On Injectivity of an Integral Operator Connected to Riemann Hypothesis

In 1993, Alcantara-Bode showed ([2]) that Riemann Hypothesisholds if and only if the integral operator on the Hilbert space L2(0; 1)having the kernel function defined by the fractional part of (y/x), isinjective. Since then, the injectivity of the integral operator used inequivalent formulation of RH has not been addressed nor has beendissociated from RH.We provided in this paper methods for investigating the injectivityof linear bounded operators on separable Hilbert spaces using theirapproximations on dense families of subspaces.On the separable Hilbert space L2(0,1), an linear bounded operator(or its associated Hermitian), strict positive definite on a dense familyof including approximation subspaces in built on simple functions, isinjective if the rate of convergence of its sequence of injectivity pa-rameters on approximation subspaces is inferior bounded by a not nullconstant, that is the case with the Beurling - Alcantara-Bode integraloperator.We applied these methods to the integral operator used in RHequivalence proving its injectivity.

Elliptic problems in Besov and Sobolev—Triebel—Lizorkin spaces of low regularity

Elliptic problems with additional unknown distributions in boundary conditions are investigated in Besov and Sobolev–Triebel–Lizorkin spaces of low regularity, specifically of an arbitrary negative order. We find that the problems induce Fredholm bounded operators on appropriate pairs of these spaces.

On Injectivity of an Integral Operator Connected to Riemann Hypothesis

In this paper we provided methods for investigating the injectivity of linear bounded operators on separable Hilbert spaces using their approximations on dense families of subspaces. As an application we proved the injectivity of the integral operator involved in the equivalent formulations of Riemann Hypothesis (see [2] and [4]) in the Example 4.2.

Levenberg–Marquardt method for ill-posed inverse problems with possibly non-smooth forward mappings between Banach spaces

In this paper, we consider a Levenberg–Marquardt method with general regularization terms that are uniformly convex on bounded sets to solve the ill-posed inverse problems in Banach spaces, where the forward mapping might not Gˆateaux differentiable and the image space is unnecessarily reflexive. The method therefore extends the one proposed by Jin and Yang in (Numer. Math. 133:655–684, 2016) for smooth inverse problem setting with globally uniformly convex regularization terms. We prove a novel convergence analysis of the proposed method under some standing assumptions, in particular, the generalized tangential cone condition and a compactness assumption. All these assumptions are fulfilled when investigating the identification of the heat source for semilinear elliptic boundary-value problems with a Robin boundary condition, a heat source acting on the boundary, and a possibly non-smooth nonlinearity. Therein, the Clarke subdifferential of the non-smooth nonlinearity is employed to construct the family of bounded operators that is a replacement for the nonexisting Gˆateaux derivative of the forward mapping. The efficiency of the proposed method is illustrated with a numerical example.

Solutions and Drift Homogenization for a Class of Viscous Lake Equations in

In this paper we study solutions and drift homogenization for a class of viscous lake equations by using the method of semigroups of bounded operators. Suppose that the initial value i.e.,for some Hölder continuous function onwith smooth function value satisfying and Then the initial value problem (2) for viscous lake equations has a unique smooth local strong solution. Using this result we study the drift homogenization for three-dimensional stationary Stokes equation in the usual sense

Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characterizes the regularity of distributions more finely than the number parameter used for the Sobolev spaces. We prove that these problems induce Fredholm bounded operators on appropriate pairs of the above spaces. Investigating generalized solutions to the problems, we prove theorems on their regularity and a priori estimates in these spaces. As an application, we find new sufficient conditions under which components of these solutions have continuous classical derivatives of given orders. We assume that the orders of boundary differential operators may be equal to or greater than the order of the relevant elliptic equation.