Trivial Extension Research Articles

Jordan centralizer maps on trivial extension algebras

AbstractThe structure of Jordan centralizer maps is investigated on trivial extension algebras. One may obtain some conditions under which a Jordan centralizer map on a trivial extension algebra is a centralizer map. As an application, we characterize the Jordan centralizer map on a triangular algebra.

Rapidity-dependent eccentricity scaling in relativistic heavy-ion collisions

There is a well-established relation between the spatial asymmetry in the initial stage of a heavy-ion collision and the final momentum anisotropy, which allows for a separation of effects from initial conditions vs. later evolution and has proved exceptionally powerful. However, until recently it has only been studied in two dimensions — either through boost-invariant simulations or studying only quantities at mid-rapidity. We explore an extension to 3 dimensions, in order to determine whether a similar understanding can be obtained for the rapidity dependence of the collision system. In particular, we introduce rapidity-dependent eccentricities and investigate a trivial extension of the 2D eccentricity scaling of elliptic and triangular flow, as well as a way to systematically improve these initial-state estimators. We then explore the dependence of the resulting new response coefficients on shear viscosity and initial total energy.

Poisson Cohomology of Trivial Extension Algebras

Let B be the trivial extension of a Poisson algebra A by a left Poisson module M. Then there is a Poisson structure on B. We define a morphism of graded algebras $$\Theta : {\text {HP}}(B)\rightarrow {\text {HP}}(A)$$ from the Poisson cohomology ring of B to that of A. In the case that both A and M are finite dimensional as vector space, there is a long exact sequence computing the Poisson cohomology of B. We study the connecting homomorphism and then obtain a necessary and sufficient condition for each $$\Theta ^n$$ to be surjective. Finally, we study the kernel of $$\Theta ^1$$ explicitly.

The ϕ-Krull dimension of some commutative extensions

In this article, we define the ϕ-Krull dimension as a generalization of Krull dimension. This property is treated here as a part of a study on some constractions of rings such as direct products, amalgamation of rings, and trivial ring extensions.

When every finitely generated prime proper ideal is regular

In this paper we introduce and investigate a class of those rings in which every finitely generated prime proper ideal is regular. We establish the transfer of this notion to the trivial ring extension, direct product and homomorphic image, and then generate new and original families of rings satisfying this property.

Third kind elliptic integrals and 1-motives

In [4] we have showed that the Generalized Grothendieck's Period Conjecture applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental conjecture involving elliptic integrals of the first and second kind, and logarithms of complex numbers.In this paper we investigate the Generalized Grothendieck's Period Conjecture in the case of a 1-motive M whose underlying semi-abelian variety is a non trivial extension of a product of elliptic curves by a torus. This will imply the introduction of elliptic integrals of the third kind for the computation of the periods of M and therefore the Generalized Grothendieck's Period Conjecture applied to M will be equivalent to a transcendental conjecture involving elliptic integrals of the first, second and third kind.

On LPI rings

In this paper, we extend the $LPI$ property (that is, every locally principal ideal in an integral domain is invertible) to rings with zero-divisors and we study the class of commutative rings in which every regular locally principal ideal is invertible called $LPI$ rings. We investigate the stability of this property under homomorphic image, and its transfer to various contexts of constructions such as direct products, amalgamation of rings and trivial ring extensions. Our results generate examples which enrich the current literature with new and original families of rings that satisfy this property.

Actions of quantum linear spaces on quantum algebras

We study actions of bosonizations of quantum linear spaces on quantum algebras. Under mild conditions, we classify actions on quantum affine spaces and quantum matrix algebras. In the former case, it is shown that all actions of generalized Taft algebras are trivial extensions of actions on quantum planes. In both cases we achieve bounds on the rank of the bosonization acting on the algebra.

On stable range one property and strongly $$\pi $$-regular rings

In this paper, we study a rings with stable range one property, strongly $$\pi $$ -regular rings and rings satisfying $$\pi $$ -regular one stable range property. We study those properties in commutative rings, and investigate their transfer to various contexts of constructions such as pullbacks, trivial ring extensions and amalgamation of rings. Our results provide new classes of commutative rings satisfying those properties.

