Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions

We study a construction algorithm for certain polynomial lattice rules modulo arbitrary polynomials. The underlying polynomial lattices are special types of digital nets as introduced by Niederreiter. Dick, Kuo, Pillichshammer and Sloan recently introduced construction algorithms for polynomial lattice rules modulo irreducible polynomials which yield a small worst-case error for integration of functions in certain weighted Hilbert spaces. Here, we generalize these results to the case where the polynomial lattice rules are constructed moduloarbitrarypolynomials.

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Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules

Journal of Complexity ◽

10.1016/j.jco.2007.02.001 ◽

2007 ◽

Vol 23 (4-6) ◽

pp. 436-453 ◽

Cited By ~ 23

Author(s):

Josef Dick ◽

Friedrich Pillichshammer

Keyword(s):

High Order ◽

Lattice Rules ◽

Multivariate Integration

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DISCRETE AND CONTINUUM APPROACHES TO THREE-DIMENSIONAL QUANTUM GRAVITY

Modern Physics Letters A ◽

10.1142/s0217732391004140 ◽

1991 ◽

Vol 06 (39) ◽

pp. 3591-3600 ◽

Cited By ~ 40

Author(s):

HIROSI OOGURI ◽

NAOKI SASAKURA

Keyword(s):

Gauge Theory ◽

Moduli Space ◽

Three Dimensional ◽

Two Dimensional ◽

Analogue Model ◽

Gauge Invariant ◽

Invariant Functions ◽

Dimensional Lattice ◽

Chern Simons ◽

Dimensional Surface

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k2, where q=e2πi/(k+2).

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The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-2006-018-7 ◽

2006 ◽

Vol 49 (2) ◽

pp. 170-184

Author(s):

Richard Atkins

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Shortest Paths ◽

Equivalence Problem ◽

Second Order ◽

System Of Differential Equations ◽

Lagrange Equations ◽

Invariant Functions ◽

Underlying Geometry ◽

The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

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Of lions and Yakshis

Semantic Web ◽

10.3233/sw-200417 ◽

2020 ◽

pp. 1-21

Author(s):

Franziska Pannach ◽

Caroline Sporleder ◽

Wolfgang May ◽

Aravind Krishnan ◽

Anusharani Sewchurran

Keyword(s):

Knowledge Base ◽

Case Studies ◽

Narrative Structure ◽

Function Class ◽

Special Focus ◽

Cultural Backgrounds ◽

Class Hierarchy ◽

Invariant Functions ◽

Narrative Functions ◽

Query System

Vladimir Propp’s theory Morphology of the Folktale identifies 31 invariant functions, subfunctions, and seven classes of folktale characters to describe the narrative structure of the Russian magic tale. Since it was first published in 1928, Propp’s approach has been used on various folktales of different cultural backgrounds. ProppOntology models Propp’s theory by describing narrative functions using a combination of a function class hierarchy and characteristic relationships between the Dramatis Personae for each function. A special focus lies on the restrictions Propp defined regarding which Dramatis Personae fulfill a certain function. This paper investigates how an ontology can assist traditional Humanities research in examining how well Propp’s theory fits for folktales outside of the Russian–European folktale culture. For this purpose, a lightweight query system has been implemented. To determine how well both the annotation schema and the query system works, twenty African tales and fifteen tales from the Kerala region in India were annotated. The system is evaluated by examining two case studies regarding the representation of characters and the use of Proppian functions in African and Indian tales. The findings are in line with traditional analogous Humanities research. This project shows how carefully modelled ontologies can be utilized as a knowledge base for comparative folklore research.

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