| 1. |
- Barack, Leor, et al.
(författare)
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Black holes, gravitational waves and fundamental physics : a roadmap
- 2019
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Ingår i: Classical and quantum gravity. - : IOP Publishing. - 0264-9381 .- 1361-6382. ; 36:14
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Forskningsöversikt (refereegranskat)abstract
- The grand challenges of contemporary fundamental physics dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on 'Black holes, Gravitational waves and Fundamental Physics'.
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| 2. |
- Craddock, Nick, et al.
(författare)
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Genome-wide association study of CNVs in 16,000 cases of eight common diseases and 3,000 shared controls
- 2010
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Ingår i: Nature. - : Springer Science and Business Media LLC. - 0028-0836 .- 1476-4687. ; 464:7289, s. 713-720
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Tidskriftsartikel (refereegranskat)abstract
- Copy number variants (CNVs) account for a major proportion of human genetic polymorphism and have been predicted to have an important role in genetic susceptibility to common disease. To address this we undertook a large, direct genome-wide study of association between CNVs and eight common human diseases. Using a purpose-designed array we typed,19,000 individuals into distinct copy-number classes at 3,432 polymorphic CNVs, including an estimated similar to 50% of all common CNVs larger than 500 base pairs. We identified several biological artefacts that lead to false-positive associations, including systematic CNV differences between DNAs derived from blood and cell lines. Association testing and follow-up replication analyses confirmed three loci where CNVs were associated with disease-IRGM for Crohn's disease, HLA for Crohn's disease, rheumatoid arthritis and type 1 diabetes, and TSPAN8 for type 2 diabetes-although in each case the locus had previously been identified in single nucleotide polymorphism (SNP)-based studies, reflecting our observation that most common CNVs that are well-typed on our array are well tagged by SNPs and so have been indirectly explored through SNP studies. We conclude that common CNVs that can be typed on existing platforms are unlikely to contribute greatly to the genetic basis of common human diseases.
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| 3. |
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| 4. |
- Byrnes, Christopher I., et al.
(författare)
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A note on the Jacobian conjecture
- 2008
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Ingår i: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 136:9, s. 3007-3011
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider the Jacobian conjecture for a map f of complex a. ne spaces of dimension n. It is well known that if f is proper, then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if f is proper onto its image.
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| 5. |
- Byrnes, Christopher I., et al.
(författare)
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IMPORTANT MOMENTS IN SYSTEMS AND CONTROL
- 2008
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Ingår i: SIAM Journal of Control and Optimization. - : Society for Industrial & Applied Mathematics (SIAM). - 0363-0129 .- 1095-7138. ; 47:5, s. 2458-2469
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Tidskriftsartikel (refereegranskat)abstract
- The moment problem matured from its various special forms in the late 19th and early 20th centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. In particular, the theory of systems and control is no exception, where the applications have historically been to circuit theory, optimal control, robust control, signal processing, spectral estimation, stochastic realization theory, and the use of the moments of a probability density. Many of these applications are also still works in progress. In this paper, we consider the generalized moment problem, expressed in terms of a basis of a finite-dimensional subspace P of the Banach space C[a, b] and a "positive" sequence c, but with a new wrinkle inspired by the applications to systems and control. We seek to parameterize solutions which are positive "rational" measures in a suitably generalized sense. Our parameterization is given in terms of smooth objects. In particular, the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a Euclidean space and which is the domain of some canonically defined functions. The analysis of these functions, and related maps, yields interesting corollaries for the moment problem and its applications, which we compare to those in the recent literature and which play a crucial role in part of our proof.
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| 6. |
- Byrnes, Christopher I., et al.
