Abstract:
Colors and their patterns are fascinating phenotypes with great importance for fitness under natural conditions. For this reason and because pigmentation is associated with diseases, much research was devoted to study the genetics of pigmentation in animals. Considerable contribution to our understanding of color phenotypes was made by studies in domesticated animals that exhibit dazzling variation in color traits. Koi strains, the ornamental variants of the common carp, are a striking example for color variability that was selected by man during a very short period on an evolutionary timescale. Among several pigmentation genes, genetic variation in Melanocrtin receptor 1 was repeatedly associated with dark pigmentation phenotypes in numerous animals. In this study, we cloned Melanocrtin receptor 1 from the common carp. We found that alleles of the gene were not associated with the development of black color in Koi. However, the mRNA expression levels of the gene were higher during dark pigmentation development in larvae and in dark pigmented tissues of adult fish, suggesting that variation in the regulation of the gene is associated with black color in Koi. These regulatory differences are reflected in both the timing of the dark-pigmentation development and the different mode of inheritance of the two black patterns associated with them. Identifying the genetic basis of color and color patterns in Koi will promote the production of this valuable ornamental fish. Furthermore, given the rich variety of colors and patterns, Koi serves as a good model to unravel pigmentation genes and their phenotypic effects and by that to improve our understanding of the genetic basis of colors also in natural populations.

Abstract:
In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the same property.

Abstract:
Following K. Mahler's suggestion for further research on intrinsic approximation on the Cantor ternary set, we obtain a Dirichlet type theorem for the limit sets of rational iterated function systems. We further investigate the behavior of these approximation functions under random translations. We connect the information regarding the distribution of rationals on the limit set encoded in the system to the distribution of rationals in reduced form by proving a Khinchin type theorem. Finally, using a result of S. Ramanujan, we prove a theorem motivating a conjecture regarding the distribution of rationals in reduced form on the Cantor ternary set.

Abstract:
Simultaneous Diophantine approximation is concerned with the approximation of a point $\mathbf x\in\mathbb R^d$ by points $\mathbf r\in\mathbb Q^d$, with a view towards jointly minimizing the quantities $\|\mathbf x - \mathbf r\|$ and $H(\mathbf r)$. Here $H(\mathbf r)$ is the so-called "standard height" of the rational point $\mathbf r$. In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examples of nonstandard height functions, computing their exponents of irrationality as well as giving more precise results. A list of open questions is also given.

Abstract:
Fix $d\in\mathbb N$, and let $S\subseteq\mathbb R^d$ be either a real-analytic manifold or the limit set of an iterated function system (for example, $S$ could be the Cantor set or the von Koch snowflake). An $extrinsic$ Diophantine approximation to a point $\mathbf x\in S$ is a rational point $\mathbf p/q$ close to $\mathbf x$ which lies $outside$ of $S$. These approximations correspond to a question asked by K. Mahler ('84) regarding the Cantor set. Our main result is an extrinsic analogue of Dirichlet's theorem. Specifically, we prove that if $S$ does not contain a line segment, then for every $\mathbf x\in S\setminus\mathbb Q^d$, there exists $C > 0$ such that infinitely many vectors $\mathbf p/q\in \mathbb Q^d\setminus S$ satisfy $\|\mathbf x - \mathbf p/q\| < C/q^{(d + 1)/d}$. As this formula agrees with Dirichlet's theorem in $\mathbb R^d$ up to a multiplicative constant, one concludes that the set of rational approximants to points in $S$ which lie outside of $S$ is large. Furthermore, we deduce extrinsic analogues of the Jarn\'ik--Schmidt and Khinchin theorems from known results.

Abstract:
We give a necessary and sufficient condition for the following property of an integer $d\in\mathbb N$ and a pair $(a,A)\in\mathbb R^2$: There exist $\kappa > 0$ and $Q_0\in\mathbb N$ such that for all $\mathbf x\in \mathbb R^d$ and $Q\geq Q_0$, there exists $\mathbf p/q\in\mathbb Q^d$ such that $1\leq q\leq Q$ and $\|\mathbf x - \mathbf p/q\| \leq \kappa q^{-a} Q^{-A}$. This generalizes Dirichlet's theorem, which states that this property holds (with $\kappa = Q_0 = 1$) when $a = 1$ and $A = 1/d$. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a comeager set. This is also true if $\mathbb R^d$ is replaced by an appropriate "Diophantine space", such as a nonsingular rational quadratic hypersurface which contains rational points. Finally, in the case $d = 1$ we describe the set of exceptions in terms of classical Diophantine conditions.

Abstract:
We propose new loss functions for learning patch based descriptors via deep Convolutional Neural Networks. The learned descriptors are compared using the L2 norm and do not require network processing of pairs of patches. The success of the method is based on a few technical novelties, including an innovative loss function that, for each training batch, computes higher moments of the score distributions. Combined with an Approximate Nearest Neighbor patch matching method and a flow interpolating method, state of the art performance is obtained on the most challenging and competitive optical flow benchmarks.

Abstract:
The tree-metric theorem provides a necessary and sufficient condition for a dissimilarity matrix to be a tree metric, and has served as the foundation for numerous distance-based reconstruction methods in phylogenetics. Our main result is an extension of the tree-metric theorem to more general dissimilarity maps. In particular, we show that a tree with n leaves is reconstructible from the weights of the m-leaf subtrees provided that n \geq 2m-1.

Abstract:
We explore and refine techniques for estimating the Hausdorff dimension of exceptional sets and their diffeomorphic images. Specifically, we use a variant of Schmidt's game to deduce the strong C^1 incompressibility of the set of badly approximable systems of linear forms as well as of the set of vectors which are badly approximable with respect to a fixed system of linear forms.

Abstract:
Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e. gives zero measure to the set of very well approximable numbers. We show on the other hand that there exist examples where the Lyapunov exponent is infinite and the invariant measure is not extremal. Finally, we answer in the negative a question posed by Kleinbock, Lindenstrauss, and Weiss, by constructing a family of measures on the real line which are Ahlfors regular and yet do not satisfy a 0-1 law for approximability.