Spin Models, Association Schemes and the Nakanishi–Montesinos Conjecture

AbstractA 3-transformation of a link is a local change which replaces two strings that are three times half twisted around each other by two untwisted strings (and vice versa). The Nakanishi–Montesinos (NM) conjecture asserts that this 3-transformation can unknot any link. We introduce the notion of the NM-spin model, which gives a link invariant preserved by 3-transformation. We try to classify such spin models and determine the corresponding link invariant. It is proved that the dimension of the Bose–Mesner algebra generated by the spin model is ≤4. For dimension 1 and 2, there is no such spin model, but for dimension 3, there exists a unique one. Its link invariant is a non-trivial specialization of the Kauffman polynomial, but does not distinguish trivial links from the others, and hence cannot disprove the NM conjecture. For dimension 4, we give a family of NM-spin models. The corresponding link invariant is identified and does not distinguish trivial links from the others. Strong regularity and triple regularity of the Bose–Mesner algebra and its fusions are studied

Clinical Management and Evolving Novel Therapeutic Strategies for Patients with Brain Tumors

A dramatic increase in knowledge regarding the molecular biology of brain tumors has been established over the past few years, and this has lead to the development of novel therapeutic strategies for these patients. In this book a review of the options available for the clinical management of patients with these tumors are outlined. In addition advances in radiology both for pre-operative diagnostic purposes along with surgical planning are described. Furthermore a review of newer developments in chemotherapy along with the evolving field of photodynamic therapy both for intra-operative management and subsequent therapy is provided. A discussion of certain surgical management issues along with tumor induced epilepsy is included. Finally a discussion of the management of certain unique problems including brain metastases, brainstem glioma, central nervous system lymphoma along with issues involving patients with a brain tumor and pregnancy is provided

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal