8,379 research outputs found

### Buser's inequality on infinite graphs

In this paper, we establish Buser type inequalities, i.e., upper bounds for
eigenvalues in terms of Cheeger constants. We prove the Buser's inequality for
an infinite but locally finite connected graph with Ricci curvature lower
bounds. Furthermore, we derive that the graph with positive curvature is
finite, especially for unbounded Laplacians. By proving Poincar\'e inequality,
we obtain a lower bound on Cheeger constant in terms of positive curvature

### Equivalent Properties of CD Inequality on Graph

We study some equivalent properties of the curvature-dimension conditions
$CD(n,K)$ inequality on infinite, but locally finite graph. These equivalences
are gradient estimate, Poincar\'e type inequalities and reverse Poincar\'e
inequalities. And we also obtain one equivalent property of gradient estimate
for a new notion of curvature-dimension conditions $CDE'(\infty, K)$ at the
same assumption of graphs.Comment: 13 page

### Monotonicity of principal eigenvalue for elliptic operators with incompressible flow: A functional approach

We establish the monotonicity of the principal eigenvalue $\lambda_1(A)$, as
a function of the advection amplitude $A$, for the elliptic operator
$L_{A}=-\mathrm{div}(a(x)\nabla)+A\mathbf{V}\cdot\nabla +c(x)$ with
incompressible flow $\mathbf{V}$, subject to Dirichlet, Robin and Neumann
boundary conditions. As a consequence, the limit of $\lambda_1(A)$ as $A\to
\infty$ always exists and is finite for Robin boundary conditions. These
results answer some open questions raised by [Berestycki, H., Hamel, F.,
Nadirashvili, N.: Elliptic eigenvalue problems with large drift and
applications to nonlinear propagation phenomena, Commun. Math. Phys. 253,
451-480 (2005)]. Our method relies upon some functional which is associated
with principal eigenfuntions for operator $L_A$ and its adjoint operator. As a
byproduct of the approach, a new min-max characterization of $\lambda_1(A)$ is
given.Comment: 13 pages, 0 figure

### The Inductive Bias of Restricted f-GANs

Generative adversarial networks are a novel method for statistical inference
that have achieved much empirical success; however, the factors contributing to
this success remain ill-understood. In this work, we attempt to analyze
generative adversarial learning -- that is, statistical inference as the result
of a game between a generator and a discriminator -- with the view of
understanding how it differs from classical statistical inference solutions
such as maximum likelihood inference and the method of moments.
Specifically, we provide a theoretical characterization of the distribution
inferred by a simple form of generative adversarial learning called restricted
f-GANs -- where the discriminator is a function in a given function class, the
distribution induced by the generator is restricted to lie in a pre-specified
distribution class and the objective is similar to a variational form of the
f-divergence. A consequence of our result is that for linear KL-GANs -- that
is, when the discriminator is a linear function over some feature space and f
corresponds to the KL-divergence -- the distribution induced by the optimal
generator is neither the maximum likelihood nor the method of moments solution,
but an interesting combination of both

### Confounding of three binary-variables counterfactual model

Confounding of three binary-variables counterfactual model is discussed in
this paper. According to the effect between the control variable and the
covariate variable, we investigate three counterfactual models: the control
variable is independent of the covariate variable, the control variable has the
effect on the covariate variable and the covariate variable affects the control
variable. Using the ancillary information based on conditional independence
hypotheses, the sufficient conditions to determine whether the covariate
variable is an irrelevant factor or a confounder in each counterfactual model
are obtained

### Asymptotic spreading of interacting species with multiple fronts II: Exponentially decaying initial data

This is part two of our study on the spreading properties of the
Lotka-Volterra competition-diffusion systems with a stable coexistence state.
We focus on the case when the initial data are exponential decaying. By
establishing a comparison principle for Hamilton-Jacobi equations, we are able
to apply the Hamilton-Jacobi approach for Fisher-KPP equation due to Freidlin,
Evans and Souganidis. As a result, the exact formulas of spreading speeds and
their dependence on initial data are derived. Our results indicate that
sometimes the spreading speed of the slower species is nonlocally determined.
Connections of our results with the traveling profile due to Tang and Fife, as
well as the more recent spreading result of Girardin and Lam, will be
discussed

