| 1. |
- Ahlgren, Scott, et al.
(author)
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SCARCITY OF CONGRUENCES FOR THE PARTITION FUNCTION
- 2023
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In: American Journal of Mathematics. - 0002-9327 .- 1080-6377. ; 145:5, s. 1509-1548
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Journal article (peer-reviewed)abstract
- The arithmetic properties of the ordinary partition function p(n) have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form p(ℓn + β) ≡ 0 (mod ℓ) for the primes ℓ = 5, 7, 11, and it is known that there are no others of this form. On the other hand, for every prime ℓ ≥ 5 there are infinitely many examples of congruences of the form p(ℓQm n + β) ≡ 0 (mod ℓ) where Q ≥ 5 is prime and m ≥ 3. This leaves open the question of the existence of such congruences when m = 1 or m = 2 (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve X(ℓQ), Galois representations and the arithmetic large sieve.
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| 2. |
- Berman, Robert, 1976
(author)
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Bergman kernels and equilibrium measures for line bundles over projective manifolds
- 2009
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In: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 131:5, s. 1485-1524
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Journal article (peer-reviewed)abstract
- Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels (i.e. orthogonal projection kernels). The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h_e associated to h, as well as in terms of the Monge-Ampere measure of h on a certain support set. It is also shown that the equilibrium metric h_e is in the class C^{1,1} on the complement of the augmented base locus of L. For L ample these results give generalizations of well-known results concerning the case when the curvature of h is globally positive (then h_e=h). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of L.
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| 3. |
- Berman, Robert, 1976, et al.
(author)
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SAMPLING OF REAL MULTIVARIATE POLYNOMIALS AND PLURIPOTENTIAL THEORY
- 2018
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In: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 140:3, s. 789-820
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Journal article (peer-reviewed)abstract
- We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety M, equipped with a weighted measure. In particular, this framework contains the well-known setting of trigonometric polynomials (when M is a torus equipped with its invariant measure), where the limit of large degree corresponds to a high frequency limit, as well as the classical setting of one-variable orthogonal algebraic polynomials (when M is the real line equipped with a suitable measure), where the sampling nodes can be seen as generalizations of the zeros of the corresponding orthogonal polynomials. It is shown that a necessary condition for sampling, in the general setting, is that the asymptotic density of the sampling points is greater than the density of the corresponding weighted equilibrium measure of M, as defined in pluripotential theory. This result thus generalizes the well-known Landau type results for sampling on the torus, where the corresponding critical density corresponds to the Nyqvist rate, as well as the classical result saying that the zeros of orthogonal polynomials become equidistributed with respect to the logarithmic equilibrium measure, as the degree tends to infinity.
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| 4. |
- Berndtsson, Bo, 1950, et al.
(author)
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COMPLEX LEGENDRE DUALITY
- 2020
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In: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 142:1, s. 323-339
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Journal article (peer-reviewed)abstract
- We introduce complex generalizations of the classical Legendre transform, operating on Kahler metrics on a compact complex manifold. These Legendre transforms give explicit local isometric symmetries for the Mabuchi metric on the space of Kaler metrics around any real analytic Kahler metric, answering a question originating in Semmes' work.
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| 5. |
- Bickel, Kelly, et al.
(author)
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Singularities of rational inner functions in higher dimensions
- 2022
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In: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 144:4, s. 1115-1157
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Journal article (peer-reviewed)abstract
- We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree (m, n, 1) RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of nontrivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.
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| 6. |
- Brändén, Petter
(author)
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The Lee-Yang and Pólya-Schur programs. III. Zero-preservers on Bargmann-Fock spaces
- 2014
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In: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 136:1, s. 241-253
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Journal article (peer-reviewed)abstract
- We characterize linear operators preserving zero-restrictions on entire functions in weighted Bargmann-Fock spaces. This extends the characterization of linear operators on polynomials preserving stability (due to Borcea and the author) to the realm of entire functions, and translates into an optimal, albeit formal, Lee-Yang theorem.
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| 7. |
- Cianchi, Andrea, et al.
(author)
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Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds
- 2013
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In: American Journal of Mathematics. - : Johns Hopkins University Press. - 0002-9327 .- 1080-6377. ; 135:3, s. 579-635
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Journal article (peer-reviewed)abstract
- We deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds M of finite volume. Sharp conditions ensuring L-q(M) and L-infinity(M) bounds for eigenfunctions are exhibited in terms of either the isoperimetric function or the isocapacitary function of M.
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| 8. |
- Gimperlein, H., et al.
(author)
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On the magnitude function of domains in Euclidean space
- 2021
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In: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 143:3, s. 939-967
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Journal article (peer-reviewed)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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| 9. |
- Granville, A., et al.
(author)
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The distribution of the zeros of random trigonometric polynomials
- 2011
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In: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 133:2, s. 295-357
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Journal article (peer-reviewed)abstract
- We study the asymptotic distribution of the number Z N of zeros of random trigonometric polynomials of degree N as N →∞. It is known that as N grows to infinity, the expected number of the zeros is asymptotic to N. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be cN for some c > 0. We prove that converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.
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| 10. |
- Hedenmalm, Håkan, 1961-
(author)
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BLOCH FUNCTIONS, ASYMPTOTIC VARIANCE, AND GEOMETRIC ZERO PACKING
- 2020
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In: American Journal of Mathematics. - : JOHNS HOPKINS UNIV PRESS. - 0002-9327 .- 1080-6377. ; 142:1, s. 267-321
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Journal article (peer-reviewed)abstract
- Motivated by a problem in quasiconformal mapping, we introduce a problem in complex analysis, with its roots in the mathematical physics of the Bose-Einstein condensates in superconductivity. The problem will be referred to as geometric zero packing, and is somewhat analogous to studying Fekete point configurations. The associated quantity is a density, denoted pc in the planar case, and pH in the case of the hyperbolic plane. We refer to these densities as discrepancy densities for planar and hyperbolic zero packing, respectively, as they measure the impossibility of atomizing the uniform planar and hyperbolic area measures. The universal asymptotic variance Sigma(2) associated with the boundary behavior of conformal mappings with quasiconformal extensions of small dilatation is related to one of these discrepancy densities: Sigma(2) = 1- rho H. We obtain the estimates 3.2 x 10(-5) < rho H <= 0.12087, where the upper estimate is derived from the estimate from below on Sigma(2) obtained by Astala, Ivrii, Perala, and Prause, and the estimate from below is more delicate. In particular, it follows that Sigma(2) < 1, which in combination with the work of ivrii shows that the maximal fractal dimension of quasicircles conjectured by Astala cannot be reached. Moreover, along the way, since the universal quasiconformal integral means spectrum has the asymptotics B(k, t) similar to 1/4 Sigma(2)vertical bar t vertical bar(2) for small t and k, the conjectured formula B(k, t) = 1/4 k(2)vertical bar t vertical bar(2) is not true. As for the actual numerical values of the discrepancy density rho(C), we obtain the estimate from above rho(C) <= 0.061203 ... by using the equilateral triangular planar zero packing, where the assertion that equality should hold can be attributed to Abrikosov. The value of pH is expected to be somewhat close to that of rho(C).
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