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In this paper, we study multiwindow discrete Gabor $(M-D-G)$ systems $\mathcal{G}(g,L,M,N)$ on discrete periodic sets $\mathbb{S}$ and give some necessary and/or sufficient matrix-conditions for a $M-D-G$ system in $\ell^2(\mathbb{S})$ to be a frame. We characterize, also, which $M-D-G$ frames are Riesz bases by the parameters $L$, $M$ and $N$. Matrix-characterizations of $M-D-G$ Parseval frames and $M-D-G$ orthonormal bases are also given. Then, we characterize the existence of $M-D-G$ frames, $M-D-G$ Parseval frames, $M-D-G$ Riesz bases and $M-D-G$ orthonormal bases for $\ell^2(\mathbb{Z})$ by the parameters $M$, $N$ and $L$. We present, also, a matrix-characterization of dual $M-D-G$ frames in $\ell^2(\mathbb{S})$. A perturbation matrix-condition of $M-D-G$ frames is also prsented. We, then, show that a pair of $M-D-G$ Bessel systems can generate pairs of M-D-G dual frames. By the Zak-transform, characterizations of complete M-D-G systems and M-D-G frames in $\ell^2(\mathbb{Z})$ are given in the case of $M=N$ and necessary conditions for a M-D-G system to be a Riesz basis/ orthonormal basis for $\ell^2(\mathbb{Z})$ are also given. We, also, study M-D-G $K$-frames in $\ell^2(\mathbb{S})$, where $K\in B(\ell^2(\mathbb{S})\,)$, and presente some sufficient matrix-conditions for a M-D-G system to form a K-frame and give a construction method of M-D-G $K$-frames which are not M-D-G frames and some examples.

The class of $N$-Koszul graded algebras of finite global dimension has gained lots of attention in recent years, especially in the study of Artin-Schelter regular algebras. While structurally rich and concrete, the only known examples of such algebras are either when $N = 2$, i.e. the algebra is Koszul, or when $N = 3$. Under a mild Hilbert series assumption, we rule out the existence of $N$-Koszul graded algebras of finite global dimension for $N$ not prime. Furthermore, we establish strong restrictions on the global dimension of such algebras. This suggests that perhaps the existence of 3-Koszul algebras with finite global dimension and `nice' Hilbert series is an anomaly.

In this work, we provide some constructions and the sum of new continuous K-g-frames in Hilbert$C^{\ast}$-Modules. We provide certain necessary and sufficient conditions for some adjointable operators on $\mathcal{H}$, under which new continuous K-g-frames can be retrieved from those that already exist. Additionally, we discuss the sum of continuous K-g-frames, discover some of their characterizations, and offer some adjointable operators to construct new continuous K-g-frames from the previous ones.

We introduce the "test space and pairing" idea for frames and apply it to the $\ell^2$ and $L^2$ spaces. First, we show that for every $J \neq \emptyset$, the notions of a classical $I$-frame with values in $H$ and a $J$-extended classical $I$-frame with values in $H$ are the same. The definition of a $J$-extended classical $I$-frame with values in $H$, $u = (u_i)_{i \in I}$, utilizes the "test space and pairing" idea by replacing the usual "test space" $H$ with $\ell^2(J;H)$ and the usual "pairing" ${P : (v ; (u_i)_{i \in I}) \in H \times \mathcal{F}_I^H \mapsto (\langle v , u_i \rangle)_{i \in I} \in \ell^2(I;\mathbb{F})}$ with ${P^J : ((v_j)_{j \in J} ; (u_i)_{i \in I}) \in \ell^2(J;H) \times \mathcal{F}_I^H \mapsto (\langle v_j , u_i \rangle)_{(i,j) \in I \times J} \in \ell^2(I \times J;\mathbb{F})}$. Secondly, we prove a similar result when the space $\ell^2(J;H)$ is replaced with the space $L^2(Y,\nu;H)$ and the frame $u = (u_x)_{x \in X}$ is $(X,\mu)$-continuous. Besides, we define the $J$-extended and $(Y,\nu)$-extended analysis, synthesis, and frame operators of the frame $u$ and note that they are just natural block-diagonal operators. After that, we generalize quite straightforwardly the well-known characterizations of Euclidean finite frames to the corresponding characterizations of Euclidean continuous frames. One tool that we use in this endeavor is some rewritings of the quotients $N(v ; (u_x)_{x \in X})$ and $N( (v_y)_{y \in Y}) ; (u_x)_{x \in X} )$. Besides, we give a simple sufficient condition for having a frame with values in $\mathbb{F}^2$ and use it to provide an example of a classical $\mathbb{N}^*$-frame with values in $\mathbb{C}^2$. Finally, we generalize some topological properties of the set of frames and Parseval frames from the Euclidean finitely indexed case to the Euclidean continuous one., Comment: 12 pages. This work is a corrected version of the previously published journal article which is why it has a different title

