On self-injectivity of the f-ring Frm(𝓟(ℝ), L)

Abstract Let 𝓕𝓟L := Frm(𝓟(ℝ), L). We show that if L is a P-frame then 𝓕𝓟L is an ℵ0-self-injective ring. We prove that a zero-dimensional frame L is extremally disconnected if and only if 𝓕𝓟L is a self-injective ring. Finally, it is shown that 𝓕𝓟L is a Baer ring if and only if 𝓕𝓟L is a continuous ring if and only if 𝓕𝓟L is a complete ring if and only if 𝓕𝓟L is a CS-ring.

Download Full-text

Теорема Артина для f-колец

Владикавказский математический журнал ◽

10.23671/vnc.2015.2.7277 ◽

2015 ◽

Author(s):

A.G. Kusraev

Keyword(s):

Rational Functions ◽

Sum Of Squares ◽

Famous Theorem ◽

The Real ◽

Complete Ring ◽

Positive Polynomial ◽

Ring Of Quotients ◽

Real Closure ◽

F Ring ◽

17Th Problem

The main result states that each positive polynomial p in N variables with coefficients in a unital Archimedean f-ring K is representable as a sum of squares of rational functions over the complete ring of quotients of K provided that p is positive on the real closure of K. This is proved by means of Boolean valued interpretation of Artin's famous theorem which answers Hilbert's 17th problem affirmatively.

Download Full-text

COMMENTS REGARDING d-IDEALS OF CERTAIN f-RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813500084 ◽

2013 ◽

Vol 12 (06) ◽

pp. 1350008 ◽

Cited By ~ 3

Author(s):

THEMBA DUBE ◽

OGHENETEGA IGHEDO

Keyword(s):

Radical Ideal ◽

Baer Ring ◽

Ring Of Fractions ◽

F Ring ◽

The Ideal

Let A be a reduced commutative f-ring with identity and bounded inversion, and let A* be its subring of bounded elements. By first observing that A is the ring of fractions of A* relative to the subset of A* consisting of elements which are units in the bigger ring, we show that the frames Did (A) and Did (A*) of d-ideals of A and A*, respectively, are isomorphic, and that the isomorphism witnessing this is precisely the restriction of the extension map I ↦ Ie which takes a radical ideal of A* to the ideal it generates in A. Specializing to the ring [Formula: see text], we show that if L is an F-frame, then the saturation quotient of [Formula: see text] is isomorphic to βL. We also investigate projectability properties of [Formula: see text] and [Formula: see text], where the latter denotes the frame of z-ideals of [Formula: see text]. We show that [Formula: see text] is flatly projectable precisely when [Formula: see text] is a feebly Baer ring. Quite easily, [Formula: see text] is projectable if and only if L is basically disconnected. Less obvious is that [Formula: see text] is projectable if and only if L is cozero-complemented.

Download Full-text

Saturn's E, G and F rings: modulated by the plasma sheet

International Astronomical Union Colloquium ◽

10.1017/s0252921100101666 ◽

1984 ◽

Vol 75 ◽

pp. 597

Author(s):

E. Grün ◽

G.E. Morfill ◽

T.V. Johnson ◽

G.H. Schwehm

Keyword(s):

Solar Wind ◽

Direct Contact ◽

Transport Processes ◽

Time Variable ◽

Dust Particles ◽

Spatial Diffusion ◽

Drag Forces ◽

Orbit Perturbation ◽

Time Variations ◽

F Ring

ABSTRACTSaturn's broad E ring, the narrow G ring and the structured and apparently time variable F ring(s), contain many micron and sub-micron sized particles, which make up the “visible” component. These rings (or ring systems) are in direct contact with magnetospheric plasma. Fluctuations in the plasma density and/or mean energy, due to magnetospheric and solar wind processes, may induce stochastic charge variations on the dust particles, which in turn lead to an orbit perturbation and spatial diffusion. It is suggested that the extent of the E ring and the braided, kinky structure of certain portions of the F rings as well as possible time variations are a result of plasma induced electromagnetic perturbations and drag forces. The G ring, in this scenario, requires some form of shepherding and should be akin to the F ring in structure. Sputtering of micron-sized dust particles in the E ring by magnetospheric ions yields lifetimes of 102to 104years. This effect as well as the plasma induced transport processes require an active source for the E ring, probably Enceladus.

