Personal income tax reforms: A genetic algorithm approach

Given a settled reduction in the present level of tax revenue, and by exploring a very large combinatorial space of tax structures, in this paper we employ a genetic algorithm in order to determine the ‘best’ structure of a real world personal income tax that allows for the maximisation of the redistributive effect of the tax, while preventing all taxpayers being worse off than with the present tax structure. We take Italy as a case study.     

Analytical determination of the regularization parameter in the retrieval of atmospheric vertical profiles

In the retrieval of vertical profiles of atmospheric constituents, regularization methods are frequently used to improve the conditioning of the solution. The regularization reduces the retrieval errors and causes the vertical resolution to deteriorate. One obtains a trade-off by tuning the strength of the regularization by way of a regularization parameter. A new analytical method for determining the regularization parameter is presented. This method is suitable for operational retrievals, for which an unattended procedure is required. The performance of the new method is compared with that of the L-curve method, and the results show that a better trade-off between retrieval errors and vertical resolution is obtained.     

Compact multireference wavefront sensor design

We present a compact optical design for a multireference Shack-Hartmann-based wavefront sensor (WFS) for multiconjugate adaptive optical systems. The key component of this WFS design is a field lenslet array that separates the exit pupil images in the sensing plane for all reference sources. An analytical method for WFS optical design is presented, and the optimal strategy for selecting optical components from a discrete set is outlined. The feasibility of the WFS design has been demonstrated for a prototype WFS system in a laboratory setup with five reference sources and two deformable mirrors representing a wavefront-distorting medium.     

Matchings in graphs of odd regularity and girth

We establish lower bounds on the matching number of graphs of given odd regularity d$d$ and odd girth g$g$, which are sharp for many values of d$d$ and g$g$. For d=g=5$d=g=5$, we characterize all extremal graphs.     

States on systems of sets that are closed under symmetric difference

We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set‐representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non‐Boolean situations. On the other hand, a difference‐closed system can be constructed such that even two‐valued states do not allow for extensions. Finally, we consider these questions in a σ‐complete setup and find a large class of such systems with rather interesting state properties.     

Infinitesimal 2-braidings and differential crossed modules

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relations, leading to six categorified relations. We prove that any infinitesimal 2-braiding gives rise to a flat and fake flat 2-connection in the configuration space of n particles in the complex plane, hence to a categorification of the Knizhnik–Zamolodchikov connection. We discuss infinitesimal 2-braidings in a certain monoidal 2-category naturally assigned to every differential crossed module, leading to the notion of a symmetric quasi-invariant tensor in a differential crossed module. Finally, we prove that symmetric quasi-invariant tensors exist in the differential crossed module associated to Wagemann's version of the String Lie-2-algebra. As a corollary, we obtain a more conceptual proof of the flatness of a previously constructed categorified Knizhnik–Zamolodchikov connection with values in the String Lie-2-algebra.     

Sub-Rayleigh imaging via N-photon detection

The Rayleigh diffraction bound sets the minimum separation for two point objects to be distinguishable in a conventional imaging system. We demonstrate sub-Rayleigh resolution by scanning a focused beam--in an arbitrary, object-covering pattern that is unknown to the imager--and using N-photon photodetection implemented with a single-photon avalanche detector array. Experiments show resolution improvement by a factor ～(N-N(max))(?) beyond the Rayleigh bound, where N(max) is the maximum average detected photon number in the image, in good agreement with theory.     

On Bernoulli decompositions for random variables, concentration bounds, and spectral localization

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: (i) an anti-concentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and (ii) a proof, based on the Bernoulli case, of spectral localization for random Schrödinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here.     

Computing equilibria of Cournot oligopoly models with mixed-integer quantities

We consider Cournot oligopoly models in which some variables represent indivisible quantities. These models can be addressed by computing equilibria of Nash equilibrium problems in which the players solve mixed-integer nonlinear problems. In the literature there are no methods to compute equilibria of this type of Nash games. We propose a Jacobi-type method for computing solutions of Nash equilibrium problems with mixed-integer variables. This algorithm is a generalization of a recently proposed method for the solution of discrete so-called “2-groups partitionable” Nash equilibrium problems. We prove that our algorithm converges in a finite number of iterations to approximate equilibria under reasonable conditions. Moreover, we give conditions for the existence of approximate equilibria. Finally, we give numerical results to show the effectiveness of the proposed method.