On the Hausdorff dimension of ultrametric subsets in R^n

For every e>0, any subset of R^n with Hausdorff dimension larger than (1-e)n must have ultrametric distortion larger than 1/(4e).Comment: 4 pages, improved layou

The Erlang Weighted Tree, A New Branching Process

In this paper, we propose a new branching process which we call Erlang Weighted Tree(EWT). EWT appears as the local weak limit of a random graph model proposed by Richard La and Maya Kabkab. We derive the main properties of EWT such as the probability of extinction, the emergence of phase transition and growth rate

A node-capacitated Okamura-Seymour theorem

The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal c > 0, if the node cut conditions are satisfied, then one can simultaneously route a c-fraction of all the demands. This answers an open question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of multi-commodity polymatroid networks introduced by Chekuri, et. al. Our approach employs a new type of random metric embedding in order to round the convex programs corresponding to these more general flow problems.Comment: 30 pages, 5 figure

Impact of Community Structure on Cascades

The threshold model is widely used to study the propagation of opinions and technologies in social networks. In this model, individuals adopt the new behavior based on how many neighbors have already chosen it. Specifically, we consider the permanent adoption model where individuals that have adopted the new behavior cannot change their state. We study cascades under the threshold model on sparse random graphs with community structure to see whether the existence of communities affects the number of individuals who finally adopt the new behavior. When seeding a small number of agents with the new behavior, the community structure has little effect on the final proportion of people that adopt it, i.e., the contagion threshold is the same as if there were just one community. On the other hand, seeding a fraction of the population with the new behavior has a significant impact on the cascade with the optimal seeding strategy depending on how strongly the communities are connected. In particular, when the communities are strongly connected, seeding in one community outperforms the symmetric seeding strategy that seeds equally in all communities. We also investigate the problem of optimum seeding given a budget constraint, and propose a gradient-based heuristic seeding strategy. Our algorithm, numerically, dispels commonly held beliefs in the literature that suggest the best seeding strategy is to seed over the nodes with the highest number of neighbors.Comment: Version to be published to EC 201

Performance Bounds for Policy-Based Average Reward Reinforcement Learning Algorithms

Many policy-based reinforcement learning (RL) algorithms can be viewed as instantiations of approximate policy iteration (PI), i.e., where policy improvement and policy evaluation are both performed approximately. In applications where the average reward objective is the meaningful performance metric, often discounted reward formulations are used with the discount factor being close to 1, which is equivalent to making the expected horizon very large. However, the corresponding theoretical bounds for error performance scale with the square of the horizon. Thus, even after dividing the total reward by the length of the horizon, the corresponding performance bounds for average reward problems go to infinity. Therefore, an open problem has been to obtain meaningful performance bounds for approximate PI and RL algorithms for the average-reward setting. In this paper, we solve this open problem by obtaining the first non-trivial error bounds for average-reward MDPs which go to zero in the limit where when policy evaluation and policy improvement errors go to zero.Comment: 30 page

Extracting reliable mechanical properties using the nanoindentation technique

PhD ThesisExtracting the mechanical properties of thin films and small volumes of bulk materials through the use of nanoindentation is a well established technique but getting good data from all types of test sample is not always easy. Factors such as surface roughness and oxidation, density/porosity of the material, adhesion/detachment of a thin film, pile-up/sink-in, the presence of the substrate, as well as grain size and its distribution have a significant effect on the observed mechanical properties (e.g. Young’s modulus and hardness). Considerable differences between predicted and observed performance can be seen depending on the material tested and how it has been prepared. This thesis concerns developing test protocols to get good nanoindentation data and reliable measurements of the properties for a range of material types (chiefly metals and ceramics). Firstly, this work highlights the effect of crystallographic anisotropy, grain size, shape and orientation on the mechanical response of metallic thin films such as copper used for semiconductor metallisation. Results obtained on highly polished semiconductor materials were compared with those from engineering surfaces with much higher roughness which show increased scatter in results across the complete range of contact scales. Further studies were carried out on hard coatings and bulk materials such as titanium carbide, zirconium nitride and tungsten. The scatter in data obtained at low tests loads is dominated by anisotropy and grain size effects but disappears at higher loads. For soft materials such as copper, the appearance of pile-up was shown to be significant when compared with harder materials which tend to sink-in. Secondly, to assess the effect of creep (time-dependent behaviour) and also grain boundary effects on the measured mechanical properties, soft materials with a range of grain sizes have been examined. Different indentation control cycles (load and displacement control, single indent and multicycling tests) have been investigated to determine what is most suitable with displacement control being essential in most cases. To study the effect of the density/porosity of the sample and its surface roughness on mechanical properties, the work was carried out on porous coatings of tin, copper and copper-tin alloy coatings with a low density. To further understand the behaviour of porous materials and their mechanical properties, finite element analysis was also used to compare the experimental results with a numerical model. The size, shape and location of porosity with respect to the indenter is critical in determining the mechanical properties of a porous material obtained from nanoindentation analysis. Finally, fully processed engineering surfaces were investigated at the component scale to compare with idealised flat plate samples. Titanium-based and cobalt-chrome alloys in the form of femoral heads and stems for replacement hips have been used to assess the effect of in service oxidation on mechanical properties. These have been studied to look at the effects of sample fixturing and support and surface contact in worn and virgin regions of the sample surface. The extent of oxidation and the mechanical properties of the oxide produced are critical in dictating performance

Evaluation of Different Types of Lasers in Surface Conditioning of Porcelains: A Review Article

To achieve proper bond strength for porcelains, adequate surface roughness is essential, which is traditionally gained by sandblasting or acid etching with hydrofluoric (HF) acid. Nowadays with the development of laser systems, serious efforts were made to apply this new instrument for surface etching of porcelains due to easy usage, safety, and more efficiency. There are different kinds of lasers and porcelains, so choosing the ones which will be good match for each other is crucial. Besides that, changing the irradiation setting can be beneficial as well. This article reviewed 33 related studies and summarized results of etching accomplished by Nd:YAG, Er:YAG, Er,Cr:YSGG and CO2 lasers on different types of porcelains considering different laser settings and evaluation methods to bring a comprehensive insight

Backward and Forward Inference in Interacting Independent-Cascade Processes: A Scalable and Convergent Message-Passing Approach

We study the problems of estimating the past and future evolutions of two diffusion processes that spread concurrently on a network. Specifically, given a known network $G=(V, \overrightarrow{E})$ and a (possibly noisy) snapshot $\mathcal{O}_n$ of its state taken at (a possibly unknown) time $W$, we wish to determine the posterior distributions of the initial state of the network and the infection times of its nodes. These distributions are useful in finding source nodes of epidemics and rumors -- $\textit{backward inference}$ -- , and estimating the spread of a fixed set of source nodes -- $\textit{forward inference}$. To model the interaction between the two processes, we study an extension of the independent-cascade (IC) model where, when a node gets infected with either process, its susceptibility to the other one changes. First, we derive the exact joint probability of the initial state of the network and the observation-snapshot $\mathcal{O}_n$. Then, using the machinery of factor-graphs, factor-graph transformations, and the generalized distributive-law, we derive a Belief-Propagation (BP) based algorithm that is scalable to large networks and can converge on graphs of arbitrary topology (at a likely expense in approximation accuracy)