We consider the stability of Schauder frames and besselian Schauder frames under perturbations. Our results are inspirit close to the results of Heil [18].

Our main goal in this paper, is to generalize to Hilbert C*-modules the concept of fusion frames. Indeed we introduce the notion of *\~nfusion frames associated to weighted sequences of orthogonally complemented submodules of a Hilbert C*-module, and prove for such a *-fusion frames some fundamental results., Comment: 38 pages

In this paper, we prove the following results. There exists a Banach space without basis which has a Schauder frame. There exists an universal Banach space $B$ (resp. $\tilde{B}$) with a basis (resp. an unconditional basis) such that, a Banach $X$ has a Schauder frame (resp. an unconditional Schauder frame ) if and only if $X$ is isomorphic to a complemented subspace of $B$ (resp. $\tilde{B}$). For a weakly sequentially complete Banach space, a Schauder frame is unconditional if and only if it is besselian. A separable Banach space $X$ has a Schauder frame if and only if it has the bounded approximation property. Consequenty, The Banach space $\mathcal{L}(\mathcal{H},\mathcal{H})$ of all bounded linear operators on a Hilbert space $\mathcal{H}$ has no Schauder frame. Also, if $X$ and $Y$ are Banach spaces with Schauder frames then, the Banach space $ X\widehat{\otimes}_{\pi}Y$ (the projective tensor product of $X$ and $Y$) has a Schauder frame. From the Faber$-$Schauder system we construct a Schauder frame for the Banach space $C[0,1]$ (the Banach space of continuous functions on the closed interval $ [0,1]$) which is not a Schauder basis of $C[0,1]$. Finally, we give a positive answer to some open problems related to the Schauder bases (In the Schauder frames setting).

In this paper, we will generelize $b$-frames; a new concept of frames for Hilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banach space and see how it can be a frame for a Hilbert space. Instead of the scalar product we will use a new product called the $b$-dual product and it is constructed via a bilinear mapping. We will introduce new results about this product, about $b$-frames, and about $K$-$b$-frames, and we will also give some examples of both $b$-frames and $K$-$b$-frames that have never been given before. We will give the expression of the reconstruction formula of the elements of the Hilbert space. We will as well study the stability and preservation of both $b$-frames and $K$-$b$-frames; and to do so, we will give the equivalent of the adjoint operator according to the $b$-dual product.

In this paper, we give a characterization and a some properties of a besselian sequences, which allows us to build some examples of a besselian Schauder frames. Also for a reflexive Banach spaces (with a besselian Schauder frames) we give some characterizations.

In this present paper we introduce weaving Hilbert space frames in the continuous case, we give new approaches for manufacturing pairs of woven continuous frames and we obtain new properties in continuous weaving frame theory related to dual frames. Also, we provide some approaches for constructing weaving continuous frames by using small perturbations.

