12,414 research outputs found
Two-Hop Walks Indicate PageRank Order
This paper shows that pairwise PageRank orders emerge from two-hop walks. The
main tool used here refers to a specially designed sign-mirror function and a
parameter curve, whose low-order derivative information implies pairwise
PageRank orders with high probability. We study the pairwise correct rate by
placing the Google matrix in a probabilistic framework, where
may be equipped with different random ensembles for
model-generated or real-world networks with sparse, small-world, scale-free
features, the proof of which is mixed by mathematical and numerical evidence.
We believe that the underlying spectral distribution of aforementioned networks
is responsible for the high pairwise correct rate. Moreover, the perspective of
this paper naturally leads to an algorithm for any single pairwise
PageRank comparison if assuming both ,
where denotes the identity matrix of order , and
are ready on hand (e.g., constructed offline in an incremental
manner), based on which it is easy to extract the top list in , thus
making it possible for PageRank algorithm to deal with super large-scale
datasets in real time.Comment: 29 pages, 2 figure
Multi-Dimensional Backward Stochastic Differential Equations of Diagonally Quadratic generators
The paper is concerned with adapted solution of a multi-dimensional BSDE with
a "diagonally" quadratic generator, the quadratic part of whose th component
only depends on the th row of the second unknown variable. Local and global
solutions are given. In our proofs, it is natural and crucial to apply both
John-Nirenberg and reverse H\"older inequalities for BMO martingales.Comment: 17 page
Multi-dimensional BSDE with Oblique Reflection and Optimal Switching
In this paper, we study a multi-dimensional backward stochastic differential
equation (BSDE) with oblique reflection, which is a BSDE reflected on the
boundary of a special unbounded convex domain along an oblique direction, and
which arises naturally in the study of optimal switching problem. The existence
of the adapted solution is obtained by the penalization method, the monotone
convergence, and the a priori estimations. The uniqueness is obtained by a
verification method (the first component of any adapted solution is shown to be
the vector value of a switching problem for BSDEs). As applications, we apply
the above results to solve the optimal switching problem for stochastic
differential equations of functional type, and we give also a probabilistic
interpretation of the viscosity solution to a system of variational
inequalities
Stochastic LQ and Associated Riccati equation of PDEs Driven by State-and Control-Dependent White Noise
The optimal stochastic control problem with a quadratic cost functional for
linear partial differential equations (PDEs) driven by a state-and
control-dependent white noise is formulated and studied. Both finite-and
infinite-time horizons are considered. The multi-plicative white noise dynamics
of the system give rise to a new phenomenon of singularity to the associated
Riccati equation and even to the Lyapunov equation. Well-posedness of both
Riccati equation and Lyapunov equation are obtained for the first time. The
linear feedback coefficient of the optimal control turns out to be singular and
expressed in terms of the solution of the associated Riccati equation. The null
controllability is shown to be equivalent to the existence of the solution to
Riccati equation with the singular terminal value. Finally, the controlled
Anderson model is addressed as an illustrating example
A thermodynamically consistent approach to describe the effect of thermal vacancy on abnormal thermodynamic behaviors of pure metals: application to body centered cubic W
In this paper, we developed a thermodynamically consistent approach to
account for the Gibbs energy of pure metallic element with thermal vacancy over
wide temperature range. Taking body centered cubic (bcc) W for a demonstration,
the strong nonlinear increase for temperature dependence of heat capacities at
high temperatures and a nonlinear Arrhenius plots of vacancy concentration in
bcc W can be nicely reproduced by the obtained Gibbs energy. The successful
description of thermal vacancy on abnormal thermodynamic behaviors in bcc W
indicates that the presently proposed thermodynamically consistent approach is
a universal one, and applicable to the other metals.Comment: 10 pages, 3 figures and 1 tabl
Mixed Deterministic and Random Optimal Control of Linear Stochastic Systems with Quadratic Costs
In this paper, we consider the mixed optimal control of a linear stochastic
system with a quadratic cost functional, with two controllers-one can choose
only deterministic time functions, called the deterministic controller, while
the other can choose adapted random processes, called the random controller.
