103 research outputs found
Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals
High-order derivatives of analytic functions are expressible as Cauchy
integrals over circular contours, which can very effectively be approximated,
e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius
of convergence is equal, numerical stability strongly depends on r. We give a
comprehensive study of this effect; in particular we show that there is a
unique radius that minimizes the loss of accuracy caused by round-off errors.
For large classes of functions, though not for all, this radius actually gives
about full accuracy; a remarkable fact that we explain by the theory of Hardy
spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and
by the saddle-point method of asymptotic analysis. Many examples and
non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature
rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat
Polynomial tuning of multiparametric combinatorial samplers
Boltzmann samplers and the recursive method are prominent algorithmic
frameworks for the approximate-size and exact-size random generation of large
combinatorial structures, such as maps, tilings, RNA sequences or various
tree-like structures. In their multiparametric variants, these samplers allow
to control the profile of expected values corresponding to multiple
combinatorial parameters. One can control, for instance, the number of leaves,
profile of node degrees in trees or the number of certain subpatterns in
strings. However, such a flexible control requires an additional non-trivial
tuning procedure. In this paper, we propose an efficient polynomial-time, with
respect to the number of tuned parameters, tuning algorithm based on convex
optimisation techniques. Finally, we illustrate the efficiency of our approach
using several applications of rational, algebraic and P\'olya structures
including polyomino tilings with prescribed tile frequencies, planar trees with
a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures,
colours. Implementation and examples are available at [1]
https://github.com/maciej-bendkowski/boltzmann-brain [2]
https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler
Sequential Probing With a Random Start
Processing user requests quickly requires not only fast servers, but also demands methods to quickly locate idle servers to process those requests. Methods of finding idle servers are analogous to open addressing in hash tables, but with the key difference that servers may return to an idle state after having been busy rather than staying busy. Probing sequences for open addressing are well-studied, but algorithms for locating idle servers are less understood. We investigate sequential probing with a random start as a method for finding idle servers, especially in cases of heavy traffic. We present a procedure for finding the distribution of the number of probes required for finding an idle server by using a Markov chain and ideas from enumerative combinatorics, then present numerical simulation results in lieu of a general analytic solution
Exact solution of the dimer model: Corner free energy, correlation functions and combinatorics
In this work, some classical results of the pfaffian theory of the dimer
model based on the work of Kasteleyn, Fisher and Temperley are introduced in a
fermionic framework. Then we shall detail the bosonic formulation of the model
{\it via} the so-called height mapping and the nature of boundary conditions is
unravelled. The complete and detailed fermionic solution of the dimer model on
the square lattice with an arbitrary number of monomers is presented, and
finite size effect analysis is performed to study surface and corner effects,
leading to the extrapolation of the central charge of the model. The solution
allows for exact calculations of monomer and dimer correlation functions in the
discrete level and the scaling behavior can be inferred in order to find the
set of scaling dimensions and compare to the bosonic theory which predict
particular features concerning corner behaviors. Finally, some combinatorial
and numerical properties of partition functions with boundary monomers are
discussed, proved and checked with enumeration algorithms.Comment: Final version to be published in Nuclear Physics B (53 pages and a
lot of figures
Limit Theorems for Stochastic Approximations Algorithms With Application to General Urn Models
In the present paper we study the multidimensional stochastic approximation algorithms where the drift function h is a smooth function and where jacobian matrix is diagonalizable over C but assuming that all the eigenvalues of this matrix are in the the region Repzq ą 0. We give results on the fluctuation of the process around the stable equilibrium point of h. We extend the limit theorem of the one dimensional Robin's Monroe algorithm [MR73]. We give also application of these limit theorem for some class of urn models proving the efficiency of this method
Cooperative Models of Stochastic Growth - On a class of reinforced processes with graph-based interactions
Consider a sequence of positive integer-valued random vectors denoted by for . Fix , and given , choose a \emph{random} coordinate . The probability that for a particular coordinate is proportional to a non-decreasing function of , where measures how strongly cooperates with . Now, on the event that , update the sequence in such a way that , where is the vector whose -th coordinate is 1 and whose other coordinates are 0. Finally, given and what can one say about
Contagion Source Detection in Epidemic and Infodemic Outbreaks: Mathematical Analysis and Network Algorithms
This monograph provides an overview of the mathematical theories and
computational algorithm design for contagion source detection in large
networks. By leveraging network centrality as a tool for statistical inference,
we can accurately identify the source of contagions, trace their spread, and
predict future trajectories. This approach provides fundamental insights into
surveillance capability and asymptotic behavior of contagion spreading in
networks. Mathematical theory and computational algorithms are vital to
understanding contagion dynamics, improving surveillance capabilities, and
developing effective strategies to prevent the spread of infectious diseases
and misinformation.Comment: Suggested Citation: Chee Wei Tan and Pei-Duo Yu (2023), "Contagion
Source Detection in Epidemic and Infodemic Outbreaks: Mathematical Analysis
and Network Algorithms", Foundations and Trends in Networking: Vol. 13: No.
2-3, pp 107-251. http://dx.doi.org/10.1561/130000006
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