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We chose the intermediate timescale with hand collecting Baidu Index’s search volume to investigate the performance of the LSTM model on stock market volatility prediction. If we have high-frequency Baidu Index’s searching volume, we may also forecast high-frequency volatility, which is much more interesting and practi... |
. By maximizing the mutual information, we can find suitable input feature set, normalization window and observation interval. When we want to expand the usage scenario of the model, such as studying the volatility of individual stock or the volatility in different financial market, we could change the input or the str... |
5. Conclusion In this work, we have successfully demonstrated that the LSTM network is able to effectively extract meaningful information from noisy financial time series data. We consider the Baidu Index’s searching volume as proxy variables. Together with (or as) the market information, they shows the power of reflec... |
# Source: arxiv 1805.11982 # Title: Skew Poincaré-Birkhoff-Witt extensions over weak $Σ$-rigid rings # Sections: all # Downloaded: 2026-03-03T02:41:49.338363+00:00 |
Skew Poincaré-Birkhoff-Witt extensions over weak [MATH] -rigid rings Abstract In this paper we introduce the notion of weak [MATH] -rigid ring which extends [MATH] -rigid rings and [MATH] -rigid rings defined for Ore extensions and skew PBW extensions, respectively. We also present the notion of weak [MATH] -skew Armen... |
Key words and phrases. Rigid rings, Armendariz rings, skew PBW extensions. 2010 Mathematics Subject Classification: 16S36, 16T20, 16S30. |
Introduction In commutative algebra, a ring [MATH] is called Armendariz (the term was introduced by Rege and Chhawchharia in ), if whenever polynomials [MATH] [MATH] such that [MATH] , then [MATH] , for every [MATH] . The interest of this notion lies in its natural and its useful role in understanding the relation betw... |
, Lemma 1, Armendariz showed that a reduced ring (a ring has no nonzero nilpotent elements) always satisfies this condition. It is well known that reduced rings are abelian (i.e., every idempotent is central). Now, following |
, a ring [MATH] is called weak Armendariz , if whenever two polynomials [MATH] and [MATH] of the polynomial ring [MATH] satisfy [MATH] , then [MATH] is a nilpotent element of [MATH] , for each [MATH] |
In the context of Ore extensions introduced by Ore in , for [MATH] an endomorphism of a ring [MATH] , Hong et al. called [MATH] an [MATH] skew Armendariz ring , if for two elements [MATH] of the Ore extension of endomorphism type [MATH] [MATH] , for every [MATH] . As a generalization of the [MATH] -skew Armendariz ring... |
defined the weak [MATH] skew Armendariz rings in the following way: a ring [MATH] is said to be weak [MATH] skew Armendariz rings , if whenever two polynomials [MATH] of [MATH] [MATH] is a nilpotent element of [MATH] , for all [MATH] . It is clear that weak [MATH] -skew Armendariz rings are more general than weak Armen... |
On the other hand, following Krempa , an endomorphism [MATH] of a ring [MATH] is called rigid , if [MATH] , for [MATH] [MATH] is called [MATH] -rigid, if there exists a rigid endomorphism [MATH] of [MATH] . It is known that any rigid endomorphism of a ring is injective and [MATH] -rigid rings are reduced (see Hong et a... |
). Several properties of [MATH] -rigid rings have been established in the literature (c.f. , and see for detailed references). With this definition in mind, Ouyang |
defined weak [MATH] rigid rings which are a generalization of [MATH] -rigid rings. More precisely, if [MATH] is en endomorphism of a ring [MATH] [MATH] is said to be weak |
[MATH] rigid , if [MATH] , where [MATH] is the set of nilpotent elements of [MATH] . Ouyang , Proposition 2.2, showed that [MATH] is [MATH] -rigid if and only if [MATH] is weak [MATH] -rigid and reduced. In this way, weak [MATH] -rigid rings are a generalization of [MATH] -rigid rings deleting the condition to be reduc... |