The optimal control is shown to exist under suitable assumptions. The optimal
control is characterized via a system of fully coupled forward-backward
stochastic differential equations (FB-SDEs) of mean-field type. We solve the
FBSDEs via solutions of two (but decoupled) Riccati equations, and give the
respective optimal feedback law for both determinis-tic and random controllers,
using solutions of both Riccati equations. The optimal state satisfies a linear
stochastic differential equation (SDE) of mean-field type. Both the singular
and infinite time-horizonal cases are also addressed
Quantum Monte Carlo studies of spinons in one-dimensional spin systems
Observing constituent particles with fractional quantum numbers in confined
and deconfined states is an interesting and challenging problem in quantum
many-body physics. Here we further explore a computational scheme [Y. Tang and
A. W. Sandvik, Phys. Rev. Lett. {\bf 107}, 157201 (2011)] based on valence-bond
quantum Monte Carlo simulations of quantum spin systems. Using several
different one-dimensional models, we characterize spinon excitations
using the spinon size and confinement length (the size of a bound state). The
spinons have finite size in valence-bond-solid states, infinite size in the
critical region, and become ill-defined in the N\'eel state. We also verify
that pairs of spinons are deconfined in these uniform spin chains but become
confined upon introducing a pattern of alternating coupling strengths
(dimerization) or coupling two chains (forming a ladder). In the dimerized
system an individual spinon can be small when the confinement length is
large---this is the case when the imposed dimerization is weak but the ground
state of the corresponding uniform chain is a spontaneously formed
valence-bond-solid (where the spinons are deconfined). Based on our numerical
results, we argue that the situation is associated with
weak repulsive short-range spinon-spinon interactions. In principle both the
length-scales can be individually tuned from small to infinite (with ) by varying model parameters. In the ladder system the two lengths
are always similar, and this is the case also in the dimerized systems when the
corresponding uniform chain is in the critical phase. In these systems the
effective spinon-spinon interactions are purely attractive and there is only a
single large length scale close to criticality, which is reflected in the
standard spin correlations as well as in the spinon characteristics.Comment: 15 pages, 15 figure
Confinement and Deconfinement of Spinons in Two Dimensions
We use Monte Carlo methods to study spinons in two-dimensional quantum spin
systems, characterizing their intrinsic size and confinement length
. We confirm that spinons are deconfined, and
finite, in a resonating valence-bond spin-liquid state. In a
valence-bond solid, we find finite and , with of a
single spinon significantly larger than the bound-state---the spinon is soft
and shrinks as the bound state is formed. Both and diverge
upon approaching the critical point separating valence-bond solid and N\'eel
ground states. We conclude that the spinon deconfinement is marginal in the
lowest-energy state in the spin-1 sector, due to weak attractive spinon
interactions. Deconfinement in the vicinity of the critical point should occur
at higher energies.Comment: 5 pages, 5 figure
A Model Predictive Control Approach for Low-Complexity Electric Vehicle Charging Scheduling: Optimality and Scalability
With the increasing adoption of plug-in electric vehicles (PEVs), it is
critical to develop efficient charging coordination mechanisms that minimize
the cost and impact of PEV integration to the power grid. In this paper, we
consider the optimal PEV charging scheduling, where the non-causal information
about future PEV arrivals is not known in advance, but its statistical
information can be estimated. This leads to an "online" charging scheduling
problem that is naturally formulated as a finite-horizon dynamic programming
with continuous state space and action space. To avoid the prohibitively high
complexity of solving such a dynamic programming problem, we provide a Model
Predictive Control (MPC) based algorithm with computational complexity
, where is the total number of time stages. We rigorously analyze
the performance gap between the near-optimal solution of the MPC-based approach
and the optimal solution for any distributions of exogenous random variables.
Furthermore, our rigorous analysis shows that when the random process
describing the arrival of charging demands is first-order periodic, the
complexity of proposed algorithm can be reduced to , which is independent
of . Extensive simulations show that the proposed online algorithm performs
very closely to the optimal online algorithm. The performance gap is smaller
than in most cases.Comment: 13 page
Summing over trajectories of stochastic dynamics with multiplicative noise
We demonstrate that the conventional path integral formulations generate
inconsistent results exemplified by the geometric Brownian motion under the
general stochastic interpretation. We thus develop a novel path integral
formulation for the overdamped Langevin equation with the multiplicative noise.
The present path integral leads to the corresponding Fokker-Planck equation,
and naturally gives a normalized transition probability consistently in
examples for general stochastic interpretations. Our result can be applied to
study the fluctuation theorems and numerical calculations based on the path
integral framework.Comment: 7 pages, 1 figur
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