Algebraic Frame Research Articles

Topology of closure systems in algebraic lattices

Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice P generates a closure system langle X rangle _P. The closure system langle Y rangle _P generated by the patch closure Y of X is the patch closure of langle X rangle _P. If X is contained in the set of nontrivial prime elements of P then langle X rangle _P is a frame and is a coherent algebraic frame if X is patch closed in P. Conversely, if the algebraic lattice P is coherent then its set of nontrivial prime elements is patch closed.

Observability Decomposition of Boolean Control Networks

This article investigates the observability decomposition of Boolean control networks (BCNs) under the algebraic frame based on the semitensor product of matrices. First, the definition for the observability decomposition of BCNs is proposed, which is consistent with the observability decomposition of traditional linear control systems. Then, by using the vertex partition method, a necessary and sufficient vertex partition condition for the observability decomposition of BCNs is proposed. Next, an algorithm is designed for the analysis and realization of observability decomposition. Finally, a biological example and a counterexample are analyzed to illustrate the obtained results.

Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations

<abstract><p>Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that extend a recently proposed SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above-mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamiltonian models based on the book and oscillator algebras, denoted by $ \mathfrak{b}_2 $ and $ \mathfrak{h}_4 $, respectively. The last generalization corresponds to an SIS system possessing the so-called two-photon algebra symmetry $ \mathfrak{h}_6 $, according to the embedding chain $ \mathfrak{b}_2\subset \mathfrak{h}_4\subset \mathfrak{h}_6 $, for which an exact solution cannot generally be found but a nonlinear superposition rule is explicitly given.</p></abstract>

Sparse Bayesian learning for complex‐valued rational approximations

AbstractSurrogate models are used to alleviate the computational burden in engineering tasks, which require the repeated evaluation of computationally demanding models of physical systems, such as the efficient propagation of uncertainties. For models that show a strongly non‐linear dependence on their input parameters, standard surrogate techniques, such as polynomial chaos expansion, are not sufficient to obtain an accurate representation of the original model response. It has been shown that for models with discontinuities or rational dependencies, for example, frequency response functions of dynamic systems, the use of a rational (Padé) approximation can significantly improve the approximation accuracy. In order to avoid overfitting issues in previously proposed standard least squares approaches, we introduce a sparse Bayesian learning approach to estimate the coefficients of the rational approximation. Therein the linearity in the numerator polynomial coefficients is exploited and the denominator polynomial coefficients as well as the problem hyperparameters are determined through type‐II‐maximum likelihood estimation. We apply a quasi‐Newton gradient‐descent algorithm to find the optimal denominator coefficients and derive the required gradients through application of ‐calculus. The method is applied to the frequency response functions of an algebraic frame structure model as well as that of an orthotropic plate finite element model.

Taking an Asset-Based Approach in the Use of a Culturally Located Task to Construct Functional Reasoning

Background: Algebra has traditionally been seen as a site of inequity and a gatekeeper for marginalized learners within both national and international studies. However, considerable research has shown that taking an asset-based approach in mathematics teaching improves learning for marginalized students, including Māori and Pāsifika learners. Purpose: The purpose of this research was to explore how a culturally located task set with a familiar Pāsifika context and within an algebraic frame was mathematized to support rich understandings. The focus was placed on how Pāsifika and Māori students drew on multimodal forms of communication (gesture, drawings, language, and symbols) to collectively make sense of a culturally located growing pattern task designed to develop functional reasoning. Research Design: The 12 student participants in the study were aged between 11 and 12 years and were of Māori and Pāsifika nations ethnicity, as were their teacher and the two researchers. The lesson reported on in this study was one of eight lessons and was representative of all lessons as the teacher and students drew on a variety of multimodal means of communication to construct functional reasoning. The design drew on both qualitative case study and design research. Data collection tools included interviews, video-recorded classroom observations, field notes, and photographs of work samples. The research design and analysis were informed through use of the Ula model as appropriate to all participants’ Pāsifika ethnicity. Similarly, storying was used, given that it is a traditional form of sharing used by indigenous peoples. Findings: Clearly evident in the results was how use of a culturally located task within a known context and the teacher’s asset-based approach supported students to engage in a challenging algebraic task. Multimodal forms of representation as reasoning and communication tools supported them to access more sophisticated forms of algebraic understandings as they began to generalize recursive, covarying, and correspondence relationships in increasingly sophisticated ways. Use of body language and gesturing was central to their communication, as was use of their first language and natural language. Conclusions/Recommendations: To change the gatekeeping role of algebra, teachers need to take a strength-based perspective and draw on what students bring to school as valued knowledge. For all students to access powerful ways of reasoning algebraically, the key role of many multimodal forms of communication needs to be recognized and affirmed.

