3,681 research outputs found
Analysis of Carries in Signed Digit Expansions
The number of positive and negative carries in the addition of two
independent random signed digit expansions of given length is analyzed
asymptotically for the -system and the symmetric signed digit
expansion. The results include expectation, variance, covariance between the
positive and negative carries and a central limit theorem.
Dependencies between the digits require determining suitable transition
probabilities to obtain equidistribution on all expansions of given length. A
general procedure is described to obtain such transition probabilities for
arbitrary regular languages.
The number of iterations in von Neumann's parallel addition method for the
symmetric signed digit expansion is also analyzed, again including expectation,
variance and convergence to a double exponential limiting distribution. This
analysis is carried out in a general framework for sequences of generating
functions
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity
We analyze combinatorial structures which play a central role in determining
spectral properties of the volume operator in loop quantum gravity (LQG). These
structures encode geometrical information of the embedding of arbitrary valence
vertices of a graph in 3-dimensional Riemannian space, and can be represented
by sign strings containing relative orientations of embedded edges. We
demonstrate that these signature factors are a special representation of the
general mathematical concept of an oriented matroid. Moreover, we show that
oriented matroids can also be used to describe the topology (connectedness) of
directed graphs. Hence the mathematical methods developed for oriented matroids
can be applied to the difficult combinatorics of embedded graphs underlying the
construction of LQG. As a first application we revisit the analysis of [4-5],
and find that enumeration of all possible sign configurations used there is
equivalent to enumerating all realizable oriented matroids of rank 3, and thus
can be greatly simplified. We find that for 7-valent vertices having no
coplanar triples of edge tangents, the smallest non-zero eigenvalue of the
volume spectrum does not grow as one increases the maximum spin \jmax at the
vertex, for any orientation of the edge tangents. This indicates that, in
contrast to the area operator, considering large \jmax does not necessarily
imply large volume eigenvalues. In addition we give an outlook to possible
starting points for rewriting the combinatorics of LQG in terms of oriented
matroids.Comment: 43 pages, 26 figures, LaTeX. Version published in CQG. Typos
corrected, presentation slightly extende
Tropicalizing the simplex algorithm
We develop a tropical analog of the simplex algorithm for linear programming.
In particular, we obtain a combinatorial algorithm to perform one tropical
pivoting step, including the computation of reduced costs, in O(n(m+n)) time,
where m is the number of constraints and n is the dimension.Comment: v1: 35 pages, 7 figures, 4 algorithms; v2: improved presentation, 39
pages, 9 figures, 4 algorithm
Solving Multi-agent MDPs Optimally with Conditional Return Graphs
In cooperative multi-agent sequential decision making under uncertainty, agents must coordinate in order find an optimal joint policy that maximises joint value. Typical solution al- gorithms exploit additive structure in the value function, but in the fully-observable multi-agent MDP setting (MMDP) such structure is not present. We propose a new optimal solver for so-called TI-MMDPs, where agents can only af- fect their local state, while their value may depend on the state of others. We decompose the returns into local returns per agent that we represent compactly in a conditional re- turn graph (CRG). Using CRGs the value of a joint policy as well as bounds on the value of partially specified joint policies can be efficiently computed. We propose CoRe, a novel branch-and-bound policy search algorithm building on CRGs. CoRe typically requires less runtime than the avail- able alternatives and is able to find solutions to problems previously considered unsolvable
Interactive Causes: Revising the Markov Condition
This paper suggests a revision of the theory of causal nets (TCN). In Section 1 we introduce an axiomatization of TCN based on a realistic understanding. It is shown that the causal Markov condition entails three independent principles. In Section 2 we analyze inde-terministic decay as the major counterexample to one of these principles: screening-off by common causes (SCC). We call (SCC)-violating common causes interactive causes. In Sec-tion 3 we develop a revised version of TCN, called TCN*, which accounts for interactive causes. It is shown that there are interactive causal models that admit of no faithful non-interactive reconstruction
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