Maximal linked systems on families of measurable rectangles

Linked and maximal linked systems (MLS) on π -systems of measurable (in the wide sense) rectangles are considered (π-system is a family of sets closed with respect to finite intersections). Structures in the form of measurable rectangles are used in measure theory and probability theory and usually lead to semi-algebra of subsets of cartesian product. In the present article, sets-factors are supposed to be equipped with π-systems with “zero” and “unit”. This, in particular, can correspond to a standard measurable structure in the form of semialgebra, algebra, or σ-algebra of sets. In the general case, the family of measurable rectangles itself forms a π -system of set-product (the measurability is identified with belonging to a π - system) which allows to consider MLS on a given π -system (of measurable rectangles). The following principal property is established: for all considered variants of π -system of measurable rectangles, MLS on a product are exhausted by products of MLS on sets-factors. In addition, in the case of infinity product, along with traditional, the “box” variant allowing a natural analogy with the base of box topology is considered. For the case of product of two widely understood measurable spaces, one homeomorphism property concerning equipments by the Stone type topologies is established.

On certain analogues of linkedness and supercompactness

Natural generalizations of properties of the family linkedness and the topological space supercompactness are considered. We keep in mind reinforced linkedness when nonemptyness of intersection of preassigned number of sets from a family is postulated. Analogously, supercompactness is modified: it is postulated the existence of an open subbasis for which, from every covering (by sets of the subbasis), it is possible to extract a subcovering with a given number of sets (more precisely, not more than a given number). It is clear that among all families having the reinforced linkedness, one can distinguish families that are maximal in ordering by inclusion. Under natural and (essentially) “minimal”' conditions imposed on the original measurable structure, among the mentioned maximal families with reinforced linkedness, ultrafilters are certainly contained. These ultrafilters form subspaces in the sense of natural topologies corresponding conceptually to schemes of Wallman and Stone. In addition, maximal families with reinforced linkedness, when applying topology of the Wallman type, have the above-mentioned property generalizing supercompactness. Thus, an analogue of the superextension of the $T_1$-space is realized. The comparability of “Wallman”' and “Stone”' topologies is established. As a result, bitopological spaces (BTS) are realized; for these BTS, under equipping with analogous topologies, ultrafilter sets are subspaces. It is indicated that some cases exist when the above-mentioned BTS is nondegenerate in the sense of the distinction for forming topologies. At that time, in the case of “usual” linkedness (this is a particular case of reinforced linkedness), very general classes of spaces are known for which the mentioned BTS are degenerate (the cases when origial set, i.e., “unit”' is equipped with an algebra of sets or a topology).

Maximal linked systems and ultrafilters: main representations and topological properties

Questions connected with representation of the ultrafilter (UF) set for widely understood measurable space are investigated; this set is considered as a subspace of bitopological space of maximal linked systems (MLS) under equipment with topologies of Wallman and Stone types (measurable structure is defined as a π -system with “zero” and “unit”). Analogous representations connected with generalized variant of cohesion is considered also; in this variant, for corresponding set family, it is postulated the nonemptyness of intersection for finite subfamilies with power not exceeding given. Conditions of identification of UF and MLS (in the above-mentioned generalized sense) are investigated. Constructions reducing to bitopological spaces with points in the form of MLS and 𝑛-supercompactness property generalizing the “usual” supercompactness are considered. Finally, some characteristic properties of MLS and their corollaries connected with the MLS contraction to a smaller π -system are being studied. The case of algebras of sets is selected separately.

