Fredholm eigenvalues and quasiconformal geometry of polygons

2020 ◽  
Vol 17 (3) ◽  
pp. 325-364
Author(s):  
Samuel Krushkal
Keyword(s):  
Potential Theory ◽  
Approximation Theory ◽  
Complex Analysis ◽  
Previous Survey ◽  
Quasiconformal Maps ◽  
Convex Polygons ◽  
Universal Teichmüller Space ◽  
Important Open Problem ◽  
Explicit Determination

An important open problem in geometric complex analysis is to establish algorithms for the explicit determination of the basic curvilinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues, and the quasireflection coefficient. This is important also for the potential theory but has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory. This survey extends our previous survey of 2005 and presents the newapproaches and recent essential progress in this field of geometric complex analysis and potential theory, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals (to which the notion of Fredholm eigenvalues also can be extended).

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

Two new measurement methods for explicit determination of complex permittivity

1998 ◽  
Vol 46 (11) ◽  
pp. 1614-1619 ◽  
Author(s):  
C. Wan ◽  
B. Nauwelaers ◽  
W. De Raedt ◽  
M. Van Rossum
Keyword(s):  
Complex Permittivity ◽  
Measurement Methods ◽  
Explicit Determination

Extensions of I-Bisimple Semigroups

1967 ◽  
Vol 19 ◽  
pp. 419-426 ◽  
Author(s):  
R. J. Warne
Keyword(s):  
Homomorphic Image ◽  
Natural Order ◽  
Isomorphism Theorem ◽  
Maximal Group ◽  
Homomorphism Theorem ◽  
Complete Determination ◽  
Explicit Determination ◽  
Usual Order

A bisimple semigroup S is called I-bisimple if Es, the set of idempotents of S, with its natural order is order-isomorphic to I, the set of integers, under the reverse of the usual order. In (9), the author completely determined the structure of I-bisimple semigroups mod groups; in this paper, he also gave an isomorphism theorem, a homomorphism theorem, an explicit determination of the maximal group homomorphic image, and a complete determination of the congruences for these semigroups.

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