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The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a k-algebra and this new “k-space” becomes a generalization of the corresponding Zariski space. We prove that a k-space is T0, quasi-compact, spectral, and connected. Moreover, we study continuous maps between such k-spaces. We conclude with a question about the construction of a sheaf of k-spaces similar to affine schemes.

The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a k-algebra and this new "k-space" becomes a generalization of the corresponding Zariski space. We prove that a k-space is T0, quasi-compact, spectral, and connected. Moreover, we study continuous maps between such k-spaces. We conclude with a question about construction of a sheaf of k-spaces similar to affine schemes. [ABSTRACT FROM AUTHOR]

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Soil water content (SWC) is an essential index that reflects water conditions in the soil surface during drought, and changes in SWC are critical in guiding agricultural irrigation management. Remote sensing is a crucial means of monitoring SWC, and the red and near-infrared bands of optical remote sensing data are often used to monitor SWC through remote sensing retrieval. With the development of remote sensing technology, red-edge band has also been used to retrieve SWC. It exhibits higher sensitivity to changes in vegetation and SWC, relatively more precise differentiation between vegetation and soil, and more accurate SWC detection. However, there has been relatively little research regarding the use of the red-edge band for retrieving SWC. In this study, we developed a novel method, based on the principle of spectral space and the calculation of modified perpendicular drought index, to calculate the soil water index using the red-edge band of the Gaofen-6 (GF-6) satellite by comparing the retrieval accuracy of GF-6 using different combinations of red, near-infrared, and red-edge bands. Based on the measured SWC in the Shijin Irrigation District of Hebei Province, an SWC retrieval model was established using linear regression. Moreover, the new method was used to retrieve SWC from GF-6 and Sentinel-2 with the same central wavelength, and the reliability of the new method was compared and analyzed. The results revealed that the combination of red band and red-edge band 2 with a central wavelength of 0.74 μm substantial. improved the retrieval accuracy of SWC. Our study provides a new method for retrieving SWC using optical remote sensing, which can effectively improve retrieval accuracy and provide data support for agricultural irrigation monitoring and water management.

The article substantiates the requirements for the quality characteristics of video information services. It is shown that it is necessary to improve the quality of video information by the following indicators: time delays for video data delivery in terms of ensuring the required completeness; integrity of video information in accordance with the requirements of the application. The consequence of this fact is an imbalance between meeting the requirements for different groups of quality characteristics of video services. A set of measures is used to reduce the load on information and communication systems. One of the key ones is the use of compression technologies to reduce the bit volume. Therefore, the development of new coding technologies in terms of localising the balance between the level of video data compression and its integrity is an urgent scientific and applied problem. The article describes the main stages of creating a method for determining the informative and positional weight for a coding system in a truncated positional basis. For this purpose, a system of mathematical relations is developed to determine the number of admissible sequences. The scientific novelty is as follows. For the first time, a system of correlations has been created to determine the informative and positional weight of the components of the diagonal sequence of a transformant based on the combinatorial configurations. On the basis of experimental studies, it is shown that the developed method has advantages in terms of ensuring the level of video data integrity under conditions of a given compression rate.

Question: Remote sensing is a fundamental tool to monitor biodiversity and spectral diversity may represent a proxy for different biodiversity facets such as taxonomic (TD) and functional diversity (FD). We used fine‐resolution multispectral imagery to explore spectral diversity (SD) patterns across spatial scales (i.e., plot, transect, area), and assess SD relationships with TD and FD along an environmental gradient. Location: Coastal sand dune, Viareggio, Italy ("Migliarino–San Rossore–Massaciuccoli" Regional Park, 43°83′ N, 10°25′ E). Methods: We measured TD as species richness, while SD and FD were computed using probability density functions based on pixel and species position in multivariate spaces. We compared SD and FD patterns in space occupation, and we tested whether these patterns are coordinated along the sea–inland gradient. We also assessed univariate relationships between SD and biodiversity facets, and we tested how these facets were partitioned across scales. Results: We found a strong correspondence between functional and spectral patterns in space occupation and along the environmental gradient, with a lack of significance when considering TD. However, TD and SD showed higher variation at broader scale while most FD variation occurred at plot level. Conclusion: By measuring FD and SD with a common methodological framework, we demonstrate that SD approximates functional patterns in plant communities. Moreover, we show that SD retrieved using high‐resolution images can capture different aspects of FD, and that the occupation of the spectral space is analogous to the occupation of the functional space. [ABSTRACT FROM AUTHOR]

