On Boundary Interpolation for Matrix Valued Schur Functions

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Publisher : American Mathematical Soc.
ISBN 13 : 9781470404604
Total Pages : 107 pages
Book Rating : 4.4/5 (46 download)

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Book Synopsis On Boundary Interpolation for Matrix Valued Schur Functions by : Vladimir Bolotnikov

Download or read book On Boundary Interpolation for Matrix Valued Schur Functions written by Vladimir Bolotnikov and published by American Mathematical Soc.. This book was released on 2006 with total page 107 pages. Available in PDF, EPUB and Kindle. Book excerpt: A number of interpolation problems are considered in the Schur class of $p\times q$ matrix valued functions $S$ that are analytic and contractive in the open unit disk. The interpolation constraints are specified in terms of nontangential limits and angular derivatives at one or more (of a finite number of) boundary points. Necessary and sufficient conditions for existence of solutions to these problems and a description of all the solutions when these conditions are met is given. The analysis makes extensive use of a class of reproducing kernel Hilbert spaces ${\mathcal{H}}(S)$ that was introduced by de Branges and Rovnyak. The Stein equation that is associated with the interpolation problems under consideration is analyzed in detail. A lossless inverse scattering problem is also considered.

On Boundary Interpolation for Matrix Valued Schur Functions

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Author :
Publisher : American Mathematical Soc.
ISBN 13 : 0821840479
Total Pages : 122 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis On Boundary Interpolation for Matrix Valued Schur Functions by : Vladimir Bolotnikov

Download or read book On Boundary Interpolation for Matrix Valued Schur Functions written by Vladimir Bolotnikov and published by American Mathematical Soc.. This book was released on 2006 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: A number of interpolation problems are considered in the Schur class of $p\times q$ matrix valued functions $S$ that are analytic and contractive in the open unit disk. The interpolation constraints are specified in terms of nontangential limits and angular derivatives at one or more (of a finite number of) boundary points. Necessary and sufficient conditions for existence of solutions to these problems and a description of all the solutions when these conditions are met is given.The analysis makes extensive use of a class of reproducing kernel Hilbert spaces ${\mathcal{H (S)$ that was introduced by de Branges and Rovnyak. The Stein equation that is associated with the interpolation problems under consideration is analyzed in detail. A lossless inverse scattering problem isalso considered.

Interpolation, Schur Functions and Moment Problems

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Publisher : Springer Science & Business Media
ISBN 13 : 3764375477
Total Pages : 304 pages
Book Rating : 4.7/5 (643 download)

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Book Synopsis Interpolation, Schur Functions and Moment Problems by : Daniel Alpay

Download or read book Interpolation, Schur Functions and Moment Problems written by Daniel Alpay and published by Springer Science & Business Media. This book was released on 2006-06-22 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Schur analysis originated with an 1917 article which associated to a function, which is analytic and contractive in the open unit disk, a sequence, finite or infinite, of numbers in the open unit disk, called Schur coefficients, often named reflection coefficients in signal processing. This volume comprises seven essays dedicated to the analysis of Schur and Carathéodory functions and to the solutions of problems for these classes.

Topics In Interpolation Theory

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Publisher : Springer Science & Business Media
ISBN 13 : 9783764357238
Total Pages : 526 pages
Book Rating : 4.3/5 (572 download)

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Book Synopsis Topics In Interpolation Theory by : Harry Dym