Homological dimension formulas for trivial extension algebras

This paper is the first one of a series of papers in which we study a finitely graded Iwanaga-Gorenstein (IG) algebras A=⨁i=0ℓAi and their representation theory. Every finitely graded algebra is graded Morita equivalent to a trivial extension algebra A=Λ⊕C with the grading deg⁡Λ=0,deg⁡C=1. Hence, in principle, representation theoretic study of finitely graded algebras is reduced to that of trivial extension algebras. The main result of this paper is a necessary and sufficient condition that a trivial extension algebra A=Λ⊕C to be an IG-algebra. We also give a description of the kernel Ker ϖ of a canonical functor ϖ:Db(modΛ)→SingZA. These results play important roles in the subsequent works.To this end, we study complexes of graded projective (injective) A-modules and establish formulas for the projective (injective) dimension of a graded A-modules in terms of Λ and C. As a byproducts, we obtain a modern expression of the classical formula for the global dimension of a trivial extension algebra by Palmer-Roos and Löfwall that is written in complicated classical derived functors.

On the Lie algebra structure of the first Hochschild cohomology of gentle algebras and Brauer graph algebras

In this paper we determine the first Hochschild homology and cohomology with different coefficients for gentle algebras and we give a geometrical interpretation of these (co)homologies using the ribbon graph of a gentle algebra as defined in [32]. We give an explicit description of the Lie algebra structure of the first Hochschild cohomology of gentle and Brauer graph algebras (with multiplicity one) based on trivial extensions of gentle algebras and we show how the Hochschild cohomology is encoded in the Brauer graph. In particular, we show that except in one low-dimensional case, the resulting Lie algebras are all solvable.

The absolutely Koszul and Backelin-Roos properties for spaces of quadrics of small codimension

Let k be a field, R a standard graded quadratic k-algebra with dimk⁡R2≤3, and let k‾ denote an algebraic closure of k. We construct a graded surjective Golod homomorphism φ:P→R⊗kk‾ such that P is a complete intersection of codimension at most 3. Furthermore, we show that R is absolutely Koszul (that is, every finitely generated R-module has finite linearity defect) if and only if R is Koszul if and only if R is not a trivial fiber extension of a standard graded k-algebra with Hilbert series (1+2t−2t3)(1−t)−1. In particular, we recover earlier results on the Koszul property of Backelin [4], Conca [7] and D'Alì [11].

Algebraic properties of expectation semirings

In this paper, we investigate the algebraic properties of the expectation semirings which are semiring version of the concept of trivial extension in ring theory. We discuss ideals, primes, maximals and primary ideals of these semirings. We also discuss the distinguished elements such as the units, idempotents, and zero-divisors of the expectations semirings. Similar to their counterparts in ring theory, we introduce pr\'{e}simplifiable, domainlike, clean, almost clean, and weakly clean semiring and see when an expectation semiring is one of these semirings.

On Divided Modules

Recall that a commutative ring R is said to be a divided ring if its each prime ideal P is comparable with each principal ideal (a), where $$a\in R.$$ In this paper, we extend the notion of divided rings to modules in two different ways: let R be a commutative ring with identity and M a unital R-module. Then M is said to be a divided (weakly divided) module if its each prime submodule N of M is comparable with each cyclic submodule Rm (rM) of M, where $$m\in M$$ ($$r\in R)$$. In addition to give many characterizations of divided modules, some topological properties of (quasi-) Zariski topology of divided modules are investigated. Also, we study the divided property of trivial extension $$R\ltimes M$$.

Radical factorization for trivial extensions and amalgamated duplication rings

Let [Formula: see text] be a commutative ring extension such that [Formula: see text] is a trivial extension of [Formula: see text] (denoted by [Formula: see text]) or an amalgamated duplication of [Formula: see text] along some ideal of [Formula: see text] (denoted by [Formula: see text]. This paper examines the transfer of AM-ring, N-ring, SSP-ring and SP-ring between [Formula: see text] and [Formula: see text]. We study the transfer of those properties to trivial ring extension. Call a special SSP-ring an SSP-ring of the following type: it is the trivial extension of [Formula: see text] by a C-module [Formula: see text], where [Formula: see text] is an SSP-ring, [Formula: see text] a von Neumann regular ring and [Formula: see text] a multiplication C-module. We show that every SSP-ring with finitely many minimal primes which is a trivial extension is in fact special. Furthermore, we study the transfer of the above properties to amalgamated duplication along an ideal with some extra hypothesis. Our results allows us to construct nontrivial and original examples of rings satisfying the above properties.

D-Representation-finite self-injective algebras

In this paper, we initiate the study of higher-dimensional Auslander–Reiten theory of self-injective algebras. We give a systematic construction of (weakly) d-representation-finite self-injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the Serre functor. The condition holds, in particular, for all fractionally Calabi-Yau algebras of global dimension at most d. This generalizes Riedtmann's classical construction of representation-finite self-injective algebras. Our method is based on an adaptation of Gabriel's covering theory for k-linear categories to the setting of higher-dimensional Auslander–Reiten theory.Applications include n-fold trivial extensions and (classical and higher) preprojective algebras, which are shown to be d-representation-finite in many cases. We also get a complete classification of all d-representation-finite self-injective Nakayama algebras for arbitrary d.

