Invariant cubature formulas for the hyperoctahedron

1997 ◽  
Vol 61 (5) ◽  
pp. 614-620 ◽  
Author(s):  
S. B. Stoyanova
Keyword(s):  
Cubature Formulas ◽  
Invariant Cubature Formulas

scholarly journals Note on Cubature Formulae and Designs Obtained from Group Orbits

2012 ◽  
Vol 64 (6) ◽  
pp. 1359-1377 ◽  
Author(s):  
Hiroshi Nozaki ◽  
Masanori Sawa
Keyword(s):  
Cubature Formula ◽  
Sufficient Conditions ◽  
Reflection Group ◽  
Alternative Proof ◽  
Invariant Polynomials ◽  
Cubature Formulas ◽  
Finite Reflection Group ◽  
Necessary And Sufficient ◽  
Cubature Formulae ◽  
Invariant Cubature Formulas

Abstract In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t . In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type B, to other finite reflection groups.

Construction of cubature formulas with fourth and fifth degrees by the reproducing kernel and Radon methods

2020 ◽  
Vol 2020 (2) ◽  
pp. 76-84
Author(s):  
G.P. Ismatullaev ◽  
S.A. Bakhromov ◽  
R. Mirzakabilov
Keyword(s):  
Reproducing Kernel ◽  
Cubature Formulas

Moderate degree cubature formulas for 3-D tetrahedral finite-element approximations

1991 ◽  
Vol 7 (6) ◽  
pp. 487-495 ◽  
Author(s):  
M. Gellert ◽  
R. Harbord
Keyword(s):  
Finite Element ◽  
Moderate Degree ◽  
Finite Element Approximations ◽  
Cubature Formulas

scholarly journals Asymptotically optimal cubature formulas on manifolds for prefixed weights

2021 ◽  
pp. 105632
Author(s):  
Martin Ehler ◽  
Ujué Etayo ◽  
Bianca Gariboldi ◽  
Giacomo Gigante ◽  
Thomas Peter
Keyword(s):  
Asymptotically Optimal ◽  
Cubature Formulas

CUBATURE FORMULAS FOR CLASSES OF FUNCTIONS WITH BOUNDED MIXED DIFFERENCE

1993 ◽  
Vol 76 (2) ◽  
pp. 283-292 ◽  
Author(s):  
V V Dubinin
Keyword(s):  
Cubature Formulas
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