Algebra Of Functions Research Articles

Algebras of symmetric and block-symmetric functions on spaces of Lebesgue measurable functions

Algebras of symmetric and block-symmetric functions on spaces of Lebesgue measurable functions

APPLICATION OF DIFFERENTIAL EQUATIONS IN VARIOUS FIELDS OF SCIENCE

The article, "Application of Differential Equations in Various Fields of Science," explores the use of differential equations for modeling economic and natural phenomena. It examines two main models of economic dynamics: the Evans model for the market of a single product, and the Solow model for economic growth.The author emphasizes the importance of proving the existence of solutions to differential equations in order to verify the accuracy of mathematical models. They also discuss the role of electronic computers in developing the theory of differential equations and its connection with other branches of mathematics such as functional analysis, algebra, and probability theory.Furthermore, the article highlights the significance of various solution methods for differential equations, including the Fourier method, Ritz method, Galerkin method, and perturbation theory.Special attention is paid to the theory of partial differential equations, the theory of differential operators, and problems arising in physics, mechanics, and technology. Differential equations are the theoretical foundation of almost all scientific and technological models and a key tool for understanding various processes in science, such as in physics, chemistry, and biology.Examples of processes described by differential equations include normalreproduction, explosive growth, and the logistic curve. Cases of using differential equations to model deterministic, finite-dimensional, and differentiable phenomena, as well as the impact of catch quotas on population dynamics, are discussed.In conclusion, the significance of differential equations for research and their role in stimulating the development of new mathematical areas is emphasized.

Von Neumann algebras of analytic functions on the unit ball

This paper is an attempt to answer the challenge posed by Ma and Zhu in [3]. We provide examples of von Neumann algebras of holomorphic functions over the unit disc as well as over the unit ball. In the end, we introduce a novel multiplication operation on the set of all holomorphic functions over the unit disc, ensuring that this collection forms a ⁎-algebra of analytic functions over D.

The wrapped Fukaya category for semi-toric Calabi–Yau

We introduce the wrapped Donaldson–Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi–Yau 3-fold. We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori–Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an open Calabi–Yau manifold carrying an SYZ fibration with nice behavior at infinity. We discuss the relation of this to the algebraic vs analytic aspects of mirror symmetry.

Tractability of sampling recovery on unweighted function classes

It is well-known that the problem of sampling recovery in the L 2 L_2 -norm on unweighted Korobov spaces (Sobolev spaces with mixed smoothness) as well as classical smoothness classes such as Hölder classes suffers from the curse of dimensionality. We show that the problem is tractable for those classes if they are intersected with the Wiener algebra of functions with summable Fourier coefficients. In fact, this is a relatively simple implication of powerful results from the theory of compressed sensing. Tractability is achieved by the use of non-linear algorithms, while linear algorithms cannot do the job.

Characters of [formula omitted] and vertex operators

In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group GLn(Fq). Green's theory of GLn(Fq) is recovered and enhanced under the realization of the Grothendieck ring of representations RG=⨁n≥0R(GLn(Fq)) as two isomorphic Fock spaces associated to two infinite-dimensional F-equivariant Heisenberg Lie algebras hˆF‾ˆq and hˆF‾q, where F is the Frobenius automorphism of the algebraically closed field F‾q. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the current approach is a simpler identification of the Fock space RG as the Hall algebra of symmetric functions via vertex operator calculus, and another is that we are able to compute in general the character table, where Green's degree formula is demonstrated as an example.

Ecosystem-based spatial modeling: Assessing the supply of hydrological services in a watershed in Southern Brazil

Hydrologic services describe the way terrestrial ecosystems affect freshwater resources. This study employed an ecosystem-based approach and spatial models to assess the significance of ecosystems in providing hydrological services in a Watershed in southern Brazil. The environmental systems and hydrological services were characterized and evaluated for the significance weights assignment by expert opinion dynamics. Spatial and tabular data generated were applied in an ecosystem services model (InVEST-Integrated Valuation of Ecosystem Services and Trade-offs) for a relative importance analysis. Map algebra was performed to integrate relative importance with runoff values, resulting in an importance index (II), which enabled the identification of the most important ecosystems for the hydrological services provision. Among the environmental systems, riparian forests (0.2457), forests (0.2455), wetlands (0.2332), and fields with livestock (0.2291) demonstrated the highest II values. Ranking environmental systems and their vegetation cover based on their performance in terms of eco-hydrological functions and hydrological services has the potential to support management actions on land use in the pursue of sustainability regarding water resources use.

