Algebra Of Functions Research Articles

Origin of model fractional Chern insulators in all topological ideal flatbands: Explicit color-entangled wave function and exact density algebra

Origin of model fractional Chern insulators in all topological ideal flatbands: Explicit color-entangled wave function and exact density algebra

The separating variety for matrix semi-invariants

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V, and let k[V]G be the corresponding algebra of invariant polynomial functions. A separating set S⊆k[V]G is a set of polynomials with the property that for all v,w∈V, if there exists f∈k[V]G separating v and w, then there exists f∈S separating v and w.In this article we consider the action of G=SL2(C)×SL2(C) on the C-vector space M2,2n of n-tuples of 2×2 matrices by multiplication on the left and the right. Minimal generating sets Sn of C[M2,2n]G are known, and |Sn|=124(n4−6n3+23n2+6n). In recent work, Domokos [8] showed that for all n≥1, Sn is a minimal separating set by inclusion, i.e. that no proper subset of Sn is a separating set. This does not necessarily mean that Sn has minimum cardinality among all separating sets for C[M2,2n]G. Our main result shows that any separating set for C[M2,2n]G has cardinality ≥5n−9. In particular, there is no separating set of size dim⁡(C[M2n]G)=4n−6 for n≥4. Further, S4 has indeed minimum cardinality as a separating set, but for n≥5 there may exist a smaller separating set than Sn. We also consider the action of G=SLl(C) on Ml,n by left multiplication. In that case the algebra of invariants has a minimum generating set of size (nl) (the l×l minors of a generic matrix) and dimension ln−l2+1. We show that a separating set for C[Ml,n]G must have size at least (2l−2)n−2(l2−l). In particular, C[Ml,n]G does not contain a separating set of size dim⁡(C[Ml,n]G) for l≥3 and n≥l+2. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem.

Exotic cyclic cohomology classes and Lipschitz algebras

We study the noncommutative geometry of algebras of Lipschitz continuous and H\"older continuous functions where non-classical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic cohomology classes that pair non-trivially with higher algebraic $K$-theory yet vanish when restricted to the algebra of smooth functions. Said cohomology classes provide additional methods to extract numerical invariants from Connes-Karoubi's relative sequence in $K$-theory.

Integrated Spatiotemporal Analysis of Vegetation Condition in a Complex Post-Mining Area: Lignite Mine Case Study

The motivation for this study arises from the need to monitor the condition of a rehabilitated post-mining areas even decades after the end of the recovery phase. This can be facilitated with satellite derived spectral vegetation indices and Geographic Information System (GIS) based spatiotemporal analysis. The study area described in this work is located in Western Poland and has unique characteristics, as it was subjected to the combined underground and open pit mining of lignite deposits that had been shaped by glaciotectonic processes. The mining ended in early 1970’ties and the area was subjected to reclamation procedures that ended in the 1980’ties. We used the Normalized Difference Vegetation Index (NDVI) and the Enhanced Vegetation Index (EVI) spectral indices derived from Sentinel-2 data for the 2015–2022. period. Then, we applied a combination of GIS-based map algebra statistics (local, zonal and combinatorial) and GI* spatial statistics (hot spot and temporal hot spot) for a complex analysis and assessment of the vegetation cover condition in a post-mining area thought to be in the rehabilitated phase. The mean values of NDVI and EVI for the post-mining study area range from 0.48 to 0.64 and 0.24 to 0.31 and are stable in the analyzed 8 year period. This indicates general good condition of the vegetation and post-recovery phase of the area of interest. However, the combination of spatiotemporal analysis allowed us to identify statistically significant clusters of higher and lower values of the vegetation indices and change of vegetation cover classes on 3% of the study area. These clusters signify the occurrence of local processes such as, the encroachment of aquatic vegetation in waterlogged subsidence basins, and growth of low vegetation in old pits filled with waste material, barren earth zones on external waste dumps, as well as present-day forest management activities. We have confirmed that significant vegetation changes related to former mining occur even five decades later. Furthermore, we identified clusters of the highest values that are associated with zones of older, healthy forest and deciduous tree species. The results confirmed applicability of Sentinel-2 derived vegetation indices for studies of post-mining environment and for the detection of local phenomena related to natural landscaping processes still taking place in the study area. The methodology adopted for this study consisting of a combination of GIS-based data mining methods can be used in combination or separately in other areas of interest, as well as aid their sustainable management.

