Radical Regularity in Differential Rings

1971 ◽  
Vol 23 (2) ◽  
pp. 197-201 ◽  
Author(s):  
Howard E. Gorman
Keyword(s):  
Jacobson Radical ◽  
Regular Ring ◽  
Prime Ideals ◽  
Differential Ring ◽  
Quotient Maps ◽  
The Stability ◽  
Definition Of ◽  
Finitely Generated Modules ◽  
Regular Rings ◽  
The Ideal

In [1], we discussed completions of differentially finitely generated modules over a differential ring R. It was necessary that the topology of the module be induced by a differential ideal of R and it was natural that this ideal be contained in J(R), the Jacobson radical of R. The ideal to be chosen, called Jd(R), was the intersection of those ideals which are maximal among the differential ideals of R. The question as to when Jd(R) ⊆ J(R) led to the definition of a class of rings called radically regular rings. These rings do satisfy the inclusion, and we showed in [1, Theorem 2] that R could always be “extended”, via localization, to a radically regular ring in such a way as to preserve all its differential prime ideals.In the present paper, we discuss the stability of radical regularity under quotient maps, localization, adjunction of a differential indeterminate, and integral extensions.

The effect of base flow variation on flow stability

2003 ◽  
Vol 476 ◽  
pp. 293-302 ◽  
Author(s):  
ALESSANDRO BOTTARO ◽  
PETER CORBETT ◽  
PAOLO LUCHINI
Keyword(s):  
Hydrodynamic Stability ◽  
Base Flow ◽  
Plane Couette Flow ◽  
Worst Case ◽  
Unconditionally Stable ◽  
Variational Techniques ◽  
Parallel Flows ◽  
The Stability ◽  
Definition Of ◽  
The Ideal

The Orr–Sommerfeld operator's eigenvalues determine the stability of exponentially growing disturbances in parallel and quasi-parallel flows. This work assesses the sensitivity of these eigenvalues to modifications of the base flow, which need not be infinitesimally small. Such base flow variations may represent differences between the laboratory flow and its ideal, theoretical counterpart. The worst case, i.e. the change in base flow with the most destabilizing effect on the eigenvalues, is found using variational techniques for the plane Couette flow. Relatively small changes in the base flow are shown to be destabilizing, although the ideal flow is unconditionally stable according to linear theory. These observations inspire a velocity-based definition of pseudospectra in the hydrodynamic stability context.

scholarly journals J-Regular Rings with Injectivities

2013 ◽  
Vol 20 (02) ◽  
pp. 343-347 ◽  
Author(s):  
Liang Shen
Keyword(s):  
Jacobson Radical ◽  
Regular Ring ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Regular Rings

Let R be a J-regular ring, i.e., R/J(R) is a von Neumann regular ring, where J(R) is the Jacobson radical of R. It is proved: (i) For every n ≥ 1, R is right n-injective if and only if every homomorphism from an n-generated small right ideal of R to RR can be extended to one from RR to RR. (ii) R is right FP-injective if and only if R is right (J,R)-FP-injective. Some known results are improved.

Some model theory of modules. II. on stability and categoricity of flat modules

1983 ◽  
Vol 48 (4) ◽  
pp. 970-985 ◽  
Author(s):  
Philipp Rothmaler
Keyword(s):  
Model Theory ◽  
Regular Ring ◽  
Main Tool ◽  
Categorical Theory ◽  
Maximal Ideals ◽  
Flat Modules ◽  
Semisimple Ring ◽  
Definable Sets ◽  
The Stability ◽  
Regular Rings

This is the second part of a study on model theory of modules begun in [RO]. Throughout, I refer to that paper as “Part I”. I observed there a coincidence between some algebraic and logical points of view in the theory of modules, which led to a convenient representation of p.p. definable sets in flat modules (§1); it becomes especially nice in the case of regular rings (cf. Remark 7 in Part I). Using this in the present paper I obtain a simplified criterion for total transcendence and superstability in the case of flat modules. This enables me to give, besides partial results for arbitrary flat modules (§2), a complete description of the stability classes for modules over regular rings (§3). Particularly, it turns out that a totally transcendental module over a regular ring can be regarded as a module over a semisimple ring (cf. Remark 10 in Part I). This is the crucial observation for the examination of categoricity here: By Morley's theorem, a countable ℵ1-categorical theory is totally transcendental. Consequently, an ℵ1-categorical module over a countable regular ring can be regarded as a module over a semisimple ring. That is why I separately treat categoricity of modules over semisimple rings (§4), even without any assumption on the power of the ring. As a consequence I obtain a complete description of ℵ1-categorical modules over countable regular rings (§5). In the investigation presented here the main tool is the technique of idempotents avoiding the commutativity assumption made in the corresponding results of Garavaglia [GA 1, pp. 86–88], who used maximal ideals for that purpose. At the end of the present paper I show how to derive these latter results in our context.On the way I simplify the known criterion for elementary equivalence for modules over regular rings (§3), which simplifies again in case of semisimple rings (§4). Such a criterion is needed in order to construct Vaughtian pairs (in the categoricity consideration) and it turns out to be useful also for another purpose treated in the third part of this series of papers.

