H\"older estimates for parabolic operators on domains with rough boundary

We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the domain, including a very weak compatibility condition between the Dirichlet boundary part and its complement, we prove H\"older continuity of the solution in space and time.Comment: 1 figur

Minimizing the Cost of Team Exploration

A group of mobile agents is given a task to explore an edge-weighted graph $G$, i.e., every vertex of $G$ has to be visited by at least one agent. There is no centralized unit to coordinate their actions, but they can freely communicate with each other. The goal is to construct a deterministic strategy which allows agents to complete their task optimally. In this paper we are interested in a cost-optimal strategy, where the cost is understood as the total distance traversed by agents coupled with the cost of invoking them. Two graph classes are analyzed, rings and trees, in the off-line and on-line setting, i.e., when a structure of a graph is known and not known to agents in advance. We present algorithms that compute the optimal solutions for a given ring and tree of order $n$, in $O(n)$ time units. For rings in the on-line setting, we give the $2$-competitive algorithm and prove the lower bound of $3/2$ for the competitive ratio for any on-line strategy. For every strategy for trees in the on-line setting, we prove the competitive ratio to be no less than $2$, which can be achieved by the $DFS$ algorithm.Comment: 25 pages, 4 figures, 5 pseudo-code

Insulin‐like growth factor 1 signaling in tenocytes is required for adult tendon growth

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/154662/1/fsb2fj201901503r.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154662/2/fsb2fj201901503r-sup-0001.pd

Ionic Liquids for Utilization of Waste Heat from Distributed Power Generation Systems

The objective of this research project was the development of ionic liquids to capture and utilize waste heat from distributed power generation systems. Ionic Liquids (ILs) are organic salts that are liquid at room temperature and they have the potential to make fundamental and far-reaching changes in the way we use energy. In particular, the focus of this project was fundamental research on the potential use of IL/CO2 mixtures in absorption-refrigeration systems. Such systems can provide cooling by utilizing waste heat from various sources, including distributed power generation. The basic objectives of the research were to design and synthesize ILs appropriate for the task, to measure and model thermophysical properties and phase behavior of ILs and IL/CO2 mixtures, and to model the performance of IL/CO2 absorption-refrigeration systems

Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems

On bounded three-dimensional domains, we consider divergence-type operators including mixed homogeneous Dirichlet and Neumann boundary conditions and discontinuous coefficient functions. We develop a geometric framework in which it is possible to prove that the operator provides an isomorphism of suitable function spaces. In particular, in these spaces, the gradient of solutions turns out to be integrable with exponent larger than the space dimension three. Relevant examples from real-world applications are provided in great detail

Tight bounds for online TSP on the line

We consider the online traveling salesperson problem (TSP), where requests appear online over time on the real line and need to be visited by a server initially located at the origin. We distinguish between closed and open online TSP, depending on whether the server eventually needs to return to the origin or not. While online TSP on the line is a very natural online problem that was introduced more than two decades ago, no tight competitive analysis was known to date. We settle this problem by providing tight bounds on the competitive ratios for both the closed and the open variant of the problem. In particular, for closed online TSP, we provide a 1.64-competitive algorithm,thus matching a known lower bound. For open online TSP, we give a new upper bound as well as a matching lower bound that establish the remarkable competitive ratio of 2.04. Additionally, we consider the online Dial-A-Ride problem on the line, where each request needs to be transported to a specified destination. We provide an improved non-preemptive lower bound of 1.75 for this setting, as well as an improved preemptive algorithm with competitive ratio 2.41.Finally, we generalize known and give new complexity results for the underlying offline problems. In particular, we give an algorithm with running time O(n2) for closed offline TSP on the line with release dates and show that both variants of offline Dial-A-Ride on the line are NP-hard for any capacity c≥2 of the server

Degree-constrained orientations of embedded graphs

Abstract. We investigate the problem of orienting the edges of an em-bedded graph in such a way that the in-degrees of both the nodes and faces meet given values. We show that the number of feasible solutions is bounded by 22g, where g is the genus of the embedding, and all so-lutions can be determined within time O(22g|E|2 + |E|3). In particular, for planar graphs the solution is unique if it exists, and in general the problem of finding a feasible orientation is fixed-parameter tractable in g. In sharp contrast to these results, we show that the problem becomes NP-complete even for a fixed genus if only upper and lower bounds on the in-degrees are specified instead of exact values.

Undirected Graph Exploration with Θ(log log n) Pebbles

We consider the fundamental problem of exploring an undi-rected and initially unknown graph by an agent with lit-tle memory. The vertices of the graph are unlabeled, and the edges incident to a vertex have locally distinct labels. In this setting, it is known that Θ(logn) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We show that this memory requirement can be decreased significantly by making a part of the mem-ory distributable in the form of pebbles. A pebble is a device that can be dropped to mark a vertex and can be collected when the agent returns to the vertex. We show that for an agent O(log logn) distinguishable pebbles and bits of mem-ory are sufficient to explore any bounded-degree graph with at most n vertices. We match this result with a lower bound exhibiting that for any agent with sub-logarithmic memory, Ω(log logn) distinguishable pebbles are necessary for explo-ration.