Water consumption remains a major concern among the world's future challenges. For applications like load monitoring and demand response, deep learning models are trained using enormous volumes of consumption data in smart cities. On the one hand, the information used is private. For instance, the precise information gathered by a smart meter that is a part of the system's IoT architecture at a consumer's residence may give details about the appliances and, consequently, the consumer's behavior at home. On the other hand, enormous data volumes with sufficient variation are needed for the deep learning models to be trained properly. This paper introduces a novel model for water consumption prediction in smart cities while preserving privacy regarding monthly consumption. The proposed approach leverages federated learning (FL) as a machine learning paradigm designed to train a machine learning model in a distributed manner while avoiding sharing the users data with a central training facility. In addition, this approach is promising to reduce the overhead utilization through decreasing the frequency of data transmission between the users and the central entity. Extensive simulation illustrate that the proposed approach shows an enhancement in predicting water consumption for different households.

A relaxation of the notion of invariant set, known as $k$-quasi-invariant set, has appeared several times in the literature in relation to group dynamics. The results obtained in this context depend on the fact that the dynamic is generated by a group. In our work, we consider the notions of invariant and 1-internally-quasi-invariant sets as applied to an action of a function $f$ on a set $I$. We answer several questions of the following type, where $k \in \{0,1\}$: what are the functions $f$ for which every finite subset of $I$ is internally-$k$-quasi-invariant? More restrictively, if $I = \mathbb{N}$, what are the functions $f$ for which every finite interval of $I$ is internally-$k$-quasi-invariant? Last, what are the functions $f$ for which every finite subset of $I$ admits a finite internally-$k$-quasi-invariant superset? This parallels a similar investigation undertaken by C. E. Praeger in the context of group actions., Comment: 27 pages

It is known that the tensor product of two sequences, in the tensor product of two separable Hilbert spaces, is a frame if and only if each component of that product is a frame. This paper proposes a sort of generalization of the aforementioned result by dealing with sequences S that are finite minimal sums of tensor products of a finite number of sequences. We prove that S is a Bessel sequence if and only if it is a sum for which each term is the tensor product of Bessel sequences. We also state necessary conditions for S to be a frame. For dimensions higher than one, we deduce several results on Gabor systems generated by finite rank square integrable functions. Meanwhile, the one dimensional versions of some of these results are surprisingly extremely difficult to prove or disapprove.

We investigate the transfer of regularity between commutative, noetherian, local rings through a class of local homomorphisms which we call basically regular. We give numerical characterizations of these maps, investigate their behavior under composition and decomposition, and compare them with Avramov-Foxby-Herzog's weakly regular local homomorphisms., Comment: 13 pages

Continuous generalized fusion frame theory was recently introduced by Rahimi and al. Several equalities and inequalities have been obtained for frame, fusion generalized fusion frame, among others. In the present paper, we continue and extend these results to obtain some important identities and inequalities in the case of continuous generalized fusion frame, Parceval continuous generalized fusion frame, $ \lambda$-tight continuous generalized fusion frame. Moreover, we obtain some new inequalities for the alternate dual continuous generalized fusion frame. Finally, we obtain frame operator of a pair of Bessel continuous generalized fusion mapping and we derive some results about resolution of identity.

If $H$ is a Hilbert space, the Stiefel manifold $St(n,H)$ is formed by all the independent $n$-tuples in $H$. In this article, we contribute to the topological study of Stiefel manifolds by proving a path-connectedness result. We prove that the intersection of translates of $St(n,H)$ is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating $n$-tuples. We rely on a lemma that we prove for the occasion., Comment: merged with arXiv:2101.00322

Controlled $\ast$-K-fusion frames are generalization of controlled fusion frames in frame theory. In this paper, we propose the notion of controlled $\ast$-k-fusions frames on Hilbert $C^{\ast}$-modules. We give some caraterizations and some of their properties are obtained. Then we study the erasures of submodules of a controlled $k$-fusion frame in Hilbert $C^{\ast}$-modules and we present some sufficient conditions under which a sequence remains a standart controlled k-fusion frame after deletion of some submodules. Finally, we introduce a perturbation for controlled $K$-fusion frames in Hilbert $C^{\ast}$-modules and it is shown that under some conditions controlled $K$-fusion frames are stable under this perturbation, and we generalize some of the results obtained for perturbations of controlled $K$-fusion frames.

