Imperial College London

ProfessorNicholasBingham

Faculty of Natural SciencesDepartment of Mathematics

Senior Research Investigator
 
 
 
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Contact

 

+44 (0)20 7594 2085n.bingham

 
 
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Location

 

702Weeks BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
to

56 results found

Bingham NH, Ostaszewski AJ, 2020, SEQUENTIAL REGULAR VARIATION: EXTENSIONS OF KENDALL'S THEOREM, QUARTERLY JOURNAL OF MATHEMATICS, Vol: 71, Pages: 1171-1200, ISSN: 0033-5606

Journal article

Bingham NH, Jablonska E, Jablonski W, Ostaszewski AJet al., 2020, On Subadditive Functions Bounded Above on a "Large" Set, RESULTS IN MATHEMATICS, Vol: 75, ISSN: 1422-6383

Journal article

Bingham NH, Ostaszewski AJ, 2020, General regular variation, Popa groups and quantifier weakening, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, Vol: 483, ISSN: 0022-247X

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Bingham N, Ostaszewski A, 2019, Beyond Haar and Cameron-Martin: the Steinhaus support, Topology and its Applications, Vol: 260, Pages: 23-56, ISSN: 0166-8641

Motivated by a Steinhaus-like interior-point property involving the Cameron-Martin space of Gaussian measure theory, we study a group-theoretic analogue, the Steinhaus triple , and construct a Steinhaus support, a Cameron-Martin-like subset, in any Polish group G corresponding to ‘sufficiently subcontinuous’ measures μ, in particular for ‘Solecki-type’ reference measures.

Journal article

Bingham NH, Symons TL, 2019, Dimension walks on Sd×R, Statistics & Probability Letters, Vol: 147, Pages: 12-17, ISSN: 0167-7152

We verify that the established one- and two-step recurrences for positive definite functions on spheres extend to the spatio-temporal case of Sd X R.

Journal article

Bingham N, Ostaszewski A, 2019, Variants on the Berz sublinearity theorem, Aequationes Mathematicae, Vol: 93, Pages: 351-369, ISSN: 0001-9054

We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice, which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function—results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.

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Bingham NH, Ostaszewski AJ, 2019, Set theory and the analyst, European Journal of Mathematics, Vol: 5, Pages: 2-48, ISSN: 2199-675X

This survey is motivated by specific questions arising in thesimilarities and contrasts between (Baire) category and (Lebesgue) measure– category-measure duality and non-duality, as it were. Thebulk of the textis devoted to a summary, intended for the working analyst, ofthe extensivebackground in set theory and logic needed to discuss such matters: to quotefrom the Preface of Kelley [Kel]: ”what every young analyst should know”.

Journal article

Kasahara Y, Bingham NH, 2018, Matricial Baxter's theorem with a Nehari sequence, Mathematical News / Mathematische Nachrichten, Vol: 291, Pages: 2590-2598, ISSN: 0025-584X

In the theory of orthogonal polynomials, (non‐trivial) probability measures on the unit circle are parametrized by the Verblunsky coefficients. Baxter's theorem asserts that such a measure is absolutely continuous and has positive density with summable Fourier coefficients if and only if its Verblusnky coefficients are summable. This note presents a version of Baxter's theorem in the matrix case from a viewpoint of the Nehari problem.

Journal article

Bingham NH, Ostaszewski AJ, 2018, Beyond Lebesgue and Baire IV: density topologies and a converse Steinhaus-Weil theorem, Topology and its Applications, Vol: 239, Pages: 274-292, ISSN: 0166-8641

The theme here is category-measure duality, in the context of a topological group. One can often handle the (Baire) category case and the (Lebesgue, or Haar) measure cases together, by working bi-topologically: switching between the original topology and a suitable refinement (a density topology). This prompts a systematic study of such density topologies, and the corresponding σ-ideals of negligibles. Such ideas go back to Weil's classic book, and to Hashimoto's ideal topologies. We make use of group norms, which cast light on the interplay between the group and measure structures. The Steinhaus–Weil interior-points theorem (‘on ’) plays a crucial role here; so too does its converse, the Simmons–Mospan theorem.

Journal article

Bingham NH, Ostaszewski AJ, 2018, Additivity, subadditivity and linearity: automatic continuity and quantifier weakening, Indagationes Mathematicae, Vol: 29, Pages: 687-713, ISSN: 0019-3577

We study the interplay between additivity (as in the Cauchy functional equation), subadditivity and linearity. We obtain automatic continuity results in which additive or subadditive functions, under minimal regularity conditions, are continuous and so linear. We apply our results in the context of quantifier weakening in the theory of regular variation, completing our programme of reducing the number of hard proofs there to zero.

