703 research outputs found
The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary
We consider the dimer-monomer problem for the rectangular lattice. By mapping
the problem into one of close-packed dimers on an extended lattice, we rederive
the Tzeng-Wu solution for a single monomer on the boundary by evaluating a
Pfaffian. We also clarify the mathematical content of the Tzeng-Wu solution by
identifying it as the product of the nonzero eigenvalues of the Kasteleyn
matrix.Comment: 4 Pages to appear in the Physical Review E (2006
Determining global mean-first-passage time of random walks on Vicsek fractals using eigenvalues of Laplacian matrices
The family of Vicsek fractals is one of the most important and
frequently-studied regular fractal classes, and it is of considerable interest
to understand the dynamical processes on this treelike fractal family. In this
paper, we investigate discrete random walks on the Vicsek fractals, with the
aim to obtain the exact solutions to the global mean first-passage time
(GMFPT), defined as the average of first-passage time (FPT) between two nodes
over the whole family of fractals. Based on the known connections between FPTs,
effective resistance, and the eigenvalues of graph Laplacian, we determine
implicitly the GMFPT of the Vicsek fractals, which is corroborated by numerical
results. The obtained closed-form solution shows that the GMFPT approximately
grows as a power-law function with system size (number of all nodes), with the
exponent lies between 1 and 2. We then provide both the upper bound and lower
bound for GMFPT of general trees, and show that leading behavior of the upper
bound is the square of system size and the dominating scaling of the lower
bound varies linearly with system size. We also show that the upper bound can
be achieved in linear chains and the lower bound can be reached in star graphs.
This study provides a comprehensive understanding of random walks on the Vicsek
fractals and general treelike networks.Comment: Definitive version accepted for publication in Physical Review
Theory of impedance networks: The two-point impedance and LC resonances
We present a formulation of the determination of the impedance between any
two nodes in an impedance network. An impedance network is described by its
Laplacian matrix L which has generally complex matrix elements. We show that by
solving the equation L u_a = lambda_a u_a^* with orthonormal vectors u_a, the
effective impedance between nodes p and q of the network is Z = Sum_a [u_{a,p}
- u_{a,q}]^2/lambda_a where the summation is over all lambda_a not identically
equal to zero and u_{a,p} is the p-th component of u_a. For networks consisting
of inductances (L) and capacitances (C), the formulation leads to the
occurrence of resonances at frequencies associated with the vanishing of
lambda_a. This curious result suggests the possibility of practical
applications to resonant circuits. Our formulation is illustrated by explicit
examples.Comment: 21 pages, 3 figures; v4: typesetting corrected; v5: Eq. (63)
correcte
Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
Virtually all questions that one can ask about the behavioral and structural
complexity of a stochastic process reduce to a linear algebraic framing of a
time evolution governed by an appropriate hidden-Markov process generator. Each
type of question---correlation, predictability, predictive cost, observer
synchronization, and the like---induces a distinct generator class. Answers are
then functions of the class-appropriate transition dynamic. Unfortunately,
these dynamics are generically nonnormal, nondiagonalizable, singular, and so
on. Tractably analyzing these dynamics relies on adapting the recently
introduced meromorphic functional calculus, which specifies the spectral
decomposition of functions of nondiagonalizable linear operators, even when the
function poles and zeros coincide with the operator's spectrum. Along the way,
we establish special properties of the projection operators that demonstrate
how they capture the organization of subprocesses within a complex system.
Circumventing the spurious infinities of alternative calculi, this leads in the
sequel, Part II, to the first closed-form expressions for complexity measures,
couched either in terms of the Drazin inverse (negative-one power of a singular
operator) or the eigenvalues and projection operators of the appropriate
transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at
http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht
Relationships between different sets involving group and Drazin projectors and nonnegativity
This paper deals with nonnegativity of matrices and their group or Drazin inverses. Firstly, the nonnegativity of a square matrix A, its group inverse A# and its group projector AA# is used to define different sets for which relationships and characterizations are given. Next, an extension of the previous results for index greater than 1 is presented. Similar sets are introduced and studied for Drazin inverses and Drazin projectors considering the core-nilpotent decomposition.
In addition, the results are applied to study the {l}-Drazin periodic matrices for l greater than or equal to 1.Herrero Debón, A.; Ramirez, FJ.; Thome, N. (2013). Relationships between different sets involving group and Drazin projectors and nonnegativity. Linear Algebra and its Applications. 438(4):1688-1699. doi:10.1016/J.LAA.2011.08.029S16881699438
Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems
We consider solution operators of linear ordinary boundary problems with "too
many" boundary conditions, which are not always solvable. These generalized
Green's operators are a certain kind of generalized inverses of differential
operators. We answer the question when the product of two generalized Green's
operators is again a generalized Green's operator for the product of the
corresponding differential operators and which boundary problem it solves.
Moreover, we show that---provided a factorization of the underlying
differential operator---a generalized boundary problem can be factored into
lower order problems corresponding to a factorization of the respective Green's
operators. We illustrate our results by examples using the Maple package
IntDiffOp, where the presented algorithms are implemented.Comment: 19 page
Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere
Using coherent-state techniques, we prove a sampling theorem for Majorana's
(holomorphic) functions on the Riemann sphere and we provide an exact
reconstruction formula as a convolution product of samples and a given
reconstruction kernel (a sinc-type function). We also discuss the effect of
over- and under-sampling. Sample points are roots of unity, a fact which allows
explicit inversion formulas for resolution and overlapping kernel operators
through the theory of Circulant Matrices and Rectangular Fourier Matrices. The
case of band-limited functions on the Riemann sphere, with spins up to , is
also considered. The connection with the standard Euler angle picture, in terms
of spherical harmonics, is established through a discrete Bargmann transform.Comment: 26 latex pages. Final version published in J. Fourier Anal. App
Necessary conditions for variational regularization schemes
We study variational regularization methods in a general framework, more
precisely those methods that use a discrepancy and a regularization functional.
While several sets of sufficient conditions are known to obtain a
regularization method, we start with an investigation of the converse question:
How could necessary conditions for a variational method to provide a
regularization method look like? To this end, we formalize the notion of a
variational scheme and start with comparison of three different instances of
variational methods. Then we focus on the data space model and investigate the
role and interplay of the topological structure, the convergence notion and the
discrepancy functional. Especially, we deduce necessary conditions for the
discrepancy functional to fulfill usual continuity assumptions. The results are
applied to discrepancy functionals given by Bregman distances and especially to
the Kullback-Leibler divergence.Comment: To appear in Inverse Problem
Direct Interactions in Relativistic Statistical Mechanics
Directly interacting particles are considered in the multitime formalism of
predictive relativistic mechanics. When the equations of motion leave a
phase-space volume invariant, it turns out that the phase average of any first
integral, covariantly defined as a flux across a -dimensional surface, is
conserved. The Hamiltonian case is discussed, a class of simple models is
exhibited, and a tentative definition of equilibrium is proposed.Comment: Plain Tex file, 26 page
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