Cevian properties in ideal lattices of Abelian ℓ-groups

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.

Stone Duality for Kolmogorov Locally Small Spaces

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.

Human Texture Vision as Multi-Order Spectral Analysis

Texture information plays a critical role in the rapid perception of scenes, objects, and materials. Here, we propose a novel model in which visual texture perception is essentially determined by the 1st-order (2D-luminance) and 2nd-order (4D-energy) spectra. This model is an extension of the dimensionality of the Filter-Rectify-Filter (FRF) model, and it also corresponds to the frequency representation of the Portilla-Simoncelli (PS) statistics. We show that preserving two spectra and randomizing phases of a natural texture image result in a perceptually similar texture, strongly supporting the model. Based on only two single spectral spaces, this model provides a simpler framework to describe and predict texture representations in the primate visual system. The idea of multi-order spectral analysis is consistent with the hierarchical processing principle of the visual cortex, which is approximated by a multi-layer convolutional network.

Features of solving the direct and inverse scattering problems for two sets of monopole scatterers

Research presented in this paper was initiated by the publication [A. D. Agaltsov and R. G. Novikov, Examples of solving the inverse scattering problem and the equations of the Veselov–Novikov hierarchy from the scattering data of point potentials, Russian Math. Surveys 74 (2019), 3, 373–386] and is based on its results. Two sets of the complex monopole scattering coefficients are distinguished among the possible values of these coefficients for nonabsorbing inhomogeneities. These sets differ in phases of the scattering coefficients. In order to analyze the features and possibilities of reconstructing the inhomogeneities of both sets, on the one hand, the inverse problem is solved for each given value of the monopole scattering coefficient using the Novikov functional algorithm. On the other hand, the scatterer is selected in the form of a homogeneous cylinder with the monopole scattering coefficient that coincides with the given one. The results obtained for the monopole inhomogeneity and for the corresponding cylindrical scatterer are compared in the coordinate and spatial-spectral spaces. The physical reasons for the similarities and differences in these results are discussed when the amplitude of the scattering coefficient changes, as well as when passing from one set to another.

CHOICE-FREE STONE DUALITY

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.