With the aim of extending the above two notions introduced by Ouyang in to a more general setting than Ore extensions, in this paper we focus on the kind of noncommutative rings known in the literature as skew Poincaré-Birkhoff-Witt extensions (briefly, skew PBW extensions). These objects were introduced by Gallego and... |
and contain strictly Ore extensions of injective type (i.e., when [MATH] is an injective endomorphism of [MATH] ; see Example 2.4 for different noncommutative rings which are skew PBW extensions but they can not be expressed as Ore extensions). As a matter of fact, skew PBW extensions generalize several families of non... |
and for a detailed reference of every one of these families): (i) universal enveloping algebras of finite dimensional Lie algebras; (ii) PBW extensions introduced by Bell and Goodearl; (iii) almost normalizing extensions defined by McConnell and Robson; (iv) solvable polynomial rings introduced by Kandri-Rody and Weisp... |
and for a considerable list of examples of all these algebras). As we see, skew PBW extensions include a lot of noncommutative rings, which means that a theory extending the two notions above for these extensions will establish general results for a lot of noncommutative rings much more general than Ore extensions, and... |
and ). The paper is organized as follows: In Section we establish some useful results about skew PBW extensions for the rest of the paper. Section contains the first concept of the paper, the weak |
[MATH] rigid rings (Definition 3.2 ). These rings are a generalization of weak [MATH] -rigid rings introduced by Ouyang and [MATH] -rigid rings defined by the first author in |
. However, as we will see in Theorem 3.4 , weak [MATH] -rigid rings and [MATH] -rigid rings coincide when the ring is assumed to be reduced. Different results of are presented in this section. In Section we present the second concept of this paper, the weak |
[MATH] skew Armendariz rings (Definition ) which are a generalization of weak [MATH] -skew Armendariz rings defined by Ouyang and [MATH] -skew Armendariz rings introduced by the first author in |
. We prove that when [MATH] is a NI ring (a ring [MATH] is called NI ring, if the set [MATH] of nilpotent elements of [MATH] forms an ideal of [MATH] ), if [MATH] is weak [MATH] -rigid ring, then [MATH] is a weak [MATH] -skew Armendariz ring (Theorem 4.4 ). We present an example which illustrates the importance of the ... |
Throughout the paper, the word ring means a ring (not necessarily commutative) with unity. [MATH] will denote the field of complex numbers. |
Skew PBW extensions In this section we establish some useful results about skew PBW extensions for the rest of the paper. Definition 2.1 |
, Definition 1) Let [MATH] and [MATH] be rings. We say that [MATH] is a skew PBW extension over [MATH] , which is denoted by [MATH] , if the following conditions hold: |
(i) [MATH] (ii) there exist elements [MATH] such that [MATH] is a left free [MATH] -module, with basis [MATH] , and [MATH] (iii) |
For each [MATH] and any [MATH] , there exists an element [MATH] such that [MATH] (iv) For any elements [MATH] , there exists [MATH] such that [MATH] |
Proposition 2.2 , Proposition 3) Let [MATH] be a skew PBW extension over [MATH] . For each [MATH] , there exist an injective endomorphism [MATH] and an [MATH] -derivation [MATH] such that [MATH] , for each [MATH] . From now on, we write [MATH] , and [MATH] |
Definition 2.3 , Definition 4) Let [MATH] be a skew PBW extension over [MATH] (i) [MATH] is called quasi-commutative if the conditions (iii) and (iv) in Definition 2.1 are replaced by the following: (iii’) for each [MATH] and all [MATH] , there exists [MATH] such that [MATH] ; (iv’) for any [MATH] , there exists [MATH]... |
