Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions

Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence $x, f_n(x), f_n(f_n(x))$, where $f_n(x) = x + n$, $n$ a nonzero integer. Thus avoiding three-term arithmetic progressions is equivalent to containing no three elements of the form $x, f_n(x), f_n(f_n(x))$ with $f_n \in\mathcal{F}_{\rm t}$, the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions ($f_n(x) = nx$ with $n > 1$ a natural number) and exponential progressions ($f_n(x) = x^n$). Progression-free sets are often constructed "greedily," including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1, and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density, however we consider several different notions of density, including the uniform and exponential densities.Comment: Version 1.0, 13 page

Bose-Einstein condensates in standing waves: The cubic nonlinear Schroedinger equation with a periodic potential

We present a new family of stationary solutions to the cubic nonlinear Schroedinger equation with a Jacobian elliptic function potential. In the limit of a sinusoidal potential our solutions model a dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided the ratio of the height of the variations of the condensate to its DC offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Numerical simulations suggest such stationary states are experimentally observable.Comment: 4 pages, 4 figure

Regular treatment with salmeterol and inhaled steroids for chronic asthma: serious adverse events

Epidemiological evidence has suggested a link between beta(2)-agonists and increased asthma mortality. There has been much debate about possible causal links for this association, and whether regular (daily) long-acting beta(2)-agonists are safe.ObjectivesThe aim of this review is to assess the risk of fatal and non-fatal serious adverse events in trials that randomised patients with chronic asthma to regular salmeterol with inhaled corticosteroids versus the same dose of inhaled corticosteroids alone.Search strategyTrials were identified using the Cochrane Airways Group Specialised Register of trials. Web sites of clinical trial registers were checked for unpublished trial data and Food and Drug Administration (FDA) submissions in relation to salmeterol were also checked. The date of the most recent search was October 2008.Selection criteriaControlled parallel design clinical trials on patients of any age and severity of asthma were included if they randomised patients to treatment with regular salmeterol and inhaled corticosteroids (in separate or combined inhalers), and were of at least 12 weeks duration.Data collection and analysisTwo authors independently selected trials for inclusion in the review. Outcome data were independently extracted by two authors. Unpublished data on mortality and serious adverse events were obtained from the sponsors, and from FDA submissions.Main resultsThe review included 30 studies (10,873 participants) in adults and adolescents, and three studies (1,173 participants) in children. The overall risk of bias was low and data on serious adverse events were obtained from all studies.Six deaths occurred in 5,710 adults on regular salmeterol with inhaled corticosteroids, and five deaths in 5,163 adults on regular inhaled corticosteroids at the same dose. The difference was not statistically significant (Peto OR 1.05; 95% CI 0.32 to 3.47) and the absolute difference between groups in risk of death of any cause was 0.00005 (95% CI -0.002 to 0.002). No deaths were reported in 1,173 children, and no deaths were reported to be asthma-related.Non-fatal serious adverse events of any cause were reported in 134 adults on regular salmeterol with inhaled corticosteroids, compared to 103 adults on regular inhaled corticosteroids; again this was not a significant increase (Peto OR 1.17; 95% CI 0.90 to 1.52). The absolute difference in the risk of non-fatal serious adverse events was 0.003 (95% CI -0.002 to 0.009).There were three of 586 children with serious adverse events on regular salmeterol with inhaled corticosteroids, compared to four out of 587 on regular inhaled corticosteroids: there was no significant difference between treatments (Peto OR 0.75; 95% CI 0.17 to 3.31).Asthma-related serious adverse events were reported in 23 and 21 adults in each group respectively, a non-significant difference (Peto OR 0.95; 95% CI 0.52 to 1.73), and only one event was reported in children.Authors' conclusionsNo significant differences have been found in fatal or non-fatal serious adverse events in trials in which regular salmeterol has been randomly allocated with inhaled corticosteroids, in comparison to inhaled corticosteroids at the same dose. Although 10,873 adults and 1,173 children have been included in trials, the number of patients suffering adverse events is too small, and the results are too imprecise to confidently rule out a relative increase in all-cause mortality or non-fatal adverse events. It is therefore not possible to determine whether the increase in all-cause non-fatal serious adverse events reported in the previous meta-analysis on regular salmeterol alone is abolished by the additional use of regular inhaled corticosteroids. The absolute difference between groups in the risk of serious adverse events was small. There were no asthma-related deaths and few asthma-related serious adverse events. Clinical decisions and information for patients regarding regular use of salmeterol have to take into account the balance between known symptomatic benefits of salmeterol and the degree of uncertainty and concern associated with its potential harmful effects

Interview of Minna F. Weinstein, Ph.D.

