scholarly journals On Injectivity of an Integral Operator Connected to Riemann Hypothesis

2021 ◽  
Author(s):  
Dumitru Adam
Keyword(s):  
Numerical Analysis ◽  
Integral Operator ◽  
Hilbert Spaces ◽  
Bounded Operator ◽  
Fractional Part ◽  
Equivalent Formulation ◽  
Linear Bounded Operator ◽  
Pure Mathematics ◽  
The One ◽  
Bode Integral

Abstract 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.

scholarly journals On Injectivity of an Integral Operator Connected to Riemann Hypothesis

2021 ◽  
Author(s):  
Dumitru Adam
Keyword(s):  
Hilbert Space ◽  
Integral Operator ◽  
Hilbert Spaces ◽  
Separable Hilbert Space ◽  
Bounded Operator ◽  
Fractional Part ◽  
Positive Definite ◽  
Its Sequence ◽  
Bounded Operators ◽  
Linear Bounded Operator

Abstract 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.

scholarly journals More on Inequalities for Weaving Frames in Hilbert Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 141 ◽  
Author(s):  
Zhong-Qi Xiang
Keyword(s):  
Hilbert Spaces ◽  
Operator Theory ◽  
Triangle Inequality ◽  
Bounded Operator ◽  
Point Of View ◽  
Real Numbers ◽  
Linear Bounded Operator ◽  
Bessel Sequences ◽  
New Inequalities

In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real numbers. It is indicated that our results are more general and cover the corresponding results recently obtained by Li and Leng. We also give a triangle inequality for weaving frames in Hilbert spaces, which is structurally different from previous ones.

scholarly journals Approximative K-atomic decompositions and frames in Banach spaces

2018 ◽  
Vol 26 (1/2) ◽  
pp. 153-166
Author(s):  
Shah Jahan
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Atomic Decomposition ◽  
Necessary Conditions ◽  
Bounded Operator ◽  
Special Kind ◽  
Frame Theory ◽  
Bessel Sequence ◽  
Atomic Decompositions ◽  
Counter Example

L. Gǎvruţa (2012) introduced a special kind of frames, named K-frames, where K is an operator, in Hilbert spaces, which is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative K-atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative K-atomic decompositions in Banach spaces. Also some results on the existence of approximative K-atomic decompositions are obtained. We discuss several methods to construct approximative K-atomic decomposition for Banach Spaces. Further, approximative d-frame and approximative d-Bessel sequence are introduced and studied. Two necessary conditions are given under which an approximative d-Bessel sequence and approximative d-frame give rise to a bounded operator with respect to which there is an approximative K-atomic decomposition. Example and counter example are provided to support our concept. Finally, a possible application is given.

scholarly journals Hume’s Fork and Mixed Mathematics

2017 ◽  
Vol 99 (1) ◽  
Author(s):  
Matias Slavov
Keyword(s):  
The Other ◽  
Epistemic Status ◽  
Causal Relations ◽  
Sharp Distinction ◽  
Other Hand ◽  
Pure Mathematics ◽  
Conservation Of Momentum ◽  
The Law ◽  
The One ◽  
Uniformity Principle

Abstract:Given the sharp distinction that follows from Hume’s Fork, the proper epistemic status of propositions of mixed mathematics seems to be a mystery. On the one hand, mathematical propositions concern the relation of ideas. They are intuitive and demonstratively certain. On the other hand, propositions of mixed mathematics, such as in Hume’s own example, the law of conservation of momentum, are also matter of fact propositions. They concern causal relations between species of objects, and, in this sense, they are not intuitive or demonstratively certain, but probable or provable. In this article, I argue that the epistemic status of propositions of mixed mathematics is that of matters of fact. I wish to show that their epistemic status is not a mystery. The reason for this is that the propositions of mixed mathematics are dependent on the Uniformity Principle, unlike the propositions of pure mathematics.

Numerical Analysis for Cavitation Intensity’s Effects on the Extend Angle of Convergence-Divergence Nozzle

2014 ◽  
Vol 937 ◽  
pp. 41-45
Author(s):  
Feng Hua Zhang ◽  
Xin Su ◽  
Chuan Lin Tang ◽  
Li Wei Shan
Keyword(s):  
Numerical Analysis ◽  
Continuity Equation ◽  
Momentum Equation ◽  
Bubbly Liquid ◽  
The One

The one of technical key of the convergence-divergence nozzle is extend Angle. To use applies continuity equation, momentum equation and Rayleigh-Plesset equation, bubbly liquid through different extend angles of converging-diverging nozzle were simulated. The results show that it is helpful to increase the vibrating width of cavitate bubble when enlarge appropriately the exit taper of nozzle, that is the convergence-divergence nozzle with extended angle is more conducive to cavitation.

