Algebra Of Functions Research Articles

Book review: “Banach Function Algebras, Arens Regularity, and BSE Norms” by Harold Garth Dales and Ali Ülger

To the memory of Garth Dales.

On monomorphisms of hypergraphic automata

Hypergraphic automata are automata, state sets and output symbol sets of which are hypergraphs, being invariant under actions of transition and output functions. Universally attracting objects in the category of hypergraphic automata are called universal hypergraphic automata. The semigroups of input symbols of such automata are derivative algebras of mappings for such automata. So their properties are interconnected with properties of the algebraic structures of the automata. This paper describes the structure of monomorphisms of such automata and their semigroups of input signals.

On the classification of function algebras on subvarieties of noncommutative operator balls

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.

Automorphism Groups of Deformations and Quantizations of Kleinian Singularities

AbstractIt is known that, for the algebra of functions on a Kleinian singularity, the parameter space of deformations and the parameter space of quantizations coincide. We prove that, for a Kleinian singularity of type $$\textbf{A}$$ A or $$\textbf{D}$$ D , isomorphisms between the quantizations are essentially the same as Poisson isomorphisms between deformations. In particular, the group of automorphisms of the deformation and the quantization corresponding to the same deformation parameter are isomorphic. We additionally describe the groups of automorphisms as abstract groups: for type $$\textbf{A}$$ A they have an amalgamated free product structure, for type $$\textbf{D}$$ D they are subgroups of the groups of Dynkin diagram automorphisms. For type $$\textbf{D}$$ D we also compute all the possible affine isomorphisms between deformations; this was not known before.

A generalization of prime ideals in MV-algebras

In this paper, we introduce a special type of prime ideals in [Formula: see text]-algebras. We define the concept of a [Formula: see text]-prime ideal, which is a proper ideal [Formula: see text] that preserves the property of being prime with respect to the ideal [Formula: see text]. If [Formula: see text] is not a subset of [Formula: see text], then we call the [Formula: see text]-prime ideal an [Formula: see text]-prime ideal. We investigate the relationships between these ideals and their connections to minimal prime ideals and maximal ideals. Additionally, we explore various properties such as extension, transitivity, meet, join, and annihilator. Finally, we study prime ideals in [Formula: see text]-algebras of continuous functions.

Differential geometry and general relativity with algebraifolds

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of algebraifold, by which we mean a commutative algebra A for which the module of derivations of A is finitely generated projective. Over R as the base ring, this class of algebras includes the algebra C∞(M) of smooth functions on a manifold M, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on M, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion M→N makes C∞(M) into an algebraifold with C∞(N) as the base ring. Over any field k of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.

Preparation of a Map of Potential Areas For the Installation of a Sewage Treatment Plant in the Porto Murtinho/MS Region

Objective: The main objective is to prepare a thematic map to identify potential areas for the installation of a Sewage Treatment Station (ETE). Theoretical Framework: This study addresses the importance of selecting locations for the installation of ETEs, which prioritize environmental and social aspects, in which geotechnologies are presented as valuable tools for efficient and integrated analysis. Method: The methodology of this research was based on the acquisition and processing of geospatial and environmental data, complemented by the generation of thematic maps such as watercourses, areas subject to flooding, roads, land use and occupation, slope, soils and distances from the urban perimeter. Data integration through map algebra was performed in the QGIS software, in addition, WRPLOT View was used for the directional analysis of the winds in Porto Murtinho/MS. Results and Discussion: The results of this study indicate that the areas with the greatest potential for installation are mostly located within 1 (one) kilometer of the urban perimeter security strip. The analysis of the prevailing winds revealed that they form a corridor in the north/south direction, suggesting that the regions east of the city are more suitable for the implementation of the treatment system. Research Implications: The multifaceted approach, considering environmental and operational factors, makes it possible to identify suitable areas for ETEs. Originality/Value: This study was developed, highlighting that limiting factors are essential in identifying areas unsuitable for installing the structure, while non-limiting maps are essential for classifying the installation potential.

