered her past was deliberately erased and her superhuman abilities deliberately created. And that's just the start: when a demolitions expert targets Cas and her friends, and the hidden conspiracy behind Cas's past starts to reappear, the past, present, and future collide in a race to save one of her dearest friends.

Critical Point Theory and Submanifold Geometry

Author: Richard S. Palais
Publisher: Springer
ISBN: 3540459960
Category : Mathematics
Languages : en
Pages : 276

Book Description

Linking Methods in Critical Point Theory

Author: Martin Schechter
Publisher: Springer Science & Business Media
ISBN: 9780817640958
Category : Mathematics
Languages : en
Pages : 320

Book Description
As is well known, The Great Divide (a.k.a. The Continental Divide) is formed by the Rocky Mountains stretching from north to south across North America. It creates a virtual "stone wall" so high that wind, rain, snow, etc. cannot cross it. This keeps the weather distinct on both sides. Since railroad trains cannot climb steep grades and tunnels through these mountains are almost formidable, the Canadian Pacific Railroad searched for a mountain pass providing the lowest grade for its tracks. Employees discovered a suitable mountain pass, called the Kicking Horse Pass, el. 5404 ft., near Banff, Alberta. (One can speculate as to the reason for the name.) This pass is also used by the Trans-Canada Highway. At the highest point of the pass the railroad tracks are horizontal with mountains rising on both sides. A mountain stream divides into two branches, one flowing into the Atlantic Ocean and the other into the Pacific. One can literally stand (as the author did) with one foot in the Atlantic Ocean and the other in the Pacific. The author has observed many mountain passes in the Rocky Mountains and Alps. What connections do mountain passes have with nonlinear partial dif ferential equations? To find out, read on ...

Minimax Methods in Critical Point Theory with Applications to Differential Equations

Author: Paul H. Rabinowitz
Publisher: American Mathematical Soc.
ISBN: 0821807153
Category : Mathematics
Languages : en
Pages : 110

Book Description
The book provides an introduction to minimax methods in critical point theory and shows their use in existence questions for nonlinear differential equations. An expanded version of the author's 1984 CBMS lectures, this volume is the first monograph devoted solely to these topics. Among the abstract questions considered are the following: the mountain pass and saddle point theorems, multiple critical points for functionals invariant under a group of symmetries, perturbations from symmetry, and variational methods in bifurcation theory. The book requires some background in functional analysis and differential equations, especially elliptic partial differential equations. It is addressed to mathematicians interested in differential equations and/or nonlinear functional analysis, particularly critical point theory.

Critical Point Theory and Hamiltonian Systems

Author: Jean Mawhin
Publisher: Springer Science & Business Media
ISBN: 1475720610
Category : Science
Languages : en
Pages : 292

Book Description
FACHGEB The last decade has seen a tremendous development in critical point theory in infinite dimensional spaces and its application to nonlinear boundary value problems. In particular, striking results were obtained in the classical problem of periodic solutions of Hamiltonian systems. This book provides a systematic presentation of the most basic tools of critical point theory: minimization, convex functions and Fenchel transform, dual least action principle, Ekeland variational principle, minimax methods, Lusternik- Schirelmann theory for Z2 and S1 symmetries, Morse theory for possibly degenerate critical points and non-degenerate critical manifolds. Each technique is illustrated by applications to the discussion of the existence, multiplicity, and bifurcation of the periodic solutions of Hamiltonian systems. Among the treated questions are the periodic solutions with fixed period or fixed energy of autonomous systems, the existence of subharmonics in the non-autonomous case, the asymptotically linear Hamiltonian systems, free and forced superlinear problems. Application of those results to the equations of mechanical pendulum, to Josephson systems of solid state physics and to questions from celestial mechanics are given. The aim of the book is to introduce a reader familiar to more classical techniques of ordinary differential equations to the powerful approach of modern critical point theory. The style of the exposition has been adapted to this goal. The new topological tools are introduced in a progressive but detailed way and immediately applied to differential equation problems. The abstract tools can also be applied to partial differential equations and the reader will also find the basic references in this direction in the bibliography of more than 500 items which concludes the book. ERSCHEIN

The Critical Point

Author: C Domb
Publisher: CRC Press
ISBN: 1482295261
Category : Science
Languages : en
Pages : 395

Book Description
The relationship between liquids and gases engaged the attention of a number of distinguished scientists in the mid 19th Century. In a definitive paper published in 1869, Thomas Andrews described experiments he performed on carbon dioxide and from which he concluded that a critical temperature exists below which liquids and gases are distinct phase

Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems

Author: Leszek Gasinski
Publisher: CRC Press
ISBN: 1420035037
Category : Mathematics
Languages : en
Pages : 790

Book Description
Starting in the early 1980s, people using the tools of nonsmooth analysis developed some remarkable nonsmooth extensions of the existing critical point theory. Until now, however, no one had gathered these tools and results together into a unified, systematic survey of these advances. This book fills that gap. It provides a complete presentation of nonsmooth critical point theory, then goes beyond it to study nonlinear second order boundary value problems. The authors do not limit their treatment to problems in variational form. They also examine in detail equations driven by the p-Laplacian, its generalizations, and their spectral properties, studying a wide variety of problems and illustrating the powerful tools of modern nonlinear analysis. The presentation includes many recent results, including some that were previously unpublished. Detailed appendices outline the fundamental mathematical tools used in the book, and a rich bibliography forms a guide to the relevant literature. Most books addressing critical point theory deal only with smooth problems, linear or semilinear problems, or consider only variational methods or the tools of nonlinear operators. Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems offers a comprehensive treatment of the subject that is up-to-date, self-contained, and rich in methods for a wide variety of problems.

Critical Point Theory and Its Applications

Author: Wenming Zou
Publisher: Springer Science & Business Media
ISBN: 0387329684
Category : Mathematics
Languages : en
Pages : 323

Book Description
This book presents some of the latest research in critical point theory, describing methods and presenting the newest applications. Coverage includes extrema, even valued functionals, weak and double linking, sign changing solutions, Morse inequalities, and cohomology groups. Applications described include Hamiltonian systems, Schrödinger equations and systems, jumping nonlinearities, elliptic equations and systems, superlinear problems and beam equations.

Cirque Du Freak: A Living Nightmare

Author: Darren Shan
Publisher: Little, Brown Books for Young Readers
ISBN: 031604184X
Category : Young Adult Fiction
Languages : en
Pages : 130

Book Description
From the Master of Horror comes the first gripping book in the twelve book New York Times bestselling Saga of Darren Shan. Start the tale from the beginning in the book that inspired the feature film The Vampire's Assistant and petrified devoted fans worldwide. A young boy named Darren Shan and his best friend, Steve, get tickets to the Cirque Du Freak, a wonderfully gothic freak show featuring weird, frightening half human/half animals who interact terrifyingly with the audience. In the midst of the excitement, true terror raises its head when Steve recognizes that one of the performers-- Mr. Crepsley-- is a vampire! Stever remains after the show finishes to confront the vampire-- but his motives are surprising! In the shadows of a crumbling theater, a horrified Darren eavesdrops on his friend and the vampire, and is witness to a monstrous, disturbing plea. As if by destiny, Darren is pulled to Mr. Crepsley and what follows is his horrifying descent into the dark and bloody world of vampires. This is the beginning of Darren's story.