Numerical analysis of the convection and diffusion of solutes to roots

Soil Research ◽  
1967 ◽  
Vol 5 (2) ◽  
pp. 149 ◽  
Author(s):  
JB Passioura ◽  
MH Frere
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Porous Medium ◽  
Diffusion Coefficient ◽  
Numerical Analysis ◽  
Average Concentration ◽  
Soil System ◽  
Dry Soil ◽  
The One ◽  
And Diffusion

A numerical method is given for solving a partial differential equation describing the radial movement of solutes through a porous medium to a root. Computer programmes based on the method were prepared and used to obtain solutions of the equation for an idealized root-soil system in which a solute is transported to the root by convection but is not taken up by the root. Various patterns of water uptake were considered, the most complex being a diurnally varying uptake from soil in which the water content is decreasing. The solutions suggest that the maximum build-up of solute at the surface of a root is trivial if the root is growing in a medium such as agar, in which the diffusion coefficient of the solute is high, but may be considerable, with a concentration up to 10 times higher than the average concentration in the soil solution, when the root is growing in a fairly dry soil. The application of the method to systems other than the one considered in detail is discussed.

Experiment and Numerical Analysis of Control Method of Natural Circulation by Injection of Helium Gas

2013 ◽  
Author(s):  
Naoto Yanagawa ◽  
Masashi Nomura ◽  
Tetsuaki Takeda ◽  
Shumpei Funatani
Keyword(s):  
Numerical Analysis ◽  
Mole Fraction ◽  
Pressure Vessel ◽  
Control Method ◽  
Natural Circulation ◽  
Circulation Flow ◽  
Helium Gas ◽  
Air Ingress ◽  
The One ◽  
Reactor Pressure

This study is to investigate a control method of the natural circulation of the air by the injection of helium gas. A depressurization is the one of the design-basis accidents of a Very High Temperature Reactor (VHTR). When the primary pipe rupture accident occurs in the VHTR, the air is predicted to enter into the reactor pressure vessel from the breach and oxidize in-core graphite structures. Finally, it seems to be probable that the natural circulation flow of the air in the reactor pressure vessel produce continuously. In order to predict or analyze the air ingress phenomenon during the depressurization accident of the VHTR, it is important to develop the method for prevention of air ingress during the accident. In this study, the air ingress process is discussed by comparing the experimental and analytical results of the reverse U-shaped channel which has parallel channels. The experiment of the natural circulation using a circular tube consisted of the reverse U-shaped type has been carried out. The vertical channel is consisted of the one side heated and the other side cooled pipe. The experimental apparatus is filled with the air and one side vertical tube is heated. A very small amount of helium gas is injected from the top of the channel. The velocity and the mole fraction of each gas are also calculated by using heat and mass transfer numerical analysis of multi-component gas. The result shows that the numerical analysis is considered to be well simulated the experiment. The natural circulation of the air has very weak velocity after the injection of helium gas. About 780 seconds later, the natural circulation suddenly produces. The natural circulation flow of the air can be controlled by the method of helium gas injection. The mechanism of the phenomenon is found that mole fraction is changed by the molecular diffusion and the very weak circulation.

scholarly journals Model Selection in Atmospheric Remote Sensing with Application to Aerosol Retrieval from DSCOVR/EPIC. Part 2: Numerical Analysis

2020 ◽  
Vol 12 (21) ◽  
pp. 3656
Author(s):  
Sruthy Sasi ◽  
Vijay Natraj ◽  
Víctor Molina García ◽  
Dmitry S. Efremenko ◽  
Diego Loyola ◽  
...  
Keyword(s):  
Numerical Analysis ◽  
Model Selection ◽  
Aerosol Layer ◽  
Angular Variation ◽  
Aerosol Model ◽  
A Value ◽  
Aerosol Retrieval ◽  
Atmospheric Remote Sensing ◽  
The One ◽  
Aerosol Parameters

An algorithm for retrieving aerosol parameters by taking into account the uncertainty in aerosol model selection is applied to the retrieval of aerosol optical thickness and aerosol layer height from synthetic measurements from the EPIC sensor onboard the Deep Space Climate Observatory. The synthetic measurements are generated using aerosol models derived from AERONET measurements at different sites, while other commonly used aerosol models, such as OPAC, GOCART, OMI, and MODIS databases are used in the retrieval. The numerical analysis is focused on the estimation of retrieval errors when the true aerosol model is unknown. We found that the best aerosol model is the one with a value of the asymmetry parameter and an angular variation of the phase function around the viewing direction that is close to the values corresponding to the reference aerosol model.

Operators with Closed Range, Pseudo-Inverses, and Perturbation of Frames for a Subspace

1999 ◽  
Vol 42 (1) ◽  
pp. 37-45 ◽  
Author(s):  
Ole Christensen
Keyword(s):  
Recent Work ◽  
Hilbert Spaces ◽  
Bounded Operator ◽  
Closed Range ◽  
The Stability

AbstractRecent work of Ding and Huang shows that if we perturb a bounded operator (between Hilbert spaces) which has closed range, then the perturbed operator again has closed range. We extend this result by introducing a weaker perturbation condition, and our result is then used to prove a theorem about the stability of frames for a subspace.

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