Bundles of Holomorphic Function Algebras on Subvarieties of the Noncommutative Ball

Bundles of Holomorphic Function Algebras on Subvarieties of the Noncommutative Ball

The BSE property for some vector-valued Banach function algebras

The BSE property for some vector-valued Banach function algebras

Soil erosion risk in the Mocache Canton applying multicriteria analysis and geographic information systems

Soil erosion is one of the main and most widespread types of soil degradation, determined by environmental and anthropogenic factors, which significantly impact the physical, chemical and biological properties of the soil, favoring its degradation. Consequently, through this research we sought to evaluate the risk of soil erosion in the canton of Mocache through the application of multicriteria analysis techniques and the use of geographic information systems (GIS). The CORINE model included several criteria for the zoning of potential and actual erosion risk, including erosivity, defined by the Modified Fournier Index (MFI) and the Bagnouls-Gaussen Aridity Index (BGI); erodibility, composed of soil texture, stoniness and depth; soil occupation; slope-orientation; and vegetation cover, through the estimation of the Normalized Difference Vegetation Index (NDVI). An expert consultation supported by Analytic Hierarchy Process (AHP) was applied to define the contribution of each criterion to erosion risk. The resulting layers were combined using a weighted relationship (map algebra) within the GIS environment. It was obtained that slope-orientation and erosivity explain a large part of the erosion risk, with a weighting of 42.46% and 27.48% in order, according to the expert evaluation and the AHP. The high erosivity was contributed by the predominant climatic factors: intense precipitation and high temperatures. In addition, it was found that the potentialand current erosion risk represent 13.1% (7,244 ha) and 10.7% (5,905 ha) of the territorial extension. It is concluded that physical-natural, meteorological factors and anthropogenic activities contribute to the risk of potential and current soil erosion in Mocache canton

Override and restricted union for partial functions

The override operation ⊔\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqcup $$\\end{document} is a natural one in computer science, and has connections with other areas of mathematics such as hyperplane arrangements. For arbitrary functions f and g, f⊔g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\sqcup g$$\\end{document} is the function with domain dom(f)∪dom(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dom}\\,}}(f)\\cup {{\\,\ extrm{dom}\\,}}(g)$$\\end{document} that agrees with f on dom(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dom}\\,}}(f)$$\\end{document} and with g on dom(g)\\dom(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dom}\\,}}(g) \\backslash {{\\,\ extrm{dom}\\,}}(f)$$\\end{document}. Jackson and the author have shown that there is no finite axiomatisation of algebras of functions of signature (⊔)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\sqcup )$$\\end{document}. But adding operations (such as update) to this minimal signature can lead to finite axiomatisations. For the functional signature (⊔,\\)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\sqcup ,\\backslash )$$\\end{document} where \\\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\backslash $$\\end{document} is set-theoretic difference, Cirulis has given a finite equational axiomatisation as subtraction o-semilattices. Define f⋎g=(f⊔g)∩(g⊔f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\curlyvee g=(f\\sqcup g)\\cap (g\\sqcup f)$$\\end{document} for all functions f and g; this is the largest domain restriction of the binary relation f∪g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\cup g$$\\end{document} that gives a partial function. Now f∩g=f\\(f\\g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\cap g=f\\backslash (f\\backslash g)$$\\end{document} and f⊔g=f⋎(f⋎g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\sqcup g=f\\curlyvee (f\\curlyvee g)$$\\end{document} for all functions f, g, so the signatures (⋎)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\curlyvee )$$\\end{document} and (⊔,∩)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\sqcup ,\\cap )$$\\end{document} are both intermediate between (⊔)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\sqcup )$$\\end{document} and (⊔,\\)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\sqcup ,\\backslash )$$\\end{document} in expressive power. We show that each is finitely axiomatised, with the former giving a proper quasivariety and the latter the variety of associative distributive o-semilattices in the sense of Cirulis.

The Peter–Weyl theorem & harmonic analysis on S^n

For finite groups, the Artin–Wedderburn theorem gives a precise decomposition of the algebra of all C-valued functions into matrix algebras. Specialized to the case of cyclic groups, this produces the classical discrete Fourier transform. In this paper, we endeavor to extend these techniques to compact topological groups, proving similar harmonic decompositions on S1, S2, and S3.