Download or read book Topics In Interpolation Theory written by Harry Dym and published by Springer Science & Business Media. This book was released on 1997 with total page 526 pages. Available in PDF, EPUB and Kindle. Book excerpt: Vladimir Petrovich Potapov, as remembered by colleagues, friends and former students.- On a minimum problem in function theory and the number of roots of an algebraic equation inside the unit disc.- On tangential interpolation in reproducing kernel Hilbert modules and applications.- Notes on a Nevanlinna-Pick interpolation problem for generalized Nevanlinna functions.- The indefinite metric in the Schur interpolation problem for analytic functions, IV.- Bitangential interpolation for upper triangular operators.- Bitangential interpolation for upper triangular operators when the Pick operator is strictly positive.- Integral representations of a pair of nonnegative operators and interpolation problems in the Stieltjes class.- On recovering a multiplicative integral from its modulus.- On Schur functions and Szegö orthogonal polynomials.- Hilbert spaces of entire functions as a J theory subject.- On transformations of Potapov's fundamental matrix inequality.- An abstract interpolation problem and the extension theory of isometric operators.- On the theory of matrix-valued functions belonging to the Smirnov class.- Integral representation of function of class Ka.- On the theory of entire matrix-functions of exponential type.- Analogs of Nehari and Sarason theorems for character-automorphic functions and some related questions.- The Blaschke-Potapov factorization theorem and the theory of nonselfadjoint operators.- Weyl matrix circles as a tool for uniqueness in the theory of multiplicative representation of J-inner functions.- On a criterion of positive definiteness.- Matrix boundary value problems with eigenvalue dependent boundary conditions (The linear case).- Weyl-Titchmarsh functions of the canonical periodical system of differential equations.- On boundary values of functions regular in a disk.

J-Contractive Matrix Valued Functions and Related Topics

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Publisher : Cambridge University Press
ISBN 13 : 0521883008
Total Pages : 576 pages
Book Rating : 4.5/5 (218 download)

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Book Synopsis J-Contractive Matrix Valued Functions and Related Topics by : Damir Z. Arov

Download or read book J-Contractive Matrix Valued Functions and Related Topics written by Damir Z. Arov and published by Cambridge University Press. This book was released on 2008-11-06 with total page 576 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive introduction to the theory of J-contractive and J-inner matrix valued functions with respect to the open upper half-plane and a number of applications of this theory. It will be of particular interest to those with an interest in operator theory and matrix analysis.

Extension and Interpolation of Linear Operators and Matrix Functions

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Publisher : Birkhäuser
ISBN 13 : 9783764325305
Total Pages : 305 pages
Book Rating : 4.3/5 (253 download)

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Book Synopsis Extension and Interpolation of Linear Operators and Matrix Functions by : I. Gohberg

Download or read book Extension and Interpolation of Linear Operators and Matrix Functions written by I. Gohberg and published by Birkhäuser. This book was released on 1990-10-01 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: The classicallossless inverse scattering (LIS) problem of network theory is to find all possible representations of a given Schur function s(z) (i. e. , a function which is analytic and contractive in the open unit disc D) in terms of an appropriately restricted class of linear fractional transformations. These linear fractional transformations corre­ spond to lossless, causal, time-invariant two port networks and from this point of view, s(z) may be interpreted as the input transfer function of such a network with a suitable load. More precisely, the sought for representation is of the form s(Z) = -{ -A(Z)SL(Z) + B(z)}{ -C(Z)SL(Z) + D(z)} -1 , (1. 1) where "the load" SL(Z) is again a Schur function and _ [A(Z) B(Z)] 0( ) (1. 2) Z - C(z) D(z) is a 2 x 2 J inner function with respect to the signature matrix This means that 0 is meromorphic in D and 0(z)* J0(z) ::5 J (1. 3) for every point zED at which 0 is analytic with equality at almost every point on the boundary Izi = 1. A more general formulation starts with an admissible matrix valued function X(z) = [a(z) b(z)] which is one with entries a(z) and b(z) which are analytic and bounded in D and in addition are subject to the constraint that, for every n, the n x n matrix with ij entry equal to X(Zi)J X(Zj )* i,j=l, . . .