Investigating Landsat data for continuous monitoring of ground electrical conductivity: implication for propagation at medium frequency in Ondo West

Ground electrical conductivity (GEC) has many applications, which makes it worthy of continuous research. Apart from the configurations of the radio transmitter and receiver, it is a major determinant of the electric field intensity of radio waves in the Medium Frequency (MF) band. It is usually measured in such a way that both the researcher and the measuring instrument are in direct contact with the ground, and measurements are made at some predetermined constant intervals to ensure good spatial coverage and even spatial distribution. The direct method becomes cumbersome when the field of study is large because the GEC over the licensed coverage area must be known such that electric field intensity can be predicted. At the planning stage, this ensures all parts of the licensed region are reached by useful signals that can suppress interference at all times and seasons. The study is motivated by the possibility of using Landsat to estimate GEC. This study utilizes Landsat images to first estimate the normalized difference salinity index and then, with the aid of map algebra, generate another raster 2 whose pixels' values are the GEC. The study covers a landmass of 967 km in the Ondo West Local Government Area (OWLGA). During analysis, the GEC was divided into five classes. The results reveal that the ranges of very low, low, moderate, high, and very high GEC are 0.124 to 0.437, 0.438 to 0.937, 0.938 to 1.635, 1.636 to 2.524, and 2.525 to 3.600 mS/m, which have mean values of 0.346 ± 0.109, 0.754 ± 0.103, 1.071 ± 0.089, 1.351 ± 2 0.067, and 1.564 ± 0.106 mS/m, and are dispersed over 176.45, 202.24, 214.14, 374.17, and 228.56 km , respectively. The values of GEC, their respective coverage, and spatial distribution recorded in this study are strong enough to sustain the propagation of useful electric field intensity over the entire landmass of Ondo West Local Government if a medium wave is established. The study recommends that the change in GEC with time in this field be studied.

Characterization of Anisotropic Salt Weathering through Nondestructive Techniques Mapping Using a GIS Environment.

Doctrinal texts on architectural heritage conservation emphasize the importance of fully understanding the structural and material characteristics and utilizing information systems. Photogrammetry allows for the generation of detailed, geo-referenced Digital Elevation Models of architectural elements at a low cost, while GIS software enables the addition of layers of material characteristic data to these models, creating different property maps that can be combined through map algebra. This paper presents the results of the mechanical characterization of materials and salt-related decay forms of the polygonal apse of the 13th-century monastery of Santa María de Bonaval (Guadalajara, Spain), which is primarily affected by salt crystallization. Rock strength is estimated using on-site nondestructive testing (ultrasound pulse velocity and Leeb hardness). They are mapped and combined through map algebra to derive a single mechanical soundness index (MSI) to determine whether the decay of the walls could be dependent on the orientation. The presented results show that salt decay in the building is anisotropic, with the south-facing side of the apse displaying an overall lower MSI than the others. The relative overheating of the south-facing side of the apse enhances the effect of salt crystallization, thereby promoting phase transitions between epsomite and hexahydrite.

LLT polynomials in the Schiffmann algebra

Abstract We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies Λ ⁢ ( X m , n ) ⊂ E \Lambda(X^{m{,}n})\subset\mathcal{E} of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for ∇ m s λ \nabla^{m}s_{\lambda} which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one.

DISCONTINUOUS HOMOMORPHISMS OF WITH

AbstractAssume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$ -morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P-generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$ -linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $ , into the Esterle algebra of formal power series. Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X, there exists a discontinuous homomorphism of $C(X)$ , the algebra of continuous real-valued functions on X.

Mixed Hodge Structures on Alexander Modules

Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let U U be a smooth connected complex algebraic variety and let f : U → C ∗ f\colon U\to \mathbb {C}^* be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C ∗ \mathbb {C}^* by f f gives rise to an infinite cyclic cover U f U^f of U U . The action of the deck group Z \mathbb {Z} on U f U^f induces a Q [ t ± 1 ] \mathbb {Q}[t^{\pm 1}] -module structure on H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) . We show that the torsion parts A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) of the Alexander modules H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) carry canonical Q \mathbb {Q} -mixed Hodge structures. We also prove that the covering map U f → U U^f \to U induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when f : U → C ∗ f\colon U\to \mathbb {C}^* is proper, we prove the semisimplicity and purity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , and we compare our mixed Hodge structure on A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) with the limit mixed Hodge structure on the generic fiber of f f .

Distributive ℓ-pregroups: Generation and decidability

We show that every distributive lattice-ordered pregroup can be embedded into a functional algebra over an integral chain, thus improving the existing Cayley/Holland-style embedding theorem. We use this result to show that the variety of all distributive lattice-ordered pregroups is generated by the functional algebra on the integers. Finally, we show that the equational theory of the variety is decidable.

Integration of Remote Sensing and Aerogeophysical Data Applied in the Geological Characterization of the Ipueira-Medrado Segment, Andorinhas/BA

The segment of the Ipueira-Medrado mines constitutes two of the main chromium mines operated by Companhia Ferro-ligas da Bahia (Ferbasa). The research area is located in the municipality of Andorinhas, in the north center of the state of Bahia, in the Piemonte Norte Do Itapicuru region. Therefore, the main objective of the present work is the use of Remote Sensing and Aerogeophysics data to elaborate the geological map of the study area and highlight new target areas for mineral research. The Digital Image Processing (PDI) stage was carried out in an environment in the ENVI® 5.3 software for the scenes of the optical Alos and radar images, Palsar. The treatment of magnetic and radiometric data was carried out in the environment of the software Oasis Montaj 9.10 by Geosoft®. In the Geographic Information System (GIS) platform of ArcGis 10.7.1®, multisource data was integrated. This integration supported the screening of information from local lithologies and the identification of four lithological units: Sienito Itiúba, Santa Luz Unit, São Bento Unit and Jacurici Valley Complex. In this last unit, a combined analysis between Map Algebra and Fuzzy logic was applied to all input raster data, in order to mark out a zoning of potential areas for the occurrence of mafic-ultramafic rocks. In the end, a good relationship was noted between the results obtained with models that already exist in the bibliography. In view of this, the application of the methodology developed in this research requires more robustness for the local geological knowledge, as it will also help in the development of new mineral research endeavors.

