37,475 research outputs found
Equidistribution of Algebraic Numbers of Norm One in Quadratic Number Fields
Given a fixed quadratic extension K of Q, we consider the distribution of
elements in K of norm 1 (denoted N). When K is an imaginary quadratic
extension, N is naturally embedded in the unit circle in C and we show that it
is equidistributed with respect to inclusion as ordered by the absolute Weil
height. By Hilbert's Theorem 90, an element in N can be written as
\alpha/\bar{\alpha} for some \alpha \in O_K, which yields another ordering of
\mathcal N given by the minimal norm of the associated algebraic integers. When
K is imaginary we also show that N is equidistributed in the unit circle under
this norm ordering. When K is a real quadratic extension, we show that N is
equidistributed with respect to norm, under the map \beta \mapsto \log| \beta |
\bmod{\log | \epsilon^2 |} where \epsilon is a fundamental unit of O_K.Comment: 19 pages, 2 figures, comments welcome
The reciprocal Mahler ensembles of random polynomials
We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree N whose Mahler measure is bounded by a constant. After a change of variables, this reduces to a generalization of Ginibre’s complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval [−2,2] on the real axis in the complex plane. In the complex (real) case, the random roots form a determinantal (Pfaffian) point process, and in both cases, the empirical measure on roots converges weakly to the arcsine distribution supported on [−2,2]. Outside this region, the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from [−2,2]. These kernels as well as the scaling limits for the kernels in the bulk (−2,2) and at the endpoints {−2,2} are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels
Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary
We investigate a two-dimensional statistical model of N charged particles
interacting via logarithmic repulsion in the presence of an oppositely charged
compact region K whose charge density is determined by its equilibrium
potential at an inverse temperature corresponding to \beta = 2. When the charge
on the region, s, is greater than N, the particles accumulate in a neighborhood
of the boundary of K, and form a determinantal point process on the complex
plane. We investigate the scaling limit, as N \to \infty, of the associated
kernel in the neighborhood of a point on the boundary under the assumption that
the boundary is sufficiently smooth. We find that the limiting kernel depends
on the limiting value of N/s, and prove universality for these kernels. That
is, we show that, the scaled kernel in a neighborhood of a point \zeta \in
\partial K can be succinctly expressed in terms of the scaled kernel for the
closed unit disk, and the exterior conformal map which carries the complement K
to the complement of the closed unit disk. When N / s \to 0 we recover the
universal kernel discovered by Doron Lubinsky in Universality type limits for
Bergman orthogonal polynomials, Comput. Methods Funct. Theory, 10:135-154,
2010.Comment: 25 pages, 11 figure
Equidistribution of Elements of Norm 1 in Cyclic Extensions
Upon quotienting by units, the elements of norm 1 in a number field form
a countable subset of a torus of dimension where and
are the numbers of real and pairs of complex embeddings. When is
Galois with cyclic Galois group we demonstrate that this countable set is
equidistributed in this torus with respect to a natural partial ordering.Comment: 7 page
Initial experimental evidence that the ability to choose between items alters attraction to familiar versus novel persons in different ways for men and women
Nonhuman species may respond to novel mates with increased sexual motivation (‘The Coolidge Effect1). In humans, novel technological advances, such as online dating platforms, are thought to result in ‘Choice Overload’2. This may undermine the goal of finding a meaningful relationship3, orienting the user toward novel possible partners versus committing to a partner. Here, we used a paradigm measuring change in attraction to familiar faces (i.e. rated on second viewing4) to investigate Coolidge-like phenomena in humans primed with choice of potential online dating partners. We examined two pre-registered hypotheses (https://osf.io/xs74r/files/). First, whether experimentally priming choice (viewing a slideshow of online dating images) directly reduces the attractiveness of familiar preferred sex faces compared to our control condition. Second, whether the predicted effect is stronger for men than women given the role of the Coolidge effect in male sexual motivation5.<br/
Sound radiation in turbulent channel flows
Lighthill’s acoustic analogy is formulated for turbulent channel flow with pressure as the acoustic variable, and integrated over the channel width to produce a two-dimensional inhomogeneous wave equation. The equivalent sources consist of a dipole distribution related to the sum of the viscous shear stresses on the two walls, together with monopole and quadrupole distributions related to the unsteady turbulent dissipation and Reynolds stresses respectively. Using a rigid-boundary Green function, an expression is found for the power spectrum of the far-field pressure radiated per unit channel area. Direct numerical simulations (DNS) of turbulent plane Poiseuille and Couette flow have been performed in large computational domains in order to obtain good resolution of the low-wavenumber source behaviour. Analysis of the DNS databases for all sound radiation sources shows that their wavenumber–frequency spectra have non-zero limits at low wavenumber. The sound power per unit channel area radiated by the dipole distribution is proportional to Mach number squared, while the monopole and quadrupole contributions are proportional to the fourth power of Mach number. Below a particular Mach number determined by the frequency and radiation direction, the dipole radiation due to the wall shear stress dominates the far field. The quadrupole takes over at Mach numbers above about 0.1, while the monopole is always the smallest term. The resultant acoustic field at any point in the channel consists of a statistically diffuse assembly of plane waves, with spectrum limited by damping to a value that is independent of Mach number in the low-M limit
Tri-N-ification
We consider a natural generalization of trinification to theories with 3N
SU(3) gauge groups. These theories have a simple moose representation and a
gauge boson spectrum that can be interpreted via the deconstruction of a 5D
theory with unified symmetry broken on a boundary. Although the matter and
Higgs sectors of the theory have no simple extra-dimensional analog, gauge
unification retains features characteristic of the 5D theory. We determine
possible assignments of the matter and Higgs fields to unified multiplets and
present theories that are viable alternatives to minimal trinified GUTs.Comment: 21 pages LaTeX, 6 eps figure
Phased-In Tax Cuts and Economic Activity
Phased-in tax reductions are a common feature of tax legislation. This paper uses a dynamic general equilibrium model to quantify the effects of delaying tax cuts. According to the analysis of the model, the phased-in tax cuts of the 2001 tax law substantially reduced employment, output, and investment during the phase-in period. In contrast, the immediate tax cuts of the 2003 tax law provided significant incentives for immediate production and investment. The paper argues that the rules and accounting procedures used by Congress for formulating tax policy have a significant impact in shaping the details of tax policy and led to the phase-ins, sunsets, and temporary tax changes in both the 2001 and 2003 tax laws.Fiscal Policy, Tax Policy
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