Compatibility of distributions in probabilistic models: an algebraic frame and some characterizations

Compatibility of distributions in probabilistic models: an algebraic frame and some characterizations

Algebraic Frames and Duality

The theory of algebraic frames for a Hilbert space $H$ is a generalization of the theory of frames and generalized frames. The paper applies the theory of unbounded operators to define the dual of algebraic frames with densely defined unbounded analysis operators. It is shown that every algebraic frame has an algebraic dual frame, and if an algebraic frame has a nonzero redundancy, then it is not Riesz-type. An example of an algebraic frame with finite redundancy is constructed which is not a Riesz-type algebraic frame. Finally, for a lower bounded analytic frame, the discreteness of its indexing measure space and the uniqueness of its algebraic dual are studied and shown to be interrelated.

Amenable and Locally Amenable Algebraic Frames

We say a prime element of an algebraic frame is amenable if it comparable to every compact element. If every prime element of an algebraic frame L is amenable, we say L is an amenable frame. If the localization of L at every prime element is amenable, we say L is locally amenable. These concepts are motivated by notions of divided and locally divided commutative rings. We show that an algebraic frame is (i) amenable precisely when its prime elements form a chain, and (ii) locally amenable precisely when its prime elements form a tree. Given any prime element p of L, we construct a certain pullback in the category of algebraic frames with the finite intersection property on compact elements, and characterize (in terms of the localization Lp and the quotient ↑p of L) when this pullback is amenable and when it is locally amenable.

On the sublocale of an algebraic frame induced by the d-nucleus

The notion of d-ideal has been abstracted by Martínez and Zenk from Riesz spaces to algebraic frames. They introduced the d-nucleus and d-elements. In this paper we extend several characterizations of d-elements that parallel similar characterizations of d-ideals in rings. For instance, calling a coherent map between algebraic frames “weakly skeletal” if it maps compact elements with equal pseudocomplements to images with equal pseudocomplements, we show that an a∈L is a d-element if and only if a=h⁎(0) for some weakly skeletal map h:L→M, where h⁎ denotes the right adjoint of h. The sublocale of L induced by the d-nucleus is denoted by dL. We characterize when d(L⊕M)≅dL⊕dM. We weaken the notion of d-element by defining eL to be the set of elements that are joins of double pseudocomplements of compact elements. We show that if L=dL and M=dM, then L⊕M=e(L⊕M). Clearly, dL⊆eL. We give an example to show that the containment can be proper. Finally, we show that dL is always a sublocale of eL.

Homogenized funtf varieties and algebraic frame completion

We introduce homogenized funtf (finite tight unit norm frames) varieties and study the degrees of their coordinate projections. These varieties compactify the affine funtf variety differently from the projectivizations studied in [12]. However, each are the closures (Zariski) of the set of finite tight unit norm frames. Our motivation comes from studying the algebraic frame completion problem.

Retraction notice to ``Fresnelet approach for image encryption in the algebraic frame'' [Applied Mathematics and Computation 334 (2018) 343–355

Retraction notice to ``Fresnelet approach for image encryption in the algebraic frame'' [Applied Mathematics and Computation 334 (2018) 343–355

RETRACTED: Fresnelet approach for image encryption in the algebraic frame

This article has been retracted: please see Elsevier Policy on Article Withdrawal https://www.elsevier.com/about/our-business/policies/article-withdrawal . This article has been retracted at the request of the Editor-in-Chief. The article is a duplicate of a paper that has already been published in PLOS One, doi.org/10.1371/journal.pone.0194343 . One of the conditions of submission of a paper for publication is that authors declare explicitly that the paper is not under consideration for publication elsewhere. As such this article represents a severe abuse of the scientific publishing system. The scientific community takes a very strong view on this matter and apologies are offered to readers of the journal that this was not detected during the submission process.