(författare)
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Interior point solutions of variational problems and global inverse function theorems
- 2007
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Ingår i: International Journal of Robust and Nonlinear Control. - : Wiley. - 1049-8923 .- 1099-1239. ; 17:5-6, s. 463-481
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Tidskriftsartikel (refereegranskat)abstract
- Variational problems and the solvability of certain nonlinear equations have a long and rich history beginning with calculus and extending through the calculus of variations. In this paper, we are interested in 'well-connected' pairs of such problems which are not necessarily related by critical point considerations. We also study constrained problems of the kind which arise in mathematical programming. We are also interested in interior minimizing points for the variational problem and in the well-posedness (in the sense of Hadamard) of solvability of the related systems of equations. We first prove a general result which implies the existence of interior points and which also leads to the development of certain generalization of the Hadamard-type global inverse function theorem, along the theme that uniqueness quite often implies existence. This result is illustrated by proving the non-existence of shock waves for certain initial data for the vector Burgers' equation. The global inverse function theorem is also illustrated by a derivation of the existence of positive definite solutions of matrix Riccati equations without first analysing the nonlinear matrix Riccati differential equation. The main results on the existence of solutions to geometrically constrained well-connected pairs are then presented and illustrated by a geometric analysis of the existence of interior points for linear programming problems.
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| 7. |
- Byrnes, Christopher I., et al.
(författare)
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The generalized moment problem with complexity constraint
- 2006
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Ingår i: Integral equations and operator theory. - : Springer Science and Business Media LLC. - 0378-620X .- 1420-8989. ; 56:2, s. 163-180
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which begins with a reformulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly convex optimization problem. Engineering applications typically demand a solution that is the ratio of functions in certain finite dimensional vector space of functions, usually the same vector space that is prescribed in the generalized moment problem. Solutions of this type are hinted at in the classical text by Krein and Nudelman and stated in the vast generalization of interpolation problems by Sarason. In this paper, formulated as generalized moment problems with complexity constraint, we give a complete parameterization of such solutions, in harmony with the above mentioned results and the engineering applications. While our previously announced results required some differentiability hypotheses, this paper uses a weak form involving integrability and measurability hypotheses that are more in the spirit of the classical treatment of the generalized moment problem. Because of this generality, we can extend the existence and well-posedness of solutions to this problem to nonnegative, rather than positive, initial data in the complexity constraint. This has nontrivial implications in the engineering applications of this theory. We also extend this more general result to the case where the numerator can be an arbitrary positive absolutely integrable function that determines a unique denominator in this finite-dimensional vector space. Finally, we conclude with four examples illustrating our results.
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| 8. |
- Byrnes, Christopher I., et al.
(författare)
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The moment problem for rational measures : convexity in the spirit of Krein
- 2009
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Ingår i: MODERN ANALYSIS AND APPLICATIONS. - Basel : Birkhäuser Verlag. - 9783764399184 ; , s. 157-169
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Konferensbidrag (refereegranskat)abstract
- The moment problem as formulated by Krein and Nudel'man is a beautiful generalization of several important classical moment problems, including the power moment problem, the trigonometric moment problem and the moment problem arising in Nevanlinna-Pick interpolation. Motivated by classical applications and examples, in both finite and infinite dimensions, we recently formulated a new version of this problem that we call the moment problem for positive rational measures. The formulation reflects the importance of rational functions in signals, systems and control. While this version of the problem is decidedly nonlinear, the basic tools still rely on convexity. In particular, we present a solution to this problem in terms of a nonlinear convex optimization problem that generalizes the maximum entropy approach used in several classical special cases.
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| 9. |
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Directions in Mathematical Systems : Theory and Optimization
- 2003
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Samlingsverk (redaktörskap) (övrigt vetenskapligt/konstnärligt)abstract
- This volume provides a compilation of recent contributions on feedback and robust control, modeling, estimation and filtering. They were presented on the occasion of the sixtieth birthday of Anders Lindquist, who has delivered fundamental contributions to the fields of systems, signals and control for more than three decades. His contributions include seminal work on the role of splitting subspaces in stochastic realization theory, on the partial realization problem for both deterministic and stochastic systems, on the solution of the rational covariance extension problem and on system identification. Lindquist's research includes the development of fast filtering algorithms, leading to a nonlinear dynamical system which computes spectral factors in its steady state, and which provide an alternate, linear in the dimension of the state space, to computing the Kalman gain from a matrix Riccati equation. He established the separation principle for stochastic function differential equations, including some fundamental work on optimal control for stochastic systems with time lags. His recent work on a complete parameterization of all rational solutions to the Nevanlinna-Pick problem is providing a new approach to robust control design.
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| 10. |
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