Operator Valued Hardy Spaces

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Publisher : American Mathematical Soc.
ISBN 13 : 0821839802
Total Pages : 78 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Operator Valued Hardy Spaces by : Tao Mei

Download or read book Operator Valued Hardy Spaces written by Tao Mei and published by American Mathematical Soc.. This book was released on 2007 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author gives a systematic study of the Hardy spaces of functions with values in the noncommutative $Lp$-spaces associated with a semifinite von Neumann algebra $\mathcal{M .$ This is motivated by matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), as well as by the recent development of noncommutative martingale inequalities. in this paper noncommutative Hardy spaces are defined by noncommutative Lusin integral function, and it isproved that they are equivalent to those defined by noncommutative Littlewood-Paley G-functions. The main results of this paper include: (i) The analogue in the author's setting of the classical Fefferman duality theorem between $\mathcal{H 1$ and $\mathrm{BMO $. (ii) The atomic decomposition of theauthor's noncommutative $\mathcal{H 1.$ (iii) The equivalence between the norms of the noncommutative Hardy spaces and of the noncommutative $Lp$-spaces $(1

Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls

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Publisher : American Mathematical Soc.
ISBN 13 : 0821839179
Total Pages : 178 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls by : Nicola Arcozzi

Download or read book Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls written by Nicola Arcozzi and published by American Mathematical Soc.. This book was released on 2006 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contents: A tree structure for the unit ball $mathbb B? n$ in $mathbb C'n$; Carleson measures; Pointwise multipliers; Interpolating sequences; An almost invariant holomorphic derivative; Besov spaces on trees; Holomorphic Besov spaces on Bergman trees; Completing the multiplier interpolation loop; Appendix; Bibliography

Entropy and Multivariable Interpolation

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Publisher : American Mathematical Soc.
ISBN 13 : 0821839128
Total Pages : 98 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Entropy and Multivariable Interpolation by : Gelu Popescu

Download or read book Entropy and Multivariable Interpolation written by Gelu Popescu and published by American Mathematical Soc.. This book was released on 2006 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: We define a new notion of entropy for operators on Fock spaces and positive multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (e.g., multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive multi-Toeplitz kernels on free semigroups (resp. multi-analytic operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra $F ninfty$. We obtain several geometric characterizations of the central intertwining lifting, a maximal principle, and a permanence principle for the noncommutative commutant lifting theorem. Under certain natural conditions, we find explicit forms for the maximal entropy solution of this multivariable commutant lifting theorem. All these results are used to solve maximal entropy interpolation problems in several variables. We obtain explicit forms for the maximal entropy solution (as well as its entropy) of the Sarason, Caratheodory-Schur, and Nevanlinna-Pick type interpolation problems for the noncommutative (resp. commutative) analytic Toeplitz algebra $F ninfty$ (resp. $W ninfty$) and their tensor products with $B({\mathcal H , {\mathcal K )$. In particular, we provide explicit forms for the maximal entropy solutions of several interpolation problems on the unit ball of $\mathbb{C n$.

Tangential Boundary Stabilization of Navier-Stokes Equations

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Publisher : American Mathematical Soc.
ISBN 13 : 0821838741
Total Pages : 146 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Tangential Boundary Stabilization of Navier-Stokes Equations by : Viorel Barbu

Download or read book Tangential Boundary Stabilization of Navier-Stokes Equations written by Viorel Barbu and published by American Mathematical Soc.. This book was released on 2006 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: In order to inject dissipation as to force local exponential stabilization of the steady-state solutions, an Optimal Control Problem (OCP) with a quadratic cost functional over an infinite time-horizon is introduced for the linearized N-S equations. As a result, the same Riccati-based, optimal boundary feedback controller which is obtained in the linearized OCP is then selected and implemented also on the full N-S system. For $d=3$, the OCP falls definitely outside the boundaries of established optimal control theory for parabolic systems with boundary controls, in that the combined index of unboundedness--between the unboundedness of the boundary control operator and the unboundedness of the penalization or observation operator--is strictly larger than $\tfrac{3}{2}$, as expressed in terms of fractional powers of the free-dynamics operator. In contrast, established (and rich) optimal control theory [L-T.2] of boundary control parabolic problems and corresponding algebraic Riccati theory requires a combined index of unboundedness strictly less than 1. An additional preliminary serious difficulty to overcome lies at the outset of the program, in establishing that the present highly non-standard OCP--with the aforementioned high level of unboundedness in control and observation operators and subject, moreover, to the additional constraint that the controllers be pointwise tangential--be non-empty; that is, it satisfies the so-called Finite Cost Condition [L-T.2].

