In existing anisotropic diffusion-based semi-supervised learning approaches, anisotropic graph Laplacian is estimated based on (potentially noisy) function evaluations. We propose to regularize the graph Laplacian estimates. We develop a framework that regularizes the Laplace-Beltrami operators on Riemannian manifolds, and discretize it to a regularizer on diffusivity operators on graphs. Isotr...

Consider an operator equation F(u) = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator F ′(u) is not boundedly invertible, and well-posed otherwise. If F is monotone C2 loc(H) operator, then we construct a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends t...

The operator Jm p (λ, )f (z) was studied by Srivastava et al. [3] and Aouf et al. [4]. From (1.2) and (1.3), we observe that J−m p (λ, )f (z) = Im p (λ, )f (z)(m > 0), so the operator Jm p (λ, )f (z) is well-defined for l ≥ 0, l ≥ 0, p Î N and m Î Z = {..., -2, -1,0,1,2,...}. El-Ashwah Journal of Inequalities and Applications 2012, 2012:35 http://www.journalofinequalitiesandapplications.com/con...

Recommended by Brigitte Forster-Heinlein For λ > −1 and μ ≥ 0, we consider a liner operator I μ λ on the class A of analytic functions in the unit disk defined by the convolution f μ −1 * fz, where f μ 1 − μz 2 F 1 a, b, c; z μzz 2 F 1 a, b, c; z , and introduce a certain new subclass of A using this operator. Several interesting properties of these classes are obtained.

The following limit result holds for the weak–type (1,1) constant of dilation-commuting singular integral operator T in Rn: for f ∈ L1(Rn), f ≥ 0, lim λ→0 λ m{x ∈ R : |Tf(x)| > λ} = 1 n ∫ Sn−1 |Ω(x)|dσ(x)‖f‖1. For the maximal operator M , the corresponding result is lim λ→0 λ m{x ∈ R : |Mf(x)| > λ} = ‖f‖1.

[1] Traditional terminology is that f → f |2kγ is the slash operator, although this name fails to suggest any meaning other than reference to the notation itself. In fact, obviously f(z)→ f(γz)(cz + d)−2k is a left translation operator, albeit complicated by the automorphy factor. That is, this is a right action of SL2(Z) on functions on H, while the group action of SL2(Z) on H is written on th...

The purpose of this paper is to derive some subordination and superordination results for functions of the form f(z) = zp + ∞ ∑ k=p+1 akz k which are p−valent in the open unit disk U = {z ∈ C : |z| < 1} by using certain differential operator Aλ,p(α, β, μ)f(z) and integral operator F m p (ρ, θ)f(z). Some special cases are also considered.

Ascending and descending Morse complexes are defined by the critical points and integral lines of a scalar field f defined on a manifold M . They induce a subdivision of M into regions of uniform gradient flow, thus providing a compact description of the topology of M and of the behavior of f over M . We represent the ascending and descending Morse complexes of f as a graph, that we call the Mo...

Many investigations have been made about of Non-Newtonian calculus and superposition operators until today. operator was defined by Sa??r Erdo?an in [9]. In this study, we *- boundedness *-locally operator. We proved that the non-Newtonian $_{N}P_{f}:c_{_{0,\alpha }}\rightarrow \ell _{1,\beta }$ is bounded if only f satisfies condition (NA??). Then shown necessary sufficient conditions for *-bo...

In this note we prove that if A and B∗ are subnormal operators and X is a bounded linear operator such that AX − XB is a Hilbert-Schmidt operator, then f(A)X −Xf(B) is also a Hilbert-Schmidt operator and ||f(A)X −Xf(B)||2 ≤ L ||AX −XB||2, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X ∈ L(H) i...