929 research outputs found
A closed form for the generalized Bernoulli polynomials via Fa\`a di Bruno's formula
We derive a closed form for the generalized Bernoulli polynomial of order
in terms of Bell polynomials and Stirling numbers of the second kind using the
Fa\`a di Bruno's formula.Comment: 4 pages; No figure
Numerical Simulation guided Lazy Abstraction Refinement for Nonlinear Hybrid Automata
This draft suggests a new counterexample guided abstraction refinement
(CEGAR) framework that uses the combination of numerical simulation for
nonlinear differential equations with linear programming for linear hybrid
automata (LHA) to perform reachability analysis on nonlinear hybrid automata. A
notion of structural robustness is also introduced which allows the
algorithm to validate counterexamples using numerical simulations.
Keywords: verification, model checking, hybrid systems, hybrid automata,
robustness, robust hybrid systems, numerical simulation, cegar, abstraction
refinement.Comment: 11 pages, 2 figure
Design of a Distributed Reachability Algorithm for Analysis of Linear Hybrid Automata
This paper presents the design of a novel distributed algorithm d-IRA for the
reachability analysis of linear hybrid automata. Recent work on iterative
relaxation abstraction (IRA) is leveraged to distribute the computational
problem among multiple computational nodes in a non-redundant manner by
performing careful infeasibility analysis of linear programs corresponding to
spurious counterexamples. The d-IRA algorithm is resistant to failure of
multiple computational nodes. The experimental results provide promising
evidence for the possible successful application of this technique.Comment: 8 page
Two new explicit formulas for the Bernoulli Numbers
In this brief note, we give two explicit formulas for the Bernoulli Numbers
in terms of the Stirling numbers of the second kind, and the Eulerian Numbers.
To the best of our knowledge, these formulas are new. We also derive two more
probably known formulas.Comment: Updated to give proofs of some necessary result
An identity involving Bernoulli numbers and the Stirling numbers of the second kind
Let denote the Bernoulli numbers, and denote the Stirling
numbers of the second kind. We prove the following identity To
the best of our knowledge, the identity is new.Comment: 3 page
Formulas for the number of -colored partitions and the number of plane partitions of in terms of the Bell polynomials
We derive closed formulas for the number of -coloured partitions and the
number of plane partitions of in terms of the Bell polynomials
Two new identities involving the Bernoulli numbers, the Euler numbers, and the Stirling numbers of the first kind
We derive two new identities involving the Bernoulli numbers, the Euler
numbers, and the Stirling numbers of the first kind using analytic continuation
of a well known identity for the Stirling numbers of the first kind.Comment: 4 Pages, no figure
A formula for the -coloured partition function in terms of the sum of divisors function and its inverse
Let denote the -coloured partition function, and
denote the sum of positive divisors of . The aim of
this note is to prove the following where , and its inverse \sigma(n) = n\,\sum_{r=1}^n
\frac{(-1)^{r-1}}{r}\, \binom{n}{r}\, p_{-r}(n). $
A formula for the number of partitions of in terms of the partial Bell polynomials
We derive a formula for (the number of partitions of ) in terms of
the partial Bell polynomials using Fa\`{a} di Bruno's formula and Euler's
pentagonal number theorem.Comment: Accepted for publication in the Ramanujan Journa
Distributed Markov Chains
The formal verification of large probabilistic models is important and
challenging. Exploiting the concurrency that is often present is one way to
address this problem. Here we study a restricted class of asynchronous
distributed probabilistic systems in which the synchronizations determine the
probability distribution for the next moves of the participating agents. The
key restriction we impose is that the synchronizations are deterministic, in
the sense that any two simultaneously enabled synchronizations must involve
disjoint sets of agents. As a result, this network of agents can be viewed as a
succinct and distributed presentation of a large global Markov chain. A rich
class of Markov chains can be represented this way.
We define an interleaved semantics for our model in terms of the local
synchronization actions. The network structure induces an independence relation
on these actions, which, in turn, induces an equivalence relation over the
interleaved runs in the usual way. We construct a natural probability measure
over these equivalence classes of runs by exploiting Mazurkiewicz trace theory
and the probability measure space of the associated global Markov chain.
It turns out that verification of our model, called DMCs (distributed Markov
chains), can often be efficiently carried out by exploiting the partial order
nature of the interleaved semantics. To demonstrate this, we develop a
statistical model checking (SMC) procedure and use it to verify two large
distributed probabilistic networks
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