Given a topological space X, let UX and ηX: X → UX denote, respectively, the Salbany compactification of X and the compactification map called the Salbany map of X. For every continuous function f: X → Y, there is a continuous function Uf: UX → UY, called the Salbany lift of f, satisfying (Uf) ◦ ηX = ηY ◦ f. If a continuous function f: X → Y has a stably compact codomain Y, then there is a Salbany extension F: UX → Y of f, not necessarily unique, such that F ◦ ηX = f. In this paper, we give a condition on a space such that its Salbany map is open. In particular, we prove that in a class of Hausdorff spaces, the spaces with open Salbany maps are precisely those that are almost discrete. We also investigate openness of the Salbany lift and a Salbany extension of a continuous function. Related to open continuous functions are initial maps as well as nearly open maps. It turns out that the Salbany map of every space is both initial and nearly open. We repeat the procedure done for openness of Salbany maps, Salbany lifts and Salbany extensions to their initiality and nearly openness. [ABSTRACT FROM AUTHOR]

This study is concerned with the collections of prime and radical subacts of free acts over a semigroup S. The collection of all prime subacts of any free act is topologized with two topologies as generalizations of the notion of the Zariski topology in rings and semigroups. Moreover, the interplay between some algebraic properties and some topological properties of the spectrum of free acts is investigated. In particular, the conditions under which the set of all prime subacts of free acts is a spectral space are studied. Also, it is shown that a certain type of Zariski space of free acts is isomorphic to a lattice of radical subacts. [ABSTRACT FROM AUTHOR]

A topological duality result is established for the category of distributive c-posets defined in this paper, as well as for some of its important full subcategories. All duality results presented extend the well-known topological duality result obtained by M. H. Stone for the category of distributive (0, 1)-lattices. [ABSTRACT FROM AUTHOR]

Soil water and salt content have been two of the most important influencing factors on crop growth and agricultural productivity all over the world nowadays. Inappropriate soil water and content can pose a potential threat to crop quality and yield. Fortunately, satellite remote sensing can be expected to monitor the soil salt content and water content, due to the largescale coverage, time- and labor-saving. Among them, the optical satellite has been the major data source for the estimation of the soil water and salt content, particularly with the high temporal return cycle, high spatial resolution, and rich spectral range. However, the soil water and salt can affect each other on the response to the spectral reflectance in the areas with their large variations, similar to their regular change in optical satellite band reflectance. There is a high demand to improve the inversion accuracy of soil water and salt content when the soil salt or water content cannot be maintained consistently. This study aims to improve the estimation accuracy of soil salt and water, and then expand the application scope of optical satellite remote sensing. A semi-analytical reflectance model-RVS model was proposed to simulate the response of vegetation spectral reflectance to the soil water and salt content in the vegetation soil. The influence of soil salt-water interaction was also considered when estimating the soil water and salt content. The RVS model was then used to simultaneously estimate the soil water and salt content in the vegetation-covered soil. The results show that the RVS model performed the better estimation accuracy in the study area in both soil water content (coefficient of determination: R 2 =0.63-0.74, root mean square error: RMSE=0.017-0.028) and soil salt content (coefficient of determination: R ² =0.68-0.75, root mean square error: RMSE=0.0525- 0.0617). There was a more outstanding response of vegetation spectral reflectance to the soil water and salt content in the deep soil during the whole stage of crop growth, compared with the shallow soil. The dominant influencing factors on the vegetation spectral reflectance were slowly shifting from the soil water content to the soil salt content and soil water-salt interactions, as the crop grew. The RVS model was then used to continuously estimate the soil water and salt content in the period of crop growth and irrigation. As such, the salt accumulation was attributed to the insufficient water that was caused by both crop growth depletion and excessive irrigation. The estimation revealed the interference of soil water content, soil salt content, and water-salt interaction on the crop spectral reflectance, considering the influence of soil water (soil salt) when estimating soil salt content (soil water content) under the concept of soil salt-water interaction. The simultaneous monitoring of soil water content and soil salt content was realized to construct the estimation models using Sentinel-2 and machine learning. The finding can provide a strong reference for the accurate monitoring of soil salt content and soil water content on the regional scale. [ABSTRACT FROM AUTHOR]

Since Hochster's work, spectral spaces have attracted increasing interest. Through this note we give a new self-contained and constructible topology-independent proof of the fact that the set of proper ideals of a ring endowed with coarse lower topology is a spectral space.