A 𝐶*-algebra of entire functions

We study the maximal abelian von Neumann algebra corresponding to L ∞ ( R ) L^\infty (\mathbb {R}) via the Bargmann transform. It is naturally an algebra of operators on the Fock space F 2 F^2 , but it can also be realized as a function algebra contained in F 2 F^2 . This provides an interesting example of a C ∗ C^* algebra whose elements are analytic functions.

Interval observer design for nonlinear discrete‐time dynamic systems

SummaryThe problem of interval observer design for systems described by nonlinear models with uncertainties is considered. The problem is solved based on special mathematical technique (the algebra of functions) which allows obtaining solution for systems containing non‐smooth nonlinearities. The relations to design interval observer insensitive or having minimal sensitivity to the disturbance and estimating the prescribed nonlinear function of the state vector of the original system are derived. Theoretical results are illustrated by example.

Derivations in disjointly complete commutative regular algebras

We show that any nonexpansive derivation on a subalgebra of a dis-jointly complete commutative regular algebra extends up to a derivation on . For an algebra of functions , continuous on a dense open subset of Stone compact X, we establish that the lack of nontrivial derivation is equivalent to σ-distributivity of the Boolean algebra of clopen subsets of X. The field is an arbitrary normed field of charachteristic zero containing a complete non-discrete subfield. Our work is motivated by two seemingly unrelated problems due to Ayupov [2] and Wickstead [32].

A mechanistic approach to weighting edge-effects in landscape connectivity assessments

ContextUnderstanding landscape functional connectivity is critical for nature conservation in fragmented landscapes. Spatially explicit graph-theoretical approaches to assessing landscape connectivity have provided a promising framework for capturing functional components driving connectivity at the landscape scale. However, existing weighting schemes used to parameterise functional connectivity in graph theory-based methods are limited with respect to their ability to capture patch-level characteristics relevant to habitat use such as edge-effects.ObjectivesWe set out to develop a new approach to weighting habitat connectivity as a function of edge-effects exerted by non-habitat patches through better delineation of edge-interior habitat transitions at the patch-level and parameterization of intra-patch movement cost at the landscape scale.MethodsWe leverage the use of raster surfaces and area-weighted exponential kernels to operationalize a mechanistic approach to computing spatially explicit edge surfaces. We integrate map algebra, graph theory and landscape resistance methods to capture connectivity for a range of species specialisms on the edge-interior spectrum. We implement our method through a set of functions in the R statistical environment.ResultThrough a real-world case study, we demonstrate that our approach, drawing on these behaviours, outperforms competing metrics when evaluating potential functional connectivity in a typically fragmented agricultural landscape. We highlight options for the optimal parameterization of graph-theoretical models.ConclusionOur method offers increased flexibility, being tuneable for interior-edge habitat transitions. This therefore represents a key opportunity that can help to re-align the fields of landscape ecology and conservation biology by reconciling patch-versus-landscape methodological stances.

Tau-functions beyond the group elements

Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a group element, i.e. satisfying the property Δ(X)=X⊗X, then their generating functions obey bilinear Hirota equations and hence are named τ-functions. However, dealing with group elements is not always easy, especially for non-commutative algebras of functions, and this slows down the development of τ-function theory and the study of integrability properties of non-perturbative functional integrals. A simple way out is to use arbitrary elements of the universal enveloping algebra, and not just the group elements. Then the Hirota equations appear to interrelate a whole system of generating functions, which one may call generalized τ-functions. It was recently demonstrated that this idea can be applicable even to a somewhat sophisticated case of the quantum toroidal algebra. We consider a number of simpler examples, including ordinary and quantum groups, to explain how the method works and what kind of solutions one can obtain.

Homological quantum mechanics

We provide a formulation of quantum mechanics based on the cohomology of the Batalin-Vilkovisky (BV) algebra. Focusing on quantum-mechanical systems without gauge symmetry we introduce a homotopy retract from the chain complex of the harmonic oscillator to finite-dimensional phase space. This induces a homotopy transfer from the BV algebra to the algebra of functions on phase space. Quantum expectation values for a given operator or functional are computed by the function whose pullback gives a functional in the same cohomology class. This statement is proved in perturbation theory by relating the perturbation lemma to Wick’s theorem. We test this method by computing two-point functions for the harmonic oscillator for position eigenstates and coherent states. Finally, we derive the Unruh effect, illustrating that these methods are applicable to quantum field theory.