| 41. |
- Rowlett, Julie, 1978, et al.
(författare)
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How to hear the corners of a drum
- 2019
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Ingår i: Matrix Annals. - Cham : Springer International Publishing. ; 2017, s. 243-278, s. 243-278
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Konferensbidrag (refereegranskat)abstract
- We prove that the existence of corners in a class of planar domain, which includes all simply connected polygonal domains and all smoothly bounded domains, is a spectral invariant of the Laplacian with both Neumann and Robin boundary conditions. The main ingredient in the proof is a locality principle in the spirit of Kac’s “principle of not feeling the boundary,” but which holds uniformly up to the boundary. Albeit previously known for Dirichlet boundary condition, this appears to be new for Robin and Neumann boundary conditions, in the geometric generality presented here. For the case of curvilinear polygons, we describe how the same arguments using the locality principle are insufficient to obtain the analogous result. However, we describe how one may be able to harness powerful microlocal methods and combine these with the locality principles demonstrated here to show that corners are a spectral invariant; this is current work-in-progress (Nursultanov et al., Preprint).
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| 42. |
- Samuelsson Kalm, Håkan, 1976
(författare)
-
The dbar-equation, duality, and holomorphic forms on a reduced complex space
- 2021
-
Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 31:2, s. 1786-1820
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Tidskriftsartikel (refereegranskat)abstract
- We solve the $\bar\partial$-equation for (p, q)-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain (p, q)-currents. In particular this gives a condition for the partial derivative-equation to be globally solvable. Our results also give information about holomorphic p-forms on singular spaces.
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| 43. |
- Caraiani, Ana, et al.
(författare)
-
Vanishing theorems for Shimura varieties at unipotent level
- 2023
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Ingår i: Journal of the European Mathematical Society. - 1435-9863 .- 1435-9855. ; 25:3, s. 869-911
-
Tidskriftsartikel (refereegranskat)abstract
- We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite Γ1.p1/-level (defined with respect to a Borel subgroup) vanishes above the middle degree, under the assumption that the group of the Shimura datum splits at p. This generalizes and strengthens the vanishing result proved in [A. Caraiani et al., Compos. Math. 156 (2020)]. As an application of this vanishing theorem, we prove a result on the codimensions of ordinary completed homology for the same groups, analogous to conjectures of Calegari–Emerton for completed (Borel–Moore) homology.
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| 44. |
- Gerlee, Philip, 1980, et al.
(författare)
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Scientific Models : Red Atoms, White Lies and Black Boxes in a Yellow Book
- 2016
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- A zebrafish, the hull of a miniature ship, a mathematical equation and a food chain - what do these things have in common? They are examples of models used by scientists to isolate and study particular aspects of the world around us. This book begins by introducing the concept of a scientific model from an intuitive perspective, drawing parallels to mental models and artistic representations. It then recounts the history of modelling from the 16th century up until the present day. The iterative process of model building is described and discussed in the context of complex models with high predictive accuracy versus simpler models that provide more of a conceptual understanding. To illustrate the diversity of opinions within the scientific community, we also present the results of an interview study, in which ten scientists from different disciplines describe their views on modelling and how models feature in their work. Lastly, it includes a number of worked examples that span different modelling approaches and techniques. It provides a comprehensive introduction to scientific models and shows how models are constructed and used in modern science. It also addresses the approach to, and the culture surrounding modelling in different scientific disciplines. It serves as an inspiration for model building and also facilitates interdisciplinary collaborations by showing how models are used in different scientific fields. The book is aimed primarily at students in the sciences and engineering, as well as students at teacher training colleges but will also appeal to interested readers wanting to get an overview of scientific modelling in general and different modelling approaches in particular.
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| 45. |
- Gerlee, Philip, 1980, et al.
(författare)
-
Scientific Models
- 2016
-
Bok (övrigt vetenskapligt/konstnärligt)abstract
- A zebrafish, the hull of a miniature ship, a mathematical equation and a food chain - what do these things have in common? They are examples of models used by scientists to isolate and study particular aspects of the world around us. This book begins by introducing the concept of a scientific model from an intuitive perspective, drawing parallels to mental models and artistic representations. It then recounts the history of modelling from the 16th century up until the present day. The iterative process of model building is described and discussed in the context of complex models with high predictive accuracy versus simpler models that provide more of a conceptual understanding. To illustrate the diversity of opinions within the scientific community, we also present the results of an interview study, in which ten scientists from different disciplines describe their views on modelling and how models feature in their work. Lastly, it includes a number of worked examples that span different modelling approaches and techniques. It provides a comprehensive introduction to scientific models and shows how models are constructed and used in modern science. It also addresses the approach to, and the culture surrounding modelling in different scientific disciplines. It serves as an inspiration for model building and also facilitates interdisciplinary collaborations by showing how models are used in different scientific fields. The book is aimed primarily at students in the sciences and engineering, as well as students at teacher training colleges but will also appeal to interested readers wanting to get an overview of scientific modelling in general and different modelling approaches in particular.
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| 46. |
- Hellström, Fredrik, 1993, et al.
