Плюригармонические определимые функции в некоторых $o$-минимальных расширениях вещественного поля

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,<)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

Organization and composition of apicomplexan kinetochores reveals plasticity in chromosome segregation across parasite modes of division

Kinetochores are multiprotein assemblies directing mitotic spindle attachment and chromosome segregation. In apicomplexan parasites, most known kinetochore components and associated regulators are apparently missing, suggesting a minimal structure with limited control over chromosome segregation. In this study, we use interactomics combined with deep homology searches to identify six divergent eukaryotic kinetochore proteins in apicomplexan parasites, in addition to a set of eight apicomplexan components (AKiTs) that bear no detectable sequence similarity to known proteins. The nanoscale organization of the apicomplexan kinetochore includes four subdomains, each displaying different evolutionary rates across the phylum. Functional analyses confirm AKiTs are essential for mitosis and reveal architectures parallel to biorientation at metaphase. Furthermore, we identify a homolog of MAD1 at the apicomplexan kinetochore, suggesting conserved spindle assembly checkpoint signaling. Finally, we show unexpected plasticity in kinetochore composition and segregation throughout the parasite lifecycle, indicating diverse requirements to maintain fidelity of chromosome segregation across apicomplexan modes of division.

Shaping of Curvilinear Steel Bar Structures for Variable Environmental Conditions Using Genetic Algorithms—Moving towards Sustainability

The successful and effective shaping of curvilinear steel bar structures is becoming an increasingly complex and difficult task, due to the growing demands to satisfy both economic and environmental requirements. However, computer software for algorithmic-aided design makes it possible to take into account many aspects affecting structures, as early as the initial design stage. In this context, the paper presents an optimization method for shaping the curvilinear steel bar canopies of hyperbolic paraboloid and cylindroid shapes, in order to obtain effective structures adapted to external environmental conditions. The best structural solutions in terms of the structure’s shape, topology and support positions are obtained as the effects of multi-criteria optimizations with the application of genetic algorithms. The following are used as the optimization criteria: minimal structure mass and minimal deflections of the structure’s members, as well as their maximal utilization. Additionally, the best canopy locations in relation to the sides of the world are determined through analyzing their shadow casts for various locations, so the structures have the least impact on the surroundings. This research, with its interdisciplinary character, aims to present the possibility of applying generative shaping tools to obtain structurally effective and environment-adaptive curvilinear steel bar structures in the first phase of their design, which can support sustainable designing.

Minimal structure approximation space and some of its application

Topological concepts play an important role in applications and solving real-life problems. Among of these concepts are neighbourhood and minimal structure. In this paper, we introduce a new space-based on a generalized system with a binary relation on a nonempty set by using the concept of a minimal structure, which is called a minimal structure approximation space (briefly, MSAS), and study some of its properties. Also, we compare the advantages of MSAS with neighbourhood approximation space which are based on the same starting point, and apply the concept of MSAS in some examples of chemistry to extraction and reduct the information. Finally, we investigate the concepts of the separation axioms on MSAS and study some of its properties in the information system as the process of approximation of information.

Counting algebraic points in expansions of o-minimal structures by a dense set

The Pila–Wilkie theorem states that if a set $X\subseteq \mathbb{R}^n$ is definable in an o-minimal structure $\mathcal{R}$ and contains ‘many’ rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal{R}}=\langle {\mathcal{R}}, P\rangle$ of ${\mathcal{R}}$ by a dense set P, which is either an elementary substructure of ${\mathcal{R}}$, or it is $\mathrm{dcl}$-independent, as follows. If X is definable in $\widetilde{\mathcal{R}}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is ${\emptyset}$-definable in $\langle \overline{\mathbb{R}}, P\rangle$, where $\overline{\mathbb{R}}$ is the real field. Along the way we introduce the notion of the ‘algebraic trace part’ $X^{{\, alg}}_t$ of any set $X\subseteq \mathbb{R}^n$, and we show that if X is definable in an o-minimal structure, then $X^{{\, alg}}_t$ coincides with the usual algebraic part of X.

Effective Cylindrical Cell Decompositions for Restricted Sub-Pfaffian Sets

The o-minimal structure generated by the restricted Pfaffian functions, known as restricted sub-Pfaffian sets, admits a natural measure of complexity in terms of a format ${{\mathcal{F}}}$, recording information like the number of variables and quantifiers involved in the definition of the set, and a degree $D$, recording the degrees of the equations involved. Khovanskii and later Gabrielov and Vorobjov have established many effective estimates for the geometric complexity of sub-Pfaffian sets in terms of these parameters. It is often important in applications that these estimates are polynomial in $D$. Despite much research done in this area, it is still not known whether cell decomposition, the foundational operation of o-minimal geometry, preserves polynomial dependence on $D$. We slightly modify the usual notions of format and degree and prove that with these revised notions, this does in fact hold. As one consequence, we also obtain the first polynomial (in $D$) upper bounds for the sum of Betti numbers of sets defined using quantified formulas in the restricted sub-Pfaffian structure.