Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let rb-index(S) denote the smallest size of a perfect rainbow polygon for a colored point set S, and let rb-index(k) be the maximum of rb-index(S) over all k-colored point sets in general position; that is, every k-colored point set S has a perfect rainbow polygon with at most rb-index(k) vertices. In this paper, we determine the values of rb-index(k) up to k=7, which is the first case where rb-index(k)¿k, and we prove that for k=5, [Formula presented] Furthermore, for a k-colored set of n points in the plane in general position, a perfect rainbow polygon with at most [Formula presented] vertices can be computed in O(nlogn) time. © 2021 Elsevier B.V

Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, $$\frac{40\lfloor (k-1)/2 \rfloor -8}{19} \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11.$$ Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time.Comment: 23 pages, 11 figures, to appear at Discrete Mathematic

Atomistic Simulations of Nanotube Fracture

The fracture of carbon nanotubes is studied by atomistic simulations. The fracture behavior is found to be almost independent of the separation energy and to depend primarily on the inflection point in the interatomic potential. The rangle of fracture strians compares well with experimental results, but predicted range of fracture stresses is marketly higher than observed. Various plausible small-scale defects do not suffice to bring the failure stresses into agreement with available experimental results. As in the experiments, the fracture of carbon nanotubes is predicted to be brittle. The results show moderate dependence of fracture strength on chirality.Comment: 12 pages, PDF, submitted to Phy. Rev.

Perfect k-Colored Matchings and (k+2)-Gonal Tilings

We derive a simple bijection between geometric plane perfect matchings on 2n points in convex position and triangulations on n+2 points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically k-colored vertices and (k+2)-gonal tilings of convex point sets. These structures are related to a generalization of Temperley–Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time

FOXP3 Expression Is Upregulated in CD4+T Cells in Progressive HIV-1 Infection and Is a Marker of Disease Severity

Understanding the role of different classes of T cells during HIV infection is critical to determining which responses correlate with protective immunity. To date, it is unclear whether alterations in regulatory T cell (Treg) function are contributory to progression of HIV infection.FOXP3 expression was measured by both qRT-PCR and by flow cytometry in HIV-infected individuals and uninfected controls together with expression of CD25, GITR and CTLA-4. Cultured peripheral blood mononuclear cells were stimulated with anti-CD3 and cell proliferation was assessed by CFSE dilution.HIV infected individuals had significantly higher frequencies of CD4(+)FOXP3(+) T cells (median of 8.11%; range 1.33%-26.27%) than healthy controls (median 3.72%; range 1.3-7.5%; P = 0.002), despite having lower absolute counts of CD4(+)FOXP3(+) T cells. There was a significant positive correlation between the frequency of CD4(+)FOXP3(+) T cells and viral load (rho = 0.593 P = 0.003) and a significant negative correlation with CD4 count (rho = -0.423 P = 0.044). 48% of our patients had CD4 counts below 200 cells/microl and these patients showed a marked elevation of FOXP3 percentage (median 10% range 4.07%-26.27%). Assessing the mechanism of increased FOXP3 frequency, we found that the high FOXP3 levels noted in HIV infected individuals dropped rapidly in unstimulated culture conditions but could be restimulated by T cell receptor stimulation. This suggests that the high FOXP3 expression in HIV infected patients is likely due to FOXP3 upregulation by individual CD4(+) T cells following antigenic or other stimulation.FOXP3 expression in the CD4(+) T cell population is a marker of severity of HIV infection and a potential prognostic marker of disease progression

Observing many researchers using the same data and hypothesis reveals a hidden universe of uncertainty

This study explores how researchers’ analytical choices affect the reliability of scientific findings. Most discussions of reliability problems in science focus on systematic biases. We broaden the lens to emphasize the idiosyncrasy of conscious and unconscious decisions that researchers make during data analysis. We coordinated 161 researchers in 73 research teams and observed their research decisions as they used the same data to independently test the same prominent social science hypothesis: that greater immigration reduces support for social policies among the public. In this typical case of social science research, research teams reported both widely diverging numerical findings and substantive conclusions despite identical start conditions. Researchers’ expertise, prior beliefs, and expectations barely predict the wide variation in research outcomes. More than 95% of the total variance in numerical results remains unexplained even after qualitative coding of all identifiable decisions in each team’s workflow. This reveals a universe of uncertainty that remains hidden when considering a single study in isolation. The idiosyncratic nature of how researchers’ results and conclusions varied is a previously underappreciated explanation for why many scientific hypotheses remain contested. These results call for greater epistemic humility and clarity in reporting scientific findings