Functional Train Algebras of Rank ≤ 3

Functional Train Algebras of Rank ≤ 3

Algebras of entire functions and representations of the twisted Heisenberg group

Algebras of entire functions and representations of the twisted Heisenberg group

Representations Of Lie Algebras of Vector Fields on Pairing Between Gauge Modules and Rudakov Modules

For an irreducible affine variety X over an algebraically closed field of characteristic zero, we define two new classes of modules over the Lie algebra of vector fields on X: gauge modules and Rudakov modules. These modules admit a compatible action of the algebra of functions on X. We prove general simplicity theorems for these two types of modules, demonstrating their irreducibility under specific conditions. Additionally, we establish a pairing between gauge modules and Rudakov modules, highlighting the connections and interactions between these two classes of modules. We have established that gauge modules and Rudakov modules, corresponding to simple glN -modules, remain irreducible as modules over the Lie algebra of vector fields unless they appear in the de Rham complex. Additionally, we have studied the irreducibility of tensor products of Rudakov modules, providing a comprehensive understanding of these module structures and their applications.

Environmental vulnerability applied to the territorial planning of a tropical semiarid basin.

Tropical semiarid regions are naturally prone to environmental damage. Human activity can worsen this situation. To understand how human actions affect the ecosystem, plan land use effectively, and establish targeted management practices, assessing environmental vulnerability is crucial. This study focuses on a sub-basin receiving water transfers from the São Francisco River in Brazil's semiarid region. Here, we map and evaluate how land use and occupation alter natural vulnerability. We also propose zoning strategies to support water resource management and implement sustainable development policies in the region. To achieve this, we conducted an integrated analysis of physical factors (soil types, geology, climate, vegetation, and landforms) and spatial land-use data using geographic information systems (GIS) and map algebra techniques. Map algebra allowed us to combine these various datasets within the GIS environment, enabling the creation of maps that synthesize both natural and environmental vulnerability across the study area. Following analysis of these vulnerability maps, our findings reveal a high level of vulnerability. The areas with high to very high degrees of natural vulnerability coincide with the places that have high slopes, high altitudes, Lithic Neosols, and thick vegetation. Furthermore, the interaction between environmental factors and human activity exacerbates vulnerability. Based on the environmental vulnerability assessment, we defined four environmental management zones. These zones require distinct protection measures and management approaches. As a method to potentially improve the basin's vulnerability scenario, soil conservation measures are recommended. This approach is highly relevant for managing land in tropical semiarid regions and, with adaptations to specific regional factors, can be applied globally.

An unbounded operator theory approach to lower frame and Riesz-Fischer sequences

Frames and orthonormal bases are important concepts in functional analysis and linear algebra. They are naturally linked to bounded operators. To describe unbounded operators those sequences might not be well suited. This has already been noted by von Neumann in the 1920ies. But modern frame theory also investigates other sequences, including those that are not naturally linked to bounded operators. The focus of this manuscript will be two such kind of sequences: lower frame and Riesz-Fischer sequences. We will discuss the inter-relation of those sequences. We will fill a hole existing in the literature regarding the classification of these sequences by their synthesis operator. We will use the idea of generalized frame operator and Gram matrix and extend it. We will use that to show properties for canonical duals for lower frame sequences, such as a minimality condition regarding its coefficients. We will also show that other results that are known for frames can be generalized to lower frame sequences. Finally, we show that the converse of a well-known result is true, i.e. that minimal lower frame sequences are equivalent to complete Riesz-Fischer sequences, without any further assumptions.To be able to tackle these tasks, we had to revisit the concept of invertibility (in particular for non-closed operators). In addition, we are able to define a particular adjoint, which is uniquely defined for any operator.

