we investigate a class of fourth-order differential operators with eigenparameter dependent boundary conditions and transmission conditions. a self-adjoint linear operator a is defined in a suitable hilbert space h such that the eigenvalues of such a problem coincide with those of a . we discuss asymptotic behavior of its eigenvalues and completeness of its eigenfunctions. finally, we obtain th...

In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if f : X × Y × Z −→ W is a bounded tri-linear mapping and h : W −→ S is a bounded linear mapping, then f is regular if and only if hof is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint D^∗∗∗∗ of a tri-derivation D is again tri-derivation.

A quantitative “fourth moment theorem” for any self-adjoint element in a homogeneous Wigner chaos is provided: the Wasserstein distance controlled by from fourth to two. The proof uses free counterpart of Stein discrepancy. On way, analogues WS inequality and WSH are established.

This paper presents an adjoint method for the optimum shape design of unsteady flows. The goal is to develop a set of discrete unsteady adjoint equations and their corresponding boundary conditions for both the time accurate and non-linear frequency domain methods. First, this paper presents the complete formulation of the time dependent optimal design problem. Second, we present the time accur...

In this paper, we consider a non-self-adjoint, singular, nonlinear fourth order boundary value problem which arises in the theory of epitaxial growth. It is possible to reduce the fourth order equation to a singular boundary value problem of second order given by w''-1/r w'=w^2/(2r^2 )+1/2 λ r^2. The problem depends on the parameter λ and admits multiple solutions. Therefore, it is difficult to...

The most general quantities of interest (called “responses”) produced by the computational model a linear physical system can depend on both forward and adjoint state functions that describe respective system. This work presents Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology (4th-CASAM) for systems, which enables efficient computation exact expressions 1st-, 2nd-, 3rd- 4th-...

In this paper, we will list up all the cases for the ray contractions of divisorial and fiber types for smooth projective varieties of dimension five. These are obtained as a corollary from the lists of n-dimensional k-th adjoint contractions f : X → Y of the same types for k = 1, 2, 3 and 4 (n ≥ 5). The lists for k = 1, 2 and 3 have previously been obtained in [Na, Proposition 1.2 and Theorem ...