COMPLEX GEOMETRIC EVALUATION OF POLYMER MATERIALS QUALITY

A computational technique of comparative evaluation of polymer material quality in a homogeneous set of samples according to a complex geometric criterion is proposed. Samples of physical and mechanical parameters of samples of industrial impact-resistant polystyrene are used for calculation. The most averaged complex of physical and mechanical properties is used as the calculation base.

Information geometry, trade-off relations, and generalized Glansdorff-Prigogine criterion for stability

We discuss a relationship between information geometry and the Glansdorff-Prigogine criterion for stability. For the linear master equation, we found a relation between the line element and the excess entropy production rate. This relation leads to a new perspective of stability in a nonequilibrium steady-state. We also generalize the Glansdorff-Prigogine criterion for stability based on information geometry. Our information-geometric criterion for stability works well for the nonlinear master equation, where the Glansdorff-Prigogine criterion for stability does not work well. We derive a trade-off relation among the fluctuation of the observable, the mean change of the observable, and the intrinsic speed. We also derive a novel thermodynamic trade-off relation between the excess entropy production rate and the intrinsic speed. These trade-off relations provide a physical interpretation of our information-geometric criterion for stability. We illustrate our information-geometric criterion for stability by an autocatalytic reaction model, where dynamics are driven by a nonlinear master equation.

A geometric criterion for the optimal spreading of active polymers in porous media

Efficient navigation through disordered, porous environments poses a major challenge for swimming microorganisms and future synthetic cargo-carriers. We perform Brownian dynamics simulations of active stiff polymers undergoing run-reverse dynamics, and so mimic bacterial swimming, in porous media. In accord with experiments of Escherichia coli, the polymer dynamics are characterized by trapping phases interrupted by directed hopping motion through the pores. Our findings show that the spreading of active agents in porous media can be optimized by tuning their run lengths, which we rationalize using a coarse-grained model. More significantly, we discover a geometric criterion for the optimal spreading, which emerges when their run lengths are comparable to the longest straight path available in the porous medium. Our criterion unifies results for porous media with disparate pore sizes and shapes and for run-and-tumble polymers. It thus provides a fundamental principle for optimal transport of active agents in densely-packed biological and environmental settings.

Generic rank of Betti map and unlikely intersections

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

Characteristic Cohomology I: Singularities of Given Type

For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of ‘type $\mathcal{V}$’ is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate sense so that $\mathcal{V}_0 = f_0^{\,-1}(\mathcal{V})$. If $\mathcal{V}$ is a hypersurface germ, then so is $\mathcal{V}_0 $, and by transversality ${\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V}_0) = {\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V})$ provided $n > {\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V})$. So $\mathcal{V}_0, 0$ will exhibit singularities of $\mathcal{V}$ up to codimension n. For singularities $\mathcal{V}_0, 0$ of type $\mathcal{V}$, we introduce a method to capture the contribution of the topology of $\mathcal{V}$ to that of $\mathcal{V}_0$. It is via the ‘characteristic cohomology’ of the Milnor fiber (for $\mathcal{V}, 0$ a hypersurface), and complement and link of $\mathcal{V}_0$ (in the general case). The characteristic cohomology of the Milnor fiber $\mathcal{A}_{\mathcal{V}}(\,f_0; R)$, and respectively of the complement $\mathcal{C}_{\mathcal{V}}(\,f_0; R)$, are subalgebras of the cohomology of the Milnor fibers, respectively the complement, with coefficients R in the corresponding cohomology. For a fixed $\mathcal{V}$, they are functorial over the category of singularities of type $\mathcal{V}$. In addition, for the link of $\mathcal{V}_0$ there is a characteristic cohomology subgroup $\mathcal{B}_{\mathcal{V}}(\,f_0, \mathbf{k})$ of the cohomology of the link over a field $\mathbf{k}$ of characteristic 0. The cohomologies $\mathcal{C}_{\mathcal{V}}(\,f_0; R)$ and $\mathcal{B}_{\mathcal{V}}(\,f_0, \mathbf{k})$ are shown to be invariant under the $\mathcal{K}_{\mathcal{V}}$-equivalence of defining germs f0, and likewise $\mathcal{A}_{\mathcal{V}}(\,f_0; R)$ is shown to be invariant under the $\mathcal{K}_{H}$-equivalence of f0 for H the defining equation of $\mathcal{V}, 0$. We give a geometric criterion involving ‘vanishing compact models’ for both the Milnor fibers and complements which detect non-vanishing subalgebras of the characteristic cohomologies, and subgroups of the characteristic cohomology of the link. Also, we consider how in the hypersurface case the cohomology of the Milnor fiber is a module over the characteristic cohomology $\mathcal{A}_{\mathcal{V}}(\,f_0; R)$. We briefly consider the application of these results to a number of cases of singularities of a given type. In part II, we specialize to the case of matrix singularities and using results on the topology of the Milnor fibers, complements and links of the varieties of singular matrices obtained in another paper allow us to give precise results for the characteristic cohomology of all three types.