Colored Non-Crossing Euclidean Steiner Forest

Given a set of $k$-colored points in the plane, we consider the problem of finding $k$ trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For $k=1$, this is the well-known Euclidean Steiner tree problem. For general $k$, a $k\rho$-approximation algorithm is known, where $\rho \le 1.21$ is the Steiner ratio. We present a PTAS for $k=2$, a $(5/3+\varepsilon)$-approximation algorithm for $k=3$, and two approximation algorithms for general~$k$, with ratios $O(\sqrt n \log k)$ and $k+\varepsilon$

Correlated disordered interactions on Potts models

Using a weak-disorder scheme and real-space renormalization-group techniques, we obtain analytical results for the critical behavior of various q-state Potts models with correlated disordered exchange interactions along d1 of d spatial dimensions on hierarchical (Migdal-Kadanoff) lattices. Our results indicate qualitative differences between the cases d-d1=1 (for which we find nonphysical random fixed points, suggesting the existence of nonperturbative fixed distributions) and d-d1>1 (for which we do find acceptable perturbartive random fixed points), in agreement with previous numerical calculations by Andelman and Aharony. We also rederive a criterion for relevance of correlated disorder, which generalizes the usual Harris criterion.Comment: 8 pages, 4 figures, to be published in Physical Review

Effect of couplings weakening and reversing in ferromagnetic Ising systems - Rigorous inequalities

We consider Ising systems where all the many-spin couplings $J_A$ are positive. We show that the absolute value of all the many-spin correlations does not increase when the value of any of the couplings is reduced, taking any value in the interval $[-J_A,J_A]$. Results of this type are motivated by work in systems such as random field Ising models.Comment: ps, 5 pages, no figure

Random field Ising systems on a general hierarchical lattice: Rigorous inequalities

Random Ising systems on a general hierarchical lattice with both, random fields and random bonds, are considered. Rigorous inequalities between eigenvalues of the Jacobian renormalization matrix at the pure fixed point are obtained. These inequalities lead to upper bounds on the crossover exponents $\{\phi_i\}$.Comment: LaTeX, 13 pages, figs. 1a,1b,2. To be published in PR

Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles

In this paper we study two geometric data structure problems in the special case when input objects or queries are fat rectangles. We show that in this case a significant improvement compared to the general case can be achieved. We describe data structures that answer two- and three-dimensional orthogonal range reporting queries in the case when the query range is a \emph{fat} rectangle. Our two-dimensional data structure uses $O(n)$ words and supports queries in $O(\log\log U +k)$ time, where $n$ is the number of points in the data structure, $U$ is the size of the universe and $k$ is the number of points in the query range. Our three-dimensional data structure needs $O(n\log^{\varepsilon}U)$ words of space and answers queries in $O(\log \log U + k)$ time. We also consider the rectangle stabbing problem on a set of three-dimensional fat rectangles. Our data structure uses $O(n)$ space and answers stabbing queries in $O(\log U\log\log U +k)$ time.Comment: extended version of a WADS'19 pape

Comparing persistence diagrams through complex vectors

The natural pseudo-distance of spaces endowed with filtering functions is precious for shape classification and retrieval; its optimal estimate coming from persistence diagrams is the bottleneck distance, which unfortunately suffers from combinatorial explosion. A possible algebraic representation of persistence diagrams is offered by complex polynomials; since far polynomials represent far persistence diagrams, a fast comparison of the coefficient vectors can reduce the size of the database to be classified by the bottleneck distance. This article explores experimentally three transformations from diagrams to polynomials and three distances between the complex vectors of coefficients.Comment: 11 pages, 4 figures, 2 table

Reducing Sexual Risk among Racial/ethnic-minority Ninth Grade Students: Using Intervention Mapping to Modify an Evidenced-based Curriculum