On the Wave Speed of Thermal Radiation Inside and Near the Boundary of an Absorbing Material

Abstract Planck's law describes thermal radiation into vacuum from a black body in thermal equilibrium. This law can be easily adapted to describe radiation into a transparent medium with a constant refractive index, and it admits a less trivial extension to radiation into a transparent medium with a nonconstant refractive index. However, this law cannot be straightforwardly generalized to describe thermal radiation into absorbing media and, in particular, to describe thermally exited electromagnetic fields inside the radiating body itself. We first analyze Planck's law and show why it cannot be straightforwardly extended to radiation into an absorbing medium. The derivation of this law relies on the assumption that a radiated field admits decomposition into normal modes, which cannot exist in absorbing media that are characterized by a complex-valued refractive index n=n′+in″, whose imaginary part describes the rate of energy dissipation. Correspondingly, the speed of electromagnetic waves in absorbing media c=c0/n, where c0 is the speed of light in vacuum, is also complex-valued, which suggests that the conventional concept of a complex valued wave speed is not suitable for modeling thermal radiation. We demonstrate that complex-valued wave speeds adequately describe waves that carry signals, such as radio waves and laser beams. Such waves decay because they pass some of their energy to the medium. The energy absorbed by the medium is eventually reradiated, but in studies focused on the transmission of signals, the reradiated fields are ignored as noise. In order to study thermal radiation in an absorbing material, one must treat the material and the radiation together as a closed system. The energy in such a system is conserved, and its distribution between the material and radiation does not change in time. This radiation admits decomposition into normal modes, which makes it possible to extend Planck's law to radiation into absorbing materials. This paper proposes a model of thermal radiation in an absorbing medium as a closed, energy conserving system. The radiation field has normal modes that correspond to an effective speed of wave propagation. Assuming that an absorbing material and the radiation in it are in thermal equilibrium, we show that deep inside the material, the average speed of photons is given by a frequency and temperature-dependent expression c∗=c0/(1+e−ℏω/κT). While this result is independent of the material, we further show that close to the boundary of the medium, the speed of thermal radiation depends in a complex way on the refractive index and the extinction coefficient of the material, as well as the direction of propagation and the distance from the material's surface.

On topological genericity of the mode-locking phenomenon

We study circle homeomorphisms extensions over a strictly ergodic homeomorphism. Under a very mild restriction, we show that the fibered rotation number is locally constant on an open and dense subset of all circle homeomorphisms extensions homotopic to the trivial extension. In the complement of this set, we find a dense subset in which every map is conjugate to a direct product. Our result provides a generalisation of Avila–Bochi–Damanik’s result on \({\mathrm{SL}}(2,{\mathbb {R}})\)-cocycles, and Jäger–Wang–Zhou’s result on quasi-periodically forced maps, to a broader setting.

On the Oseen Fundamental Solution and the Asymptotic Profile of Flows Past a Translating Object

The Oseen equations are associated with flow past an object which moves with small translational velocity $$\zeta \in {{\mathbb {R}}}^{d}$$ within a viscous liquid. In this paper, we are interested in the fundamental solution of the 2- and 3-d Oseen equations and its role in explaining the behavior at infinity of such fluid flows. After a review of the fundamental solutions of several differential operators, we obtain a general expression for the Oseen fundamental solution without making any restrictions on the magnitude and orientation of $$\zeta $$. The existence of a wake region and the anisotropic behavior of the fluid flow are then explained by simplified expressions and precise estimates of the velocity component. Using a trivial extension of the velocity and pressure to the whole space and the framework of tempered distributions, we also deduce the integral representation and the asymptotic expansion of weak solutions of the Oseen exterior problem.

A novel selective naïve Bayes algorithm

Naïve Bayes is one of the most popular data mining algorithms. Its efficiency comes from the assumption of attribute independence, although this might be violated in many real-world data sets. Many efforts have been done to mitigate the assumption, among which attribute selection is an important approach. However, conventional efforts to perform attribute selection in naïve Bayes suffer from heavy computational overhead. This paper proposes an efficient selective naïve Bayes algorithm, which adopts only some of the attributes to construct selective naïve Bayes models. These models are built in such a way that each one is a trivial extension of another. The most predictive selective naïve Bayes model can be selected by the measures of incremental leave-one-out cross validation. As a result, attributes can be selected by efficient model selection. Empirical results demonstrate that the selective naïve Bayes shows superior classification accuracy, yet at the same time maintains the simplicity and efficiency.