The Hilbert Function of a Level Algebra

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Publisher : American Mathematical Soc.
ISBN 13 : 0821839403
Total Pages : 154 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis The Hilbert Function of a Level Algebra by : A. V. Geramita

Download or read book The Hilbert Function of a Level Algebra written by A. V. Geramita and published by American Mathematical Soc.. This book was released on 2007 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let $R$ be a polynomial ring over an algebraically closed field and let $A$ be a standard graded Cohen-Macaulay quotient of $R$. The authors state that $A$ is a level algebra if the last module in the minimal free resolution of $A$ (as $R$-module) is of the form $R(-s)a$, where $s$ and $a$ are positive integers. When $a=1$ these are also known as Gorenstein algebras. The basic question addressed in this paper is: What can be the Hilbert Function of a level algebra? The authors consider the question in several particular cases, e.g., when $A$ is an Artinian algebra, or when $A$ is the homogeneous coordinate ring of a reduced set of points, or when $A$ satisfies the Weak Lefschetz Property. The authors give new methods for showing that certain functions are NOT possible as the Hilbert function of a level algebra and also give new methods to construct level algebras. In a (rather long) appendix, the authors apply their results to give complete lists of all possible Hilbert functions in the case that the codimension of $A = 3$, $s$ is small and $a$ takes on certain fixed values.

Ramanujan's Forty Identities for the Rogers-Ramanujan Functions

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Publisher : American Mathematical Soc.
ISBN 13 : 082183973X
Total Pages : 110 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Ramanujan's Forty Identities for the Rogers-Ramanujan Functions by : Bruce C. Berndt

Download or read book Ramanujan's Forty Identities for the Rogers-Ramanujan Functions written by Bruce C. Berndt and published by American Mathematical Soc.. This book was released on 2007 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sir Arthur Conan Doyle's famous fictional detective Sherlock Holmes and his sidekick Dr. Watson go camping and pitch their tent under the stars. During the night, Holmes wakes his companion and says, ``Watson, look up at the stars and tell me what you deduce.'' Watson says, ``I see millions of stars, and it is quite likely that a few of them are planets just like Earth. Therefore there may also be life on these planets.'' Holmes replies, ``Watson, you idiot. Somebody stole ourtent.'' When seeking proofs of Ramanujan's identities for the Rogers-Ramanujan functions, Watson, i.e., G. N. Watson, was not an ``idiot.'' He, L. J. Rogers, and D. M. Bressoud found proofs for several of the identities. A. J. F. Biagioli devised proofs for most (but not all) of the remaining identities.Although some of the proofs of Watson, Rogers, and Bressoud are likely in the spirit of those found by Ramanujan, those of Biagioli are not. in particular, Biagioli used the theory of modular forms. Haunted by the fact that little progress has been made into Ramanujan's insights on these identities in the past 85 years, the present authors sought ``more natural'' proofs. Thus, instead of a missing tent, we have had missing proofs, i.e., Ramanujan's missing proofs of his forty identities for theRogers-Ramanujan functions. in this paper, for 35 of the 40 identities, the authors offer proofs that are in the spirit of Ramanujan. Some of the proofs presented here are due to Watson, Rogers, and Bressoud, but most are new. Moreover, for several identities, the authors present two or threeproofs. For the five identities that they are unable to prove, they provide non-rigorous verifications based on an asymptotic analysis of the associated Rogers-Ramanujan functions. This method, which is related to the 5-dissection of the generating function for cranks found in Ramanujan's lost notebook, is what Ramanujan might have used to discover several of the more difficult identities. Some of the new methods in this paper can be employed to establish new identities for the Rogers-Ramanujanfunctions.