On the sobriety of the inverse topology

An algebraic frame L with the finite intersection property (FIP) on compact elements is said to be polarised if every minimal prime element in it is complemented. In this note, we give a necessary and sufficient condition for the inverse topology on the set of minimal prime elements of such a frame to be sober. We also establish some sufficient conditions for sobriety when the polarisation condition is relaxed.

On the hull-kernel and inverse topologies as frames

We consider some frame-theoretic properties of the hull-kernel and the inverse topologies on the set of minimal prime ideals of an algebraic frame with the finite intersection property on its compact elements. Denote by Alg do the subcategory of Frm consisting of such frames together with dense onto coherent maps. We construct a functor \({{\sf T} : {\bf Alg}_{\rm do} \rightarrow {\bf Frm}}\) and a natural transformation \({\tau : {\sf E} \rightarrow {\sf T}}\), where E is the inclusion functor from Alg do to Frm.

Conrad frames

A Conrad frame is a frame which is isomorphic to the frame C ( G ) of all convex ℓ-subgroups of some lattice-ordered group G. It has long been known that Conrad frames have the disjointification property. In this paper a number of properties are considered that strengthen the disjointification property; they are referred to as the Conrad conditions. A particularly strong form of the disjointification property, the C-frame condition, is studied in detail. The class of lattice-ordered groups G for which C ( G ) is a C-frame is shown to coincide with the class of pairwise splitting ℓ-groups. The arguments are mostly frame-theoretic and Choice-free, until one tackles the question of whether C-frames are Conrad frames. They are, but the proof is decidedly not point-free. This proof actually does more: it shows that every algebraic frame with the FIP and disjointification can be coherently embedded in a C-frame. When the discussion is restricted to normal-valued lattice-ordered groups, one is able to produce examples of coherent frames having disjointification, which are not Conrad frames.

Sublattices Generated by Polars

In an algebraic frame L, the complete sublattice CP(L) generated by the polars of L is studied, in comparison with FP(L), the subframe generated by the polars. It is shown, by an example from the theory of ℓ-groups, that these are distinct, in general. The relationship between FP(L), CP(L), and other established constructs, closely related to the boolean algebra of polars, is also studied.

Feebly Projectable Algebraic Frames and Multiplicative Filters of Ideals

In the article (Martinez and Zenk, Algebra Universalis, 50, 231–257, 2003.), the authors studied several conditions on an algebraic frame L. In particular, four properties called Reg(1), Reg(2), Reg(3), and Reg(4) were considered. There it was shown that Reg(3) is equivalent to the more familiar condition known as projectability. In this article we show that there is a nice property, which we call feebly projectable, that is between Reg(3) and Reg(4). In the main section of the article we apply our notions to the frame of multiplicative filters of ideals in a commutative ring with unit and give characterizations of several well-known classes of commutative rings.

Scopes and limits of modality in quantum mechanics

AbstractWe develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum systems.

Supersymmetric Källen-Lehmann representation

We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum systems.

Dimension in algebraic frames

In an algebraic frame L the dimension, dim(L), is defined, as in classical ideal theory, to be the maximum of the lengths n of chains of primes p 0 < p 1 < ... < p n , if such a maximum exists, and ∞ otherwise. A notion of “dominance” is then defined among the compact elements of L, which affords one a primefree way to compute dimension. Various subordinate dimensions are considered on a number of frame quotients of L, including the frames dL and zL of d-elements and z-elements, respectively. The more concrete illustrations regarding the frame convex ℓ-subgroups of a lattice-ordered group and its various natural frame quotients occupy the second half of this exposition. For example, it is shown that if A is a commutative semiprime f-ring with finite ℓ-dimension then A must be hyperarchimedean. The d-dimension of an ℓ-group is invariant under formation of direct products, whereas ℓ-dimension is not. r-dimension of a commutative semiprime f-ring is either 0 or infinite, but this fails if nilpotent elements are present. sp-dimension coincides with classical Krull dimension in commutative semiprime f-rings with bounded inversion.