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 ⟨ X ⟩ P . The closure system ⟨ Y ⟩ P generated by the patch closure Y of X is the patch closure of ⟨ X ⟩ P . If X is contained in the set of nontrivial prime elements of P then ⟨ X ⟩ 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. [ABSTRACT FROM AUTHOR]

The rings considered in this article are commutative with identity 1 6= 0. In this paper, we introduce and study commutative ring with Artinian spectrum, that is commutative ring with descending chain condition on radical ideals. A characterization via prime ideals of such commutative ring is given. As part of the study, we introduce a topology τc on SpecR for which the open subsets are exactly the Zariski closed subsets. For instance, it is shown that for a commutative ring R with Artinian spectrum, (SpecR, τc) is a spectral space if and only if R has Noetherian spectrum. [ABSTRACT FROM AUTHOR]

The article proves that the quality of video information services determines the current state of informatization of society and personality. The requirements to the qualitative characteristics of the provision of video services are substantiated. It is shown that they are regulated by the QoS system. The article systematically proves that the correspondence of the real characteristics of the provision of video information services to the required level depends mainly on the presence of an imbalance. This imbalance is determined by the overload of infocommunication networks. Based on a thorough analysis, it is concluded that in order to eliminate this imbalance, it is necessary to create technologies for reducing the bit volume of video data. At the same time, it was revealed that modern technologies have many significant shortcomings. To eliminate such shortcomings, it is necessary to develop new technologies for compression coding of video data. Therefore, there is an urgent scientific and applied problem that concerns the creation of new compression coding technologies in the context of localization of the balance between the speed of video data delivery and their integrity. One of the basic directions here is the further improvement of video segment coding technologies in spectral space. At the same time, it is necessary to establish new types of patterns in the structural and semantic content of the video image. The perspective approach concerns taking into account the regularities caused by the combinatorial configuration of the transformer in an unevenly diagonal texture. This approach is proposed in the works. At the same time, the most problematic here is the formation of informative and positional weight. Such components are key in the process of determining the code value of the corresponding diagonal sequence of the transformer. A method for determining the informative-positional weight (uneven-weight component of the code value) of the truncated-positional number is developed. It is defined as the number of valid subsequences that precede the corresponding subsequence of the diagonal and satisfy the constraints according to the features of the combinatorial configuration of the transformer in a two-dimensional unevenly diagonal format. [ABSTRACT FROM AUTHOR]

We consider the set of all the ideals of a ring, endowed with the coarse lower topology. The aim of this paper is to study topological properties of distinguished subspaces of this space and detect the spectrality of some of them. [ABSTRACT FROM AUTHOR]

It is well known that the real spectrum of any commutative unital ring, and the ℓ-spectrum of any Abelian lattice-ordered group with order-unit, are all completely normal spectral spaces. We complete the existing list of containments and non-containments between the associated spectral spaces and their spectral subspaces, by proving the following results: (1) Every real spectrum can be embedded, as a spectral subspace, into some ℓ-spectrum. (2) Not every real spectrum is an ℓ-spectrum. (3) A spectral subspace of a real spectrum may not be a real spectrum. (4) Not every ℓ-spectrum can be embedded, as a spectral subspace, into a real spectrum. The commutative unital rings and Abelian lattice-ordered groups constructed in Eqs. 2, 3, and 4 all have cardinality ℵ1. Moreover, Eq. 3 solves a problem stated in 2012 by Mellor and Tressl. [ABSTRACT FROM AUTHOR]

Since Hochster’s work, spectral spaces have attracted increasing interest. Through this note we give a new self-contained and constructible topology-independent proof of the fact that the set of proper ideals of a ring endowed with coarse lower topology is a spectral space. [ABSTRACT FROM AUTHOR]