In [Appl. Comput. Harmon. Anal., 46(3):664-673, 2019], O. Christensen and M. Hasannasab observed that assuming the existence of an operator $T$ sending $e_n$ to $e_{n+1}$ for all $n \in \mathbb{N}$ (where $(e_n)_{n \in \mathbb{N}}$ is a sequence of vectors) guarantees that $(e_n)_{n \in \mathbb{N}}$ is linearly independent if and only if $\dim(\text{span}\{e_n\}_{n \in \mathbb{N}}) = \infty$. In this article, we recover this result as a particular case of a general order-theory-based model-theoretic result. We then return to the context of vector spaces to show that, if we want to use a condition like $T(e_i)=e_{\phi(i)}$ for all $i \in I$ where $I$ is countable as a replacement of the previous one, the conclusion will only stay true if $\phi : I \to I$ is conjugate to the successor function $succ : n \mapsto n+1$ defined on $\mathbb{N}$. We finally prove a tentative generalization of the result, where we replace the condition $T(e_i)=e_{\phi(i)}$ for all $i \in I$ where $\phi$ is conjugate to the successor function with a more sophisticated one, and to which we have not managed to find a new application yet., Comment: 12 pages

We proved in a previous article a path-connectedness property of the intersection of translates of the space of finite-dimensional Hilbert space frames, which we formulated in the language of Stiefel manifolds $St(n,H)$. Yet, the notion of a Hilbert space frame has already been successfully extended to $C^*$-algebras and Hilbert modules. In this article, we prove a new property of the intersection of translates of the space of frames both in the finite-dimensional Hilbert space and finite-dimensional $C^*$-algebraic cases. This property expresses that there are many almost-vector spaces in these intersections, and it relies crucially on the hypothesis of finite dimension. When the translating $l$-tuple is made of frames, we see that the intersection almost contains the vector space spanned by these frames. We finally conjecture that the property and its consequence are true in a finite-dimensional left-Hilbert module because of the simple and related structure of the latter., Comment: 9 pages. arXiv admin note: text overlap with arXiv:2101.00322, arXiv:2107.10290

If $H$ is a Hilbert space, the non-compact Stiefel manifold $St(n,H)$ consists of independent $n$-tuples in $H$. In this article, we contribute to the topological study of non-compact Stiefel manifolds, mainly by proving two results on the path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold. In the first part, after introducing and proving an essential lemma, we prove that $\bigcap_{j \in J} \left( U(j) + St(n,H) \right)$ is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating $J$-family. Then, in the second part, we show that the topological closure of $St(n,H) \cap S$ contains all polynomial paths contained in $S$ and passing through a point in $St(n,H)$. As a consequence, we prove that $St(n,H)$ is relatively dense in a certain class of subsets which we illustrate with many examples from frame theory coming from the study of the solutions of some linear and quadratic equations which are finite-dimensional continuous frames. Since $St(n,L^2(X,\mu;\mathbb{F}))$ is isometric to $\mathcal{F}_{(X,\Sigma,\mu),n}^\mathbb{F}$, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames., Comment: 25 pages

In this work, we introduce a new concept of integral $K$-operator frame for the set of all adjointable operators from Hilbert $C^{\ast}$-modules $\mathcal{H}$ to it self noted $End_{\mathcal{A}}^{\ast}(\mathcal{H}) $. We give some propertis relating some construction of integral $K$-operator frame and operators preserving integral $K$-operator frame and we establish some new results.