Background: Racial/ethnic-minority 9th graders are at increased risk for teen pregnancy, HIV, and STIs compared to their White peers. Yet, few effective sexual health education programs exist for this population. Purpose: To apply IM Adapt—a systematic theory- and evidence-based approach to program adaptation—to modify an effective middle school sexual health education curriculum, It’s Your Game…Keep It Real! (IYG), for racial/ethnic-minority 9th graders. Methods: Following the six steps of IM Adapt, we conducted a needs assessment to describe the health problems and risk behaviors of the new population; reviewed existing evidence-based programs; assessed the fit of IYG for the new population regarding behavioral outcomes, determinants, change methods, delivery, and implementation; modified materials and activities; planned for implementation and evaluation. Results: Needs assessment findings indicated that IYG targeted relevant health and risk behaviors for racial/ethnic-minority 9th graders but required additional focus on contraceptive use, dating violence prevention, active consent, and access to healthcare services. Behavioral outcomes and matrices of change objectives for IYG were modified accordingly. Theoretical methods and practical applications were identified to address these behavioral outcomes, and new activities developed. Youth provided input on activity modifications. School personnel guided modifications to IYG’s scope and sequence, and delivery. The adapted program, Your Game, Your Life, comprised fifteen 30-minute lessons targeting determinants of sexual behavior and healthy dating relationships. Pilot-test data from 9th graders in two urban high schools indicate promising results. Conclusion: IM Adapt provides a systematic theory- and evidence-based approach for adapting existing evidence-based sexual health education curricula for a new population whilst retaining essential elements that made the original program effective. Youth and school personnel input ensured that the adapted program was age-appropriate, culturally sensitive, and responsive to the needs of the new population. IM Adapt contributes to the limited literature on systematic approaches to program adaptation

Conflict-Free Coloring Made Stronger

In FOCS 2002, Even et al. showed that any set of $n$ discs in the plane can be Conflict-Free colored with a total of at most $O(\log n)$ colors. That is, it can be colored with $O(\log n)$ colors such that for any (covered) point $p$ there is some disc whose color is distinct from all other colors of discs containing $p$. They also showed that this bound is asymptotically tight. In this paper we prove the following stronger results: \begin{enumerate} \item [(i)] Any set of $n$ discs in the plane can be colored with a total of at most $O(k \log n)$ colors such that (a) for any point $p$ that is covered by at least $k$ discs, there are at least $k$ distinct discs each of which is colored by a color distinct from all other discs containing $p$ and (b) for any point $p$ covered by at most $k$ discs, all discs covering $p$ are colored distinctively. We call such a coloring a {\em $k$-Strong Conflict-Free} coloring. We extend this result to pseudo-discs and arbitrary regions with linear union-complexity. \item [(ii)] More generally, for families of $n$ simple closed Jordan regions with union-complexity bounded by $O(n^{1+\alpha})$, we prove that there exists a $k$-Strong Conflict-Free coloring with at most $O(k n^\alpha)$ colors. \item [(iii)] We prove that any set of $n$ axis-parallel rectangles can be $k$-Strong Conflict-Free colored with at most $O(k \log^2 n)$ colors. \item [(iv)] We provide a general framework for $k$-Strong Conflict-Free coloring arbitrary hypergraphs. This framework relates the notion of $k$-Strong Conflict-Free coloring and the recently studied notion of $k$-colorful coloring. \end{enumerate} All of our proofs are constructive. That is, there exist polynomial time algorithms for computing such colorings

The Role of Vortices in the Mutual Coupling of Superconducting and Normal-Metal Films

I propose a possible explanation to a recently observed ``cross-talk'' effect in metal-insulator-metal trilayers, indicating a sharp peak near a superconducting transition in one of the metal films. Coulomb interactions are excluded as a dominant coupling mechanism, and an alternative is suggested, based on the local fluctuating electric field induced by mobile vortices in the superconducting layer. This scenario is compatible with the magnitude of the peak signal and its shape; most importantly, it addresses the {\it non-reciprocity} of the effect in exchanging the roles of the films.Comment: 13 pages, RevTe