Non-Rectangular Convolutions and (Sub-)Cadences with Three Elements

The discrete acyclic convolution computes the 2n-1 sums sum_{i+j=k; (i,j) in [0,1,2,...,n-1]^2} (a_i b_j) in O(n log n) time. By using suitable offsets and setting some of the variables to zero, this method provides a tool to calculate all non-zero sums sum_{i+j=k; (i,j) in (P cap Z^2)} (a_i b_j) in a rectangle P with perimeter p in O(p log p) time. This paper extends this geometric interpretation in order to allow arbitrary convex polygons P with k vertices and perimeter p. Also, this extended algorithm only needs O(k + p(log p)^2 log k) time. Additionally, this paper presents fast algorithms for counting sub-cadences and cadences with 3 elements using this extended method

The \mu-Calculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity

It is known that the alternation hierarchy of least and greatest fixpoint operators in the mu-calculus is strict. However, the strictness of the alternation hierarchy does not necessarily carry over when considering restricted classes of structures. A prominent instance is the class of infinite words over which the alternation-free fragment is already as expressive as the full mu-calculus. Our current understanding of when and why the mu-calculus alternation hierarchy is not strict is limited. This paper makes progress in answering these questions by showing that the alternation hierarchy of the mu-calculus collapses to the alternation-free fragment over some classes of structures, including infinite nested words and finite graphs with feedback vertex sets of a bounded size. Common to these classes is that the connectivity between the components in a structure from such a class is restricted in the sense that the removal of certain vertices from the structure's graph decomposes it into graphs in which all paths are of finite length. Our collapse results are obtained in an automata-theoretic setting. They subsume, generalize, and strengthen several prior results on the expressivity of the mu-calculus over restricted classes of structures.Comment: In Proceedings GandALF 2012, arXiv:1210.202

On Maximal Repeats in Compressed Strings

This paper presents and proves a new non-trivial upper bound on the number of maximal repeats of compressed strings. Using Theorem 1 of Raffinot\u27s article "On Maximal Repeats in Strings", this upper bound can be directly translated into an upper bound on the number of nodes in the Compacted Directed Acyclic Word Graphs of compressed strings. More formally, this paper proves that the number of maximal repeats in a string with z (self-referential) LZ77-factors and without q-th powers is at most 3q(z+1)^3-2. Also, this paper proves that for 2000 <= z <= q this upper bound is tight up to a constant factor

On Extensions of Maximal Repeats in Compressed Strings

This paper provides an upper bound for several subsets of maximal repeats and maximal pairs in compressed strings and also presents a formerly unknown relationship between maximal pairs and the run-length Burrows-Wheeler transform. This relationship is used to obtain a different proof for the Burrows-Wheeler conjecture which has recently been proven by Kempa and Kociumaka in "Resolution of the Burrows-Wheeler Transform Conjecture". More formally, this paper proves that a string $S$ with $z$ LZ77-factors and without $q$-th powers has at most $73(\log_2 |S|)(z+2)^2$ runs in the run-length Burrows-Wheeler transform and the number of arcs in the compacted directed acyclic word graph of $S$ is bounded from above by $18q(1+\log_q |S|)(z+2)^2$

Diverse outcomes of homologous recombination in the human Y chromosome

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Biology, 2008.Includes bibliographical references.Mammalian sex chromosomes began diverging from an ordinary pair of autosomes roughly 300 million years ago. Inversions in the evolving Y chromosome sequentially suppressed recombination with the X chromosome. While pseudoautosomal regions in the human Y chromosome still participate regularly in allelic homologous recombination, the male-specific region of the Y (MSY) - the only haploid portion of the nuclear genome - does not. It does, however, engage in non-allelic homologous recombination. In this thesis, I examine modes and outcomes of non-allelic homologous recombination in the MSY. The predictions presented here are based on the double-strand break repair model of recombination between homologous chromosomes, in which a double-strand break (DSB) is the common precursor to crossing over and gene conversion. First, I show that massive MSY-specific palindromes, which maintain arm-to-arm sequence identity via gene conversion, are also the targets of crossing over. Crossover events in palindromes can lead to isochromosome formation and diverse reproductive disorders including sex reversal, male infertility, and Turner syndrome. Second, I demonstrate that a region of the MSY - thought to be recombinationally suppressed with the X chromosome - does undergo extensive X-Y gene conversion. This region encompasses hotspots of ectopic crossover events that lead to X-Y translocations associated with sex reversal syndromes. Although sequences in the MSY engage in productive recombination via gene conversion, alternative resolution of DSBs by crossing over can produce evolutionary "dead ends".by Julian H. Lange.Ph.D

How to Design Learning Applications that Support Learners in their Moment of Need – Didactic Requirements of Micro Learning

The COVID-19 pandemic is showing the limits of our traditional education systems that mainly build on classroom lectures with face-to-face interaction between teachers or trainers and learners. Now more than ever, there is a growing need for digital learning formats that make it possible to maintain teaching in universities, schools, and enterprises despite the spatial distance from the learners. To address these new conditions of learning, short and small learning units are a promising approach when it comes to demand-oriented learning solutions. However, the question of how to design didactically appropriate micro content is not yet answered by research. To close this research gap, we conducted a qualitative interview study with professionals in the field of instructional design and technology-enhanced learning-design. With this information, we were able to derive 20 requirements for designing effective micro content

Pricing High Growth Firms: Arbitrage Opportunities in the Inc. 100

The ability of the market to price high growth stocks is examined by analyzing the returns to simple investment portfolio strategies based on public information. The portfolios consist of shares in the firms listed in the Inc. 100 Ranking of the fastest growing public companies in America. The results indicate that significant abnormal returns are generated by these strategies, even after adjusting for risk. Although the tests could potentially be affected by a form of survivorship bias, supplementary analyses indicate that this is unlikely to be the case here. These results support the assumption that markets have difficulties pricing high-growth entities, leaving significant arbitrage opportunities in these stocks and validating the use of various market timing practices