Certain fractional calculus and integral transform results of incomplete ℵ-functions with applications

In this paper, we study certain interesting and useful properties of incomplete $\aleph$-functions. The incomplete $\aleph$-function is an extension of the $\aleph$-function. We find several useful classical integral transforms of these functions. Further, we examine the fractional calculus with the incomplete $\aleph$-functions and point out several special cases. Finally, we give the applications of incomplete $\aleph$-functions in detecting glucose supply in human blood.     

Entropy solutions for nonlinear elliptic anisotropic problems with homogeneous Neumann boundary condition

This study is about a nonlinear anisotropic problem with homogeneous Neumann boundary condition. We first prove, by using the technic of monotone operators in Banach spaces, the existence of weak solution, and by approximation methods, we achieve a result of existence and uniqueness of entropy solution     

Sobolev estimates for (pseudo)‐differential operators and applications

In this work we show that if is a linear differential operator of order ν with smooth complex coefficients in from a complex vector space E to a complex vector space F, the Sobolev a priori estimate holds locally at any point if and only if is elliptic and the constant coefficient homogeneous operator is canceling in the sense of Van Schaftingen for every which means that Here is the homogeneous part of order ν of and is the principal symbol of . This result implies and unifies the proofs of several estimates for complexes and pseudo‐complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.     

Multilinear forms and measures of dependence between random variables

Based on an idea of Rosenblatt, the methods of interpolation theory are used to establish moment inequalities and equivalence relations for measures of dependence between two or more families of random variables. A couple of “interpolation” theorems proved here appear to be new.     

混合单调算子的某些不动点定理与锥上的正不动点

给出在半序距离空间以迭代而得到的某些不动点定理的混合单调算子和在锥上某些非零不动点定理。     

Nakai's conjecture for varieties smoothed by normalization

The notion of D-simplicity is used to give a short proof that varieties whose normalization is smooth satisfy Ishibashi's extension of Nakai's conjecture to arbitrary characteristic. This gives a new proof of Nakai's conjecture for curves and Stanley-Reisner rings.

     

The heat kernel weighted Hodge Laplacian on noncompact manifolds

On a compact orientable Riemannian manifold, the Hodge Laplacian has compact resolvent, therefore a spectral gap, and the dimension of the space of harmonic -forms is a topological invariant. By contrast, on complete noncompact Riemannian manifolds, is known to have various pathologies, among them the absence of a spectral gap and either ``too large' or ``too small' a space . In this article we use a heat kernel measure to determine the space of square-integrable forms and to construct the appropriate Laplacian . We recover in the noncompact case certain results of Hodge's theory of in the compact case. If the Ricci curvature of a noncompact connected Riemannian manifold is bounded below, then this ``heat kernel weighted Laplacian' acts on functions on in precisely the manner we would wish, that is, it has a spectral gap and a one-dimensional kernel. We prove that the kernel of on -forms is zero-dimensional on , as we expect from topology, if the Ricci curvature is nonnegative. On Euclidean space, there is a complete Hodge theory for . Weighted Laplacians also have a duality analogous to Poincaré duality on noncompact manifolds. Finally, we show that heat kernel-like measures give desirable spectral properties (compact resolvent) in certain general cases. In particular, we use measures with Gaussian decay to justify the statement that every topologically tame manifold has a strong Hodge decomposition.

     

Baskakov算子加权逼近的收敛阶

本文讨论了Ｂａｓｋａｋｏｖ算子加Ｊａｃｏｂｉ权逼近的收敛性，首先指出了按通常的加权范数，Ｂａｓｋａｋｏｖ算子是无界的。然后引入一种新的范数，在此范数下Ｂａｓｋａｋｏｖ算子具有压缩性，最后借助于Ｋ－泛函，我们着重讨论了它的特征刻划问题。     

APPROXIMATE PARTIAL NOETHER OPERATORS OF THE SCHWARZSCHILD SPACETIME

The objective of this paper is twofold: (a) to find a natural example of a perturbed Lagrangian that has different partial Noether operators with symmetries different from those of the underlying Lagrangian. First we regard the Schwarzschild spacetime as a perturbation of the Minkowski spacetime and investigate the approximate partial Noether operators for this perturbed spacetime. It is shown that the Minkowski spacetime has 12 partial Noether operators, 10 of which are different from the 17 Noether symmetries for this spacetime. It is found that for the perturbed Schwarzschild spacetime we recover the exact partial Noether operators as trivial first-order approximate partial Noether operators and there is no non-trivial approximate partial Noether operator as for the Noether case. As a consequence we state a conjecture. (b) Then we prove a conjecture that the approximate symmetries of a perturbed Lagrangian form a subalgebra of the approximate symmetries of the corresponding perturbed Euler–Lagrange equations and illustrate it by our examples. This is in contrast to approximate partial Noether operators.     

Some remarks on the operator of Foias and Williams

In this paper we study the Foias-Williams operator

where , and is a Hankel operator with symbol . We exhibit a relationship between the similarity of to a contraction and the rate of decay of , the absolute values of the Fourier coefficients of the symbol .