Following the theory of tensor triangular support introduced by Sanders, which generalizes the Balmer-Favi support, we prove the local version of the result of Zou that the Balmer spectrum being Hochster weakly scattered implies the local-to-global principle. That is, given an object t of a tensor triangulated category T we show that if the tensor triangular support Supp (t) is a weakly scattered subset with respect to the inverse topology of the Balmer spectrum Spc (T c) , then the local-to-global principle holds for t. As immediate consequences, we have the analogue adaptations of the well-known statements that the Balmer spectrum being noetherian or Hausdorff scattered implies the local-to-global principle. We conclude with an application of the last result to the examination of the support of injective superdecomposable modules in the derived category of an absolutely flat ring which is not semi-artinian. • Topological properties on the support imply the local-to-global principle. • Support weakly scattered in the inverse topology implies local-to-global principle. • Noetherian support implies the local-to-global principle. • Scattered support in a profinite Balmer spectrum implies local-to-global. • Example of a injective superdecomposable module with perfect support. [ABSTRACT FROM AUTHOR]

Soil water content (SWC) is an essential index that reflects water conditions in the soil surface during drought, and changes in SWC are critical in guiding agricultural irrigation management. Remote sensing is a crucial means of monitoring SWC, and the red and near-infrared bands of optical remote sensing data are often used to monitor SWC through remote sensing retrieval. With the development of remote sensing technology, red-edge band has also been used to retrieve SWC. It exhibits higher sensitivity to changes in vegetation and SWC, relatively more precise differentiation between vegetation and soil, and more accurate SWC detection. However, there has been relatively little research regarding the use of the red-edge band for retrieving SWC. In this study, we developed a novel method, based on the principle of spectral space and the calculation of modified perpendicular drought index, to calculate the soil water index using the red-edge band of the Gaofen-6 (GF-6) satellite by comparing the retrieval accuracy of GF-6 using different combinations of red, near-infrared, and red-edge bands. Based on the measured SWC in the Shijin Irrigation District of Hebei Province, an SWC retrieval model was established using linear regression. Moreover, the new method was used to retrieve SWC from GF-6 and Sentinel-2 with the same central wavelength, and the reliability of the new method was compared and analyzed. The results revealed that the combination of red band and red-edge band 2 with a central wavelength of 0.74 μm substantial. improved the retrieval accuracy of SWC. Our study provides a new method for retrieving SWC using optical remote sensing, which can effectively improve retrieval accuracy and provide data support for agricultural irrigation monitoring and water management. • SWC sensitivity is evaluated using various band combinations in spectral space. • A novel Soil Water Index (SWI) is developed using three specific band combinations. • SWI reliability is validated by demonstrating with Sentinel-2, which has the same central wavelength as GF6. [ABSTRACT FROM AUTHOR]

We continue our research plan of developing the theory of small and locally small spaces, proposing this theory as a realisation of Grothendieck's idea of tame topology on the level of general topology. In this paper, we develop the theory of Heyting small spaces and prove a new version of Esakia Duality for such spaces. To do this, we notice that spectral spaces may be seen as sober small spaces with all smops compact and introduce the method of the standard spectralification. This helps to understand open continuous definable mappings between definable spaces over o-minimal structures. [ABSTRACT FROM AUTHOR]

As a generalization of the Artinian module, we introduce the class of pseudo Artinian modules. We explore some algebraic properties of this class, and we study some topological properties of the prime spectrum of pseudo Artinian modules. [ABSTRACT FROM AUTHOR]

The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander's Subbase Theorem, which asserts that a topological space X is compact if every subbasic open cover of X admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem—whose proof depends upon Zorn's Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces; choice-free in the sense that our representation avoids use of Alexander's Subbase Theorem, along with its associated nonconstructive choice principles. We then introduce a new subclass of spectral spaces which we call upper Vietoris orthospaces in order to characterize up to homeomorphism (and isomorphism with respect to their orthospace reducts) the spectral spaces of proper lattice filters used in our representation. It is then shown how our constructions give rise to a choice-free dual equivalence of categories between the category of orthocomplemented lattices and the dual category of upper Vietoris orthospaces. Our duality combines Bezhanishvili and Holliday's choice-free spectral space approach to Stone duality for Boolean algebras with Goldblatt and Bimbó's choice-dependent orthospace approach to Stone duality for orthocomplemented lattices. [ABSTRACT FROM AUTHOR]