The method of linear-logical operators and logical equations in information extraction tasks

Relational and logical methods of knowledge representation play a key role in creating a mathematical basis for information systems. Predicate algebra and predicate operators are among the most effective tools for describing information in detail. These tools make it easy to formulate formalized information, create database queries, and simulate human activity. In the context of the new need for reliable and efficient data selection, a problem arises in deeper analysis. Subject of the study is the theory of quantum linear equations based on the algebra of linear predicate operations, the formal apparatus of linear logic operators and methods for solving logical equations in information extraction tasks. Aim of the study is a developing of a method for using linear logic operators and logical equations to extract information. This approach can significantly optimize the process of extracting the necessary information, even in huge databases. Main tasks: analysis of existing approaches to information extraction; consideration of the theory of linear logic operators; study of methods for reducing logic to an algebraic form; analysis of logical spaces and the algebra of finite predicate actions and the theory of linear logic operators. The research methods involve a systematic analysis of the mathematical structure of the algebra of finite predicates and predicate functions to identify the key elements that affect the query formation process. The method of using linear logic operators and logical equations for information extraction is proposed. The results of the study showed that the method of using linear logic operators and logical equations is a universal and adaptive tool for working with algebraic data structures. It can be applied in a wide range of information extraction tasks and prove its value as one of the possible methods of information processing. Conclusion. The paper investigates formal methods of intelligent systems, in particular, ways of representing knowledge in accordance with the peculiarities of the field of application and the language that allows encoding this knowledge for storage in computer memory. The proposed method can be implemented in the development of language interfaces for automated information access systems, in search engine algorithms, for logical analysis of information in databases and expert systems, as well as in performing tasks related to object recognition and classification.

On Singularity Properties of Word Maps and Applications to Probabilistic Waring Type Problems

We study singularity properties of word maps on semisimple Lie algebras, semisimple algebraic groups and matrix algebras and obtain various applications to random walks induced by word measures on compact p p -adic groups. Given a word w w in a free Lie algebra L r \mathcal {L}_{r} , it induces a word map φ w : g r → g \varphi _{w}:\mathfrak {g}^{r}\rightarrow \mathfrak {g} for every semisimple Lie algebra g \mathfrak {g} . Given two words w 1 ∈ L r 1 w_{1}\in \mathcal {L}_{r_{1}} and w 2 ∈ L r 2 w_{2}\in \mathcal {L}_{r_{2}} , we define and study the convolution of the corresponding word maps φ w 1 ∗ φ w 2 ≔ φ w 1 + φ w 2 : g r 1 + r 2 → g \varphi _{w_{1}}*\varphi _{w_{2}}≔\varphi _{w_{1}}+\varphi _{w_{2}}:\mathfrak {g}^{r_{1}+r_{2}}\rightarrow \mathfrak {g} . By introducing new degeneration techniques, we show that for any word w ∈ L r w\in \mathcal {L}_{r} of degree d d , and any simple Lie algebra g \mathfrak {g} with φ w ( g r ) ≠ 0 \varphi _{w}(\mathfrak {g}^{r})\neq 0 , one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking O ( d 4 ) O(d^{4}) self-convolutions of φ w \varphi _{w} . Similar results are obtained for matrix word maps. We deduce that a group word map of length ℓ \ell becomes (FRS), locally around identity, after O ( ℓ 4 ) O(\ell ^{4}) self-convolutions, for every semisimple algebraic group G _ \underline {G} . We furthermore provide uniform lower bounds on the log canonical threshold of the fibers of Lie algebra, matrix and group word maps. For the commutator word w 0 = [ X , Y ] w_{0}=[X,Y] , we show that φ w 0 ∗ 4 \varphi _{w_{0}}^{*4} is (FRS) for any semisimple Lie algebra, improving a result of Aizenbud-Avni, and obtaining applications in representation growth of compact p p -adic and arithmetic groups. The singularity properties we consider, such as the (FRS) property, are intimately connected to the point count of fibers over finite rings of the form Z / p k Z \mathbb {Z}/p^{k}\mathbb {Z} . This allows us to relate them to properties of some natural families of random walks on finite and compact p p -adic groups. We explore these connections, characterizing some of the singularity properties discussed in probabilistic terms, and provide applications to p p -adic probabilistic Waring type problems.

Quantum van Est isomorphism

Motivated by the fact that the Hopf-cyclic (co)homologies of function algebras over Lie groups and universal enveloping algebras over Lie algebras capture the Lie group and Lie algebra (co)homologies, we hereby upgrade the classical van Est isomorphism to ones between the Hopf-cyclic (co)homologies of quantized algebras of functions and quantized universal enveloping algebras, both in h -adic and q -deformation frameworks.