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Mathematics, Volume 2, Issue 4 (December 2014) – 3 articles , Pages 196-239

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Open AccessArticle
A Conjecture of Han on 3-Cores and Modular Forms
Mathematics 2014, 2(4), 232-239; https://doi.org/10.3390/math2040232 - 19 Dec 2014
Viewed by 1635
Abstract
In his study of Nekrasov–Okounkov type formulas on “partition theoretic” expressions for families of infinite products, Han discovered seemingly unrelated q-series that are supported on precisely the same terms as these infinite products. In collaboration with Ono, Han proved one instance of [...] Read more.
In his study of Nekrasov–Okounkov type formulas on “partition theoretic” expressions for families of infinite products, Han discovered seemingly unrelated q-series that are supported on precisely the same terms as these infinite products. In collaboration with Ono, Han proved one instance of this occurrence that exhibited a relation between the numbers a(n) that are given in terms of hook lengths of partitions, with the numbers b(n) that equal the number of 3-core partitions of n. Recently Han revisited the q-series with coefficients a(n) and b(n), and numerically found a third q-series whose coefficients appear to be supported on the same terms. Here we prove Han’s conjecture about this third series by proving a general theorem about this phenomenon. Full article
Open AccessArticle
Characteristic Variety of the Gauss–Manin Differential Equations of a Generic Parallelly Translated Arrangement
Mathematics 2014, 2(4), 218-231; https://doi.org/10.3390/math2040218 - 16 Oct 2014
Cited by 1 | Viewed by 1875
Abstract
We consider a weighted family of \(n\) generic parallelly translated hyperplanes in \(\mathbb{C}^k\) and describe the characteristic variety of the Gauss–Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined [...] Read more.
We consider a weighted family of \(n\) generic parallelly translated hyperplanes in \(\mathbb{C}^k\) and describe the characteristic variety of the Gauss–Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the Plücker coordinates of the associated point in the Grassmannian Gr\((k,n)\). The Laurent polynomials are in involution. Full article
Open AccessArticle
The Second-Order Shape Derivative of Kohn–Vogelius-Type Cost Functional Using the Boundary Differentiation Approach
Mathematics 2014, 2(4), 196-217; https://doi.org/10.3390/math2040196 - 26 Sep 2014
Cited by 4 | Viewed by 2487
Abstract
A shape optimization method is used to study the exterior Bernoulli free boundaryproblem. We minimize the Kohn–Vogelius-type cost functional over a class of admissibledomains subject to two boundary value problems. The first-order shape derivative of the costfunctional is recalled and its second-order shape [...] Read more.
A shape optimization method is used to study the exterior Bernoulli free boundaryproblem. We minimize the Kohn–Vogelius-type cost functional over a class of admissibledomains subject to two boundary value problems. The first-order shape derivative of the costfunctional is recalled and its second-order shape derivative for general domains is computedvia the boundary differentiation scheme. Additionally, the second-order shape derivative ofJ at the solution of the Bernoulli problem is computed using Tiihonen’s approach. Full article
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)
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