The article provides justification for the need to eliminate the imbalance between the growing information load on infocommunications and their bandwidth. It is shown that there is a destructive influence on the functioning of critical infrastructure systems and a complex of key areas of society for life. It is argued that during the functioning of infocommunication systems, the solution of issues of information overload of the network is achieved through the use of an integrated approach. It is shown that one of the key here is the use of management technologies and the coordination of the intensity of information traffic. Based on the research, it is sealed that in the case of a critical level of video information flow, the existing regulatory mechanisms do not ensure the redistribution of traffic according to the quality system of network functioning. Therefore, it is necessary to create new technologies for processing and transmitting video data. The direction of improving video traffic management technologies at the level of information presentation has been approved. The need to create a new method of compression coding of the video stream has been proven. A comprehensive analysis of the latest research is given. On the basis of which it is shown that the most effective results are achieved for coding methods, taking into account the patterns of video segments in spectral space. The presence of inherent shortcomings is justified. They relate to the dependence of their effectiveness on the level of loss of reliability of information. To localize these shortcomings, the article introduces the creation of a process for reducing redundancy to organize based on the identification of new patterns in the structure of transformed video segments. One of these directions is an approach based on the coding of unevenly diagonal sequences of transformants. This takes into account structural dependencies. Accordingly, cases of additional loss of the level of reliability of video information are excluded. A model for assessing the informativeness of the transformant as a combinatorial object is being created, taking into account its structural and topological features and features of the psychovisual perception of the reconstructed video segments. [ABSTRACT FROM AUTHOR]

Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempotent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the prime k-spectrum of an idempotent semiring. In fact, we enrich Hochster's theorem by constructing a subcategory of idempotent semirings which is antiequivalent to the category of spectral spaces. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the space of valuations and the space of prime congruences on an idempotent semiring are spectral, and there is a natural bijection of sets between the two; this shows a stark difference between rings and idempotent semirings. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of closure operations for semirings, and provide several examples. In particular, we introduce an integral closure operation and a Frobenius closure operation for idempotent semirings. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum Spec (R) of a unital commutative ring R is always a spectral (= coherent) topological space. In this generalization, which includes several other known ones, the role of ideals of R is played by elements of an abstract complete lattice L equipped with a binary multiplication with x y ⩽ x ∧ y for all x , y ∈ L . In fact when no further conditions on L are required, the resulting space can be and is only shown to be sober, and we discuss further conditions sufficient to make it spectral. This discussion involves establishing various comparison theorems on so-called prime, radical, solvable, and locally solvable elements of L; we also make short additional remarks on semiprime elements. We consider categorical and universal-algebraic applications involving general theory of commutators, and an application to ideals in what we call the commutative world. The cases of groups and of non-commutative rings are briefly considered separately. [ABSTRACT FROM AUTHOR]

We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation. [ABSTRACT FROM AUTHOR]

The partially integrated transport modeling (PITM) method first introduced in Refs. Schiestel and Dejoan (Theor Comput Fluid Dyn 18:443, 2005) and in Chaouat and Schiestel (Phys Fluids 17:065106, 2005) provides a continuous approach for hybrid RANS-LES simulations. Inspired from the multiple scale approach, the basis of the development of the method is the spectral space in quasi-homogeneous turbulence. The PITM method embodies a partitioning control function that monitors the ratio of subfilter energy to total turbulent energy by reference to the cutoff wavenumber location. How this procedure behaves in inhomogeneous flows is an important question. The present paper demonstrates that the same control function can be used both in homogeneous and in non-homogeneous flows as well. Further on, an analysis of the effect of anisotropic filters generally used for wall flows is conducted for computing the equivalent cutoff wavenumber suitable for determining the subfilter energy of the spectrum and see how it interferes with control function. Illustrations in the turbulent plane channel flow are given that confirm the efficiency of the procedure and DNS data have been used to support and supplement the discussion. [ABSTRACT FROM AUTHOR]

We continue our research plan of developing the theory of small and locally small spaces, proposing this theory as a realisation of Grothendieck’s idea of tame topology on the level of general topology. In this paper, we develop the theory of Heyting small spaces and prove a new version of Esakia Duality for such spaces. To do this, we notice that spectral spaces may be seen as sober small spaces with all smops compact and introduce the method of the standard spectralification. This helps to understand open continuous definable mappings between definable spaces over o-minimal structures.

