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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