UNIT-REGULAR MODULES

2017 ◽  
Vol 60 (1) ◽  
pp. 1-15
Author(s):  
H. CHEN ◽  
W. K. NICHOLSON ◽  
Y. ZHOU
Keyword(s):  
Regular Element ◽  
Regular Ring ◽  
Stable Range ◽  
Morita Context ◽  
Regular Elements ◽  
Regular Module ◽  
Natural Extensions ◽  
Definition Of ◽  
Regular Rings ◽  
Study Unit

AbstractIn 2014, the first two authors proved an extension to modules of a theorem of Camillo and Yu that an exchange ring has stable range 1 if and only if every regular element is unit-regular. Here, we give a Morita context version of a stronger theorem. The definition of regular elements in a module goes back to Zelmanowitz in 1972, but the notion of a unit-regular element in a module is new. In this paper, we study unit-regular elements and give several characterizations of them in terms of “stable” elements and “lifting” elements. Along the way, we give natural extensions to the module case of many results about unit-regular rings. The paper concludes with a discussion of when the endomorphism ring of a unit-regular module is a unit-regular ring.

scholarly journals Two New Types of Rings Constructed from Quasiprime Ideals

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Manal Ghanem ◽  
Hassan Al-Ezeh
Keyword(s):  
Positive Characteristic ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Von Neumann ◽  
Differential Ring ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

Keigher showed that quasi-prime ideals in differential commutative rings are analogues of prime ideals in commutative rings. In that direction, he introduced and studied new types of differential rings using quasi-prime ideals of a differential ring. In the same sprit, we define and study two new types of differential rings which lead to the mirrors of the corresponding results on von Neumann regular rings and principally flat rings (PF-rings) in commutative rings, especially, for rings of positive characteristic.

On the lattice of biorder ideals of regular rings

2019 ◽  
Vol 13 (06) ◽  
pp. 2050103
Author(s):  
R. Akhila ◽  
P. G. Romeo
Keyword(s):  
Regular Semigroup ◽  
Significant Role ◽  
Modular Lattice ◽  
Regular Ring ◽  
Biordered Set ◽  
Definition Of ◽  
Regular Rings

The study of biordered set plays a significant role in describing the structure of a regular semigroup and since the definition of regularity involves only the multiplication in the ring, it is natural that the study of semigroups plays a significant role in the study of regular rings. Here, we extend the biordered set approach to study the structure of the regular semigroup [Formula: see text] of a regular ring [Formula: see text] by studying the idempotents [Formula: see text] of the regular ring and show that the principal biorder ideals of the regular ring [Formula: see text] form a complemented modular lattice and certain properties of this lattice are studied.

Postmodern More

Moreana ◽  
2003 ◽  
Vol 40 (Number 153- (1-2) ◽  
pp. 219-239
Author(s):  
Anne Lake Prescott
Keyword(s):  
Human Language ◽  
Thomas More ◽  
The Stability ◽  
Definition Of ◽  
Recent Critic

Thomas More is often called a “humanist,” and rightly so if the word has its usual meaning in scholarship on the Renaissance. “Humanist” has by now acquired so many different and contradictory meanings, however, that it needs to be applied carefully to the likes of More. Many postmodernists tend to use the word, pejoratively, to mean someone who believes in an autonomous self, the stability of words, reason, and the possibility of determinable meanings. Without quite arguing that More was a postmodernist avant la lettre, this essay suggests that he was not a “humanist” who stalks the pages of much recent postmodernist theory and that in fact even while remaining a devout Catholic and sensible lawyer he was quite as aware as any recent critic of the slipperiness of human selves and human language. It is time that literary critics tightened up their definition of “humanist,” especially when writing about the Renaissance.

scholarly journals Influence of study model, baseline catalytic concentrations and analytical system on the stability of serum alanine aminotransferase

2020 ◽  
Vol 1 (2) ◽  
Author(s):  
Josep Miquel Bauça ◽  
Andrea Caballero ◽  
Carolina Gómez ◽  
Débora Martínez-Espartosa ◽  
Isabel García del Pino ◽  
...  
Keyword(s):  
Alanine Aminotransferase ◽  
Experimental Models ◽  
Individual Variability ◽  
Serum Samples ◽  
Multicentric Study ◽  
Different Temperatures ◽  
The Stability ◽  
Analytical System ◽  
Definition Of ◽  
Analytical Platforms