Minna F. Weinstein (1933-2008) was born in Baltimore, Maryland. Her parents were both deaf and met at a school for the deaf in Western Maryland. Her father was a major proponent of education, and both she and her brother became teachers. She went on to college and graduate school at the University of Maryland, where she earned her B.A. in History, 1955, an M.A. in History, 1957, and a Ph.D. in History in 1965. During her time in the PhD program, she was a history instructor at Temple University, from 1961 to 1964, becoming an Assistant Professor in 1965. In Spring, 1966, she was hired at LaSalle College for the 1967-68 school year where she became the first full-time woman professor in LaSalle’s history. At La Salle, Dr. Weinstein was promoted from Assistant Professor to Associate Professor in 1970 and to full Professor in 1974. She earned the Lindback Award, for distinguished teaching in 1969. In the Fall of 1972, she led a successful effort to establish a Women’s Center on campus. Later, her colleagues elected her to the Faculty Senate. Dr. Weinstein left LaSalle in 1980 for a position as an Assistant Director at Middle States. She was with that organization until 2000, when she left to begin a consulting practice for schools in the accreditation process. She has one daughter, Alfia, who she adopted in 1988. At the time of the interview, Alfia was a freshman at Penn State University. This interview covers her experiences as a teacher and a woman at LaSalle, her perspectives on teaching History in particular, her career at Middle States, her consulting practice, and the adoption of her daughter

Simulations of multi-field ultralight axion-like dark matter

As constraints on ultralight axion-like particles (ALPs) tighten, models with multiple species of ultralight ALP are of increasing interest. We perform simulations of two-ALP models with particles in the currently supported range [arXiv:1307.1705] of plausible masses. The code we modified, UltraDark.jl, not only allows for multiple species of ultralight ALP with different masses, but also different self-interactions and inter-field interactions. This allows us to perform the first three-dimensional simulations of two-field ALPs with self-interactions and inter-field interactions. Our simulations show that having multiple species and interactions introduces different phenomenological effects as compared to a single field, non-interacting scenarios. In particular, we explore the dynamics of solitons. Interacting multi-species ultralight dark matter has different equilibrium density profiles as compared to single-species and/or non-interacting ultralight ALPs. As seen in earlier work [arXiv:2011.09510], attractive interactions tend to contract the density profile while repulsive interactions spread out the density profile. We also explore collisions between solitons comprised of distinct axion species. We observe a lack of interference patterns in such collisions, and that resulting densities depend on the relative masses of the ALPs and their interactions.Comment: 16 pages, 11 figure

Biological Sequence Kernels with Guaranteed Flexibility

Applying machine learning to biological sequences - DNA, RNA and protein - has enormous potential to advance human health, environmental sustainability, and fundamental biological understanding. However, many existing machine learning methods are ineffective or unreliable in this problem domain. We study these challenges theoretically, through the lens of kernels. Methods based on kernels are ubiquitous: they are used to predict molecular phenotypes, design novel proteins, compare sequence distributions, and more. Many methods that do not use kernels explicitly still rely on them implicitly, including a wide variety of both deep learning and physics-based techniques. While kernels for other types of data are well-studied theoretically, the structure of biological sequence space (discrete, variable length sequences), as well as biological notions of sequence similarity, present unique mathematical challenges. We formally analyze how well kernels for biological sequences can approximate arbitrary functions on sequence space and how well they can distinguish different sequence distributions. In particular, we establish conditions under which biological sequence kernels are universal, characteristic and metrize the space of distributions. We show that a large number of existing kernel-based machine learning methods for biological sequences fail to meet our conditions and can as a consequence fail severely. We develop straightforward and computationally tractable ways of modifying existing kernels to satisfy our conditions, imbuing them with strong guarantees on accuracy and reliability. Our proof techniques build on and extend the theory of kernels with discrete masses. We illustrate our theoretical results in simulation and on real biological data sets

Constraining bosonic asymmetric dark matter with neutron star mass-radius measurements

Neutron stars can accumulate asymmetric dark matter (ADM) in their interiors, which affects the neutron star's measurable properties and makes compact objects prime targets to search for ADM. In this work, we use Bayesian inference to explore potential neutron star mass-radius measurements, from current and future x-ray telescopes, to constrain the bosonic ADM parameters for the case where bosonic ADM has accumulated in the neutron star interior. We find that the current uncertainties in the baryonic equation of state do not allow for constraints on the ADM parameter space to be made. However, we also find that ADM cannot be excluded and the inclusion of bosonic ADM in neutron star cores relaxes the constraints on the baryonic equation of state space. If the baryonic equation of state were more tightly constrained independent of ADM, we find that statements about the ADM parameter space could be made. In particular, we find that the high bosonic ADM particle mass ($m_\chi$) and low effective self-interaction strength ($g_\chi/m_\phi)$ regime is disfavored due to the observationally and theoretically motivated constraint that neutron stars must have at least a mass of $1 \, \mathrm{M_\odot}$. However, within the remaining parameter space, $m_\chi$ and $g_\chi/m_\phi$ are individually unconstrained. On the other hand, the ADM mass-fraction, i.e., the fraction of ADM mass inside the neutron star, can be constrained by such neutron star measurements.Comment: 19 pages, 7 figures. This paper was made to be as similar to the PRD version as possibl