Regular Elements Research Articles

The monoid of regular elements in commutative rings with zero divisors

Let R be a commutative ring with identity, be the multiplicative monoid of regular elements in R, t be the so-called t-operation on R or A Marot ring is a ring whose regular ideals are generated by their regular elements. Marot rings were introduced by J. Marot in 1969 and have been playing a key role in the study of rings with zero divisors. The notion of Marot rings can be extended to t-Marot rings such that Marot rings are t-Marot rings. In this paper, we study some ideal-theoretic relationships between a t-Marot ring R and the monoid We first construct an example of a t-Marot ring that is not Marot. This also serves as an example of a rank-one DVR of reg-dimension Let R be a t-Marot ring, t- (resp., t-) be the set of regular prime t-ideals of R (resp., the set of non-empty prime t-ideals of ), and be the class group of A for A = R or Then, among other things, we prove that the map -- given by is bijective; and R is a factorial ring if and only if is a factorial monoid.

Geometrical constructions of equilibrium states

In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic systems—center isometries—that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying on symbolic dynamics) and even provide new information in the classical case.

Asymptotics of the powers in finite reductive groups

AbstractLet 𝐺 be a connected reductive group defined overFq\mathbb{F}_{q}. Fix an integerM≥2M\geq 2, and consider the power mapx↦xMx\mapsto x^{M}on 𝐺. We denote the image ofG⁢(Fq)G(\mathbb{F}_{q})under this map byG⁢(Fq)MG(\mathbb{F}_{q})^{M}and estimate what proportion of regular semisimple, semisimple and regular elements ofG⁢(Fq)G(\mathbb{F}_{q})it contains. We prove that, asq→∞q\to\infty, the set of limits for each of these proportions is the same and provide a formula. This generalizes the well-known results forM=1M=1where all the limits take the same value 1. We also compute this more explicitly for the groupsGL⁢(n,q)\mathrm{GL}(n,q)andU⁢(n,q)\mathrm{U}(n,q)and show that the set of limits are the same for these two group, in fact, in bijection underq↦-qq\mapsto-qfor a fixed 𝑀.

“The Human Was Created Out of Haste.” On Prophecy and the Problem of Human Nature in the Qur’an

In this article it is argued that the Qur’an’s doctrine of divine mercy is best understood in light of its pessimistic anthropology, an aspect of the text that is often underappreciated. The so-called “primordial covenant” verse (Q 7:172) of the Qur’an holds humans responsible for submission to God. The Qur’anic language on “signs” in the natural world suggests that humans should recognize God (and be grateful to Him) by reflection on nature alone. Yet, according to the Qur’an they do not. The Qur’an refers frequently to humans as “ungrateful” and “hasty”. It also makes divine punishments a regular element of human history, suggesting that rebellion is endemic to human nature. It is, I argue, precisely the rebelliousness of humans that makes God’s initiative in sending prophets merciful. The ministry of prophets in the Qur’an is an unmerited manifestation of divine compassion for a sinful humanity.

N-chained semigroups and -perspective modules and rings

We introduce and study a new class of modules and rings we call -perspective, which can either be described in terms of perspective direct summands, associate idempotents, or generalized inverses. When n is small (), we recover existing class of modules and rings: (endo)abelian, strongly IC, and perspective ones. And 3/2-perspective rings are characterized by all their regular elements being special clean. Standard constructions are also discussed and examples are provided.

On quasi Steinberg characters of symmetric and alternating groups and their double covers

An irreducible character of a finite group [Formula: see text] is called quasi [Formula: see text]-Steinberg character for a prime [Formula: see text] if it takes a nonzero value on every [Formula: see text]-regular element of [Formula: see text]. In this paper, we classify the quasi [Formula: see text]-Steinberg characters of Symmetric ([Formula: see text]) and Alternating ([Formula: see text]) groups and their double covers. In particular, an existence of a nonlinear quasi [Formula: see text]-Steinberg character of [Formula: see text] implies [Formula: see text] and of [Formula: see text] implies [Formula: see text].

Perspectivity and von Neumann regularity

We investigate connections between von Neumann regularity of endomorphisms and perspectivity of direct summands in modules. This leads to a new classification of those rings whose regular elements are strongly regular, which turn out to be exactly the rings R whose idempotents are central modulo the Jacobson radical J(R). An important component of our work is an investigation of the left and right associate relations on idempotents, as well as chains of these relations. As applications we give new characterizations of strongly regular elements and of idempotents that are central modulo the Jacobson radical. We also introduce a new class of regular elements that we call pc-regular elements, related to perspectivity in complement summands. These pc-regular elements are exactly the special clean elements. Generalizing the well-known fact that unit-regular rings are special clean, we then show that the unit-regular elements of any regular ring satisfying general comparability are special clean. Consequently, unit-regular endomorphisms of quasi-continuous modules are special clean, answering, in the positive, a conjecture of Lam.