A method for quantifying, visualising, and analysing gastropod shell form

Quantitative analysis of organismal form is an important component for almost every branch of biology. Although generally considered an easily-measurable structure, the quantification of gastropod shell form is still a challenge because shells lack homologous structures and have a spiral form that is difficult to capture with linear measurements. In view of this, we adopt the idea of theoretical modelling of shell form, in which the shell form is the product of aperture ontogeny profiles in terms of aperture growth trajectory that is quantified as curvature and torsion, and of aperture form that is represented by size and shape. We develop a workflow for the analysis of shell forms based on the aperture ontogeny profile, starting from the procedure of data preparation (retopologising the shell model), via data acquisition (calculation of aperture growth trajectory, aperture form and ontogeny axis), and data presentation (qualitative comparison between shell forms) and ending with data analysis (quantitative comparison between shell forms). We evaluate our methods on representative shells of the genus Opisthostoma, which exhibit great variability in shell form. The outcome suggests that our method is a robust, reproducible, and versatile approach for the analysis of shell form. Finally, we propose several potential applications of our methods in functional morphology, theoretical modelling, taxonomy, and evolutionary biology.

A method for quantifying, visualising, and analysing gastropod shell form

Quantitative analysis of organismal form is an important component for almost every branch of biology. Although generally considered an easily-measurable structure, the quantification of gastropod shell form is still a challenge because shells lack homologous structures and have a spiral form that is difficult to capture with linear measurements. In view of this, we adopt the idea of theoretical modelling of shell form, in which the shell form is the product of aperture ontogeny profiles in terms of aperture growth trajectory that is quantified as curvature and torsion, and of aperture form that is represented by size and shape. We develop a workflow for the analysis of shell forms based on the aperture ontogeny profile, starting from the procedure of data preparation (retopologising the shell model), via data acquisition (calculation of aperture growth trajectory, aperture form and ontogeny axis), and data presentation (qualitative comparison between shell forms) and ending with data analysis (quantitative comparison between shell forms). We evaluate our methods on representative shells of the genus Opisthostoma, which exhibit great variability in shell form. The outcome suggests that our method is a robust, reproducible, and versatile approach for the analysis of shell form. Finally, we propose several potential applications of our methods in functional morphology, theoretical modelling, taxonomy, and evolutionary biology.

A method for quantifying, visualising, and analysing gastropod shell form

Quantitative analysis of organismal form is an important component for almost every branch of biology. Although generally considered an easily-measurable structure, the quantification of gastropod shell form is still a challenge because shells lack homologous structures and have a spiral form that is difficult to capture with linear measurements. In view of this, we adopt the idea of theoretical modelling of shell form, in which the shell form is the product of aperture ontogeny profiles in terms of aperture growth trajectory that is quantified as curvature and torsion, and of aperture form that is represented by size and shape. We develop a workflow for the analysis of shell forms based on the aperture ontogeny profile, starting from the procedure of data preparation (retopologising the shell model), via data acquisition (calculation of aperture growth trajectory, aperture form and ontogeny axis), and data presentation (qualitative comparison between shell forms) and ending with data analysis (quantitative comparison between shell forms). We evaluate our methods on representative shells of the genus Opisthostoma, which exhibit great variability in shell form. The outcome suggests that our method is a robust, reproducible, and versatile approach for the analysis of shell form. Finally, we propose several potential applications of our methods in functional morphology, theoretical modelling, taxonomy, and evolutionary biology.

A method for quantifying, visualising, and analysing gastropod shell form

Quantitative analysis of organismal form is an important component for almost every branch of biology. Although generally considered an easily-measurable structure, the quantification of gastropod shell form is still a challenge because shells lack homologous structures and have a spiral form that is difficult to capture with linear measurements. In view of this, we adopt the idea of theoretical modelling of shell form, in which the shell form is the product of aperture ontogeny profiles in terms of aperture growth trajectory that is quantified as curvature and torsion, and of aperture form that is represented by size and shape. We develop a workflow for the analysis of shell forms based on the aperture ontogeny profile, starting from the procedure of data preparation (retopologising the shell model), via data acquisition (calculation of aperture growth trajectory, aperture form and ontogeny axis), and data presentation (qualitative comparison between shell forms) and ending with data analysis (quantitative comparison between shell forms). We evaluate our methods on representative shells of the genus Opisthostoma, which exhibit great variability in shell form. The outcome suggests that our method is more robust, reproducible, and versatile than the conventional traditional and geometric morphometric approaches for the analysis of shell form. Finally, we propose several potential applications of our methods in functional morphology, theoretical modelling, taxonomy, and evolutionary biology.