Computing Lempel-Ziv Factorization Online

We present an algorithm which computes the Lempel-Ziv factorization of a word $W$ of length $n$ on an alphabet $\Sigma$ of size $\sigma$ online in the following sense: it reads $W$ starting from the left, and, after reading each $r = O(\log_{\sigma} n)$ characters of $W$, updates the Lempel-Ziv factorization. The algorithm requires $O(n \log \sigma)$ bits of space and O(n \log^2 n) time. The basis of the algorithm is a sparse suffix tree combined with wavelet trees

On the dependence of the reflection operator on boundary conditions for biharmonic functions

The biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the Stokes flows, as well as in a radar imaging problem. We discuss the reflection formulas for the biharmonic functions $u(x,y)\in\mathbb{R}^2$ subject to different boundary conditions on a real-analytic curve in the plane. The obtained formulas, generalizing the celebrated Schwarz symmetry principle for harmonic functions, have different structures. In particular, in the special case of the boundary, $\Gamma_0 :=\{y=0\}$, reflections are point to point when the given on $\Gamma_0$ conditions are $u=\partial_nu=0$, $u=\Delta u=0$ or $\partial_nu=\partial n\Delta u=0$, and point to a continuous set when $u=\partial_n\Delta u=0$ or $\partial_nu=\Delta u=0$ on $\Gamma_0$.Comment: 18 page

Mexican Dance Group: Breaking Barriers One Tap at a Time (Chapter in Emerging Issues and Trends in Education)

Excerpt: With a growing number of Hispanic students in schools (Díaz-Rico & Weed, 2014; Oregon Department of Education, 2012), many educators wonder how to make the school community more accessible to Hispanic parents (González, Moll, & Amanti, 2005; Gorski & Pothini, 2014; Valdés, 1996; Wink, 2005). The dance group described in this chapter demonstrates a natural way in which one teacher has accomplished Hispanic parental accessibility and, in doing so, positively impacted her school community. Rosa Floyd, the director of Nellie Muir’s Dance Group, has been teaching in Spanish-English bilingual classrooms for more than nineteen years. She came from Mexico as an adult and learned English, becoming an instructional assistant and subsequently a teacher. She has chosen to work with Hispanic students and regards her work as a bridge between Mexican parents and schools. Bilingual and bicultural, Rosa understands the Mexican community as well as the Anglo-dominated school culture. For several years, she has efffectively facilitated cross-cultural relationships between parents and teachers through the use of traditional dance groups. The months of practice preparing for the Cinco de Mayo presentations provide a catalyst for change as the teachers and parents address the invisible barriers that have kept Hispanic parents separated from the school community. Rosa’s effforts have led to a more welcoming and respectful school environment that embraces Mexican parents and reinforces students’ sense of cultural identity and heritage pride