Biophysical regulation of stem cell behavior within the niche.

Stem cells reside within most tissues throughout the lifetimes of mammalian organisms. To maintain their capacities for division and differentiation and thereby build, maintain, and regenerate organ structure and function, these cells require extensive and precise regulation, and a critical facet of this control is the local environment or niche surrounding the cell. It is well known that soluble biochemical signals play important roles within such niches, and a number of biophysical aspects of the microenvironment, including mechanical cues and spatiotemporally varying biochemical signals, have also been increasingly recognized to contribute to the repertoire of stimuli that regulate various stem cells in various tissues of both vertebrates and invertebrates. For example, biochemical factors immobilized to the extracellular matrix or the surface of neighboring cells can be spatially organized in their placement. Furthermore, the extracellular matrix provides mechanical support and regulatory information, such as its elastic modulus and interfacial topography, which modulate key aspects of stem cell behavior. Numerous examples of each of these modes of regulation indicate that biophysical aspects of the niche must be appreciated and studied in conjunction with its biochemical properties

Twisted Blanchfield pairings, twisted signatures and Casson-Gordon invariants

This paper decomposes into two main parts. In the algebraic part, we prove an isometry classification of linking forms over $\mathbb{R}[t^{\pm 1}]$ and $\mathbb{C}[t^{\pm 1}]$. Using this result, we associate signature functions to any such linking form and thoroughly investigate their properties. The topological part of the paper applies this machinery to twisted Blanchfield pairings of knots. We obtain twisted generalizations of the Levine-Tristram signature function which share several of its properties. We study the behavior of these twisted signatures under satellite operations. In the case of metabelian representations, we relate our invariants to the Casson-Gordon invariants and obtain a concrete formula for the metabelian Blanchfield pairings of satellites. Finally, we perform explicit computations on certain linear combinations of algebraic knots, recovering a non-slice result of Hedden, Kirk and Livingston.Comment: 81 pages, 1 figur

On the growth rate of 1324-avoiding permutations

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function in this case does not have an algebraic singularity. Rather, the number of 1324-avoiding permutations of length $n$ behaves as $$B\cdot \mu^n \cdot \mu_1^{n^{\sigma}} \cdot n^g.$$ We estimate $\mu=11.60 \pm 0.01,$ $\sigma=1/2,$ $\mu_1 = 0.0398 \pm 0.0010,$ $g = -1.1 \pm 0.2$ and $B =9.5 \pm 1.0.$Comment: 20 pages, 10 figure