This book gives an introduction to the very active field of combinatorics of affine Schubert calculus, explains the current state of the art, and states the current open problems. Affine Schubert calculus lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory. The other direction is the study of the Schubert bases of the (co)homology of the affine Grassmannian, an algebro-topological formulation of a problem in enumerative geometry. This is the first introductory text on this subject. It contains many examples in Sage, a free open source general purpose mathematical software system, to entice the reader to investigate the open problems. This book is written for advanced undergraduate and graduate students, as well as researchers, who want to become familiar with this fascinating new field.
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.
The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian ${\rm Gr}$ associated with $SL(n,\mathbb{C})$.Their main results are: Pieri rules for the Schubert bases of $H^*({\rm Gr})$ and $H_*({\rm Gr})$, which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. A new combinatorial definition for $k$-Schur functions, which represent the Schubert basis of $H_*({\rm Gr})$. A combinatorial interpretation of the pairing $H^*({\rm Gr})\times H_*({\rm Gr}) \rightarrow\mathbb Z$ induced by the cap product.
40 million American women of marriageable age are single. This approachable essay addresses many of their concerns in a profound and delightful way. Inspired by the author's own experiences as well as by 18th century philosophers, and literary and historical references, it offers insights and the courage to help us revel in the game of life, the delight of reading, the art of the journey, and the right to say "no" to chains of obligations and family.
Murray and Nadel’s Textbook of Respiratory Medicine has long been the definitive and comprehensive pulmonary disease reference. Robert J. Mason, MD now presents the fifth edition in full color with new images and highlighted clinical elements. The fully searchable text is also online at www.expertconsult.com, along with regular updates, video clips, additional images, and self-assessment questions. This new edition has been completely updated and remains the essential tool you need to care for patients with pulmonary disease. Consult this title on your favorite e-reader, conduct rapid searches, and adjust font sizes for optimal readability. Compatible with Kindle®, nook®, and other popular devices. Master the scientific principles of respiratory medicine and its clinical applications. Work through differential diagnosis using detailed explanations of each disease entity. Learn new subjects in Pulmonary Medicine including Genetics, Ultrasound, and other key topics. Grasp the Key Points in each chapter. Search the full text online at expertconsult.com, along with downloadable images, regular updates, more than 50 videos, case studies, and self-assessment questions. Consult new chapters covering Ultrasound, Innate Immunity, Adaptive Immunity, Deposition and Clearance, Ventilator-Associated Pneumonia. Find critical information easily using the new full-color design that enhances teaching points and highlights challenging concepts. Apply the expertise and fresh ideas of three new editors—Drs. Thomas R. Martin, Talmadge E. King, Jr., and Dean E. Schraufnagel. Review the latest developments in genetics with advice on how the data will affect patient care.
The Patient-Centered Clinical Method (PCCM) has been a core tenet of the practice and teaching of medicine since the first edition of Patient-Centered Medicine - Transforming the Clinical Method was published in 1995. This timely fourth edition continues to define the principles underpinning the patient-centered clinical method using four major components, clarifying its evolution and consequent development, and it brings the reader fully up to date. It reinforces the relevance of the method in the current much-changed realities of health care in a world where virtual care will remain common, dependence on technology is rising, and societal changes away from compassion, equity, and relationships toward confrontation, inequity, and self-absorption. Fully revised by its highly experienced author team ensuring wide interest and written for those practising now and for the practitioners of the future, this new edition will be welcomed by a wide international audience comprising all health professionals from medicine, nursing, social work, occupational therapy, physical therapy, pharmacy, veterinary medicine, and other fields.