In this paper we study the concept of controlled $\ast$-operator frmae for $End_{\mathcal{A}}^{\ast}(\mathcal{H})$. Also we discuss characterizations of controlled $\ast$-operator frames and we give some properties, Comment: It is my honor to send you this paper for publication in your journal best regards. arXiv admin note: text overlap with arXiv:1806.03993, arXiv:2008.05950

The notion of controlled frames for Hilbert spaces were introduced by Balazs, Antoine and Grybos to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Controlled Frame Theory has a great revolution in recent years. This Theory have been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. In this paper we introduce and study the extension of this notion to integral frame for Hilbert $C^{\ast}$-module. Also we give some characterizations between integral frame in Hilbert $C^{\ast}$-module

Frame Theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. In this paper, we introduce the concept of Controlled K-operator frame for the space $End_{\mathcal{A}}^{\ast}(\mathcal{H})$ of all adjointable operators on a Hilbert $\mathcal{A}$-module $\mathcal{H}$ and we establish some results., Comment: arXiv admin note: text overlap with arXiv:2006.08354

In this paper, we will introduce a new notion, that of $K$-Integral operator frames in the set of all bounded linear operators noted $\mathcal{B}(H)$, where $H$ is a separable Hilbert space. Also, we prove some results of integral $K$-operator frame. Lastly we will establish some new properties for the perturbation and stability for an integral $K$-operator frames for $\mathcal{B}(H)$

$K$-fusion frames are a generalization of fusion frames in frame theory. In this paper, we extend the concept of controlled fusion frames to controlled $K$-fusion frames, and we develop some results on the controlled $K$-fusion frames for Hilbert spaces, which generalized some well known of controlled fusion frames case. also we discuss some characterizations of controlled Bessel $K$-fusion sequences and of controlled Bessel $K$-fusion. Further, we analyse stability conditions of controlled $K$-fusion frames under perturbation.

Frame Theory has a great revolution for recent years. This Theory has been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. The purpose of this paper is the introduction and the study of the new concept that of Continuous Controlled K-g-Frame for Hilbert $C^{\ast}$-Modules wich is a generalizations of discrete Controlled K-g-Frames in Hilbert $C^{\ast}$-Modules. Also we establish some results., Comment: arXiv admin note: substantial text overlap with arXiv:2006.08354

In this paper, we introduce and we study the concept of Continuous Controlled K-Frame for Hilbert $C^{\ast}$-Modules wich are generalizations of discrete Controlled K-Frames.

Frame theory is recently an active research area in mathematics, computer science and engineering with many exciting applications in a variety of different fields. This theory has been generalized rapidly and various generalizations of frames in Hilbert spaces. In this papers we study the notion of dual continuous $K$-frames in Hilbert spaces. Also we etablish some new properties.

Frame theory is an exciting, dynamic and fast paced subject with applications in numerous fields of mathematics and engineering. In this paper we study Continuous Frame and introduce Continuous Frame with $C^{\ast}$-valued bounds. Also, we establich some properties., Comment: arXiv admin note: text overlap with arXiv:1712.02450, arXiv:1902.05970

In this paper we define Gevrey polyanalytic classes of order N on the unit disk D and we obtain for these classes a characteristic expansion into N-analytic polynomials on suitable neighborhoods of D. As an application of our main theorem, we perform for the Gevrey polyanalytic classes of order N on the unit disk D, an analogue to E. M. Dyn'kin's theorem. We also derive, for these classes, their characteristic degree of the best uniform approximation on D by N-analytic polynomials., Comment: 23 pages

Frame Theory has a great revolution in recent years, this Theory have been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. The purpose of this paper is the introduction and the study of the concept of Controlled Continuous g-Frames in Hilbert $C^{\ast}$-Modules. Also we give some properties.

