Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass

Riemann, Georg Friedrich Bernhard

doi:10.1007/978-3-663-10149-9_1

Cited by 59 publications

(46 citation statements)

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“…In 1877, Moens [23] and Kortweg [20] independently published analyses of flow in thin-walled elastic vessels, deriving what is now known as the Moens-Kortweg equation for the wave speed. Riemann [25] (1860) provided the analytical tools for the general equations when he introduced the method of characteristics, which was first applied to arterial flow more than 50 years ago, most notably by Anliker and co-workers [30,31] and Skalak [27].…”

Section: Reduced Modelling Of Arterial Networkmentioning

confidence: 99%

Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network

Xiu

Sherwin

2007

Journal of Computational Physics

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Reduced models of human arterial networks are an efficient approach to analyse quantitative macroscopic features of human arterial flows. The justification for such models typically arise due to the significantly long wavelength associated with the system in comparison to the lengths of arteries in the networks. Although these types of models have been employed extensively and many issues associated with their implementations have been widely researched, the issue of data uncertainty has received comparatively little attention. Similar to many biological systems, a large amount of uncertainty exists in the value of the parameters associated with the models. Clearly reliable assessment of the system behaviour can not be made unless the effect of such data uncertainty is quantified.In this paper we present a study of parametric data uncertainty in reduced modelling of human arterial networks which is governed by a hyperbolic system. The uncertain parameters are modelled as random variables and the governing equations for the arterial network therefore become stochastic. This type stochastic hyperbolic systems have not been previously systematically studied due to the difficulties introduced by the uncertainty such as a potential change in the mathematical character of the system and imposing boundary conditions. We demonstrate how the application of a high-order stochastic collocation method based on the generalized polynomial chaos expansion, combined with a discontinuous Galerkin spectral/hp element discretisation in physical space, can successfully simulate this type of hyperbolic system subject to restrictions on the uncertainty interval. Building upon a numerical study of propagation of uncertainty and sensitivity in a simplified model with a single bifurcation, a systematical parameter sensitivity analysis is conducted on the wave dynamics in a multiple bifurcating human arterial network. Using the physical understanding of the dynamics of pulse waves in these types of networks we are able to provide an insight into the results of the stochastic simulations, thereby demonstrating the effects of uncertainty in physiologically accurate human arterial networks.

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Section: Reduced Modelling Of Arterial Networkmentioning

confidence: 99%

Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network

Xiu

Sherwin

2007

Journal of Computational Physics

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“…where a, b, and c are given real-valued continuous functions on T. This classical problem considered by Riemann in his dissertation [83], and its extension to non-simply connected domains and bordered Riemann surfaces, is one of the main boundary value problems of analytic function theory. Geometrically speaking, (3.1) demands that the boundary value of f at any point z ∈ T lies in a certain affine real line l z ⊂ C depending on z.…”

Section: The Riemann-hilbert Problem and Proper Holomorphic Maps Of Bmentioning

confidence: 99%

Complex Geometry and Dynamics

Fornæss¹,

Irgens²,

Wold³

2015

Abel Symposia

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In this paper we survey some recent contributions by the authors [5,6,7] to the theory of null holomorphic curves in the complex Euclidean space C 3 , as well as their applications to null holomorphic curves in the special linear group SL 2 (C), minimal surfaces in the Euclidean space R 3 , and constant mean curvature one surfaces (Bryant surfaces) in the hyperbolic space H 3 . The paper is an expanded version of the lecture given by the second named author at the Abel Symposium 2013 in Trondheim.

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“…Riemann showed that the connectedness number of S was independent of the choice of maximal family C. The key to his assertion is the following result, which is often called "Riemann's Lemma" [Riem,p. 85]: Suppose that A, B and C are three families of curves on S such that A and B form the complete boundary of one region of S, and A and C form the complete boundary of a second region of S. Then B and C together must also form the boundary of a third region, obtained as the symmetric difference of the other two regions (obtained by adding the regions together, and then subtracting any part where they overlap).…”

Section: Riemann and Bettimentioning

confidence: 99%

“…Riemann also considered the effect of making cuts (Querschnitte) in S. By making each cut q j transverse to a curve C j (see [Riem,p. 89]), he showed that the number of cuts needed to make S simply connected equals the connectivity number.…”

Section: Riemann and Bettimentioning

confidence: 99%

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History of Homological Algebra

Weibel

1999

History of Topology

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Homological algebra had its origins in the 19th century, via the work of Riemann (1857) and Betti (1871) on "homology numbers," and the rigorous development of the notion of homology numbers by Poincaré in 1895. A 1925 observation of Emmy Noether [N25] shifted the attention to the "homology groups" of a space, and algebraic techniques were developed for computational purposes in the 1930's. Yet homology remained a part of the realm of topology until about 1945.During the period 1940-1955, these topologically-motivated techniques for computing homology were applied to define and explore the homology and cohomology of several algebraic systems: Tor and Ext for abelian groups, homology and cohomology of groups and Lie algebras, and the cohomology of associative algebras. In addition, Leray introduced sheaves, sheaf cohomology and spectral sequences.At this point Cartan and Eilenberg's book [CE] crystallized and redirected the field completely. Their systematic use of derived functors, defined via projective and injective resolutions of modules, united all the previously disparate homology theories. It was a true revolution in mathematics, and as such it was also a new beginning. The search for a general setting for derived functors led to the notion of abelian categories, and the search for nontrivial examples of projective modules led to the rise of algebraic K-theory. Homological algebra was here to stay.Several new fields of study grew out of the Cartan-Eilenberg Revolution. The importance of regular local rings in algebra grew out of results obtained by homological methods in the late 1950's. The study of injective resolutions led to Grothendieck's theory of sheaf cohomology, the discovery of Gorenstein rings and Local Duality in both ring theory and algebraic geometry. In turn, cohomological methods played a key role in Grothendieck's rewriting of the foundations of algebraic geometry, including the development of derived categories. Number theory was infused with new results from Galois cohomology, which in turn led to the development ofétale cohomology and the eventual solution of the Weil Conjectures by Deligne.Simplicial methods were introduced in the 1950's by Dold, Kan, Moore and Puppe. They led to the rise of homotopical algebra and nonabelian derived functors in the 1960's. Among its many applications, perhaps André-Quillen homology for commutative rings and higher algebraic K-theory are the most noteworthy. Simplicial methods also played a more recent role in the development of Hochschild homology, topological Hochschild homology and cyclic homology.This completes a quick overview of the history we shall discuss in this article. Now let us turn to the beginnings of the subject.

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Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass

Cited by 59 publications

References 0 publications

Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network

Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network

Complex Geometry and Dynamics

History of Homological Algebra

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