Download Full-text

£-Single Valued Extremally Disconnected Ideal Neutrosophic Topological Spaces

Symmetry ◽

10.3390/sym13010053 ◽

2020 ◽

Vol 13 (1) ◽

pp. 53

Author(s):

Fahad Alsharari

Keyword(s):

Topological Spaces ◽

Neutrosophic Sets ◽

Extremally Disconnected ◽

Fundamental Properties ◽

Separation Axioms ◽

New Concepts ◽

Definition Of

This paper aims to mark out new concepts of r-single valued neutrosophic sets, called r-single valued neutrosophic £-closed and £-open sets. The definition of £-single valued neutrosophic irresolute mapping is provided and its characteristic properties are discussed. Moreover, the concepts of £-single valued neutrosophic extremally disconnected and £-single valued neutrosophic normal spaces are established. As a result, a useful implication diagram between the r-single valued neutrosophic ideal open sets is obtained. Finally, some kinds of separation axioms, namely r-single valued neutrosophic ideal-Ri (r-SVNIRi, for short), where i={0,1,2,3}, and r-single valued neutrosophic ideal-Tj (r-SVNITj, for short), where j={1,2,212,3,4}, are introduced. Some of their characterizations, fundamental properties, and the relations between these notions have been studied.

Download Full-text

Synthesis of 20-substituted chlorophyll derivatives with F-ring and optical properties of their less distorted chlorin π-systems

Tetrahedron ◽

10.1016/j.tet.2021.132260 ◽

2021 ◽

pp. 132260

Author(s):

Yuki Takarada ◽

Marie Doi ◽

Shin Ogasawara ◽

Hitoshi Tamiaki

Keyword(s):

Optical Properties ◽

F Ring ◽

Chlorophyll Derivatives

Download Full-text

Intracorneal circular ring implant with femtosecond laser: Pocket versus tunnel

European Journal of Ophthalmology ◽

10.1177/1120672121994729 ◽

2021 ◽

pp. 112067212199472

Author(s):

Luis Izquierdo ◽

Ana M Rodríguez ◽

Ramón A Sarquis ◽

Diego Altamirano ◽

Maria A Henriquez

Keyword(s):

Visual Acuity ◽

Femtosecond Laser ◽

Randomized Study ◽

Circular Ring ◽

Continuous Ring ◽

Refractive Outcomes ◽

Scheimpflug Imaging ◽

Ring Segment ◽

Intracorneal Ring Segment ◽

Student’S T

Purpose: To evaluate and compare visual and refractive outcomes after implantation of the intracorneal continuous ring 360° arc (ICCR) versus the intracorneal ring segment 340° arc (ICRS) using femtosecond laser for central keratoconus. Setting: Research Department, Oftalmosalud, Instituto de Ojos, Lima, Peru. Methods: Randomized study that included 40 eyes of 32 patients diagnosed with central keratoconus between November 2014 and March 2015. Twenty eyes had an implantation of ICCR (MyoRing, Dioptex GmbH, Austria) through an intrastromal pocket and 20 eyes had an implantation of ICRS (Keraring, Mediphacos, Brazil) through an intrastromal tunnel. Both procedures were performed with a femtosecond laser (LDV Z6 model, Ziemer Ophthalmic Systems AG). Visual acuity (VA), refraction, and Scheimpflug imaging analysis were performed pre- and postoperatively at 1 month and 1 year. Comparisons of means were performed using the Student’s t-test. Results: At 1 year, uncorrected VA improved 0.77 LogMAR ( p < 0.001) in the ICCR group and 0.79 LogMAR ( p = 0.01) in the ICRS group; mean sphere improvement was 5.13 Diopters (D) in the ICCR group and 6.27 D in the ICRS group ( p < 0.001 both); mean Steeper Keratometry improvement was 4.24 D in the ICCR group and 5.53 D in the ICRS group ( p < 0.001 both). In the ICCR group, mean decrease in the pachymetry at the thinnest point of the cornea was 32.16 µm ( p = 0.01), and in the ICRS group, mean increase was 4.2 µm at 1 year ( p = 0.61). Conclusion: Intracorneal continuous ring 360° arc (ICCR) and intracorneal ring segment 340° (ICRS) are effective treatments for central keratoconus. No significant differences between rings were found on visual acuity, refraction, and keratometry improvement.

Download Full-text