AbstractObjectivesThe stability of the analytes most commonly used in routine clinical practice has been the subject of intensive research, with varying and even conflicting results. Such is the case of alanine aminotransferase (ALT). The purpose of this study was to determine the stability of serum ALT according to different variables.MethodsA multicentric study was conducted in eight laboratories using serum samples with known initial catalytic concentrations of ALT within four different ranges, namely: <50 U/L (<0.83 μkat/L), 50–200 U/L (0.83–3.33 μkat/L), 200–400 U/L (3.33–6.67 μkat/L) and >400 U/L (>6.67 μkat/L). Samples were stored for seven days at two different temperatures using four experimental models and four laboratory analytical platforms. The respective stability equations were calculated by linear regression. A multivariate model was used to assess the influence of different variables.ResultsCatalytic concentrations of ALT decreased gradually over time. Temperature (−4%/day at room temperature vs. −1%/day under refrigeration) and the analytical platform had a significant impact, with Architect (Abbott) showing the greatest instability. Initial catalytic concentrations of ALT only had a slight impact on stability, whereas the experimental model had no impact at all.ConclusionsThe constant decrease in serum ALT is reduced when refrigerated. Scarcely studied variables were found to have a significant impact on ALT stability. This observation, added to a considerable inter-individual variability, makes larger studies necessary for the definition of stability equations.

scholarly journals Reliable Learning with PDE-Based CNNs and DenseNets for Detecting COVID-19, Pneumonia, and Tuberculosis from Chest X-Ray Images

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 434
Author(s):  
Anca Nicoleta Marginean ◽  
Delia Doris Muntean ◽  
George Adrian Muntean ◽  
Adelina Priscu ◽  
Adrian Groza ◽  
...  
Keyword(s):  
Transfer Learning ◽  
Negative Impact ◽  
X Ray ◽  
Domain Specific ◽  
Direct Training ◽  
Public Data ◽  
Normal Chest ◽  
Chest X Ray ◽  
The Stability ◽  
Definition Of

It has recently been shown that the interpretation by partial differential equations (PDEs) of a class of convolutional neural networks (CNNs) supports definition of architectures such as parabolic and hyperbolic networks. These networks have provable properties regarding the stability against the perturbations of the input features. Aiming for robustness, we tackle the problem of detecting changes in chest X-ray images that may be suggestive of COVID-19 with parabolic and hyperbolic CNNs and with domain-specific transfer learning. To this end, we compile public data on patients diagnosed with COVID-19, pneumonia, and tuberculosis, along with normal chest X-ray images. The negative impact of the small number of COVID-19 images is reduced by applying transfer learning in several ways. For the parabolic and hyperbolic networks, we pretrain the networks on normal and pneumonia images and further use the obtained weights as the initializers for the networks to discriminate between COVID-19, pneumonia, tuberculosis, and normal aspects. For DenseNets, we apply transfer learning twice. First, the ImageNet pretrained weights are used to train on the CheXpert dataset, which includes 14 common radiological observations (e.g., lung opacity, cardiomegaly, fracture, support devices). Then, the weights are used to initialize the network which detects COVID-19 and the three other classes. The resulting networks are compared in terms of how well they adapt to the small number of COVID-19 images. According to our quantitative and qualitative analysis, the resulting networks are more reliable compared to those obtained by direct training on the targeted dataset.

Measurement of Angular Acceleration of a Rigid Body Using Linear Accelerometers

1975 ◽  
Vol 42 (3) ◽  
pp. 552-556 ◽  
Author(s):  
A. J. Padgaonkar ◽  
K. W. Krieger ◽  
A. I. King
Keyword(s):  
Experimental Data ◽  
Rigid Body ◽  
Simple Procedure ◽  
Three Dimensional ◽  
Angular Acceleration ◽  
Theory And Practice ◽  
Body Segments ◽  
Impact Biomechanics ◽  
The Stability ◽  
Definition Of

The computation of angular acceleration of a rigid body from measured linear accelerations is a simple procedure, based on well-known kinematic principles. It can be shown that, in theory, a minimum of six linear accelerometers are required for a complete definition of the kinematics of a rigid body. However, recent attempts in impact biomechanics to determine general three-dimensional motion of body segments were unsuccessful when only six accelerometers were used. This paper demonstrates the cause for this inconsistency between theory and practice and specifies the conditions under which the method fails. In addition, an alternate method based on a special nine-accelerometer configuration is proposed. The stability and superiority of this approach are shown by the use of hypothetical as well as experimental data.

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