arXiv subjects

Saarik Kalia

Publications and source records attributed to Saarik Kalia.

21 records · Page 2Linked to original sources

arXiv2017

Modeling Smooth Backgrounds and Generic Localized Signals with Gaussian Processes

We describe a procedure for constructing a model of a smooth data spectrum using Gaussian processes rather than the historical parametric description. This approach considers a fuller space of possible functions, is robust at increasing luminosity, and allows us to incorporate our understanding of the underlying physics. We demonstrate the application of this approach to modeling the background to searches for dijet resonances at the Large Hadron Collider and describe how the approach can be used in the search for generic localized signals.

physics.data-an↗

arXiv2016

Number Density of Peaks in a Chi-Squared Field

We investigate the statistics of stationary points in the sum of squares of $N$ Gaussian random fields, which we call a "chi-squared" field. The behavior of such a field at a point is investigated, with particular attention paid to the formation of topological defects. An integral to compute the number density of stationary points at a given field amplitude is constructed. We compute exact expressions for the integral in various limits and provide code to evaluate it numerically in the general case. We investigate the dependence of the number density of stationary points on the field amplitude, number of fields, and power spectrum of the individual Gaussian random fields. This work parallels the work of Bardeen, Bond, Kaiser and Szalay, who investigated the statistics of peaks of Gaussian random fields. A number of results for integrating over matrices are presented in appendices.

math-ph↗

arXiv2015

Generalizations of the Szemerédi-Trotter Theorem

We generalize the Szemerédi-Trotter incidence theorem, to bound the number of complete \emph{flags} in higher dimensions. Specifically, for each $i=0,1,\ldots,d-1$, we are given a finite set $S_i$ of $i$-flats in $\R^d$ or in $\C^d$, and a (complete) flag is a tuple $(f_0,f_1,\ldots,f_{d-1})$, where $f_i\in S_i$ for each $i$ and $f_i\subset f_{i+1}$ for each $i=0,1,\ldots,d-2$. Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in $\R^3$ such that among the lines incident to a point, at most $O(1)$ of them can be coplanar, (ii) incidences with Legendrian lines in $\R^3$, a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in $\R^3$ (involving points, lines, and planes), where no given line can contain too many points or lie on too many planes. The bound that we obtain in (iii) is nearly tight in the worst case. Finally, we explore a group theoretic interpretation of flags, a generalized version of which leads us to new incidence problems.

math.CO↗