| 1. |
- Wang, Siyang, et al.
(författare)
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CONVERGENCE OF FINITE DIFFERENCE METHODS FOR THE WAVE EQUATION IN TWO SPACE DIMENSIONS
- 2018
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 87:314, s. 2737-2763
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Tidskriftsartikel (refereegranskat)abstract
- When using a finite difference method to solve an initial-boundary-value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many research works for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis.
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| 2. |
- Bogfjellmo, Geir, 1987, et al.
(författare)
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A Numerical Algorithm for C-2-Splines on Symmetric Spaces
- 2018
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Ingår i: SIAM Journal on Numerical Analysis. - : Siam Publications. - 1095-7170 .- 0036-1429. ; 56:4, s. 2623-2647
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Tidskriftsartikel (refereegranskat)abstract
- Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized .Bezier curves. To construct C-2-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C-2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the n-sphere, complex projective space, and the real Grassmannian.
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| 3. |
- Jensen, M., et al.
(författare)
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Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions
- 2022
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Ingår i: Ima Journal of Numerical Analysis. - : Oxford University Press (OUP). - 0272-4979 .- 1464-3642. ; 42:1, s. 199-228
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Tidskriftsartikel (refereegranskat)abstract
- We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the spatial discretization and the backward Euler scheme for the temporal discretization. Furthermore, we prove uniqueness and higher regularity of the solution on creased domains and additional regularity in the interior of the domain. Due to a variational formulation with a cut-off functional, the convergence analysis does not require a discrete maximum principle, permitting approximation spaces suitable for adaptive mesh refinement, responding to the difference in regularity within the domain.
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| 4. |
- Meisner, Patrick, et al.
(författare)
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Low-lying zeros in families of elliptic curve L-functions over function fields
- 2022
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Ingår i: Finite Fields and their Applications. - : Elsevier BV. - 1071-5797 .- 1090-2465. ; 84
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Tidskriftsartikel (refereegranskat)abstract
- We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over Fq(T). In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions.
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| 5. |
- Atai, F., et al.
(författare)
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Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation
- 2021
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 388:1, s. 435-468
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Tidskriftsartikel (refereegranskat)abstract
- The super-Macdonald polynomials, introduced by Sergeev andVeselov (Commun Math Phys 288: 653-675, 2009), generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald-Ruijsenaars operators introduced by the same authors in Sergeev and Veselov (CommunMath Phys 245: 249-278, 2004). We introduce a Hermitian form on the algebra spanned by the super-Macdonald polynomials, prove their orthogonality, compute their (quadratic) norms explicitly, and establish a corresponding Hilbert space interpretation of the super-Macdonald polynomials and deformed MacdonaldRuijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformedMacdonald-Ruijsenaars operators. Motivated by recent results in the nonrelativistic (q -> 1) case, we propose that these models describe the particles and anti-particles of an underlying relativistic quantum field theory, thus providing a natural generalisation of the trigonometric Ruijsenaars model.
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| 6. |
- Kow, P. Z., et al.
(författare)
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Quadrature Domains for the Helmholtz Equation with Applications to Non-scattering Phenomena
- 2023
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 60, s. 387-424
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps.
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| 7. |
- Alexandersson, Per, et al.
(författare)
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P-partitions and p-positivity
- 2019
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Ingår i: FPSAC 2019 - 31st International Conference on Formal Power Series and Algebraic Combinatorics. - : Formal Power Series and Algebraic Combinatorics. ; :82B
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Konferensbidrag (refereegranskat)abstract
- Using the combinatorics of a-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse P-partitions expand positively into quasisymmetric power sums. Consequently any nonnegative linear combination of such functions is p-positive whenever it is symmetric. We apply this method to derive positivity results for chromatic quasisymmetric functions and unicellular LLT polynomials.
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| 8. |
- Foda, O., et al.
(författare)
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Overlaps of partial Neel states and Bethe states
- 2016
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Ingår i: Journal of Statistical Mechanics. - : IOP Publishing. - 1742-5468.
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Tidskriftsartikel (refereegranskat)abstract
- Partial Neel states are generalizations of the ordinary Neel (classical anti-ferromagnet) state that can have arbitrary integer spin. We study overlaps of these states with Bethe states. We first identify this overlap with a partial version of reflecting-boundary domain-wall partition function, and then derive various determinant representations for off-shell and on-shell Bethe states.
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| 9. |
- Ivanenko, Yevhen, et al.
(författare)
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Quasi-Herglotz functions and convex optimization
- 2020
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Ingår i: Royal Society Open Science. - : The Royal Society Publishing. - 2054-5703. ; 7:1, s. 1-15
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Tidskriftsartikel (refereegranskat)abstract
- We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modelling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions and we show that several of the important properties and modelling perspectives are inherited by the new set of quasi-Herglotz functions. In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory. Numerical examples are included to demonstrate the modelling of a non-passive gain medium formulated as a convex optimization problem, where the generating measure is modelled by using a finite expansion of B-splines and point masses.
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| 10. |
- Cheraghchi, M., et al.
(författare)
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Approximating linear threshold predicates
- 2012
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Ingår i: ACM Transactions on Computation Theory. - : Association for Computing Machinery (ACM). - 1942-3462 .- 1942-3454. ; 4:1
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Tidskriftsartikel (refereegranskat)abstract
- We study constraint satisfaction problems on the domain {-1, 1}, where the given constraints are homogeneous linear threshold predicates, that is, predicates of the form sgn(w 1 x 1 + · · · + w n x n ) for some positive integer weights w 1 , . . . , w n . Despite their simplicity, current techniques fall short of providing a classification of these predicates in terms of approximability. In fact, it is not easy to guess whether there exists a homogeneous linear threshold predicate that is approximation resistant or not. The focus of this article is to identify and study the approximation curve of a class of threshold predicates that allow for nontrivial approximation. Arguably the simplest such predicate is the majority predicate sgn(x 1 + · · · + x n ), for which we obtain an almost complete understanding of the asymptotic approximation curve, assuming the Unique Games Conjecture. Our techniques extend to a more general class of "majority-like" predicates and we obtain parallel results for them. In order to classify these predicates, we introduce the notion of Chow-robustness that might be of independent interest.
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