(ii) [MATH] is called bijective , if [MATH] is bijective for each [MATH] , and [MATH] is invertible, for any [MATH] (iii) [MATH] is called of endomorphism type , if [MATH] , for every [MATH] . In addition, if every [MATH] is bijective, [MATH] is a skew PBW extension of automorphism type |
Examples 2.4 If [MATH] is an iterated Ore extension where [MATH] is injective, for [MATH] [MATH] [MATH] , for every [MATH] and [MATH] |
[MATH] , for [MATH] , and [MATH] , where [MATH] has a left inverse; [MATH] , for [MATH] then [MATH] , p. 1212). Note that skew PBW extensions of endomorphism type are more general than iterated Ore extensions [MATH] , and in general, skew PBW extensions are more general than Ore extensions of injective type. More preci... |
or for the reference of every example). Examples of these extensions appearing in noncommutative algebraic geometry and theoretical physics can be found in |
and (a) Let [MATH] be a commutative ring and [MATH] a finite dimensional Lie algebra over [MATH] with basis [MATH] . The universal enveloping algebra of [MATH] , denoted [MATH] , is a skew PBW extension over [MATH] , since [MATH] [MATH] [MATH] , for [MATH] . In particular, the universal enveloping algebra |
of a Kac-Moody Lie algebra is a skew PBW extension over a polynomial ring. (b) The universal enveloping ring [MATH] , where [MATH] is a [MATH] -algebra, and [MATH] is a [MATH] -vector space which is also a Lie ring containing [MATH] and [MATH] as Lie ideals with suitable relations. The enveloping ring [MATH] is a finit... |
(c) Let [MATH] [MATH] [MATH] and [MATH] be as in the previous example; let [MATH] be a [MATH] -algebra containing [MATH] . The tensor product |
[MATH] is a skew PBW extension over [MATH] , and it is a particular case of crossed product [MATH] of [MATH] by [MATH] , which is a skew PBW extension over [MATH] |
(d) The twisted or smash product differential operator ring [MATH] , where [MATH] is a finite-dimensional Lie algebra acting on [MATH] by derivations, and [MATH] is Lie 2-cocycle with values in [MATH] |
(e) Diffusion algebras arise in physics as a possible way to understand a large class of [MATH] -dimensional stochastic process. A diffusion algebra |
[MATH] with parameters [MATH] for [MATH] , is an algebra over [MATH] [MATH] subject to relations [MATH] , whenever [MATH] [MATH] for all [MATH] [MATH] admits a [MATH] -basis of standard monomials [MATH] , that is, [MATH] is a diffusion algebra if these standard monomials are a [MATH] -vector space basis for [MATH] . Fr... |
Definition 2.5 , Definition 6) Let [MATH] be a skew PBW extension over [MATH] . Then: (i) for [MATH] [MATH] [MATH] . If [MATH] , then |
[MATH] (ii) For [MATH] [MATH] [MATH] , and [MATH] . The symbol [MATH] will denote a total order defined on [MATH] (a total order on [MATH] ). For an element [MATH] [MATH] . If |
[MATH] but [MATH] , we write [MATH] . Every element [MATH] can be expressed uniquely as [MATH] , with [MATH] , and [MATH] (eventually, we will use expressions as [MATH] , with [MATH] , and [MATH] ). With this notation, we define [MATH] , the leading monomial of [MATH] [MATH] , the leading coefficient of [MATH] [MATH] ,... |
[MATH] . Note that [MATH] . Finally, if [MATH] , then [MATH] [MATH] [MATH] . We also consider [MATH] for any [MATH] . For a detailed description of monomial orders in skew PBW extensions, see |
, Section 3. Proposition 2.6 , Proposition 2.9) If [MATH] and [MATH] is an element of [MATH] , then [EQUATION] Remark 2.7 , Remark 2.10)) |
About Proposition 2.6 , we have the following observation: If [MATH] and [MATH] , then when we compute every summand of [MATH] we obtain products of the coefficient [MATH] with several evaluations of [MATH] in [MATH] ’s and [MATH] ’s depending of the coordinates of [MATH] . This assertion follows from the expression: |