Stationary phase method, powers of functions, and applications to functional analysis

The utility of the (weighted) van der Corput inequalities or of the stationary phase method is illustrated with various examples borrowed from: differentiability issues (Riemann’s function and related); functional analysis on Banach spaces or algebras of analytic functions (composition operators); and local Banach space geometry (Schäffer’s problem).

Detection of small-scale landscape elements with remote sensing

Abstract. Landscape elements located on agricultural fields or on their edges play a crucial role in the biodiversity of agricultural land. The landscape elements’ database in Estonia is updated in accordance with the applications of the field owners, and usually it does not represent a real situation of the landscape elements on the field. Hence, the analysis and control over landscape elements are limited. The main aim of this study is to create a methodology to map landscape elements in Estonia with remote sensing data. The first method was created considering the importance of computational efficiency and therefore fast and non-complex map algebra solution was developed. The second, more precise but more computationally expensive way to map landscape elements, was the object-based image analysis method utilizing machine learning classification. Both methods displayed high overall accuracies, but users’ and producers’ accuracies were lower. Taking into account the computational time and accuracy, it was concluded that the map algebra method is better suitable for fast landscape elements’ detection. However, the object-based image analysis method is more suitable for identifying more exact classes of landscape elements.

DreamCoder: growing generalizable, interpretable knowledge with wake-sleep Bayesian program learning.

Expert problem-solving is driven by powerful languages for thinking about problems and their solutions. Acquiring expertise means learning these languages-systems of concepts, alongside the skills to use them. We present DreamCoder, a system that learns to solve problems by writing programs. It builds expertise by creating domain-specific programming languages for expressing domain concepts, together with neural networks to guide the search for programs within these languages. A 'wake-sleep' learning algorithm alternately extends the language with new symbolic abstractions and trains the neural network on imagined and replayed problems. DreamCoder solves both classic inductive programming tasks and creative tasks such as drawing pictures and building scenes. It rediscovers the basics of modern functional programming, vector algebra and classical physics, including Newton's and Coulomb's laws. Concepts are built compositionally from those learned earlier, yielding multilayered symbolic representations that are interpretable and transferrable to new tasks, while still growing scalably and flexibly with experience. This article is part of a discussion meeting issue 'Cognitive artificial intelligence'.

Analysis of Morphometric Traits and Ecological Threats in Stubbs Creek Forest Reserve, Akwa Ibom State, Nigeria

This study was conducted in Stubbs Creek Forest Reserve, (SCFR) Akwa Ibom State, Nigeria. The aim was to analyse the morphometric traits of this area, using some relevant terrain attributes (elevation, slope and aspect) in ascertaining areas threatened by significant ecological problems such as flood, erosion and others. In terms of method of study, a digital elevation model (DEM) was purchased as a raster with longitude and latitude; geo-referenced, but re-projected to WGS, UTM zone 32 N in order to enhance data integration into the GIS interfaces. The surface operation tool of ArcGIS 9.3 spatial analyst extension was used to extract the different layers being elevation, slope and aspect. These were subjected to reclassification function in the same software to harmonise the units of measurement for integration into the same analysis environment. After reclassification, the layers were taken to map algebra environment to model erosion, flood hazards and possibility of siltation of streams, tributaries and wetlands in this area. Consequently, slope, elevation and aspect were the surface layers utilized for the single output map algebra. The result confirmed the elevation of this area to be low-lying altitude above the mean sea level with Open Water, Sea Level Terrain, Beach Ridge Complex, Alluvial Plain Sands with Mangroves and Gently Undulating Coastal Plain Sands as the five significant terrain classes in SCFR. The result further revealed that areas without adequate vegetation cover recorded series of ecological hitches than areas with adequate vegetation cover. Specifically, the identified problems included water pollution, coastal erosion, land degradation, soil contamination, flood, habitat fragmentation, distortion of wetlands and siltation of streams, rills and tributaries which restriction of anthropogenic activities and adequate restoration measures in SCFR are strongly recommended for its sustainability.