In this paper we introduce the notion of generalized Gavr\`uta stability of functional equations in order to study, in the framework of a nonquasianalytic Carleman class, the stability of a class of cohomological equations., Comment: 11 pages

In this papers we investigate the g-frame and Bessel g-sequence related to a linear bounded operator $K$ in Hilbert $C^{\ast}$-module and we establish some results.

Frame theory has a great revolution in recent years. This new Theory have been extended from Hilbert spaces to Hilbert C*-modules. In this paper, we introduce the notion of dual *-K-g-frames in Hilbert A-modules. Lastly we study *-K-g-frames in tensor product of Hilbert C*-Modules and we establish some new results.

In this paper, a new notion of frames is introduced: $\ast$-operator frame as generalization of $\ast$-frames in Hilbert $C^{\ast}$-modules introduced by A. Alijani and M. A. Dehghan \cite{Ali} and we establish some results., Comment: arXiv admin note: text overlap with arXiv:1712.00806

Frames play significant role in various areas of science and engineering. In this paper, we introduce the concepts of frames for $End_{\mathcal{A}}^{\ast}(\mathcal{H, K})$ and their generalizations. Moreover, we obtain some new results for generalized frames in Hilbert modules.

In this paper, we introduce the concept of Continuous $\ast$-g-Frame in Hilbert $C^{\ast}$-Modules and we establish some results. We also discuss the stability problem for Continuous $\ast$-g-Frame.

In this paper, we study $\ast$-g-frames in tensor products of Hilbert $C^{\ast}$-modules. We show that a tensor product of two $\ast$-g-frames is a $\ast$-g-frames, and we get some result.

Let k be a field. This paper investigates the embedding dimension and codimension of Noetherian local rings arising as localizations of tensor products of k-algebras. We use results and techniques from prime spectra and dimension theory to establish an analogue of the "special chain theorem" for the embedding dimension of tensor products, with effective consequence on the transfer or defect of regularity as exhibited by the (embedding) codimension., Comment: To appear in: Rings, Polynomials, and Modules, Springer, New York, pp. 1-25

This paper investigates t-reductions of ideals in pullback constructions. Section 2 examines the correlation between the notions of reduction and t-reduction in pseudo-valuation domains. Section 3 solves an open problem on whether the finite t-basic and v-basic ideal properties are distinct. We prove that these two notions coincide in any arbitrary domain. Section 4 features the main result, which establishes the transfer of the finite t-basic ideal property to pullbacks in line with Fontana-Gabelli's result on PvMDs and Gabelli-Houston's result on v-domains. This allows us to enrich the literature with new families of examples, which put the class of domains subject to the finite t-basic ideal property strictly between the two classes of v-domains and integrally closed domains., Comment: 12 pages. New York Journal of Mathematics 2016

In this paper, we prove an extension of Zaks' conjecture on integral domains with semi-regular proper homomorphic images (with respect to finitely generated ideals) to arbitrary rings (i.e., possibly with zero-divisors). The main result extends and recovers Levy's related result on Noetherian rings and Matlis' related result on Prufer domains. It also globalizes Couchot's related result on chained rings. New examples of rings with semi-regular proper homomorphic images stem from the main result via trivial ring extensions., Comment: The condition C2 is slightly modified and Corollary 2.9 and Example 3.2 are added. 14 pages. To appear in Journal of Algebra

This paper contributes to the study of homological aspects of trivial ring extensions (also called Nagata idealizations). Namely, we investigate the transfer of the notion of (Matlis') semi-regular ring (also known as IF-ring) along with related concepts, such as coherence, in trivial ring extensions issued from integral domains. All along the paper, we put the new results in use to enrich the literature with new families of examples subject to semi-regularity., Comment: 15 pages, to appear in Colloquium Mathematicum

Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is t-integral over I if there is an equation x^{n} + a_{1}x^{n-1} +...+ a_{n-1}x + a_{n} = 0 with a_{i} in (I^{i})_{t} for I = 1,...,n. The set of all elements that are t-integral over I is called the t-integral closure of I. This paper investigates the t-reductions and t-integral closure of ideals. Our objective is to establish satisfactory t-analogues of well-known results, in the literature, on the integral closure of ideals and its correlation with reductions. Namely, Section 2 identifies basic properties of t-reductions of ideals and features explicit examples discriminating between the notions of reduction and t-reduction. Section 3 investigates the concept of t-integral closure of ideals, including its correlation with t-reductions. Section 4 studies the persistence and contraction of t-integral closure of ideals under ring homomorphisms. All along the paper, the main results are illustrated with original examples., Comment: 13 pages. Rocky Mountain Journal of Mathematics 2016

Let R be a commutative ring and I an ideal of R. A sub-ideal J of I is a reduction of I if JI^n = I^n+1 for some positive integer n. The ring R has the (finite) basic ideal property if (finitely generated) ideals of R do not have proper reductions. Hays characterized (one-dimensional) Prufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays' results to Prufer v-multiplication domains by replacing "basic" with "w-basic," where w is a particular star operation. We also investigate relations among star-basic properties for certain star operations., Comment: 9 pages. Journal of Commutative Algebra 2016

The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by t-multiplication. This paper investigates integral domains with Boolean t-class semigroup with an emphasis on the GCD and stability conditions. The main results establish t-analogues for well-known results on Prufer domains and Bezout domains of finite character., Comment: 8 pages

AF-rings are algebras over a field k which satisfy the Altitude Formula over k. This paper surveys a few works in the literature on the Krull and valuative dimensions of tensor products of AF-rings. The first section extends Wadsworth's classical results on the Krull dimension of AF-domains to the larger class of AF-rings. It also provides formulas for computing the valuative dimension with effect on the transfer of the (locally) Jaffard property. The second section studies tensor products of AF-rings over a zero-dimensional ring. Most results on algebras over a field are extended to these general constructions. The third section establishes formulas for the Krull and valuative dimensions of tensor products of pullbacks issued from AF-domains. Throughout, examples are provided to illustrate the scope and limits of the results., Comment: 18 pages, Palestine Journal of Mathematics 2016

In 1969, Osofsky proved that a chained ring (i.e., local arithmetical ring) with zero divisors has infinite weak global dimension; that is, the weak global dimension of an arithmetical ring is 0, 1, or infinite. In 2007, Bazzoni and Glaz studied the homological aspects of Prufer-like rings, with a focus on Gaussian rings. They proved that Osofsky's aforementioned result is valid in the context of coherent Gaussian rings (and, more generally, in coherent Prufer rings). They closed their paper with a conjecture sustaining that ``the weak global dimension of a Gaussian ring is 0, 1, or infinite. In 2010, Bakkari-Kabbaj-Mahdou provided an example of a Gaussian ring which is neither arithmetical nor coherent and has an infinite weak global dimension. In 2011, Abuhlail-Jarrar-Kabbaj introduced and investigated the new class of fqp-rings which stands strictly between the two classes of arithmetical rings and Gaussian rings. Then, they proved the Bazzoni-Glaz conjecture for fqp-rings. This paper surveys a few recent works in the literature on the weak global dimension of Prufer-like rings making this topic accessible and appealing to a broad audience. As a prelude to this, the first section of this paper provides full details for Osofsky's proof of the existence of a module with infinite projective dimension on a chained ring. Numerous examples -arising as trivial ring extensions- are provided to illustrate the concepts and results involved in this paper., Comment: 17 pages

This paper investigates ideal-theoretic as well as homological extensions of the Prufer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prufer conditions., Comment: 11 pages

This paper establishes necessary and sufficient conditions for a bi-amalgamation to inherit the arithmetical property, with applications on the weak global dimension and transfer of the semihereditary property. The new results compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations. All results are backed with new and illustrative examples arising as bi-amalgamations., Comment: 11 pages, Journal of Algebra and its Applications 2016