14,450 research outputs found
Generalized Erdos Numbers for network analysis
In this paper we consider the concept of `closeness' between nodes in a
weighted network that can be defined topologically even in the absence of a
metric. The Generalized Erd\H{o}s Numbers (GENs) satisfy a number of desirable
properties as a measure of topological closeness when nodes share a finite
resource between nodes as they are real-valued and non-local, and can be used
to create an asymmetric matrix of connectivities. We show that they can be used
to define a personalized measure of the importance of nodes in a network with a
natural interpretation that leads to a new global measure of centrality and is
highly correlated with Page Rank. The relative asymmetry of the GENs (due to
their non-metric definition) is linked also to the asymmetry in the mean first
passage time between nodes in a random walk, and we use a linearized form of
the GENs to develop a continuum model for `closeness' in spatial networks. As
an example of their practicality, we deploy them to characterize the structure
of static networks and show how it relates to dynamics on networks in such
situations as the spread of an epidemic
Tractrices, Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old Conjecture
Geometry of the tracks left by a bicycle is closely related with the
so-called Prytz planimeter and with linear fractional transformations of the
complex plane. We describe these relations, along with the history of the
problem, and give a proof of a conjecture made by Menzin in 1906.Comment: 20 pages, 18 figure
Symmetries of differential-difference dynamical systems in a two-dimensional lattice
Classification of differential-difference equation of the form
are considered
according to their Lie point symmetry groups. The set represents the
point and its six nearest neighbors in a two-dimensional triangular
lattice. It is shown that the symmetry group can be at most 12-dimensional for
abelian symmetry algebras and 13-dimensional for nonsolvable symmetry algebras.Comment: 24 pages, 1 figur
Oscillations and stability of numerical solutions of the heat conduction equation
The mathematical model and results of numerical solutions are given for the one dimensional problem when the linear equations are written in a rectangular coordinate system. All the computations are easily realizable for two and three dimensional problems when the equations are written in any coordinate system. Explicit and implicit schemes are shown in tabular form for stability and oscillations criteria; the initial temperature distribution is considered uniform
Loop space homology associated with the mod 2 Dickson invariants
Peer reviewedPublisher PD
Bayesian inference of a negative quantity from positive measurement results
In this paper the Bayesian analysis is applied to assign a probability
density to the value of a quantity having a definite sign. This analysis is
logically consistent with the results, positive or negative, of repeated
measurements. Results are used to estimate the atom density shift in a caesium
fountain clock. The comparison with the classical statistical analysis is also
reported and the advantages of the Bayesian approach for the realization of the
time unit are discussed.Comment: 10 pages, 4 figures, submitted to Metrologi
Lie discrete symmetries of lattice equations
We extend two of the methods previously introduced to find discrete
symmetries of differential equations to the case of difference and
differential-difference equations. As an example of the application of the
methods, we construct the discrete symmetries of the discrete Painlev\'e I
equation and of the Toda lattice equation
- …