| 1. |
- Andersson, Fredrik K., et al.
(författare)
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The mathematics of internet search engines
- 2008
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Ingår i: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications. - : Springer. - 0167-8019 .- 1572-9036. ; 104:2, s. 211-242
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Tidskriftsartikel (refereegranskat)abstract
- This article presents a survey of techniques for ranking results in search engines, with emphasis on link-based ranking methods and the PageRank algorithm. The problem of selecting, in relation to a user search query, the most relevant documents from an unstructured source such as the WWW is discussed in detail. The need for extending classical information retrieval techniques such as boolean searching and vector space models with link-based ranking methods is demonstrated. The PageRank algorithm is introduced, and its numerical and spectral properties are discussed. The article concludes with an alternative means of computing PageRank, along with some example applications of this new method.
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| 2. |
- Arnlind, Joakim, et al.
(författare)
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Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras
- 2011
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Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 52:12, s. 123502-
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Tidskriftsartikel (refereegranskat)abstract
- As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions. (C) 2011 American Institute of Physics. [doi:10.1063/1.3653197]
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| 3. |
- Eisenmann, Monika, et al.
(författare)
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Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations
- 2022
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Ingår i: Bit Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 62:3, s. 803-48
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we derive error estimates of the backward Euler-Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler-Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in Nochetto et al. (Commun Pure Appl Math 53(5):525-589, 2000). We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation.
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| 4. |
- Olsson, Carl, 1978, et al.
(författare)
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Relaxations for Non-Separable Cardinality/Rank Penalties
- 2021
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Ingår i: Proceedings of the IEEE International Conference on Computer Vision. - 1550-5499. ; 2021-October, s. 162-171
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Konferensbidrag (refereegranskat)abstract
- Rank and cardinality penalties are hard to handle in optimization frameworks due to non-convexity and discontinuity. Strong approximations have been a subject of intense study and numerous formulations have been proposed. Most of these can be described as separable, meaning that they apply a penalty to each element (or singular value) based on size, without considering the joint distribution. In this paper we present a class of non-separable penalties and give a recipe for computing strong relaxations suitable for optimization. In our analysis of this formulation we first give conditions that ensure that the global ly optimal solution of the relaxation is the same as that of the original (unrelaxed) objective. We then show how a stationary point can be guaranteed to be unique under the restricted isometry property (RIP) assumption.1
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| 5. |
- 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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| 6. |
- Silvestrov, Sergei, 1970-
(författare)
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Foundations of Santilli's Lie-Isotopic Theory
- 2010
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Ingår i: Numerical Analysis and Applied Mathematics (AIP Conference Proceedings). - : American Institute of Physics (AIP). ; 1281, s. 860-863
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Konferensbidrag (refereegranskat)abstract
- The need and benefits of new universal mathematical and especially algebraic and operator structures and methods in all branches and levels of modem physics from experiments to fundamentals is nowdays widely recognized. The far reaching pioneering discoveries of R. M. Santilli extend significantly the Lie analysis and operator methods and models of modem physics opening new areas of applications with possible essential improvements and revisions in models in particle physics and cosmology. In this article, some fundamental aspects of Lie-Santilli isotopic theory and relevant references are presented.
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| 7. |
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| 8. |
- Gerosa, Daniele, et al.
(författare)
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Bias Versus Non-Convexity in Compressed Sensing
- 2022
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Ingår i: Journal of Mathematical Imaging and Vision. - : Springer Science and Business Media LLC. - 1573-7683 .- 0924-9907. ; 64:4, s. 379-394
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Tidskriftsartikel (refereegranskat)abstract
- Cardinality and rank functions are ideal ways of regularizing under-determined linear systems, but optimization of the resulting formulations is made difficult since both these penalties are non-convex and discontinuous. The most common remedy is to instead use the ℓ1- and nuclear norms. While these are convex and can therefore be reliably optimized, they suffer from a shrinking bias that degrades the solution quality in the presence of noise. This well-known drawback has given rise to a fauna of non-convex alternatives, which usually features better global minima at the price of maybe getting stuck in undesired local minima. We focus in particular penalties based on the quadratic envelope, which have been shown to have global minima which even coincide with the “oracle solution,” i.e., there is no bias at all. So, which one do we choose, convex with a definite bias, or non-convex with no bias but less predictability? In this article, we develop a framework which allows us to interpolate between these alternatives; that is, we construct sparsity inducing penalties where the degree of non-convexity/bias can be chosen according to the specifics of the particular problem.
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| 9. |
- Gimperlein, H., et al.
(författare)
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On the magnitude function of domains in Euclidean space
- 2021
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Ingår i: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 143:3, s. 939-967
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Tidskriftsartikel (refereegranskat)abstract
- We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain X subset of R2m-1, we find geometric significance in the function M-X(R) = mag(R . X). The function M-X extends from the positive half-line to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit R -> infinity, M-X admits an asymptotic expansion. The three leading terms of M-X at R = +infinity are proportional to the volume, surface area and integral of the mean curvature. In particular, for convex X the leading terms are proportional to the intrinsic volumes, and we obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton, with corrected coefficients.
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| 10. |
- Goffeng, Magnus C H T, 1987, et al.
(författare)
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Clusters of eigenvalues for the magnetic Laplacian with Robin condition
- 2016
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Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 57:6
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Tidskriftsartikel (refereegranskat)abstract
- We study the Schrodinger operator with a constant magnetic field in the exterior of a compact domain in Euclidean space. Functions in the domain of the operator are subject to a boundary condition of the third type (a magnetic Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which is independent of the boundary condition. Published by AIP Publishing.
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