In this paper, we use the generalized notions of Riemann–Liouville (fractional calculus with respect to a regular function) to extend the definitions of fractional integration and derivative from the ...functional sense to the distributional sense. First, we give some properties of fractional integral and derivative for the functions infinitely differentiable with compact support. Then, we define the weak derivative, as well as the integral and derivative of a distribution with compact support, and the integral and derivative of a distribution using the convolution product. Then, we generalize those concepts from the unidimensional to the multidimensional case. Finally, we propose the definitions of some usual differential operators.
In this paper, we introduce and analyze a lowest-order locking-free weak Galerkin (WG) finite element scheme for the grad-div formulation of linear elasticity problems. The scheme uses linear ...functions in the interior of mesh elements and constants on edges (2D) or faces (3D), respectively, to approximate the displacement. An H(div)-conforming displacement reconstruction operator is employed to modify test functions in the right-hand side of the discrete form, in order to eliminate the dependence of the Lame´ parameter λ in error estimates, i.e., making the scheme locking-free. The method works without requiring λ‖∇⋅u‖1 to be bounded. We prove optimal error estimates, independent of λ, in both the H1-norm and the L2-norm. Numerical experiments validate that the method is effective and locking-free.
In this paper, we propose a new unbiased stochastic derivative estimator in a framework that can handle discontinuous sample performances with structural parameters. This work extends the three most ...popular unbiased stochastic derivative estimators: (1) infinitesimal perturbation analysis (IPA), (2) the likelihood ratio (LR) method, and (3) the weak derivative method, to a setting where they did not previously apply. Examples in probability constraints, control charts, and financial derivatives demonstrate the broad applicability of the proposed framework. The new estimator preserves the single-run efficiency of the classic IPA-LR estimators in applications, which is substantiated by numerical experiments.
The online appendix is available at
https://doi.org/10.1287/opre.2017.1674
.
Significance Twister is a small self-cleaving ribozyme similar in size to the hammerhead ribozyme but uses an orthogonal fold for a similar catalytic rate constant. However, the mechanistic source of ...the catalytic rate increase generated by twister was unknown. We present crystal structures of twister from Orzyza sativa as well as a twister sequence from an organism that has not been cultured in isolation and identify RNA nucleotides that are vital for self-cleavage, suggest their catalytic roles, and update twister’s conserved secondary structure model.
A discontinuous Galerkin method with skeletal multipliers (DGSM) is developed for diffusion problem. Skeletal multiplier is introduced on the edge/face of each element through the definition of a ...weak divergence and a weak derivative in the method. The local weak formulation is derived by weakly imposing the Dirichlet boundary condition and continuity of fluxes and solutions on the edges/faces. The global weak formulation is then obtained by adding all the local problems. Equivalence of the weak formulation and the original problem is proved. Stability of DGSM is shown and an error estimate is derived in a broken norm. A DGSM for linear convection–diffusion–reaction problems is also derived. An explanation on algorithmic aspects is given. Some numerical results are presented. Singularities due to discontinuities in the diffusion coefficients are accurately approximated. Internal/boundary layers are well captured without showing spurious oscillations. Robustness of the method in increasingly small diffusivity is demonstrated on the whole domain.
We investigate some spectral properties of differential–difference operators, which are symmetrizations of differential operators of the form
(
d
†
d
)
k
and
(
d
d
†
)
k
,
k
⩾
1
. Here,
d
=
p
d
d
x
+
...q
and
d
†
stands for the formal adjoint of
d
on
L
2
(
(
0
,
b
)
,
w
d
x
)
. In the simpliest case
k
=
1
, this symmetrization brings in the operator
-
D
2
, which can be seen as a ‘Laplacian’, and
D
f
:
=
D
d
f
=
d
(
f
even
)
-
d
†
(
f
odd
)
, a skew-symmetric operator in
L
2
(
I
,
w
d
x
)
,
I
=
(
-
b
,
0
)
∪
(
0
,
b
)
, is the symmetrization of
d
. Investigated spectral properties include self-adjoint extensions, among them the Friedrichs extensions, of the symmetrized operators.
In this paper, we introduce the HK-Sobolev space and establish a fundamental theorem of calculus and an integration by parts formula, then we give sufficient conditions for the existence and ...uniqueness of a solution to a variational problem associated with a Sturm–Liouville type equation involving Henstock-Kurzweil integrable functions as source terms.
Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer–Itô, applies to one dimensional ...semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes where the Meyer–Itô assumptions are weakened even further. We study a version of Itô's formula for multi-dimensional finite variation Lévy processes assuming that the underlying function is continuous and admits weak derivatives. We also discuss some applications of this extension, particularly in finance.
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