English

In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes the form of a differential graded Lie algebra of graphs, denoted $\mathsf{fGC}_2$, together with an injective morphism towards the Chevalley-Eilenberg complex associated with the Schouten algebra. The latter morphism is given by explicit local formulas making implicit use of the supergeometric interpretation of the Schouten algebra as the algebra of functions on a graded symplectic manifold of degree $1$. The ambition of the present work is to generalise Kontsevich's construction to graded symplectic manifolds of arbitrary degree $n\geq1$. The corresponding graph model is given by the full Kontsevich graph complex $\mathsf{fGC}_d$ where $d=n+1$ stands for the dimension of the associated AKSZ type $\sigma$-model. This generalisation is instrumental to classify universal structures on graded symplectic manifolds. In particular, the zeroth cohomology of the full graph complex $\mathsf{fGC}_{d}$ is shown to act via $\mathsf{Lie}_\infty$-automorphisms on the algebra of functions on graded symplectic manifolds of degree $n$. This generalises the known action of the Grothendieck-Teichm\"{u}ller algebra $\mathfrak{grt}_1\simeq H^0(\mathsf{fGC}_2)$ on the space of polyvector fields. This extended action can in turn be used to generate new universal deformations of Hamiltonian functions, generalising Kontsevich flows on the space of Poisson manifolds to differential graded manifolds of higher degrees. As an application of the general formalism, universal deformations of Courant algebroids via trivalent graphs are presented.

Graphs and Algebras of Symmetric Functions

We describe an algebraic technique for operating with power series whose coefficients are represented by integrals of symmetric functions fn defined on the Cartesian powers Ωn of a set Ω with a measure μ. Moreover, each of the coefficient functions fn is obtained by means of a special mapping from graphs with n labeled vertices belonging to a fixed class. This technique has application to equilibrium statistical mechanics and to problems of enumeration of graphs.

Poisson Trace Orders

Abstract The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley–Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3D and 4D Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras.

On isomorphisms of algebras of entire symmetric functions on Banach spaces

We consider the problem of isomorphism of Fréchet algebras of symmetric entire functions of bounded type on complex Banach spaces. We show that two such Fréchet algebras are isomorphic if semigroups of symmetries on underlying Banach spaces satisfy some natural conditions. We apply this result to Fréchet algebras of symmetric entire functions on Cartesian powers of complex Banach spaces of Lebesgue integrable in a power p∈[1,+∞) functions and of Lebesgue measurable essentially bounded functions on [0,1].

Fuzzy hyperspheres via confining potentials and energy cutoffs

We simplify and complete the construction of fully O(D)-equivariant fuzzy spheres , for all dimensions , initiated in Fiore and Pisacane (2018 J. Geom. Phys. 132 423–51). This is based on imposing a suitable energy cutoff on a quantum particle in subject to a confining potential well V(r) with a very sharp minimum on the sphere of radius r = 1; the cutoff and the depth of the well diverge with . As a result, the noncommutative Cartesian coordinates generate the whole algebra of observables on the Hilbert space ; applying polynomials in the to any we recover the whole . The commutators of the are proportional to the angular momentum components, as in Snyder noncommutative spaces. , as carrier space of a reducible representation of O(D), is isomorphic to the space of harmonic homogeneous polynomials of degree Λ in the Cartesian coordinates of (commutative) , which carries an irreducible representation of . Moreover, is isomorphic to . We resp. interpret , as fuzzy deformations of the space of (square integrable) functions on S d and of the associated algebra of observables, because they resp. go to as Λ diverges (with fixed). With suitable , in the same limit goes to the (algebra of functions on the) Poisson manifold ; more formally, yields a fuzzy quantization of a coadjoint orbit of that goes to the classical phase space .

An accurate and efficient multiphase solver based on THINC scheme and adaptive mesh refinement

This paper presents an accurate and efficient multiphase solver which is developed based on THINC/QQ (THINC method with quadratic surface representation and Gauss quadrature) scheme and AMR (adaptive mesh refinement) technique in the framework of OpenFOAM. In order to make consistent interpolation after the mesh refinement, a novel algorithm is proposed for the algebraic mapping of the volume fraction field in the child cells, which significantly improves the geometric faithfulness of mapped interface particularly for curved surface. Second-order schemes are used for the spatial and temporal discretization of momentum equations which further enhances the solution accuracy compared with interFoam. Using THINC/QQ scheme with AMR technique in the multiphase simulation, the so-called thincFoam delivers high fidelity solutions in the vicinity of interface while largely improving the computational efficiency. As verified by various benchmark tests in 2D and 3D, such as bubble rising, dam-break and propagation of solitary wave, it convinces that the numerical results predicted by thincFoam are in excellent agreement with experimental data and other reference solutions. Thus, the present solver possesses a promising prospect in capturing complicated interfaces undergoing large deformations and topological changes for the simulation of nonlinear interaction between multiphase flows with solid structures.

