Ueber die Entwickelung reeller Functionen in Reihen mittelst der Methode der kleinsten Quadrate.

Gram, J. P.

doi:10.1515/crll.1883.94.41

Cited by 117 publications

(14 citation statements)

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“…Excited states are obtained by Gram-Schmidt orthogonalisation: 39,40 choosing eigenfunctions within a degenerate subspace to be orthogonal to the system's basis. For example, consider a converged, orthonormal ground state |ψ 0 and an initial, non-orthonormal guess for the first excited state |ψ 1 .…”

Section: Methodsmentioning

confidence: 99%

Delocalized Oxygen as the Origin of Two-Level Defects in Josephson Junctions

et al. 2013

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One of the key problems facing superconducting qubits and other Josephson junction devices is the decohering effects of bistable material defects. Although a variety of phenomenological models exist, the true microscopic origin of these defects remains elusive. For the first time we show that these defects may arise from delocalization of the atomic position of the oxygen in the oxide forming the Josephson junction barrier. Using a microscopic model, we compute experimentally observable parameters for phase qubits. Such defects are charge neutral but have nonzero response to both applied electric field and strain. This may explain the observed long coherence time of two-level defects in the presence of charge noise, while still coupling to the junction electric field and substrate phonons.

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Section: Methodsmentioning

confidence: 99%

Delocalized Oxygen as the Origin of Two-Level Defects in Josephson Junctions

et al. 2013

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“…Inspired from the shapelet reconstructions of galaxy images presented by Ref. [37], we expand our PDF around a Gaussian in a so called Gram-Charlier Series [38][39][40][41] or Edgeworth Expansion [42]. In Ref.…”

Section: Estimatormentioning

confidence: 99%

Information gains from cosmological probes

Grandis

Seehars

Réfrégier

et al. 2016

J. Cosmol. Astropart. Phys.

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Abstract. In light of the growing number of cosmological observations, it is important to develop versatile tools to quantify the constraining power and consistency of cosmological probes. Originally motivated from information theory, we use the relative entropy to compute the information gained by Bayesian updates in units of bits. This measure quantifies both the improvement in precision and the 'surprise', i.e. the tension arising from shifts in central values. Our starting point is a WMAP9 prior which we update with observations of the distance ladder, supernovae (SNe), baryon acoustic oscillations (BAO), and weak lensing as well as the 2015 Planck release. We consider the parameters of the flat ΛCDM concordance model and some of its extensions which include curvature and Dark Energy equation of state parameter w. We find that, relative to WMAP9 and within these model spaces, the probes that have provided the greatest gains are Planck (10 bits), followed by BAO surveys (5.1 bits) and SNe experiments (3.1 bits). The other cosmological probes, including weak lensing (1.7 bits) and H 0 measures (1.7 bits), have contributed information but at a lower level. Furthermore, we do not find any significant surprise when updating the constraints of WMAP9 with any of the other experiments, meaning that they are consistent with WMAP9. However, when we choose Planck15 as the prior, we find that, accounting for the full multidimensionality of the parameter space, the weak lensing measurements of CFHTLenS produce a large surprise of 4.4 bits which is statistically significant at the 8 σ level. We discuss how the relative entropy provides a versatile and robust framework to compare cosmological probes in the context of current and future surveys.

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“…Consider the non-empty orthogonal bases of loads F and states U of length c. One can investigate the linear dependency of a load f (e.g. a physical load f or adjoint load ∂g ∂u ) with respect to F by applying the last step of the well known Gram-Schmidt orthogonalisation procedure 2 (Laplace 1820;Gram 1883;Schmidt 1907). The residual r is obtained via…”

Section: Orthogonalisation and Reconstructionmentioning

confidence: 99%

Efficient computation of states and sensitivities for compound structural optimisation problems using a Linear Dependency Aware Solver (LDAS)

Koppen¹,

Kolk²,

Boom³

et al. 2022

Preprint

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Real-world structural optimisation problems involve multiple loading conditions and design constraints, with responses typically depending on states of discretised governing equations. Generally, one uses gradientbased nested analysis and design approaches to solve these problems. Herein, solving both physical and adjoint problems dominates the overall computational effort. Although not commonly detected, real world problems can contain linear dependencies between encountered physical and adjoint loads. Manually keeping track of such dependencies becomes tedious as design problems become increasingly involved. To detect and exploit such dependencies, this work proposes the use of a Linear Dependency Aware Solver (LDAS), which is able to efficiently detect linear dependencies between all loads to avoid unnecessary solves entirely and automatically. Illustrative examples are provided that demonstrate the need and benefits of using an LDAS, including a run-time experiment.

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Ueber die Entwickelung reeller Functionen in Reihen mittelst der Methode der kleinsten Quadrate.

Cited by 117 publications

References 0 publications

Delocalized Oxygen as the Origin of Two-Level Defects in Josephson Junctions

Delocalized Oxygen as the Origin of Two-Level Defects in Josephson Junctions

Information gains from cosmological probes

Efficient computation of states and sensitivities for compound structural optimisation problems using a Linear Dependency Aware Solver (LDAS)

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