scholarly journals Computable Bounds on the Spectral Gap for Unreliable Jackson Networks

2015 ◽  
Vol 47 (02) ◽  
pp. 402-424 ◽  
Author(s):  
Paweł Lorek ◽  
Ryszard Szekli
Keyword(s):  
Convergence Rates ◽  
Spectral Gap ◽  
Service Rate ◽  
Spectral Gaps ◽  
Birth And Death Processes ◽  
Marginal Distributions ◽  
Hazard Functions ◽  
Jackson Networks ◽  
One Dimensional ◽  
The One

The goal of this paper is to identify exponential convergence rates and to find computable bounds for them for Markov processes representing unreliable Jackson networks. First, we use the bounds of Lawler and Sokal (1988) in order to show that, for unreliable Jackson networks, the spectral gap is strictly positive if and only if the spectral gaps for the corresponding coordinate birth and death processes are positive. Next, utilizing some results on birth and death processes, we find bounds on the spectral gap for network processes in terms of the hazard and equilibrium functions of the one-dimensional marginal distributions of the stationary distribution of the network. These distributions must be in this case strongly light-tailed, in the sense that their discrete hazard functions have to be separated from 0. We relate these hazard functions with the corresponding networks' service rate functions using the equilibrium rates of the stationary one-dimensional marginal distributions. We compare the obtained bounds on the spectral gap with some other known bounds.

scholarly journals Computable Bounds on the Spectral Gap for Unreliable Jackson Networks

2015 ◽  
Vol 47 (2) ◽  
pp. 402-424 ◽  
Author(s):  
Paweł Lorek ◽  
Ryszard Szekli
Keyword(s):  
Convergence Rates ◽  
Spectral Gap ◽  
Service Rate ◽  
Spectral Gaps ◽  
Birth And Death Processes ◽  
Marginal Distributions ◽  
Hazard Functions ◽  
Jackson Networks ◽  
One Dimensional ◽  
The One

The goal of this paper is to identify exponential convergence rates and to find computable bounds for them for Markov processes representing unreliable Jackson networks. First, we use the bounds of Lawler and Sokal (1988) in order to show that, for unreliable Jackson networks, the spectral gap is strictly positive if and only if the spectral gaps for the corresponding coordinate birth and death processes are positive. Next, utilizing some results on birth and death processes, we find bounds on the spectral gap for network processes in terms of the hazard and equilibrium functions of the one-dimensional marginal distributions of the stationary distribution of the network. These distributions must be in this case strongly light-tailed, in the sense that their discrete hazard functions have to be separated from 0. We relate these hazard functions with the corresponding networks' service rate functions using the equilibrium rates of the stationary one-dimensional marginal distributions. We compare the obtained bounds on the spectral gap with some other known bounds.

Dependent Random Variables with Independent Subsets - II

1990 ◽  
Vol 33 (1) ◽  
pp. 24-28 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Joint Distribution ◽  
Random Variables ◽  
Marginal Distributions ◽  
Dependent Random Variables ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we consolidate into one two separate problems - dependent random variables with independent subsets and construction of a joint distribution with given marginals. Let N = {1,2,3,...} and X = {Xn; n ∊ N} be a sequence of random variables with nondegenerate one-dimensional marginal distributions {Fn; n ∊ N}. An example is constructed to show that there exists a sequence of random variables Y = {Yn; n ∊ N} such that the components of a subset of Y are independent if and only if its size is ≦ k, where k ≧ 2 is a prefixed integer. Furthermore, the one-dimensional marginal distributions of Y are those of X.

scholarly journals Damped Oscillations of the Energy of a Bosonic Bath due to Spectral Gaps and Special Initial Correlations

2018 ◽  
Vol 25 (03) ◽  
pp. 1850011 ◽  
Author(s):  
Filippo Giraldi
Keyword(s):  
Open System ◽  
Spectral Gap ◽  
Initial Conditions ◽  
Low Frequency ◽  
Power Laws ◽  
Spectral Gaps ◽  
Damped Oscillations ◽  
Different Temperatures ◽  
Long Time ◽  
The One

The energy of the bosonic bath and the flow of quantum information in local dephasing channels are studied over short and long times in case the distribution of frequency modes of the bosonic bath exhibits a low-frequency gap. The initial conditions consist in special correlations between the qubit and the bosonic bath or are factorized, and involve thermal states of the whole system or of the bath. The low-frequency gap generates damped oscillations of the bath energy around the asymptotic value, for the correlated initial conditions, and induces the open system to alternately loose and gain information, for the factorized initial configurations. The long-time oscillations of the bath energy become regular and the frequency of the oscillations coincides with the upper cut-off frequency of the spectral gap. Regular long-time sequences of intervals are found over which the bath energy increases (decreases), for the correlated initial conditions, and information is lost (gained) by the open system, for the factorized initial configurations, even at different temperatures. This relation is reversed, if compared to the one obtained without the low-frequency gap, and can fail if the spectral density is tailored near the spectral gap according to power laws with odd natural powers.

scholarly journals On the number of eigenvalues in the spectral gap of a Dirac system

1986 ◽  
Vol 29 (3) ◽  
pp. 367-378 ◽  
Author(s):  
D. B. Hinton ◽  
A. B. Mingarelli ◽  
T. T. Read ◽  
J. K. Shaw
Keyword(s):  
Differential Operators ◽  
Spectral Gap ◽  
Equivalence Classes ◽  
Value Functions ◽  
Complex Vector ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Integrable Functions ◽  
The One ◽  
Image Position

We consider the one-dimensional operator,on 0<x<∞ with. The coefficientsp,V1andV2are assumed to be real, locally Lebesgue integrable functions;c1andc2are positive numbers. The operatorLacts in the Hilbert spaceHof all equivalence classes of complex vector-value functionssuch that.Lhas domainD(L)consisting of ally∈Hsuch thatyis locally absolutely continuous andLy∈H; thus in the language of differential operatorsLis a maximal operator. Associated withLis the minimal operatorL0defined as the closure ofwhereis the restriction ofLto the functions with compact support in (0,∞).

scholarly journals A theory of spectral partitions of metric graphs

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
James B. Kennedy ◽  
Pavel Kurasov ◽  
Corentin Léna ◽  
Delio Mugnolo
Keyword(s):  
Spectral Gaps ◽  
Qualitative Properties ◽  
Metric Graphs ◽  
One Dimensional ◽  
Graph Partitions ◽  
Planar Domains ◽  
Open Questions ◽  
The One ◽  
Abstract Framework ◽  
Special Case

AbstractWe introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815–838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic—rather than numerical—results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45–72, 2005), Helffer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:101–138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

scholarly journals On ℤ2-indices for ground states of fermionic chains

2020 ◽  
Vol 32 (09) ◽  
pp. 2050028 ◽  
Author(s):  
Chris Bourne ◽  
Hermann Schulz-Baldes
Keyword(s):  
Spectral Gap ◽  
Ground States ◽  
Higher Order ◽  
Spectral Flow ◽  
One Dimensional ◽  
Finite Systems ◽  
Car Algebra ◽  
The One ◽  
Definition Of ◽  
Topological Obstruction

For parity-conserving fermionic chains, we review how to associate [Formula: see text]-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the [Formula: see text]-valued spectral flow provides a topological obstruction for two systems to have the same [Formula: see text]-index. A rudimentary definition of a [Formula: see text]-phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

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