Convergence bounds for empirical nonlinear least-squares

We consider best approximation problems in a nonlinear subset [[EQUATION]] of a Banach space of functions [[EQUATION]] . The norm is assumed to be a generalization of the [[EQUATION]] -norm for which only a weighted Monte Carlo estimate [[EQUATION]] can be computed. The objective is to obtain an approximation [[EQUATION]] of an unknown function [[EQUATION]] by minimizing the empirical norm [[EQUATION]] . We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the nonlinear least squares setting. Several model classes are examined where analytical statements can be made about the RIP and the results are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.

Calculating the primary Lund Jet Plane density

The Lund-jet plane has recently been proposed as a powerful jet substructure tool with a broad range of applications. In this paper, we provide an all-order single logarithmic calculation of the primary Lund-plane density in Quantum Chromodynamics, including contributions from the running of the coupling, collinear effects for the leading parton, and soft logarithms that account for large-angle and clustering effects. We also identify a new source of clustering logarithms close to the boundary of the jet, deferring their resummation to future work. We then match our all-order results to exact next-to-leading order predictions. For phenomenological applications, we supplement our perturbative calculation with a Monte Carlo estimate of non-perturbative corrections. The precision of our final predictions for the Lund-plane density is 5−7% at high transverse momenta, worsening to about 20% at the lower edge of the perturbative region, corresponding to transverse momenta of about 5 GeV. We compare our results to a recent measurement by the ATLAS collaboration at the Large-Hadron Collider, revealing good agreement across the perturbative domain, i.e. down to about 5 GeV.

Examining the Schelling Model Simulation through an Estimation of its Entropy

The Schelling model of segregation allows for a general description of residential movements in an environment modeled by a lattice. The key factor is that occupants change positions until they are surrounded by a designated minimum number of similarly labeled residents. An analogy to the Ising model has been made in previous research, primarily due the assumption of state changes being dependent upon the adjacent cell positions. This allows for concepts produced in statistical mechanics to be applied to the Schelling model. Here is presented a methodology to estimate the entropy of the model for different states of the simulation. A Monte Carlo estimate is obtained for the set of macrostates defined as the different aggregate homogeneity satisfaction values across all residents, which allows for the entropy value to be produced for each state. This produces a trace of the estimated entropy value for the states of the lattice configurations to be displayed with each iteration. The results show that the initial random placements of residents have larger entropy values than the final states of the simulation when the overall homogeneity of the residential locality is increased.

Examining the Schelling Model Simulation through an Estimation of Its Entropy

The Schelling model of segregation allows for a general description of residential movements in an environment modeled by a lattice. The key factor is that occupants change positions until they are surrounded by a designated minimum number of similarly labeled residents. An analogy to the Ising model has been made in previous research, primarily due the assumption of state changes being dependent upon the adjacent cell positions. This allows for concepts produced in statistical mechanics to be applied to the Schelling model. Here is presented a methodology to estimate the entropy of the model for different states of the simulation. A Monte Carlo estimate is obtained for the set of macrostates defined as the different aggregate homogeneity satisfaction values across all residents, which allows for the entropy value to be produced for each state. This produces a trace of the estimated entropy value for the states of the lattice configurations to be displayed with each iteration. The results show that the initial random placements of residents have larger entropy values than the final states of the simulation when the overall homogeneity of the residential locality is increased.

Examining the Schelling Model Simulation through an Estimation of its Entropy

The Schelling model of segregation allows for a general description of residential movements in an environment modeled by a lattice. The key factor is that occupants change positions until they are surrounded by a designated minimum number of similarly labeled residents. An analogy to the Ising model has been made in previous research, primarily due the assumption of state changes being dependent upon the adjacent cell positions. This allows for concepts produced in statistical mechanics to be applied to the Schelling model. Here is presented a methodology to estimate the entropy of the model for different states of the simulation. A Monte Carlo estimate is obtained for the set of macrostates defined as the different aggregate homogeneity satisfaction values across all residents, which allows for the entropy value to be produced for each state. This produces a trace of the estimated entropy value for the states of the lattice configurations to be displayed with each iteration. The results show that the initial random placements of residents have larger entropy values than the final states of the simulation when the overall homogeneity of the residential locality is increased.

Difference between two species of emu hides a test for lepton flavour violation

We argue that an LHC measurement of some simple quantities related to the ratio of rates of e + μ − to e − μ + events is surprisingly sensitive to as-yet unexcluded R-parity violating supersymmetric models with non-zero λ 231 ′ couplings. The search relies upon the approximate lepton universality in the Standard Model, the sign of the charge of the proton, and a collection of favourable detector biases. The proposed search is unusual because: it does not require any of the displaced vertices, hadronic neutralino decay products, or squark/gluino production relied upon by existing LHC RPV searches; it could work in cases in which the only light sparticles were smuons and neutralinos; and it could make a discovery (though not necessarily with optimal significance) without requiring the computation of a leading-order Monte Carlo estimate of any background rate. The LHC has shown no strong hints of post-Higgs physics and so precision Standard Model measurements are becoming ever more important. We argue that in this environment growing profits are to be made from searches that place detector biases and symmetries of the Standard Model at their core — searches based around ‘controls’ rather than around signals.

Bayesian optimal design of fixed knockout tournament brackets

We present a methodology for finding globally optimal knockout tournament designs when partial information is known about the strengths of the players. Our approach involves maximizing an expected utility through a Bayesian optimal design framework. Given the prohibitive computational barriers connected with direct computation, we compute a Monte Carlo estimate of the expected utility for a fixed tournament bracket, and optimize the expected utility through simulated annealing. We demonstrate our method by optimizing the probability that the best player wins the tournament. We compare our approach to other knockout tournament designs, including brackets following the standard seeding. We also demonstrate how our approach can be applied to a variety of other utility functions, including whether the best two players meet in the final, the consistency between the number of wins and the player strengths, and whether the players are matched up according to the standard seeding.