# Source Themes

## Moving sidewinding forward: optimizing contact patterns for limbless robots via geometric mechanics

Contact planning is crucial to the locomotion performance of limbless robots. Typically; the pattern by which contact is made and broken between the mechanism and its environment determines the motion of the robot. The design of these patterns; often …

## Reconstruction of Backbone Curves for Snake Robots

Snake robots composed of alternating single-axis pitch and yaw joints have many internal degrees of freedom, which make them capable of versatile three-dimensional locomotion. In motion planning process, snake robot motions are often designed …

## Optimizing Coordinate Choice for Locomotion Systems with Toroidal Shape Spaces

In a geometric mechanics framework, the configuration space is decomposed into a shape space and a position space. The internal motion of the system is prescribed by a closed loop in the shape space, which causes net motion in the position space. If …

## Bounding The Number of Linear Regions in Local Area for Neural Networks with ReLU Activations

The number of linear regions is one of the distinct properties of the neural networks using piecewise linear activation functions such as ReLU, comparing with those conventional ones using other activation functions. Previous studies showed this …

## Tropical optimal transport and Wasserstein distances

We study the problem of optimal transport in tropical geometry and define the Wasserstein-p distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric—a combinatorial metric that has been …

## Two-player incentive compatible outcome functions are affine maximizers

In mechanism design, for a given type space, there may be incentive compatible outcome functions which are not affine maximizers. We prove that for two-player games on a discrete type space, any given outcome function can be turned into an affine …

## Tropical Fermat-Weber Points

In a metric space, the Fermat-Weber points of a sample are statistics to measure the central tendency of the sample and it is well known that the Fermat--Weber point of a sample is not necessarily unique in the metric space. We investigate the …

## Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective

Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. As data objects, they are characterized by the challenges associated with "big data," as well as the complication that their discrete geometric …

## Towards a Tropical Hodge Bundle

The moduli space $M_g^{trop}$ of tropical curves of genus $g$ is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus $g$. For each such graph $\Gamma$, the associated canonical linear system $\vert K_\Gamma\vert$ has …

## Convexity in Tree Spaces

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics. The $CAT-0$ metric of Billera-Holmes-Vogtman arises from the theory of orthant …