This study aims to investigate the Zariski topology on the prime ideals of a commutative semigroup S, denoted by Spec(S). First, we show that a topological space X is homeomorphic to Spec(S) for some commutative semigroup S if and only if X is an SS-space that can be described purely in topological terms. Next, we show that an adjunction exists between the category of commutative semigroups and that of SS-spaces. We further show that the category of commutative idempotent semigroups is dually equivalent to that of SS-spaces. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian ℓ -groups. As a first main result, we show that a topological space X is the prime spectrum of an MV-algebra if and only if X is spectral, and the lattice K(X) of compact open subsets of X is a closed epimorphic image of the lattice of “cylinder rational polyhedra” (a natural generalization of rational polyhedra) of [ 0 , 1 ] Y for some set Y. As a second main result we extend our results to Abelian ℓ -groups. That is, a topological space X is the prime spectrum of an Abelian ℓ -group if and only if X is generalized spectral, and the lattice K(X) is a closed epimorphic image of the lattice of “cylinder rational cones” (a generalization of rational cones) in R Y for some set Y. Finally, we axiomatize, in monadic second order logic, the Belluce lattices of free MV-algebras (equivalently, the lattice of cylinder rational polyhedra) of dimension 1, 2 and infinite, and we study the problem of describing Belluce lattices in certain fragments of second order logic. [ABSTRACT FROM AUTHOR]

We correct three issues in the paper "The Zariski topology on sets of semistar operations without finite-type assumptions". [ABSTRACT FROM AUTHOR]

Hyperspectral images (HSIs) restoration is an important preprocessing step. The spectral vectors in HSI can be separated into different classification based on the land‐covers, which means the spectral space can be regarded as the union of several low‐rank subspaces. Subspace low‐rank representation (SLRR) is powerful in exploring the inner low‐rank structure and has been applied for HSI restoration. However, the traditional SLRR only seek for the rank‐minimum representation under a given dictionary, which may treat the structured sparse noise as inherent low‐rank components. In addition, the SLRR framework cannot make full use of the spatial information. In this study, a framework named subspace representation with low‐rank constraint and spatial‐spectral total variation is proposed for HSI restoration. In which, an artificial rank constraint is introduced to control the rank of the representation result, which can improve the removal of the structured sparse noise and exploit the intrinsic structure of spectral space more effectively. Meanwhile, the spatial‐spectral total variation regularisation is applied to enhance the spatial and spectral smoothness. Several experiments conducted in simulated and real HSI datasets demonstrate that the proposed method can achieve a state‐of‐the‐art performance both in visual quality and quantitative assessments. [ABSTRACT FROM AUTHOR]

Extending pioneering work by Weinberg, Conrad, McCleary, and others, we provide a systematic way of relating spaces of right orders on a partially ordered group, on the one hand, and spectral spaces of free lattice-ordered groups, on the other. The aim of the theory is to pave the way for further fruitful interactions between the study of right orders on groups and that of lattice-groups. Special attention is paid to the important case of orders on groups. [ABSTRACT FROM AUTHOR]

In this paper, we prove new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of (quasi-) compact open sets as objects and spectral mappings respecting those decent lumps and satisfying a boundedness condition as morphisms. Furthermore, it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. This helps to understand Kolmogorov locally small spaces and morphisms between them. We comment also on spectralifications of topological spaces.

The second spectrum Specs(M) is the collection of all second elements of M. In this paper, we study the topology on Specs(M), which is a generalization of the Zariski topology on the prime spectrum of lattice modules. Besides some properties, Specs(M) is characterized and the interrelations between the topological properties of Specs(M) and the algebraic properties of M, are studied.