Classical Function Theory, Operator Dilation Theory, and Machine Computation on Multiply-Connected Domains

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Publisher : American Mathematical Soc.
ISBN 13 : 0821840460
Total Pages : 176 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Classical Function Theory, Operator Dilation Theory, and Machine Computation on Multiply-Connected Domains by : Jim Agler

Download or read book Classical Function Theory, Operator Dilation Theory, and Machine Computation on Multiply-Connected Domains written by Jim Agler and published by American Mathematical Soc.. This book was released on 2008 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work begins with the presentation of generalizations of the classical Herglotz Representation Theorem for holomorphic functions with positive real part on the unit disc to functions with positive real part defined on multiply-connected domains. The generalized Herglotz kernels that appear in these representation theorems are then exploited to evolve new conditions for spectral set and rational dilation conditions over multiply-connected domains. These conditions form the basis for the theoretical development of a computational procedure for probing a well-known unsolved problem in operator theory, the so called rational dilation conjecture. Arbitrary precision algorithms for computing the Herglotz kernels on circled domains are presented and analyzed. These algorithms permit an effective implementation of the computational procedure which results in a machine generated counterexample to the rational dilation conjecture.

Invariant Differential Operators for Quantum Symmetric Spaces

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Publisher : American Mathematical Soc.
ISBN 13 : 0821841319
Total Pages : 104 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Invariant Differential Operators for Quantum Symmetric Spaces by : Gail Letzter

Download or read book Invariant Differential Operators for Quantum Symmetric Spaces written by Gail Letzter and published by American Mathematical Soc.. This book was released on 2008 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper studies quantum invariant differential operators for quantum symmetric spaces in the maximally split case. The main results are quantum versions of theorems of Harish-Chandra and Helgason: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Moreover, the image of the center under this map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not of four exceptional types. In the process, the author finds a particularly nice basis for the quantum invariant differential operators that provides a new interpretation of difference operators associated to Macdonald polynomials.

The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra

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Publisher : American Mathematical Soc.
ISBN 13 : 0821840541
Total Pages : 98 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra by : Michael Kapovich

Download or read book The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra written by Michael Kapovich and published by American Mathematical Soc.. This book was released on 2008 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper the authors apply their results on the geometry of polygons in infinitesimal symmetric spaces and symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over $\mathbb{Q}$ and its complex Langlands' dual. The authors give a new proof of the Saturation Conjecture for $GL(\ell)$ as a consequence of their solution of the corresponding saturation problem for the Hecke structure constants for all split reductive algebraic groups over $\mathbb{Q}$.

Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories

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Publisher : American Mathematical Soc.
ISBN 13 : 0821841424
Total Pages : 208 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories by : Dominic Verity

Download or read book Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories written by Dominic Verity and published by American Mathematical Soc.. This book was released on 2008 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: The primary purpose of this work is to characterise strict $\omega$-categories as simplicial sets with structure. The author proves the Street-Roberts conjecture in the form formulated by Ross Street in his work on Orientals, which states that they are exactly the ``complicial sets'' defined and named by John Roberts in his handwritten notes of that title (circa 1978). On the way the author substantially develops Roberts' theory of complicial sets itself and makes contributions to Street's theory of parity complexes. In particular, he studies a new monoidal closed structure on the category of complicial sets which he shows to be the appropriate generalisation of the (lax) Gray tensor product of 2-categories to this context. Under Street's $\omega$-categorical nerve construction, which the author shows to be an equivalence, this tensor product coincides with those of Steiner, Crans and others.

Semisolvability of Semisimple Hopf Algebras of Low Dimension

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Publisher : American Mathematical Soc.
ISBN 13 : 0821839489
Total Pages : 138 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Semisolvability of Semisimple Hopf Algebras of Low Dimension by : Sonia Natale

Download or read book Semisolvability of Semisimple Hopf Algebras of Low Dimension written by Sonia Natale and published by American Mathematical Soc.. This book was released on 2007 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author proves that every semisimple Hopf algebra of dimension less than $60$ over an algebraically closed field $k$ of characteristic zero is either upper or lower semisolvable up to a cocycle twist.