170 research outputs found

### Poisson trees, succession lines and coalescing random walks

We give a deterministic algorithm to construct a graph with no loops (a tree
or a forest) whose vertices are the points of a d-dimensional stationary
Poisson process S, subset of R^d. The algorithm is independent of the origin of
coordinates. We show that (1) the graph has one topological end --that is, from
any point there is exactly one infinite self-avoiding path; (2) the graph has a
unique connected component if d=2 and d=3 (a tree) and it has infinitely many
components if d\ge 4 (a forest); (3) in d=2 and d=3 we construct a bijection
between the points of the Poisson process and Z using the preorder-traversal
algorithm. To construct the graph we interpret each point in S as a space-time
point (x,r)\in\R^{d-1}\times R. Then a (d-1) dimensional random walk in
continuous time continuous space starts at site x at time r. The first jump of
the walk is to point x', at time r'>r, (x',r')\in S, where r' is the minimal
time after r such that |x-x'|<1. All the walks jumping to x' at time r'
coalesce with the one starting at (x',r'). Calling (x',r') = \alpha(x,r), the
graph has vertex set S and edges {(s,\alpha(s)), s\in S}. This enables us to
shift the origin of S^o = S + \delta_0 (the Palm version of S) to another point
in such a way that the distribution of S^o does not change (to any point if d =
2 and d = 3; point-stationarity).Comment: 15 pages. Second version with minor correction

### Relational reasoning via probabilistic coupling

Probabilistic coupling is a powerful tool for analyzing pairs of
probabilistic processes. Roughly, coupling two processes requires finding an
appropriate witness process that models both processes in the same probability
space. Couplings are powerful tools proving properties about the relation
between two processes, include reasoning about convergence of distributions and
stochastic dominance---a probabilistic version of a monotonicity property.
While the mathematical definition of coupling looks rather complex and
cumbersome to manipulate, we show that the relational program logic pRHL---the
logic underlying the EasyCrypt cryptographic proof assistant---already
internalizes a generalization of probabilistic coupling. With this insight,
constructing couplings is no harder than constructing logical proofs. We
demonstrate how to express and verify classic examples of couplings in pRHL,
and we mechanically verify several couplings in EasyCrypt

### Sharing emotions and space - empathy as a basis for cooperative spatial interaction

Boukricha H, Nguyen N, Wachsmuth I. Sharing emotions and space - empathy as a basis for cooperative spatial interaction. In: Kopp S, Marsella S, Thorisson K, Vilhjalmsson HH, eds. Proceedings of the 11th International Conference on Intelligent Virtual Agents (IVA 2011). LNAI. Vol 6895. Berlin, Heidelberg: Springer; 2011: 350-362.Empathy is believed to play a major role as a basis for humans’ cooperative behavior. Recent research shows that humans empathize with each other to different degrees depending on several modulation factors including, among others, their social relationships, their mood, and the situational context. In human spatial interaction, partners share and sustain a space that is equally and exclusively reachable to them, the so-called interaction space. In a cooperative interaction scenario of relocating objects in interaction space, we introduce an approach for triggering and modulating a virtual humans cooperative spatial behavior by its degree of empathy with its interaction partner. That is, spatial distances like object distances as well as distances of arm and body movements while relocating objects in interaction space are modulated by the virtual human’s degree of empathy. In this scenario, the virtual human’s empathic emotion is generated as a hypothesis about the partner’s emotional state as related to the physical effort needed to perform a goal directed spatial behavior

### The strong weak convergence of the quasi-EA

In this paper, we investigate the convergence of a novel simulation scheme to the target diffusion process. This scheme, the Quasi-EA, is closely related to the Exact Algorithm (EA) for diffusion processes, as it is obtained by neglecting the rejection step in EA. We prove the existence of a myopic coupling between the Quasi-EA and the diffusion. Moreover, an upper bound for the coupling probability is given. Consequently we establish the convergence of the Quasi-EA to the diffusion with respect to the total variation distance

### Aspects of nitrogen and mineral nutrition in Icelandic reindeer, Rangifer tarandus

Nitrogen and mineral (Fe, Mg, Na, K, and Ca) compositions of foodstuffs consumed by and dung produced by male, pregnant and lactating female adult Icelandic reindeer and calves were determined during May of 1992. Iron levels in foodstuffs are consistently above the reported toxicity level for similar-sized sheep (0.5 ppt) and may lead to iron toxicity when consumed by reindeer during periods of high lean body mass catabolism. Male and female reindeer meet nutrient requirements for all measured elements and nitrogen with the possible exception of calcium for males during antler growth.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31343/1/0000253.pd