This textbook provides a comprehensive yet accessible treatment of weather and climate prediction, for graduate students, researchers and professionals. It teaches the strengths, weaknesses and best practices for the use of atmospheric models. It is ideal for the many scientists who use such models across a wide variety of applications. The book describes the different numerical methods, data assimilation, ensemble methods, predictability, land-surface modeling, climate modeling and downscaling, computational fluid-dynamics models, experimental designs in model-based research, verification methods, operational prediction, and special applications such as air-quality modeling and flood prediction. This volume will satisfy everyone who needs to know about atmospheric modeling for use in research or operations. It is ideal both as a textbook for a course on weather and climate prediction and as a reference text for researchers and professionals from a range of backgrounds: atmospheric science, meteorology, climatology, environmental science, geography, and geophysical fluid mechanics/dynamics.
Be refreshed by the power and beauty of faithful Christ-centered teaching. Engage the issues of today. Build a firm foundation for this generation—and the next. To be an evangelical is to believe in life-changing truth; it is to believe that the Bible not only gives us the key to eternal life in Jesus but also the way to live today. This is news worth sharing. But how do we do this in a rapidly changing culture that appears to be shifting from a general disinterest in the gospel to a more active opposition to it? When it feels like our faith is being challenged in every area, how should we respond? The Evangelical Study Bible will rejuvenate your faith through the unchanging truth of the gospel. With verse-by-verse commentary and interesting sidebars that dig deeper into the historical and contextual background of Scripture, it will help you grow in your understanding and appreciation of the life-changing truths of the Christian faith. Also included are more than fifty articles designed to help you better understand your faith and effectively share it with others. Many of these articles address the most pressing issues of our day, equipping you to engage others with confidence, and laying a foundation of truth for the next generation. Features include: Verse-by-verse commentary on the entire Bible from trusted scholars Over 200 short articles on Christian teachings and doctrine 100 articles to equip readers to navigate the cultural engagement issues of today 100 cultural background notes 100 archeological notes 550 Word Studies to explore key terms of the Bible Biographies of 150 biblical figures Full-color maps, charts, and family trees Book introductions, with outlines and reading plans Topical indexes Cross references Concordance Clear and readable NKJV Comfort Print® typeface in a 10.5-point print size
We give a combinatorial expansion of a Schubert homology class in the affine Grassmannian GrSLk into Schubert homology classes in GrSLk+1. This is achieved by studying the combinatorics of a new class of partitions called k-shapes, which interpolates between k k-cores and kk+1-cores. We define a symmetric function for each k-shape, and show that they expand positively in terms of dual k-Schur functions. We obtain an explicit combinatorial description of the expansion of an ungraded k k-Schur function into k+1-Schur functions. As a corollary, we give a formula for the Schur expansion of an ungraded k-Schur function.
This is a first-person account of an overseas Chinese who was involved in the armed struggle for the independence of Malaya and Singapore and who evolved through several roles during those tumultuous times — from an anti-Japanese agent in World War II, to a communist guerrilla leader during the Emergency and subsequently to a citizen who worked for the Special Branch to defend the nation during Confrontation.The book also offers an in-depth first-person account of the history, operational methodology, struggles and ultimate failure of the Malayan Communist Party in the armed struggle and provides an in-depth analysis from a direct participant of those events. It describes the zeitgeist of the time amongst overseas Chinese in Malaya and Singapore — in the prelude to and the aftermath of World War II and the subsequent struggle of independence in a world where they had hitherto been second-class residents. Due to the inherent secrecy of the clandestine Malayan Communist Party, many of the events described in the book were not previously known; events that the protagonist in his old age now wishes to record for posterity.
Selected as a Doody’s Core Title for 2023! Featuring what is perhaps the largest single collection of Aunt Minnie-like cases in any one publication, Aunt Minnie’s Atlas and Imaging-Specific Diagnosis, 5th Edition, written by well-known subspecialists and edited by Dr. Thomas L. Pope, Jr., is an excellent study tool for radiology board examinations. More than 1,100 high-quality images highlight over 380 cases that depict radiologic findings so specific and compelling that no realistic differential diagnosis exists. These classic, unforgettable images represent all modalities and subspecialties in diagnostic imaging, making this unique reference an ideal review tool for cased-based and image-heavy radiology board and oral board examinations.
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