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.

Idempotent Fourier multipliers acting contractively on $$H^p$$ spaces

We describe the idempotent Fourier multipliers that act contractively on $$H^p$$ H p spaces of the d-dimensional torus $$\mathbb {T}^d$$ T d for $$d\ge 1$$ d ≥ 1 and $$1\le p \le \infty $$ 1 ≤ p ≤ ∞ . When p is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on $$L^p$$ L p spaces, which in turn can be described by suitably combining results of Rudin and Andô. When $$p=2(n+1)$$ p = 2 ( n + 1 ) , with n a positive integer, contractivity depends in an interesting geometric way on n, d, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on $$H^p(\mathbb {T}^\infty )$$ H p ( T ∞ ) for every $$1 \le p \le \infty $$ 1 ≤ p ≤ ∞ and that extends to a bounded operator if and only if $$p=2,4,\ldots ,2(n+1)$$ p = 2 , 4 , … , 2 ( n + 1 ) .

On Injectivity of an Integral Operator Connected to Riemann Hypothesis

Using the equivalent formulation of RH given by Beurling ([4],1955), Alcantara-Bode showed ([2], 1993) that Riemann Hypothesisholds if and only if the integral operator on the Hilbert space L2(0; 1)having the kernel defined by fractional part function of the expressionbetween brackets {y/x}, is injective.Since then, the injectivity of the integral operator used in equivalentformulation of RH has not been addressed nor has been dissociatedfrom RH and, a pure mathematics solution for RH is not ready yet.Here is a numerical analysis approach of the injectivity of the linearbounded operators on separable Hilbert spaces addressing the problemslike the one presented in [2]. Apart of proving the injectivity of theBeurling - Alcantara-Bode integral operator, we obtained the followingresult: every linear bounded operator (or its associated Hermitian),strict positive definite on a dense family of including approximationsubspaces in L2(0,1) built on simple functions, is injective if the rateof convergence to zero of its unbounded sequence of inverse conditionnumbers on approximation subspaces is o(n-s) for some s ≥ 0. Whens = 0, the sequence is inferior bounded by a not null constant, that isthe case in the Beurling - Alcantara-Bode integral operator.In the Theorem 4.1 we addressed with numerical analysis toolsthe injectivity of the integral operator in [2] claiming that - even if asolution in pure mathematics is desired, together with the Theorem 1,pg. 153 in [2], the RH holds.

A theory of higher-order subtyping with type intervals

The calculus of Dependent Object Types (DOT) has enabled a more principled and robust implementation of Scala, but its support for type-level computation has proven insufficient. As a remedy, we propose F ·· ω , a rigorous theoretical foundation for Scala’s higher-kinded types. F ·· ω extends F <: ω with interval kinds , which afford a unified treatment of important type- and kind-level abstraction mechanisms found in Scala, such as bounded quantification, bounded operator abstractions, translucent type definitions and first-class subtyping constraints. The result is a flexible and general theory of higher-order subtyping. We prove type and kind safety of F ·· ω , as well as weak normalization of types and undecidability of subtyping. All our proofs are mechanized in Agda using a fully syntactic approach based on hereditary substitution.

Inverse Problem for the Sobolev Type Equation of Higher Order

The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with the Cauchy and overdetermination conditions. This problem was reduced to two equivalent problems in the aggregate: regular and singular. For these problems, the theory of polynomially bounded operator pencils is used. The unknown coefficient of the original equation is restored using the method of successive approximations. The main result of this work is a theorem on the unique solvability of the original problem. This study continues and generalizes the authors’ previous research in this area. All the obtained results can be applied to the mathematical modeling of various processes and phenomena that fit the problem under study.

Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces

Among the many different definitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan–Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag–Leffler functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan–Nersesyan derivative in the case of relative p-boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan–Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables.

On totally global solvability of evolutionary equation with unbounded operator

Let $X$ be a Hilbert space, $U$ be a Banach space, $G\colon X\to X$ be a linear operator such that the operator $B_\lambda=\lambda I-G$ is maximal monotone with some (arbitrary given) $\lambda\in\mathbb{R}$. For the Cauchy problem associated with controlled semilinear evolutionary equation as follows \[x^\prime(t)=Gx(t)+f\bigl( t,x(t),u(t)\bigr),\quad t\in[0;T];\quad x(0)=x_0\in X,\] where $u=u(t)\colon[0;T]\to U$ is a control, $x(t)$ is unknown function with values in $X$, we prove the totally (with respect to a set of admissible controls) global solvability subject to global solvability of the Cauchy problem associated with some ordinary differential equation in the space $\mathbb{R}$. Solution $x$ is treated in weak sense and is sought in the space $\mathbb{C}_w\bigl([0;T];X\bigr)$ of weakly continuous functions. In fact, we generalize a similar result having been proved by the author formerly for the case of bounded operator $G$. The essence of this generalization consists in that postulated properties of the operator $B_\lambda$ give us the possibility to construct Yosida approximations for it by bounded linear operators and thus to extend required estimates from “bounded” to “unbounded” case. As examples, we consider initial boundary value problems associated with the heat equation and the wave equation.

AN UNBOUNDED OPERATOR WITH SPECTRUM IN A STRIP AND MATRIX DIFFERENTIAL OPERATORS

Let A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$ , assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.