Combinatorial and Automated Proofs of Certain Identities

This paper focuses on two binomial identities. The proofs illustrate the power and elegance in enumerative/algebraic combinatorial arguments, modern machine-assisted techniques of Wilf-Zeilberger and the classical tools of generatingfunctionology.United States. National Security Agency (Grant H98230-10-1-0222

Entangled Polymer Rings in 2D and Confinement

The statistical mechanics of polymer loops entangled in the two-dimensional array of randomly distributed obstacles of infinite length is discussed. The area of the loop projected to the plane perpendicular to the obstacles is used as a collective variable in order to re-express a (mean field) effective theory for the polymer conformation. It is explicitly shown that the loop undergoes a collapse transition to a randomly branched polymer with $R\propto lN^\frac{1}{4}$.Comment: 17 pages of Latex, 1 ps figure now available upon request, accepted for J.Phys.A:Math.Ge

Apolipoprotein E4 has extensive conformational heterogeneity in lipid-free and lipid-bound forms

The ε4-allele variant of apolipoprotein E (ApoE4) is the strongest genetic risk factor for Alzheimer\u27s disease, although it only differs from its neutral counterpart ApoE3 by a single amino acid substitution. While ApoE4 influences the formation of plaques and neurofibrillary tangles, the structural determinants of pathogenicity remain undetermined due to limited structural information. Previous studies have led to conflicting models of the C-terminal region positioning with respect to the N-terminal domain across isoforms largely because the data are potentially confounded by the presence of heterogeneous oligomers. Here, we apply a combination of single-molecule spectroscopy and molecular dynamics simulations to construct an atomically detailed model of monomeric ApoE4 and probe the effect of lipid association. Importantly, our approach overcomes previous limitations by allowing us to work at picomolar concentrations where only the monomer is present. Our data reveal that ApoE4 is far more disordered and extended than previously thought and retains significant conformational heterogeneity after binding lipids. Comparing the proximity of the N- and C-terminal domains across the three major isoforms (ApoE4, ApoE3, and ApoE2) suggests that all maintain heterogeneous conformations in their monomeric form, with ApoE2 adopting a slightly more compact ensemble. Overall, these data provide a foundation for understanding how ApoE4 differs from nonpathogenic and protective variants of the protein

On the connection of the generalized nonlinear sigma model with constrained stochastic dynamics

The dynamics of a freely jointed chain in the continuous limit is described by a field theory which closely resembles the nonlinear sigma model. The generating functional $\Psi[J]$ of this field theory contains nonholonomic constraints, which are imposed by inserting in the path integral expressing $\Psi[J]$ a suitable product of delta functions. The same procedure is commonly applied in statistical mechanics in order to enforce topological conditions on a system of linked polymers. The disadvantage of this method is that the contact with the stochastic process governing the diffusion of the chain is apparently lost. The main goal of this work is to reestablish this contact. To this purpose, it is shown here that the generating functional $\Psi[J]$ coincides with the generating functional of the correlation functions of the solutions of a constrained Langevin equation. In the discrete case, this Langevin equation describes as expected the Brownian motion of beads connected together by links of fixed length.Comment: LaTeX+RevTeX, 12 pages, no figure

A randomized, double-blind, placebo-controlled study of milk oral immunotherapy for cow's milk allergy

Background Orally administered, food-specific immunotherapy appears effective in desensitizing and potentially permanently tolerizing allergic individuals. Objective We sought to determine whether milk oral immunotherapy (OIT) is safe and efficacious in desensitizing children with cow's milk allergy. Methods Twenty children were randomized to milk or placebo OIT (2:1 ratio). Dosing included 3 phases: the build-up day (initial dose, 0.4 mg of milk protein; final dose, 50 mg), daily doses with 8 weekly in-office dose increases to a maximum of 500 mg, and continued daily maintenance doses for 3 to 4 months. Double-blind, placebo-controlled food challenges; end-point titration skin prick tests; and milk protein serologic studies were performed before and after OIT. Results Nineteen patients, 6 to 17 years of age, completed treatment: 12 in the active group and 7 in the placebo group. One dropped out because of persistent eczema during dose escalation. Baseline median milk IgE levels in the active (n = 13) versus placebo (n = 7) groups were 34.8 kUa/L (range, 4.86–314 kUa/L) versus 14.6 kUa/L (range, 0.93–133.4 kUa/L). The median milk threshold dose in both groups was 40 mg at the baseline challenge. After OIT, the median cumulative dose inducing a reaction in the active treatment group was 5140 mg (range 2540-8140 mg), whereas all patients in the placebo group reacted at 40 mg (P = .0003). Among 2437 active OIT doses versus 1193 placebo doses, there were 1107 (45.4%) versus 134 (11.2%) total reactions, with local symptoms being most common. Milk-specific IgE levels did not change significantly in either group. Milk IgG levels increased significantly in the active treatment group, with a predominant milk IgG4 level increase. Conclusions Milk OIT appears to be efficacious in the treatment of cow's milk allergy. The side-effect profile appears acceptable but requires further study

Perceptions About Work/Life Balance Among DU Community Members with Young Children