(författare)
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Generalization Bounds via Information Density and Conditional Information Density
- 2020
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Ingår i: IEEE Journal on Selected Areas in Information Theory. - 2641-8770. ; 1:3, s. 824-839
-
Tidskriftsartikel (refereegranskat)abstract
- We present a general approach, based on an exponential inequality, to derive bounds on the generalization error of randomized learning algorithms. Using this approach, we provide bounds on the average generalization error as well as bounds on its tail probability, for both the PAC-Bayesian and single-draw scenarios. Specifically, for the case of sub-Gaussian loss functions, we obtain novel bounds that depend on the information density between the training data and the output hypothesis. When suitably weakened, these bounds recover many of the information-theoretic bounds available in the literature. We also extend the proposed exponential-inequality approach to the setting recently introduced by Steinke and Zakynthinou (2020), where the learning algorithm depends on a randomly selected subset of the available training data. For this setup, we present bounds for bounded loss functions in terms of the conditional information density between the output hypothesis and the random variable determining the subset choice, given all training data. Through our approach, we recover the average generalization bound presented by Steinke and Zakynthinou (2020) and extend it to the PAC-Bayesian and singledraw scenarios. For the single-draw scenario, we also obtain novel bounds in terms of the conditional α-mutual information and the conditional maximal leakage.
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| 47. |
- Holroyd, Alexander E., et al.
(författare)
-
Minimal matchings of point processes
- 2022
-
Ingår i: Probability Theory and Related Fields. - : Springer Science and Business Media LLC. - 0178-8051 .- 1432-2064. ; 184:1-2, s. 571-611
-
Tidskriftsartikel (refereegranskat)abstract
- Suppose that red and blue points form independent homogeneous Poisson processes of equal intensity in R-d. For a positive (respectively, negative) parameter gamma we consider red-blue matchings that locally minimize (respectively, maximize) the sum of gamma th powers of the edge lengths, subject to locally minimizing the number of unmatched points. The parameter can be viewed as a measure of fairness. The limit gamma -> -infinity is equivalent to Gale-Shapley stable matching. We also consider limits as gamma approaches 0, 1-, 1+ and infinity. We focus on dimension d = 1. We prove that almost surely no such matching has unmatched points. (This question is open for higher d). For each gamma < 1 we establish that there is almost surely a unique such matching, and that it can be expressed as a finitary factor of the points. Moreover, its typical edge length has finite rth moment if and only if r < 1 /2. In contrast, for gamma = 1 there are uncountably many matchings, while for gamma > 1 there are countably many, but it is impossible to choose one in a translation-invariant way. We obtain existence results in higher dimensions (covering many but not all cases). We address analogous questions for one-colour matchings also.
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| 48. |
- Meisner, Patrick, et al.
(författare)
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Low-lying zeros in families of elliptic curve L-functions over function fields
- 2022
-
Ingår i: Finite Fields and their Applications. - : Elsevier BV. - 1071-5797 .- 1090-2465. ; 84
-
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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| 49. |
- Johansson, Moa, 1981, et al.
(författare)
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Conjectures, tests and proofs: An overview of theory exploration
- 2021
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Ingår i: Electronic Proceedings in Theoretical Computer Science, EPTCS. - 2075-2180. ; 341, s. 1-16
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Konferensbidrag (refereegranskat)abstract
- A key component of mathematical reasoning is the ability to formulate interesting conjectures about a problem domain at hand. In this paper, we give a brief overview of a theory exploration system called QuickSpec, which is able to automatically discover interesting conjectures about a given set of functions. QuickSpec works by interleaving term generation with random testing to form candidate conjectures. This is made tractable by starting from small sizes and ensuring that only terms that are irreducible with respect to already discovered conjectures are considered. QuickSpec has been successfully applied to generate lemmas for automated inductive theorem proving as well as to generate specifications of functional programs. We give an overview of typical use-cases of QuickSpec, as well as demonstrating how to easily connect it to a theorem prover of the user’s choice.
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| 50. |
- Vasilis, Jonatan, 1981
(författare)
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Harmonic measures
- 2010
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis uses both analytic and probabilistic methods to study continuous and discrete problems. The main areas of study are the asymptotic properties of p-harmonic measure, and various aspects of the square root of the Poisson kernel. Fix a domain and a boundary point, subject to certain regularity conditions. Consider the part of the boundary that lies within a disc, centered at the fixed boundary point. It is shown that as the radius of the disc tends to zero, the p-harmonic measure of the boundary set decays as an explicitly given power of the radius. The square root of the Poisson kernel is studied in both continuous and discrete settings. In the continuous case the domain is the unit disc, and a Hardy space related to the square root of the Poisson kernel is defined. The main result is that, as opposed to the classical Hardy space, the positive functions do not admit a characterization in terms of an Orlicz space. Similar results are given also in the discrete case, where the domain is instead a regular tree. Further results in the discrete setting include the construction of a nearest neighbor random walk on the tree with exit distribution determined by powers of the Poisson kernel. The minimally thin sets of these random walks are characterized. Finally, we suggest a generalization of a two-dimensional geometric result – the ring lemma – to three dimensions.
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