### The Lorentz factor distribution and luminosity function of relativistic jets in AGNs

The observed apparent velocities and luminosities of the relativistic jets in
AGNs are significantly different from their intrinsic values due to strong
special relativistic effects. We adopt the maximum likelihood method to
determine simultaneously the intrinsic luminosity function and the Lorentz
factor distribution of a sample of AGNs. The values of the best estimated
parameters are consistent with the previous results, but with much better
accuracy. In previous study, it was assumed that the shape of the observed
luminosity function of Fanaroff-Riley type II radio galaxies is the same with
the intrinsic luminosity function of radio loud quasars. Our results prove the
validity of this assumption. We also find that low and high redshift groups
divided by z=0.1 are likely to be from different parent populations.Comment: 18 pages, 7 figures. The original version of this paper was submitted
to ApJL on January 19, 2007. The current version was submitted to ApJ on
April 15, 2007 and accepted on May 16, 200

### Orbit Tracking Control of Quantum Systems

The orbit tracking of free-evolutionary target system in closed quantum
systems is studied in this paper. Based on the concept of system control
theory, the unitary transformation is applied to change the time-dependent
target function into a stationary target state so that the orbit tracking
problem is changed into the state transfer one. A Lyapunov function with
virtual mechanical quantity P is employed to design a control law for such a
state transferring. The target states in density matrix are grouped into two
classes: diagonal and non-diagonal. The specific convergent conditions for
target state of diagonal mixed-states are derived. In the case that the target
state is a non-diagonal superposition state, we propose a non-diagonal P
construction method; if the target state is a non-diagonal mixed-state we use a
unitary transformation to change it into a diagonal state and design a diagonal
P. In such a way, the orbit tracking problem with arbitrary initial state is
properly solved. The explicit expressions of P are derived to obtain a
convergent control law. At last, the system simulation experiments are
performed on a two-level quantum system and the tracking process is illustrated
on the Bloch sphere.Comment: 20 pages, 5 figure

### Ultracontractivity and functional inequalities on infinite graphs

In this paper, we prove the equivalent of ultracontractive bound of heat
semigroup or the uniform upper bound of the heat kernel with the Nash
inequality, Log-Sobolev inequalities on graphs. We also show that under the
assumption of volume growth and nonnegative curvature $CDE'(n,0)$ the Sobolev
inequality, Nash inequality, Faber-Krahn inequality, Log-Sobolev inequalities,
discrete and continuous-time uniform upper estimate of heat kernel are all true
on graph.Comment: 13 page

### A gradient estimate for positive functions on graphs

We derive a gradient estimate for positive functions, in particular for
positive solutions to the heat equation, on finite or locally finite graphs.
Unlike the well known Li-Yau estimate, which is based on the maximum principle,
our estimate follows from the graph structure of the gradient form and the
Laplacian operator. Though our assumption on graphs is slightly stronger than
that of Bauer, Horn, Lin, Lippner, Mangoubi, and Yau (J. Differential Geom. 99
(2015) 359-405), our estimate can be easily applied to nonlinear differential
equations, as well as differential inequalities.
As applications, we estimate the greatest lower bound of Cheng's eigenvalue
and an upper bound of the minimal heat kernel, which is recently studied by
Bauer, Hua and Yau (Preprint, 2015) by the Li-Yau estimate. Moreover,
generalizing an earlier result of Lin and Yau (Math. Res. Lett. 17 (2010)
343-356), we derive a lower bound of nonzero eigenvalues by our gradient
estimate.Comment: 11 page

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