Virtual Sensors for Nonlinear Discrete-Time Dynamic Systems

The problem of virtual sensor design for nonlinear systems under the disturbance is investigated. Two different mathematical techniques are used to solve the problem: the algebra of functions and the logic-dynamic approach. The first one allows obtaining a general solution while the second one produces a solution for nonlinear systems by linear algebra methods. The virtual sensors are designed to be insensitive to the disturbance based on invariant functions. They estimate the prescribed function of the original system state vector. The practical example illustrates theoretical results.

Homomorphisms on algebras of analytic functions on non-symmetrically regular spaces

Homomorphisms on algebras of analytic functions on non-symmetrically regular spaces

Algebras of Smooth Functions and Holography of Traversing Flows

Let X be a smooth compact manifold and v a vector field on X which admits a smooth function f : X ! R such that df(v) > 0. Let @X be the boundary of X. We denote by C1(X) the algebra of smooth functions on X and by C1(@X) the algebra of smooth functions on @X. With the help of (v; f), we introduce two subalgebras A(v) and B(f) of C1(@X) and prove (under mild hypotheses) that C1(X) _ A(v) ^B(f), the topological tensor product. Thus the topological algebras A(v) and B(f), viewed as boundary data, allow for a reconstruction of C1(X). As a result, A(v) and B(f) allow for the recovery of the smooth topological type of the bulk X.

Daugavet property of Banach algebras of holomorphic functions and norm-attaining holomorphic functions

We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball BE of a Banach space E lack weak⁎-strongly exposed points. Consequently, we obtain that some Banach algebras of holomorphic functions on an arbitrary Banach space have the Daugavet property which extends the observation of P. Wojtaszczyk [56]. Moreover, we present a new denseness result by proving that the set of norm-attaining vector-valued holomorphic functions on the open unit ball of a dual Banach space is dense provided that its predual space has the metric π-property. Besides, we obtain several equivalent statements for the Banach space of vector-valued homogeneous polynomials to be reflexive, which improves the result of J. Mujica [47], J. A. Jaramillo and L. A. Moraes [39]. As a byproduct, we generalize some results on polynomial reflexivity due to J. Farmer [35].

A simulator for charged particle guns and lenses

An algorithm is presented that extends the application of ray optics to charged particle guns, where particles are emitted with a wide range of angles and relative energies. The extension is made possible by replacing the ray tangent as a coordinate with the directional sine, which remains finite for all take-off angles, and if necessary, by simulating the ray optics for multiple segments of the energy distribution. Consequently, an electron gun comprising a cathode, an extractor, and an anode, for example, may be characterized by its focal length, the location of its image plane, the magnification of the emission spot at this plane, and the primary and higher-order aberrations of the module, which reshape this spot. This description is realized by modeling the particle trajectories as a power series in their initial conditions and by using standard algebraic techniques, namely, the binomial and Taylor expansions, to transform the relativistic Newton’s equation, which describes the evolution of the trajectories in the electromagnetic fields of the gun, into a differential equation for the evolution of the series coefficients. In addition to providing the optical parameters, the series coefficients produce an algebraic map between the particle coordinates along the gun axis and their initial values at the plane of emission, thereby circumventing the need to launch a multitude of trajectories in a particle simulator to model the source that the gun presents to the optical elements downstream. The algorithm may be enhanced in the future by incorporating some effects of space charge from the emission current.

Finite-Dimensional Approximations and Semigroup Coactions for Operator Algebras

Abstract The residual finite-dimensionality of a ${\textrm{C}}^{\ast }$-algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this paradigm to general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge in this greater generality, we exhibit novel tools for constructing finite-dimensional approximations. One such tool is a notion of a residually finite-dimensional coaction of a semigroup on an operator algebra, which allows us to construct finite-dimensional approximations for operator algebras of functions and operator algebras of semigroups. Our investigation is intimately related to the question of when residual finite-dimensionality of an operator algebra is inherited by its maximal ${\textrm{C}}^{\ast }$-cover, which we establish in many cases of interest.