Expansion Methods for the Isoperimetric Problem of Bolza in Non-Parametric Form

1949 ◽  
Vol 71 (4) ◽  
pp. 946 ◽  
Author(s):  
William T. Reid
Keyword(s):  
Isoperimetric Problem ◽  
Parametric Form ◽  
Problem Of Bolza ◽  
Non Parametric

scholarly journals Model-independent constraints on the hydrogen-ionizing emissivity at z > 6

2019 ◽  
Vol 489 (2) ◽  
pp. 2669-2676 ◽  
Author(s):  
Charlotte A Mason ◽  
Rohan P Naidu ◽  
Sandro Tacchella ◽  
Joel Leja
Keyword(s):  
Cosmic Microwave Background ◽  
Bayesian Analysis ◽  
Optical Depth ◽  
Total Output ◽  
Rapid Decline ◽  
Parametric Form ◽  
Microwave Background ◽  
Model Independent ◽  
Dark Pixel ◽  
Non Parametric

ABSTRACT Modelling reionization often requires significant assumptions about the properties of ionizing sources. Here, we infer the total output of hydrogen-ionizing photons (the ionizing emissivity, $\dot{N}_\textrm {ion}$) at z = 4–14 from current reionization constraints, being maximally agnostic to the properties of ionizing sources. We use a Bayesian analysis to fit for a non-parametric form of $\dot{N}_\textrm {ion}$, allowing us to flexibly explore the entire prior volume. We infer a declining $\dot{N}_\textrm {ion}$ with redshift at z > 6, which can be used as a benchmark for reionization models. Model-independent reionization constraints from the cosmic microwave background (CMB) optical depth and Ly α and Ly β forest dark pixel fraction produce $\dot{N}_\textrm {ion}$ evolution ($\mathrm{ d}\log _{10}\dot{\mathbf {N}}_{\bf ion}/\mathrm{ d}z|_{z=6\rightarrow 8} = -0.31\pm 0.35$ dex) consistent with the declining UV luminosity density of galaxies, assuming constant ionizing photon escape fraction and efficiency. Including measurements from Ly α damping of galaxies and quasars produces a more rapid decline: $\mathrm{ d}\log _{10}\dot{\mathbf {N}}_{\bf ion}/\mathrm{ d}z|_{z=6\rightarrow 8} =-0.44\pm 0.22$ dex, steeper than the declining galaxy luminosity density (if extrapolated beyond $M_\rm{\small UV}\gtrsim -13$), and constrains the mid-point of reionization to z = 6.93 ± 0.14.

scholarly journals Modelling Burglary in Chicago using a self-exciting point process with isotropic triggering

2021 ◽  
pp. 1-23
Author(s):  
CRAIG GILMOUR ◽  
DESMOND J. HIGHAM
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Distance Measure ◽  
Predictive Accuracy ◽  
Synthetic Data ◽  
Public Domain ◽  
Spatial Distance ◽  
Parametric Form ◽  
Space And Time ◽  
Non Parametric

Self-exciting point processes have been proposed as models for the location of criminal events in space and time. Here we consider the case where the triggering function is isotropic and takes a non-parametric form that is determined from data. We pay special attention to normalisation issues and to the choice of spatial distance measure, thereby extending the current methodology. After validating these ideas on synthetic data, we perform inference and prediction tests on public domain burglary data from Chicago. We show that the algorithmic advances that we propose lead to improved predictive accuracy.

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