[EQUATION] Weak [MATH] -rigid rings For a ring [MATH] with a ring endomorphism [MATH] , an [MATH] -derivation [MATH] , considering the Ore extension [MATH] , Krempa in |
defined [MATH] as a rigid endomorphism if [MATH] implies [MATH] , for [MATH] . Krempa called [MATH] [MATH] -rigid if there exists a rigid endomorphism [MATH] of [MATH] . Since Ore extensions of injective type are particular examples of skew PBW extensions, the first author introduced the following definition with the p... |
Definition 3.1 , Definition 3.2) Let [MATH] be a ring and [MATH] a family of endomorphisms of [MATH] [MATH] is called a rigid endomorphisms family if [MATH] implies [MATH] , for every [MATH] and [MATH] . A ring [MATH] is called to be [MATH] rigid if there exists a rigid endomorphisms family [MATH] of [MATH] |
Note that if [MATH] is a rigid endomorphisms family, then every element [MATH] is a monomorphism. In fact, [MATH] -rigid rings are reduced rings: if [MATH] is a [MATH] -rigid ring and [MATH] for [MATH] , then [MATH] , i.e., [MATH] and so [MATH] , that is, [MATH] is reduced (note that there exists an endomorphism of a r... |
, Example 9). With this in mind, we consider the family of injective endomorphisms [MATH] and the family [MATH] of [MATH] -derivations in a skew PBW extension [MATH] over a ring [MATH] (see Proposition 2.2 ). Remarkable examples of [MATH] -rigid rings can be found in |
, Examples 3.3, , Examples 2.9 or , Example 2. Now, following the ideas presented by Ouyang for Ore extensions, we present the following definition which extends [MATH] -rigid rings. |
Definition 3.2 Let [MATH] and [MATH] be a family of endomorphisms and [MATH] -derivations of [MATH] , respectively. [MATH] is called a weak |
[MATH] rigid ring , if [MATH] , for each element [MATH] and every [MATH] Remark 3.3 It is clear that [MATH] -rigid rings are weak [MATH] -rigid. However, the converse is false as we can appreciated in the following example taken from |
, Example 2.1. Let [MATH] be an endomorphism of a ring [MATH] which is an [MATH] -rigid ring. Consider the ring [EQUATION] If we extend the endomorphism [MATH] of [MATH] to the endomorphism [MATH] defined by [MATH] , then [MATH] is a weak [MATH] -rigid ring but [MATH] is not [MATH] -rigid. Therefore, weak [MATH] -rigid... |
The next theorem gives an equivalence between the notions of [MATH] -rigid rings and weak [MATH] -rigid rings. This result extends |
, Proposition 2.2. Theorem 3.4 Let [MATH] and [MATH] be a family of endomorphisms and [MATH] -derivations of [MATH] , respectively. [MATH] is [MATH] -rigid if and only if [MATH] is weak [MATH] -rigid and reduced. |
Proof. Suppose that [MATH] is [MATH] -rigid. As we saw above, [MATH] is reduced. Let us see that [MATH] is weak [MATH] -rigid. If [MATH] , then [MATH] , since [MATH] is reduced, whence [MATH] , for all [MATH] and [MATH] . Now, if [MATH] , for [MATH] and every [MATH] , then [MATH] , for all [MATH] , since [MATH] is redu... |
Conversely, suppose that [MATH] is weak [MATH] -rigid and reduced, and let [MATH] , for [MATH] and [MATH] . Then [MATH] , since [MATH] is weak [MATH] -rigid, and so [MATH] because [MATH] is reduced. Therefore [MATH] is [MATH] -rigid. |
The next proposition extends , Proposition 2.3 (compare also with , Lemma 3.3). Proposition 3.5 If [MATH] is a NI ring which is weak [MATH] -rigid, then we have the following assertions: |
(1) If [MATH] , then [MATH] , for every elements [MATH] (2) If [MATH] , for some element [MATH] , then [MATH] (3) If [MATH] , for some element [MATH] , then [MATH] |
Proof. (1) Let [MATH] . Using that [MATH] , for every [MATH] , where [MATH] is an ideal of [MATH] , we obtain [MATH] which shows that [MATH] whence [MATH] , for all [MATH] . If we consider repeatedly this argument, then we obtain that [MATH] , for all [MATH] . In a similar way, if [MATH] , then [MATH] , and so [MATH] w... |