Journal article

Bingham NH, Ostaszewski AJ, 2017, Category-measure duality: convexity, midpoint convexity and Berz sublinearity, Aequationes Mathematicae, Vol: 91, Pages: 801-836, ISSN: 0001-9054

Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice, DC rather than the Axiom of Choice, AC. Berz used the Hahn–Banach theorem over ℚ to prove that the graph of a measurable sublinear function that is ℚ+-homogeneous consists of two half-lines through the origin. We give a category form of the Berz theorem. Our proof is simpler than that of the classical measure-theoretic Berz theorem, our result contains Berz’s theorem rather than simply being an analogue of it, and we use only DC rather than AC. Furthermore, the category form easily generalizes: the graph of a Baire sublinear function defined on a Banach space is a cone. The results are seen to be of automatic-continuity type. We use Christensen Haar null sets to extend the category approach beyond the locally compact setting where Haar measure exists. We extend Berz’s result from Euclidean to Banach spaces, and beyond. Passing from sublinearity to convexity, we extend the Bernstein–Doetsch theorem and related continuity results, allowing our conditions to be ‘local’—holding off some exceptional set.

Journal article

Kasahara Y, Bingham NH, 2017, Coefficient stripping in the matricial Nehari problem, Journal of Approximation Theory, Vol: 220, Pages: 1-11, ISSN: 0021-9045

This note deals with a matricial Schur function arising from a completely indeterminate Nehari problem. The Schur algorithm is characterized by a unilateral shift for a Nehari sequence.

Journal article

Bingham NH, Gashi B, 2016, Voronoi means, moving averages, and power series, Journal of Mathematical Analysis and Applications, Vol: 449, Pages: 682-696, ISSN: 1096-0813

We introduce a non-regular generalisation of the Nörlund mean, and show its equivalence with a certain moving average. The Abelian and Tauberian theorems establish relations with convergent sequences and certain power series. A strong law of large numbers is also proved.

Journal article

Bingham NH, 2016, Probability unfolding, 1965‒2015, Advances in Applied Probability, Vol: 48, Pages: 1-13, ISSN: 0001-8678

We give a personal (and we hope, not too idiosyncratic) view of how our subject of probability theory has developed during the last half-century, and the author in tandem with it.

Journal article

Bingham NH, Ostaszewski AJ, 2015, Beurling moving averages and approximate homomorphisms, Indagationes Mathematicae, Vol: 27, Pages: 601-633, ISSN: 0019-3577

Journal article

Bingham NH, Ostaszewski AJ, 2015, Cauchy's functional equation and extensions: Goldie's equation and inequality, the Golab-Schinzel equation and Beurling's equation, AEQUATIONES MATHEMATICAE, Vol: 89, Pages: 1293-1310, ISSN: 0001-9054

Journal article

Bingham NH, 2015, Hardy, Littlewood and probability, BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, Vol: 47, Pages: 191-202, ISSN: 0024-6093

Journal article

Bingham NH, Gashi B, 2015, Logarithmic moving averages, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, Vol: 421, Pages: 1790-1802, ISSN: 0022-247X

Journal article

Bingham NH, 2015, ON SCALING AND REGULAR VARIATION, PUBLICATIONS DE L INSTITUT MATHEMATIQUE-BEOGRAD, Vol: 97, Pages: 161-174, ISSN: 0350-1302

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Bingham NH, Missaoui B, 2014, ASPECTS OF PREDICTION, JOURNAL OF APPLIED PROBABILITY, Vol: 51, Pages: 189-201, ISSN: 0021-9002

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Bingham NH, Ostaszewski AJ, 2014, Beurling slow and regular variation, Transactions of the London Mathematical Society, Vol: 1, Pages: 29-56, ISSN: 2052-4986

Journal article

Bingham NH, 2014, Modelling and Prediction of Financial Time Series, COMMUNICATIONS IN STATISTICS-THEORY AND METHODS, Vol: 43, Pages: 1351-1361, ISSN: 0361-0926

Journal article

Kasahara Y, Bingham NH, 2014, VERBLUNSKY COEFFICIENTS AND NEHARI SEQUENCES, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 366, Pages: 1363-1378, ISSN: 0002-9947

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Bingham NH, 2014, THE WORLDWIDE INFLUENCE OF THE WORK OF B. V. GNEDENKO, THEORY OF PROBABILITY AND ITS APPLICATIONS, Vol: 58, Pages: 17-24, ISSN: 0040-585X

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Bingham NH, Ostaszewski AJ, 2013, The Steinhaus theorem and regular variation: de Bruijn and after, INDAGATIONES MATHEMATICAE-NEW SERIES, Vol: 24, Pages: 679-692, ISSN: 0019-3577

Journal article

Bingham NH, Inoue A, Kasahara Y, 2012, An explicit representation of Verblunsky coefficients, STATISTICS & PROBABILITY LETTERS, Vol: 82, Pages: 403-410, ISSN: 0167-7152

Journal article

Bingham NH, 2012, Szegö’s theorem and its probabilistic descendants, Probability Surveys, Vol: 9, ISSN: 1549-5787

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Bingham NH, 2012, Multivariate prediction and matrix Szegö theory, Probability Surveys, Vol: 9, ISSN: 1549-5787

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Bingham NH, Ostaszewski AJ, 2011, Homotopy and the Kestelman-Borwein-Ditor Theorem, CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, Vol: 54, Pages: 12-20, ISSN: 0008-4395

Journal article

Bingham NH, Ostaszewski AJ, 2011, Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski, MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, Vol: 150, Pages: 1-22, ISSN: 0305-0041

Journal article

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