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| 2. |
- Hedenmalm, Håkan, et al.
(författare)
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Weighted Bergman spaces and the integral means spectrum of conformal mappings
- 2005
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Ingår i: Duke mathematical journal. - 0012-7094 .- 1547-7398. ; 127:2, s. 341-393
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Tidskriftsartikel (refereegranskat)abstract
- The classical theory of conformal mappings involves best possible pointwise estimates of the derivative, thus supplying a measure of the extremal expansion/contraction possible for a conformal mapping. It is natural to consider also the integral means of \phi'\(t) along circles \z\ = r, where phi is the conformal mapping in question and t is a real parameter (0 < r < 1 if phi is defined in the unit disk, while 1 < r < +infinity if phi is defined in the exterior disk). The extremal growth rate as r --> 1 of the integral means which follows from the classical pointwise estimates is by far too fast. Better estimates were found by Clunie, Makarov, Pommerenke, Bertilsson, Shimorin, and others. Here we introduce a new method-based on area-type estimates-which discards as little as possible of the information supplied by the area methods. The result is a considerable improvement in the estimates of the integral means spectrum known tip to this point.
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| 4. |
- Maz´ya, Vladimir G.
(författare)
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The Wiener test for higher order elliptic equations
- 2002
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Ingår i: Duke mathematical journal. - 0012-7094 .- 1547-7398. ; 115:3, s. 479-512
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Tidskriftsartikel (refereegranskat)abstract
- We deal with strongly elliptic differential operators of an arbitrary even order 2m with constant real coefficients and introduce a notion of the regularity of a boundary point with respect to the Dirichlet problem which is equivalent to that given by N. Wiener in the case of m = 1. It is shown that a capacitary Wiener's type criterion is necessary and sufficient for the regularity if n = 2m. In the case of n > 2m, the same result is obtained for a subclass of strongly elliptic operators.
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| 5. |
- Passare, Mikael, et al.
(författare)
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Amoebas, Monge-Ampère measures and triangulations of the Newton polytope
- 2004
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Ingår i: Duke Mathematical Journal. - 0012-7094 .- 1547-7398. ; 121:3, s. 481-507
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Tidskriftsartikel (refereegranskat)abstract
- The amoeba of a holomorphic function $f$ is, by definition, the image in $\mathbf{R}^n$ of the zero locus of $f$ under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function $N_f$ with the property that its Monge-Ampére mass is concentrated to the amoeba of $f$ We obtain results of two kinds; by approximating $N_f$ with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of $f$; by computing the Monge-Ampére measure, we find sharp bounds for the area of amoebas in $\mathbf{R}^n$. We also consider systems of functions $f_{1},\dots,f_{n}$ and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations.
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| 7. |
- Ameur, Yacin, et al.
(författare)
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FLUCTUATIONS OF EIGENVALUES OF RANDOM NORMAL MATRICES
- 2011
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Ingår i: Duke mathematical journal. - : Duke University Press. - 0012-7094 .- 1547-7398. ; 159:1, s. 31-81
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane-the "droplet." We prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field, and we discuss various ramifications of this result.
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| 8. |
- Ayyer, Arvind, et al.
(författare)
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Goe fluctuations for the maximum of the top path in alternating sign matrices
- 2023
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Ingår i: Duke mathematical journal. - : Duke University Press. - 0012-7094 .- 1547-7398. ; 172:10, s. 1961-2014
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Tidskriftsartikel (refereegranskat)abstract
- The six-vertex model is an important toy-model in statistical mechanics for twodimensional ice with a natural parameter A. When A = 0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all A, there has been very little progress in understanding its statistics in the scaling limit for other values. In this work, we focus on the six-vertex model with domain wall boundary conditions at A = 1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We show that the maximum of the topmost level line for a uniformly random ASMs has the Gaussian orthogonal ensemble (GOE) Tracy-Widom distribution after appropriate rescaling. A key ingredient in our proof is Zeilberger's proof of the ASM conjecture. As far as we know, this is the first edge fluctuation result away from the tangency points for the domain-wall six-vertex model when we are not in the free-fermion case.
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| 9. |
- Bank, Efrat, et al.
(författare)
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PRIME POLYNOMIALS IN SHORT INTERVALS AND IN ARITHMETIC PROGRESSIONS
- 2015
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Ingår i: Duke mathematical journal. - : Duke University Press. - 0012-7094 .- 1547-7398. ; 164:2, s. 277-295
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x x(is an element of)] is about x(is an element of)/logx. The second says that the number of primes p < x in the arithmetic progression p equivalent to a (mod d), for d < x(1-delta), is about pi(x)/phi(d), where phi is the Euler totient function. More precisely, for short intervals we prove: Let k be a fixed integer Then pi(q)(I(f,is an element of)) similar to #I(f,is an element of)/k, q -> infinity holds uniformly for all prime powers q, degree k monic polynomials is an element of F-a[t] and is an element of(0)(f, q) <= is an element of, where is an element of(0) is either 1/k, or 2/k if p vertical bar k(k-1), or 3/k if further p = 2 and deg f ' <= 1. Here I (f, is an element of) = {g is an element of F-q[t] vertical bar deg( f - g) <= is an element of deg f }, and pi(q)(I(f, is an element of)) denotes the number of prime polynomials in I (f, 6). We show that this estimation fails in the neglected cases. For arithmetic progressions we prove: let k be a fixed integer. Then pi(q)(k; D,f) similar to pi(q)(k)/phi(D), q -> infinity, holds uniformly for all relatively prime polynomials D, f is an element of F-q[t] satisfying parallel to D parallel to <= q(k)(1-(delta 0)) where delta(0) is either 3/k or 4/k if p = 2 and (f /D)' is a constant. Here pi(q)(k) is the number of degree k prime polynomials and pi(a)(k; D, f) is the number of such polynomials in the arithmetic progression P equivalent to f (mod D). We also generalize these results to arbitrary factorization types.
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| 10. |
- Baranov, Anton, et al.
(författare)
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Boundary properties of Green functions in the plane
- 2008
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Ingår i: Duke mathematical journal. - : Duke University Press. - 0012-7094 .- 1547-7398. ; 145:1, s. 1-24
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Tidskriftsartikel (refereegranskat)abstract
- We study the boundary properties of the Green function of bounded simply connected domains in the plane. Essentially, this amounts to studying the conformal mapping taking the unit disk onto the domain in question. Our technique is inspired by a 1995 article of Jones and Makarov [11]. The main tools are an integral identity as well as a uniform Sobolev embedding theorem. The latter is in a sense dual to the exponential integrability of Marcinkiewicz-Zygmund integrals. We also develop a Grunsky identity, which contains the information of the classical Grunsky inequality. This Grunsky identity is the case where p = 2 of a more general Grunsky identily for L-p-spaces.
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