Inverse complements and strongly unit regular elements

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

Equivalent generating pairs of an ideal of a commutative ring

Let R be a commutative ring with identity and let I be a two-generated ideal of R. We denote by SL2(R) the group of 2×2 matrices over R with determinant 1. We study the action of SL2(R) by matrix right-multiplication on V2(I), the set of generating pairs of I. Let Fitt1(I) be the second Fitting ideal of I. Our main result asserts that V2(I)/SL2(R) identifies with a group of units of R/Fitt1(I) via a natural generalization of the determinant if I can be generated by two regular elements. This result is illustrated in several Bass rings for which we also show that SLn(R) acts transitively on Vn(I) for every n>2. As an application, we derive a formula for the number of cusps of a modular group over a quadratic order.

NMR assignments and secondary structure distribution of emfourin, a novel proteinaceous protease inhibitor.

Emfourin (M4in) from Serratia proteamaculans is a new proteinaceous inhibitor of protealysin-like proteases (PLPs), a subgroup of the well-known and widely represented metallopeptidase M4 family. Although the biological role of PLPs is debatable, data published indicate their involvement in pathogenesis, including bacterial invasion into eukaryotic cells, suppression of immune defense of some animals, and destruction of plant cell walls. Gene colocalization into a bicistronic operon observed for some PLPs and their inhibitors (as in the case of M4in) implies a mutually consistent functioning of both entities. The originality of the amino acid sequence of M4in suggests it belongs to a previously unknown protein family and this encourages structural studies. In this work, we report a near-complete assignment of 1H, 13C, and 15N resonances of recombinant M4in and its structural-dynamic properties derived from the chemical shifts. According the NMR data analysis, the M4in molecule comprises 3-5 helical elements and 4-6 β-strands, at least two of which are apparently antiparallel, ascribing this obviously globular protein to the α + β structural class. Besides, two disordered regions also exist in the central loops between the regular secondary structural elements. The obtained data provide the basis for determining the high-resolution structure as well as functioning mechanism of M4in that can be used for development of new antibacterial therapeutic strategies.

Cohomological representations for real reductive groups

For a connected reductive group $G$ over ${\mathbb R}$, we study cohomological $A$-parameters, which are Arthur parameters with the infinitesimal character of a finite-dimensional representation of $G({\mathbb C})$. We prove a structure theorem for such $A$-parameters, and deduce from it that a morphism of $L$-groups which takes a regular unipotent element to a regular unipotent element respects cohomological $A$-parameters. This is used to give complete understanding of cohomological $A$-parameters for all classical groups. We review the parametrization of Adams-Johnson packets of cohomological representations of $G({\mathbb R})$ by cohomological $A$-parameters and discuss various examples. We prove that the sum of the ranks of cohomology groups in a packet on any real group (and with any infinitesimal character) is independent of the packet under consideration, and can be explicitly calculated. This result has a particularly nice form when summed over all pure inner forms.

Decomposition of exterior and symmetric squares in characteristic two

Let V be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element e∈sl(V), we describe the Jordan normal form of e on the sl(V)-modules ∧2(V) and S2(V). In the case where e is a regular nilpotent element, we are able to give a closed formula.We also consider the closely related problem of describing, for every unipotent element u∈SL(V), the Jordan normal form of u on ∧2(V) and S2(V). A recursive formula for the Jordan block sizes of u on ∧2(V) was given by Gow and Laffey [J. Group Theory 9 (2006), 659–672]. We show that their proof can be adapted to give a similar formula for the Jordan block sizes of u on S2(V).

Evaluating the Voice Type Component Distributions of Excised Larynx Phonations at Three Subglottal Pressures.

Purpose The excised canine larynx provides an advantageous experimental framework in the study of voice physiology. In recent years, signal processing methods have been applied to analyze phonations in excised canine larynx experiments. However, phonations have a highly complex and nonstationary nature corresponding to different proportions of regular and chaotic signal elements. Current nonlinear dynamic methods that are used to assess the degree of irregularity in the voice fail to recognize the distribution of voice type components (VTCs). Method Based on measures of intrinsic dimension, this article presents a method to analyze the VTC distribution of phonations in excised canine larynx experiments. Thirty-nine phonation samples from 13 excised canine larynges at three different subglottal pressures were analyzed. Results Phonation produced with subglottal pressures above phonation instability pressure (PIP) and below phonation threshold pressure (PTP) resulted in high proportions of Voice Types 3 and 4, characterized by chaotic and noisy signals. Phonation produced with pressure between PTP and PIP contained mostly Type 1 voice, characterized by a regular and nearly periodic signal. Mean proportions of all VTCs varied significantly in comparisons of phonations produced with Sub-PTP and PTP as well as in comparisons of phonations produced with PTP and PIP. Conclusions Across all VTCs, the VTC profiles of normal and abnormal phonation differ significantly. Normal phonation is strongly associated with VTC1 (Voice Type Component 1), whereas abnormal phonation exhibits increased VTC4 (Voice Type Component 4). The study further demonstrates the ability of intrinsic dimension to successfully detect multiple voice types in an acoustic signal and highlights the need for expanded use of intrinsic dimension in human voice. Supplemental Material https://doi.org/10.23641/asha.14417585.

