Given a domain
we introduce a class of
plurisubharmonic (psh) functions
and Monge–Ampère
operators
,
, on
that
extend the Bedford–Taylor–Demailly Monge–Ampère operators.
Here
is a closed positive ...current of bidegree
that dominates
the non-pluripolar Monge–Ampère current
.
We prove that
is the limit of Monge–Ampère currents of
certain natural regularizations of
On a compact Kähler manifold
we introduce a notion of
non-pluripolar energy
and a corresponding finite energy class
that is a global version of the class
.
From the local construction
we get global Monge–Ampère currents
for
that only depend on the current
.
The limits of Monge–Ampère currents of certain natural
regularizations of φ can be expressed in terms of
for
.
We get a mass formula involving the currents
that describes the loss of mass of the
non-pluripolar Monge–Ampère measure
.
The class
includes ω-psh functions with
analytic singularities and the class
of
ω-psh functions of finite energy and certain other convex energy
classes, although it is not convex itself.
This paper reports a cellular automata (CA) study of the transient dynamic responses of anti-plane shear Lamb’s problems on random fields (RFs) with fractal and Hurst effects. Both Cauchy and Dagum ...random field models are employed to capture the combined effects of spatial randomness in both mass density and stiffness tensor fields. First, with a dyadic representation, we formulate a second-rank anti-plane stiffness tensor random field (TRF) model with full anisotropy. Its statistical, fractal, and Hurst properties are investigated, leading to introduction of a so-called
MOSP model
of TRF. Then, we generalize the CA approach to incorporate the inhomogeneity in mass density as well as stiffness fields. Through parametric studies for both Cauchy and Dagum TRFs, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. In general, the mean response amplitude is lowered by the presence of randomness, and the Hurst parameter (especially, for
β
<
0.5
) is found to have a stronger influence than the fractal dimension on the response. The results are compared with two simpler random fields: (1) randomness is present only in the mass density field; (2) randomness is present in the mass density field and in a locally isotropic stiffness tensor field. Overall, the results show that a second-rank anti-plane stiffness TRF with full anisotropy leads to the strongest fluctuation in displacement responses followed by a locally isotropic RF model.
Cyclic Adams operations Brown, Michael K.; Miller, Claudia; Thompson, Peder ...
Journal of pure and applied algebra,
07/2017, Letnik:
221, Številka:
7
Journal Article
Recenzirano
Odprti dostop
Let Q be a commutative, Noetherian ring and Z⊆Spec(Q) a closed subset. Define K0Z(Q) to be the Grothendieck group of those bounded complexes of finitely generated projective Q-modules that have ...homology supported on Z. We develop “cyclic” Adams operations on K0Z(Q) and we prove these operations satisfy the four axioms used by Gillet and Soulé in 9. From this we recover a shorter proof of Serre's Vanishing Conjecture. We also show our cyclic Adams operations agree with the Adams operations defined by Gillet and Soulé in certain cases.
Our definition of the cyclic Adams operators is inspired by a formula due to Atiyah 1. They have also been introduced and studied before by Haution 10.
The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and ...isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in
$\mathbb{R}^d$
of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed-form expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.
DOLFIN Logg, Anders; Wells, Garth N.
ACM transactions on mathematical software,
2010, Letnik:
37, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library ...combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which low-level code is automatically generated, compiled, and seamlessly integrated with efficient implementations of computational meshes and high-performance linear algebra. Easy-to-use object-oriented interfaces to the library are provided in the form of a C++ library and a Python module. This article discusses the mathematical abstractions and methods used in the design of the library and its implementation. A number of examples are presented to demonstrate the use of the library in application code.
Abstract
We prove that the class of trilinear multiplier forms with singularity over a one-dimensional subspace, including the bilinear Hilbert transform, admits bounded $L^p$-extension to triples of ...intermediate $\operatorname{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\operatorname{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\textrm{UMD}$-valued setting. This is then employed to obtain appropriate single-tree estimates by appealing to the $\textrm{UMD}$-valued bound for bilinear Calderón–Zygmund operators recently obtained by the same authors.
We study the optimal control for stochastic differential equations (SDEs) of mean-field type, in which the coefficients depend on the state of the solution process as well as of its expected value. ...Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. For a general action space a Peng’s-type stochastic maximum principle (Peng, S.: SIAM J. Control Optim.
2
(4), 966–979,
1990
) is derived, specifying the necessary conditions for optimality. This maximum principle differs from the classical one in the sense that here the first order adjoint equation turns out to be a linear mean-field backward SDE, while the second order adjoint equation remains the same as in Peng’s stochastic maximum principle.
The main goal of this paper is to study for the local existence
and decay estimates results for a high-order viscoelastic wave equation with
logarithmic nonlinearity. We obtain several results: ...Firstly, by using Faedo-
Galerkin method and a logaritmic Sobolev inequality, we proved local existence
of solutions. Later, we proved general decay results of solutions.
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