Helly numbers of Algebraic Subsets of Rd\mathbb R^d

We study $S$-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in $\mathbb R^d$ with a proper subset $S\subset \mathbb R^d$. We contribute new results about their $S$-Helly numbers. We extend prior work for $S=\mathbb R^d$, $\mathbb Z^d$, and $\mathbb Z^{d-k}\times\mathbb R^k$; we give sharp bounds on the $S$-Helly numbers in several new cases. We considered the situation for low-dimensional $S$ and for sets $S$ that have some algebraic structure, in particular when $S$ is an arbitrary subgroup of $\mathbb R^d$ or when $S$ is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz method we obtain colorful versions of many monochromatic Helly-type results, including several colorful versions of our own results.Comment: 13 pages, 3 figures. This paper is a revised version of what was originally the first half of arXiv:1504.00076v

Beyond Chance-Constrained Convex Mixed-Integer Optimization: A Generalized Calafiore-Campi Algorithm and the notion of SS-optimization

The scenario approach developed by Calafiore and Campi to attack chance-constrained convex programs utilizes random sampling on the uncertainty parameter to substitute the original problem with a representative continuous convex optimization with $N$ convex constraints which is a relaxation of the original. Calafiore and Campi provided an explicit estimate on the size $N$ of the sampling relaxation to yield high-likelihood feasible solutions of the chance-constrained problem. They measured the probability of the original constraints to be violated by the random optimal solution from the relaxation of size $N$. This paper has two main contributions. First, we present a generalization of the Calafiore-Campi results to both integer and mixed-integer variables. In fact, we demonstrate that their sampling estimates work naturally for variables restricted to some subset $S$ of $\mathbb R^d$. The key elements are generalizations of Helly's theorem where the convex sets are required to intersect $S \subset \mathbb R^d$. The size of samples in both algorithms will be directly determined by the $S$-Helly numbers. Motivated by the first half of the paper, for any subset $S \subset \mathbb R^d$, we introduce the notion of an $S$-optimization problem, where the variables take on values over $S$. It generalizes continuous, integer, and mixed-integer optimization. We illustrate with examples the expressive power of $S$-optimization to capture sophisticated combinatorial optimization problems with difficult modular constraints. We reinforce the evidence that $S$-optimization is "the right concept" by showing that the well-known randomized sampling algorithm of K. Clarkson for low-dimensional convex optimization problems can be extended to work with variables taking values over $S$.Comment: 16 pages, 0 figures. This paper has been revised and split into two parts. This version is the second part of the original paper. The first part of the original paper is arXiv:1508.02380 (the original article contained 24 pages, 3 figures

Pattern formation in a predator-prey system characterized by a spatial scale of interaction

We describe pattern formation in ecological systems using a version of the classical Lotka-Volterra model characterized by a spatial scale which controls the predator-prey interaction range. Analytical and simulational results show that patterns can emerge in some regions of the parameters space where the instability is driven by the range of the interaction. The individual-based implementation captures realistic ecological features. In fact, spatial structures emerge in an erratic oscillatory regime which can contemplate predators' extinction.Comment: 5 pages, 4 figure

Magneto--Acoustic Energetics Study of the Seismically Active Flare of 15 February 2011

Multi--wavelength studies of energetic solar flares with seismic emissions have revealed interesting common features between them. We studied the first GOES X--class flare of the 24th solar cycle, as detected by the Solar Dynamics Observatory (SDO). For context, seismic activity from this flare (SOL2011-02-15T01:55-X2.2, in NOAA AR 11158) has been reported in the literature (Kosovichev, 2011; Zharkov et al., 2011). Based on Dopplergram data from the Helioseismic and Magnetic Imager (HMI), we applied standard methods of local helioseismology in order to identify the seismic sources in this event. RHESSI hard X-ray data are used to check the correlation between the location of the seismic sources and the particle precipitation sites in during the flare. Using HMI magnetogram data, the temporal profile of fluctuations in the photospheric line-of-sight magnetic field is used to estimate the magnetic field change in the region where the seismic signal was observed. This leads to an estimate of the work done by the Lorentz-force transient on the photosphere of the source region. In this instance this is found to be a significant fraction of the acoustic energy in the attendant seismic emission, suggesting that Lorentz forces can contribute significantly to the generation of sunquakes. However, there are regions in which the signature of the Lorentz-force is much stronger, but from which no significant acoustic emission emanates.Comment: Submitted to Solar Physic

The Impact of Single vs. Poly Victimization by Maltreatment Type on Pre-Treatment PTSD Scores

Studies show that an estimated 21-50% of children who have experienced maltreatment will develop post-traumatic stress disorder (PTSD) within their lifetime (Schuck & Widom 2019). Research suggests that the type and number of exposures to trauma may influence symptom severity. Further, trauma symptom scores of children who experienced child sexual abuse (CSA) were higher among the children who endorsed poly-victimization, meaning when CSA was combined with another form of maltreatment (Racine et al 2022). This study seeks to examine the impact of poly-victimization and combined types of maltreatment on children’s PTSD scores in an archival dataset from a child advocacy center (CAC) serving children exposed to various forms of trauma. Participants from the overall child sample (N = 721) who have pre-treatment PTSD scores (n = 290) will be analyzed. The analyzed sample includes 83 minors exposed to poly-victimization, 175 exposed to single victimization, and 32 where the trauma type was missing from the dataset. Specifically, the following hypotheses will be tested: (1) victims of poly-victimization will have higher PTSD scores than victims of single victimization; (2) among participants with poly-victimization, those with a combination of sexual abuse and physical abuse will have the highest PTSD scores compared to other combinations. Results will be discussed in the context of current referral pathways for child advocacy centers

Childhood Trauma and PTSD symptoms: Disentangling the Roles of Emotion Regulation and Distress Tolerance

Background: Research documents that individuals with a history of childhood trauma experience greater emotion regulation difficulties (ED), as well as lower distress tolerance (DT), two constructs that independent lines of research implicate as transdiagnostic factors in post-traumatic stress disorder (PTSD). To help elucidate how such mechanisms may influence the etiology, maintenance, and treatment of PTSD, the current study investigates the distinct mediating roles of ED and DT and explores which accounts for more unique variance. Methods: Participants (N = 385, aged 18-48) with a childhood trauma history provided self-ratings of cumulative trauma exposure, and endorsed current ED, DT, and PTSD symptoms. Single and dual mediation analyses were conducted to assess the roles of ED and DT in the relation between childhood trauma exposure and current PTSD symptoms. Results: ED and DT were inversely related. Higher childhood trauma exposure predicted higher PTSD symptoms in adulthood. ED and DT each mediated/explained the association between childhood trauma exposure and current PTSD symptoms; ED accounted for a larger effect size. Limitations: Cross-sectional design precludes determining causality. Retrospective self-report in measurement is subject to bias and findings may not generalize to behavioral measures. Conclusions: The current study results provide preliminary evidence of ED and DT as distinct mechanisms. Of clinical relevance, current findings support post-trauma processing theories that contend individuals’ recovery requires accepting and learning to modulate trauma-related emotional states. Implications for potential pathways from childhood trauma to PTSD, as well as methods of treatment and prevention are discussed