(2) Suppose that [MATH] , for some element [MATH] . We have [MATH] , by part (1). Since [MATH] is an ideal of [MATH] [MATH] , whence we obtain [MATH] by definition of weak [MATH] -rigid ring. Continuing in this way we can prove that [MATH] . Again, continuing this procedure we can see that [MATH] |
(3) The proof uses a similar argument to the considered in part (2). The next proposition generalizes , Proposition 2.4 (compare also with |
, Proposition 3.5). Proposition 3.6 If [MATH] is a NI and weak [MATH] -rigid ring, then [MATH] , for every central idempotent element [MATH] |
Proof. Consider [MATH] a central idempotent of [MATH] . It is clear that [MATH] . By Proposition 3.5 (1) we obtain [MATH] , for [MATH] . This means that there exists some positive integer [MATH] such that [MATH] (for a fixed [MATH] ). In this way [MATH] , for all [MATH] . In a similar way, [MATH] , and so [MATH] , when... |
With the aim of establishing the following proposition, an ideal [MATH] of [MATH] will be called weak [MATH] -rigid, if [MATH] , for every element [MATH] and each [MATH] . Our Proposition 3.7 extends |
, Proposition 2.5. Proposition 3.7 If [MATH] is an abelian ring with [MATH] [MATH] , for every idempotent element [MATH] of [MATH] , then the following assertions are equivalent: |
(1) [MATH] is weak [MATH] -rigid. (2) [MATH] and [MATH] are weak [MATH] -rigid ideals. Proof. [MATH] It is clear since [MATH] and [MATH] are subrings of [MATH] |
[MATH] Let [MATH] be a nilpotent element of [MATH] . Then [MATH] . Having in mind that [MATH] and [MATH] are weak [MATH] -rigid, there exist positive integers [MATH] with [MATH] and [MATH] , for a fixed [MATH] . If we take [MATH] , then [MATH] . Therefore [MATH] , that is, [MATH] , for all [MATH] |
Conversely, suppose that [MATH] , for [MATH] . Then [MATH] and [MATH] . So, [MATH] and [MATH] , since [MATH] and [MATH] are weak [MATH] -rigid ideals. Hence [MATH] , that is, [MATH] is weak [MATH] -rigid. |
Weak [MATH] -skew Armendariz rings In the literature we find the following notions about Armendariz rings in commutative and noncommutative case concerning Ore extensions. |
Definition 4.1 (i) , Definition 2.1) A ring [MATH] is called weak Armendariz , if whenever polynomials [MATH] and [MATH] satisfy [MATH] , then [MATH] , for each [MATH] |
(ii) , p. 104) [MATH] is called [MATH] skew Armendariz , if whenever [MATH] and [MATH] with [MATH] , then [MATH] , for every [MATH] |
(iii) , p. 110) [MATH] is called weak [MATH] -skew Armendariz, if whenever [MATH] and [MATH] satisfy [MATH] , then [MATH] In the context of skew PBW extensions, the authors have defined the following Armendariz notions: |
Definition 4.2 Let [MATH] be a skew PBW extension over a ring [MATH] . Then: (i) , Definition 3.4) [MATH] is called a [MATH] skew Armendariz ring , if whenever [MATH] [MATH] with [MATH] , then [MATH] , for every value of [MATH] and [MATH] |
(ii) , Definition 3.1) [MATH] is called a [MATH] skew Armendariz ring , if for elements [MATH] and [MATH] in [MATH] , the equality [MATH] implies [MATH] , for all [MATH] and [MATH] , where [MATH] |
(iii) , Definition 4.1) [MATH] is a skew-Armendariz ring, if for polynomials [MATH] and [MATH] in [MATH] [MATH] implies [MATH] , for each [MATH] |
(iv) , Definition 3.1) [MATH] is called a skew [MATH] Armendariz ring , if for elements [MATH] of [MATH] [MATH] implies that [MATH] , for every [MATH] and [MATH] |
Several relations about these four notions of Armendariz rings for coefficient rings of skew PBW extensions can be found in , Section 3, |
, Sections 3 and 4, and , Section 3. Now, with the aim of extending Definition 4.1 (iii) from Ore extensions of endomorphism type to skew PBW extensions of endomorphism type (which are more general, see Examples 2.4 ), and also [MATH] -skew Armendariz rings defined by the first author in |
, Definition 3.1, (at least in the endomorphism case), we present the following definition. Definition 4.3 Let [MATH] be a skew PBW extension of endomorphism type over a ring [MATH] [MATH] is called a weak |