### A probabilistic approach to Zhang's sandpile model

The current literature on sandpile models mainly deals with the abelian
sandpile model (ASM) and its variants. We treat a less known - but equally
interesting - model, namely Zhang's sandpile. This model differs in two aspects
from the ASM. First, additions are not discrete, but random amounts with a
uniform distribution on an interval $[a,b]$. Second, if a site topples - which
happens if the amount at that site is larger than a threshold value $E_c$
(which is a model parameter), then it divides its entire content in equal
amounts among its neighbors. Zhang conjectured that in the infinite volume
limit, this model tends to behave like the ASM in the sense that the stationary
measure for the system in large volumes tends to be peaked narrowly around a
finite set. This belief is supported by simulations, but so far not by
analytical investigations.
We study the stationary distribution of this model in one dimension, for
several values of $a$ and $b$. When there is only one site, exact computations
are possible. Our main result concerns the limit as the number of sites tends
to infinity, in the one-dimensional case. We find that the stationary
distribution, in the case $a \geq E_c/2$, indeed tends to that of the ASM (up
to a scaling factor), in agreement with Zhang's conjecture. For the case $a=0$,
$b=1$ we provide strong evidence that the stationary expectation tends to
$\sqrt{1/2}$.Comment: 47 pages, 3 figure

### Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology

This paper is a brief overview of the concepts involved in measuring the
degree of contextuality and detecting contextuality in systems of binary
measurements of a finite number of objects. We discuss and clarify the main
concepts and terminology of the theory called "contextuality-by-default," and
then discuss a possible generalization of the theory from binary to arbitrary
measurements.Comment: Lecture Notes in Computer Science 9535 (with the corrected list of
authors) (2016

### Billiards in a general domain with random reflections

We study stochastic billiards on general tables: a particle moves according
to its constant velocity inside some domain ${\mathcal D} \subset {\mathbb
R}^d$ until it hits the boundary and bounces randomly inside according to some
reflection law. We assume that the boundary of the domain is locally Lipschitz
and almost everywhere continuously differentiable. The angle of the outgoing
velocity with the inner normal vector has a specified, absolutely continuous
density. We construct the discrete time and the continuous time processes
recording the sequence of hitting points on the boundary and the pair
location/velocity. We mainly focus on the case of bounded domains. Then, we
prove exponential ergodicity of these two Markov processes, we study their
invariant distribution and their normal (Gaussian) fluctuations. Of particular
interest is the case of the cosine reflection law: the stationary distributions
for the two processes are uniform in this case, the discrete time chain is
reversible though the continuous time process is quasi-reversible. Also in this
case, we give a natural construction of a chord "picked at random" in
${\mathcal D}$, and we study the angle of intersection of the process with a
$(d-1)$-dimensional manifold contained in ${\mathcal D}$.Comment: 50 pages, 10 figures; To appear in: Archive for Rational Mechanics
and Analysis; corrected Theorem 2.8 (induced chords in nonconvex subdomains

### Distances in random graphs with finite variance degrees

In this paper we study a random graph with $N$ nodes, where node $j$ has
degree $D_j$ and $\{D_j\}_{j=1}^N$ are i.i.d. with \prob(D_j\leq x)=F(x). We
assume that $1-F(x)\leq c x^{-\tau+1}$ for some $\tau>3$ and some constant
$c>0$. This graph model is a variant of the so-called configuration model, and
includes heavy tail degrees with finite variance.
The minimal number of edges between two arbitrary connected nodes, also known
as the graph distance or the hopcount, is investigated when $N\to \infty$. We
prove that the graph distance grows like $\log_{\nu}N$, when the base of the
logarithm equals \nu=\expec[D_j(D_j -1)]/\expec[D_j]>1. This confirms the
heuristic argument of Newman, Strogatz and Watts \cite{NSW00}. In addition, the
random fluctuations around this asymptotic mean $\log_{\nu}{N}$ are
characterized and shown to be uniformly bounded. In particular, we show
convergence in distribution of the centered graph distance along exponentially
growing subsequences.Comment: 40 pages, 2 figure

### Metric properties of discrete time exclusion type processes in continuum

A new class of exclusion type processes acting in continuum with synchronous
updating is introduced and studied. Ergodic averages of particle velocities are
obtained and their connections to other statistical quantities, in particular
to the particle density (the so called Fundamental Diagram) is analyzed
rigorously. The main technical tool is a "dynamical" coupling applied in a
nonstandard fashion: we do not prove the existence of the successful coupling
(which even might not hold) but instead use its presence/absence as an
important diagnostic tool. Despite that this approach cannot be applied to
lattice systems directly, it allows to obtain new results for the lattice
systems embedding them to the systems in continuum. Applications to the traffic
flows modelling are discussed as well.Comment: 27 pages, 4 figures; minor errors corrected; details added to proofs
of Theorems 4.1 and 5.

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