Background: In the past fifty years, families in the USA have changed in configuration, size and dynamics. The percentage of families that do not conform to the traditional family unit (married mother and father with children) has increased as there are more single-parent families, LGBTQ families and interracial families. The proportion of unmarried or divorced families has also increased, as it has the number of married and unmarried couples that opt to not have children and, additionally, more couples are opting for adoption and foster parenting (Pew Research Center 2010). Furthermore, the percentage of households where all the adults work has increased, which impacts the amount and quality of time available for family activities and household chores (Bianchi, Robinson and Milkie 2006). These and other trends have led to the identification of “work-family balance” as an important challenge of our times, one that families have been facing for decades and that institutions are only starting to pay attention to (Hochschild 2013). Although there are many aspects of family life that are challenging to balance with workplace demands, childcare has been specifically identified as one that needs attention (Desilver 2014). Methods: Study goal: To describe the perceptions that some DU community members with children have about work-family balance with attention to challenges, difficulties and institutional responses. Study design: Descriptive, cross-sectional, qualitative study. Population and sample: We recruited 63 University of Denver students (13), staff (14) and faculty (36) who are responsible of parenting at least one child under 10 years of age. We used purposive sampling. which consists in actively finding individuals who meet the criteria. Data collection: Semi structured interviews (January 23-February 8, 2017), in person, audio recorded and transcribed within one week. Participants’ autonomy, confidentiality and anonymity were protected throughout the process. Data analysis: Thematic analysis, which consists in the systematic identification of themes in the interview transcripts, followed by their conceptual organization and hierarchization. Research team: sixty-six undergraduate students taking Cultural Anthropology (ANTH 2010) in winter 2017, four graduate teaching assistants and one course instructor. Findings: Student participants portrayed work/life balance as set of interconnected situations and relations that go from the deeply personal to the interpersonal, communal and institutional. Aiming at capturing such complexity, we organized our findings in four themes: work/life balance, family dynamics, personal challenges and support. Participants told us about their struggles when negotiating work and life responsibilities which often lead to feelings of guilt, which are mediated by their colleagues’ reactions, schedule flexibility, their job situation and the presence or absence of maternity leave. Family dynamics reflected a tension between a narrative of independence and one of dependence in raising children, highlighting the importance of social networks, both of which are also affected by immigration status and intra-household negotiations particularly, Perceptions about work/life balance among DU community members with young children Cultural Anthropology (ANTH 2010) winter 2017 4 with their partners. Personal challenges relate primarily with time management and establishing clear boundaries between work and family, which related to managing emails, organization and scheduling of activities, maintaining a financial balance, and solving transportation needs, all of which were mediated the ability parents have of controlling a flexible work schedule, an ability greatly diminished among students. Support parents need related to child care goes from the one that happens in interpersonal interactions with neighbors, friends, relatives and colleagues, to the institutionalized forms of support, where participants expressed their frustration for the insufficiency of accessible options in Denver, the lack of options at DU, and the inaccessibility of DU’s Fisher Early Learning Center. Conclusions and recommendations: Participant’s ability to control their schedules together with their financial and social capital seem to shape important differences in the ability that parents have for balancing work and life. Students, single parents and recent immigrants seem to have a combination of elements that add to the challenges. At the interpersonal level, simple acts of kindness, sympathy and empathy in the everyday interactions seem to make an important difference to parents. The perception that many of the student participants expressed about the academy not being comfortable with children, families or parents could be addressed by making it normal to talk about all these aspects of life. At the institutional level, efforts could be made at reaching out to parents, especially students and single parents, to offer them guidance and support that is already in place at DU, such as counselling and wellbeing resources, as well as orientation related to institutional policies. Policies related to maternity and paternity leave should be refined to ensure that they do not negatively affect those they are supposed to support. Convenient, affordable and sustainable on-campus child care options should be seriously considered given that they would enhance the possibilities for parents to participate in activities at DU. Events should be organized where members of the DU community have the opportunity to share not as students, staff or faculty, but as members of families

A method of constructing invariant measures at fixed mass

Invariant measures are a useful tool in constructing and analyzing solutions u(t,x) to nonlinear dispersive partial differential equations, especially when a deterministic well-posedness result is not known, and have been studied extensively since the formative work of Bourgain in the 1990s. Due to a wealth of research in recent years, most dispersive PDEs are known to admit an invariant measure with little or no restriction placed on the profile of the initial data u_0 = u(0,x). This prompts one to analyze the ergodicity of such a measure. Unfortunately there has been little progress on decomposition of these measures into invariant measures supported on smaller subsets of the initial data space. Oh and Quastel constructed an invariant measure for each fixed mass and momentum for the NLS equation on T, but there has only been this isolated result. In this thesis we present a more general method of constructing invariant measures supported on H^(1/2-)(T) ∩ {|u|_{L^2}^2 =m} for a fixed mass m. Typically one constructs an invariant measure by multiplying a Gaussian measure µ by a density function Ѱ(u) to produce a new probability measure ρ with density containing the exponential of the remainder terms of a conservation law of the PDE. Combining Ѱ with the Gaussian base ensures that ρ is proportional to the exponential of the conservation law, making ρ invariant with respect to the PDE flow. One can only define ρ to satisfy the Radon-Nikodym derivative dρ(u) = |Ѱ(u)|_{L^1(µ)}^{-1} Ѱ(u) dµ(u) if |Ѱ(u)|_{L^1(µ)} is finite, therefore bounding |Ѱ(u)|_{L^1(µ)} is the main obstacle to constructing the invariant measure. For each m>0 we will construct a base measure µ_m that is supported on the set of functions of mass m and decompose this measure as a sum dµ_m = m^{-1} Σ_{k ≥ 0} c_{k,m} dν_m^k for a sequence {ν_m^k : k ≥ 0} of measures derived from scaling the Fourier coefficients of u. We then compute a specific formula for the integral of a function F with respect to each ν_m^k, allowing us to utilize this measure decomposition to integrate functions with respect to µ_m. In the introduction we will demonstrate that this computation is motivated by an alternative proof of the divergence theorem. We use this method to construct an invariant measure at fixed mass supported on H^{1/2-}(T) for two PDEs: the Benjamin-Ono equation, for which L^2(T) well-posedness is already known, and the derivative non-linear Schrodinger equation, for which well-posedness has only been proven at H^{1/2}(T) level