[MATH] skew Armendariz ring , if for elements [MATH] and [MATH] in [MATH] , the equality [MATH] implies [MATH] , for all [MATH] and [MATH] , where [MATH] |
The following theorem extends , Theorem 3.3. We need to assume that the elements [MATH] in Definition 3.5 (iv) are both central and invertible in [MATH] . We denote [MATH] |
Theorem 4.4 If [MATH] is a NI and weak [MATH] -rigid ring, then [MATH] is a weak [MATH] -skew Armendariz ring. Proof. Suppose that [MATH] , where [MATH] and [MATH] , with the monomial order [MATH] and [MATH] , respectively (Definition 2.5 (ii)). Since [MATH] , then [MATH] . By assumption, the elements [MATH] (Definitio... |
[EQUATION] By assumption we know that [MATH] , for [MATH] . So, Proposition 3.5 (1) guarantees that the product [EQUATION] is an element of [MATH] . Proposition 3.5 guarantees that [MATH] is also an element of [MATH] . In this way, multiplying ( 4.1 ) by [MATH] , and using the fact that the elements [MATH] in Definitio... |
[EQUATION] whence, [MATH] . Since [MATH] and [MATH] , then [MATH] , so [MATH] whence [MATH] by Proposition 3.5 . Therefore, we now have to study the expression ( 4.1 ) for [MATH] and [MATH] . If we multiply ( 4.2 ) by [MATH] we obtain |
[EQUATION] Using a similar reasoning as above, we can see that [MATH] , and using the assumptions on the elements [MATH] . Now, since [MATH] , and [MATH] , Proposition 3.5 imply that [MATH] . Continuing in this way, we prove that [MATH] , for [MATH] . Therefore [MATH] , for [MATH] and [MATH] |
Remark 4.5 The importance of the condition NI on [MATH] in Theorem 4.4 can be appreciated in the following example taken from , Example 3.4, which presents a noncommutative ring which is weak [MATH] -rigid but not weak [MATH] -skew Armendariz. Let [MATH] be a ring and [MATH] be the [MATH] matrix ring over [MATH] . Let |
[EQUATION] It is clear that [MATH] is a ring with usual matrix operations. If we consider the endomorphism [MATH] defined by [EQUATION] |
then [MATH] is weak [MATH] -rigid but not weak [MATH] -skew Armendariz. Future work Having in mind that [MATH] -rigid rings have been studied in several papers concerning ring theoretical properties such as Armendariz, Baer, quasi-Baer, p.p. and p.q.-Baer rings, zip, McCoy, invariant ideals, ascending chain condition o... |
and ), there is a considerable number of results about [MATH] -rigid rings which can be extended to the more general setting of weak [MATH] -rigid rings. This will be our line of thinking in future papers. |
Acknowledgements The first author was supported by the research fund of Facultad de Ciencias, Universidad Nacional de Colombia, Bogotá, Colombia, HERMES CODE 41535. |
# Source: arxiv 1805.12026 # Title: Projection operators in the Weihrauch lattice # Sections: all # Downloaded: 2026-03-03T02:36:43.096835+00:00 |
Projection Operators in the Weihrauch lattice Abstract. In this paper we study, for [MATH] , the projection operators over [MATH] , that is the multi-valued functions that associate to [MATH] |
and [MATH] closed, the points of [MATH] which are closest to [MATH] We also deal with approximate projections, where we content ourselves with points of [MATH] which are almost the closest to [MATH] . We use the tools of Weihrauch reducibility to classify these operators depending on the representation of [MATH] and th... |
Marcone’s research partially supported by PRIN 2012 Grant “Logica, Modelli e Insiemi” and by the departmental PRID funding “HiWei — The higher levels of the Weihrauch hierarchy”. |
1. Introduction Projecting a point over a non-empty subset of a Euclidean space is an operation deeply grounded in our geometrical intuition of the spatial continuum and has many important applications in higher mathematics. More precisely, given [MATH] and [MATH] we seek [MATH] such that [MATH] (when [MATH] is closed,... |
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