Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebr

Little k has an adventure with items beginning with his letter's sound, such as kittens, kingbirds, a koala, and kids in a kindergarten. Additional features to aid comprehension include rebus icons for word recognition, a word list for review, activities for further learning, a note to parents and educators, and an introduction to the author and illustrator.

A NEW YORK TIMES NOTABLE BOOK Once in a great while a writer comes along who can truly capture the drama and passion of the life of a family. David James Duncan, author of the novel The River Why and the collection River Teeth, is just such a writer. And in The Brothers K he tells a story both striking and in its originality and poignant in its universality. This touching, uplifting novel spans decades of loyalty, anger, regret, and love in the lives of the Chance family. A father whose dreams of glory on a baseball field are shattered by a mill accident. A mother who clings obsessively to religion as a ward against the darkest hour of her past. Four brothers who come of age during the seismic upheavals of the sixties and who each choose their own way to deal with what the world has become. By turns uproariously funny and deeply moving, and beautifully written throughout, The Brothers K is one of the finest chronicles of our lives in many years. Praise for The Brothers K “The pages of The Brothers K sparkle.”—The New York Times Book Review “Duncan is a wonderfully engaging writer.”—Los Angeles Times “This ambitious book succeeds on almost every level and every page.”—USA Today “Duncan’s prose is a blend of lyrical rhapsody, sassy hyperbole and all-American vernacular.”—San Francisco Chronicle “The Brothers K affords the . . . deep pleasures of novels that exhaustively create, and alter, complex worlds. . . . One always senses an enthusiastic and abundantly talented and versatile writer at work.”—The Washington Post Book World “Duncan . . . tells the larger story of an entire popular culture struggling to redefine itself—something he does with the comic excitement and depth of feeling one expects from Tom Robbins.”—Chicago Tribune

These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.

The authors give a combinatorial expansion of a Schubert homology class in the affine Grassmannian $\mathrm{Gr}_{\mathrm{SL}_k}$ into Schubert homology classes in $\mathrm{Gr}_{\mathrm{SL}_{k+1}}$. This is achieved by studying the combinatorics of a new class of partitions called $k$-shapes, which interpolates between $k$-cores and $k+1$-cores. The authors define a symmetric function for each $k$-shape, and show that they expand positively in terms of dual $k$-Schur functions. They obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, they give a formula for the Schur expansion of an ungraded $k$-Schur function.

Volume 2 of two - also available in a set of both volumes.

We apply the notion of single-valued neutrosophic sets to K-algebras. We develop the concept of single-valued neutrosophic K-subalgebras, and present some of their properties. Moreover, we study the behavior of single-valued neutrosophic K-subalgebras under homomorphism.