Coupled Thermo-Hydro-Mechanical Numerical Modeling of Evolving Fractures in Rocks

We present a numerical technique capable of handling evolving fractures in rocks triggered by coupled thermo-hydro-mechanical (THM) phenomena. The approach is formulated in the context of the finite-element method (FEM) and consists in introducing especial (high-aspect ratio) finite elements in-between the regular (bulk) finite elements. We called this method the mesh fragmentation technique (MFT). The MFT has been successfully used to model mechanical and hydro-mechanical problems related to drying cracks in soils, fractures in concrete, and hydraulic fractures in rocks. In this paper, we extend the MFT for tackling non-isothermal problems in porous media. We present the main components of the mathematical formulation together with its implementation in a fully coupled THM computer code. The proposed method is verified and validated using available analytical, experimental, and numerical results. A very satisfactory performance of the proposed method is observed in all the analyzed cases. These results are encouraging and show the potential of the MFT to tackle THM applications involving fractured rocks. A clear advantage of the proposed framework is that it can be easily implemented in existing numerical FEM codes for continuous porous media to upgrade them to tackle THM engineering problems with evolving discontinuities.

Turbulent boundary-layer flow over regular multiscale roughness

Abstract

On commutator fibers over regular semisimple and central elements of SLn

We study fibers of the commutator map on general linear maps using representation theory. Especially, we give detailed information on dimension and geometric connectedness of those fibers over regular semisimple and central elements.

Left uniquely generated elements in rings

An element a in a ring R is left uniquely generated (or left UG) if, whenever Ra = Rb, a = ub for a unit u in R, and is left strongly UG if, whenever Ra = Rb, a = ub = bu for a unit u in R. A ring is a left UG (resp. left strongly UG) ring if each of its elements is left UG (resp. left strongly UG). Motivated by a result of Marks [15] that a (von Neumann) regular ring is left UG iff it is unit-regular, we establish characterizations of a regular left UG element and, respectively, a regular left strongly UG element. As applications, many notable consequences are presented and connections with commutativity of rings are discussed.

A study of non-commutative Von-Neumann regular rings

In this paper, we study the Von-Neumann regular elements of non-commutative rings M2(ℤ2) and M2(ℤ3). We prove that M2(ℤ2) and M3(ℤ3) are Von-Neumann rings. We also give code in C++to compute Von-Neumann regular elements of M2(ℤn).

Green’s Relations on Regular Elements of Semigroup of Relational Hypersubstitutions for Algebraic Systems of Type ((m), (n))

Any relational hypersubstitution for algebraic systems of type (τ,τ′) = ((mi)i∈I,(nj)j∈J) is a mapping which maps any mi-ary operation symbol to an mi-ary term and maps any nj - ary relational symbol to an nj-ary relational term preserving arities, where I,J are indexed sets. Some algebraic properties of the monoid of all relational hypersubstitutions for algebraic systems of a special type, especially the characterization of its order and the set of all regular elements, were first studied by Phusanga and Koppitz[13] in 2018. In this paper, we study the Green’srelationsontheregularpartofthismonoidofaparticulartype(τ,τ′) = ((m),(n)), where m, n ≥ 2.

Some results on semigroups of transformations with restricted range

Abstract Let X X be a non-empty set and Y Y a non-empty subset of X X . Denote the full transformation semigroup on X X by T ( X ) T\left(X) and write f ( X ) = { f ( x ) ∣ x ∈ X } f\left(X)=\{f\left(x)| x\in X\} for each f ∈ T ( X ) f\in T\left(X) . It is well known that T ( X , Y ) = { f ∈ T ( X ) ∣ f ( X ) ⊆ Y } T\left(X,Y)=\{f\in T\left(X)| f\left(X)\subseteq Y\} is a subsemigroup of T ( X ) T\left(X) and R T ( X , Y ) RT\left(X,Y) , the set of all regular elements of T ( X , Y ) T\left(X,Y) , also forms a subsemigroup of T ( X , Y ) T\left(X,Y) . Green’s ∗ \ast -relations and Green’s ∼ <mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> \sim </mml:mpadded>\hspace{0.08em} -relations (with respect to a non-empty subset U U of the set of idempotents) were introduced by Fountain in 1979 and Lawson in 1991, respectively. In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup T ( X , Y ) T\left(X,Y) . This investigation proves that the semigroup T ( X , Y ) T\left(X,Y) is always a left Ehresmann semigroup, and R T ( X , Y ) RT\left(X,Y) is orthodox (resp. completely regular) if and only if Y Y contains at most two elements.