Simple Geometric Criteria Research Articles

Critical Edges in Rips Complexes and Persistence

We consider persistent homology obtained by applying homology to the open Rips filtration of a compact metric space (X, d). We show that each decrease in zero-dimensional persistence and each increase in one-dimensional persistence is induced by local minima of the distance function d. When d attains local minimum at only finitely many pairs of points, we prove that each above mentioned change in persistence is induced by a specific critical edge in Rips complexes, which represents a local minimum of d. We use this fact to develop a theory (including interpretation) of critical edges of persistence. The obtained results include upper bounds for the rank of one-dimensional persistence and a corresponding reconstruction result. Of potential computational interest is a simple geometric criterion recognizing local minima of d that induce a change in persistence. We conclude with a proof that each locally isolated minimum of d can be detected through persistent homology with selective Rips complexes. The results of this paper offer the first interpretation of critical scales of persistent homology (obtained via Rips complexes) for general compact metric spaces.

The Mechanism of Selective Recognition of Lipid Substrate by hDHHC20 Enzyme

S-acylation is a post-translational linkage of long chain fatty acids to cysteines, playing a key role in normal physiology and disease. In human cells, the reaction is catalyzed by a family of 23 membrane DHHC-acyltransferases (carrying an Asp-His-His-Cys catalytic motif) in two stages: (1) acyl-CoA-mediated autoacylation of the enzyme; and (2) further transfer of the acyl chain to a protein substrate. Despite the availability of a 3D-structure of human acyltransferase (hDHHC20), the molecular aspects of lipid selectivity of DHHC-acyltransferases remain unclear. In this paper, using molecular dynamics (MD) simulations, we studied membrane-bound hDHHC20 right before the acylation by C12-, C14-, C16-, C18-, and C20-CoA substrates. We found that: (1) regardless of the chain length, its terminal methyl group always reaches the "ceiling" of the enzyme's cavity; (2) only for C16, an optimal "reactivity" (assessed by a simple geometric criterion) permits the autoacylation; (3) in MD, some key interactions between an acyl-CoA and a protein differ from those in the reference crystal structure of the C16-CoA-hDHHS20 mutant complex (probably, because this structure corresponds to a non-native dimer). These features of specific recognition of full-size acyl-CoA substrates support our previous hypothesis of "geometric and physicochemical selectivity" derived for simplified acyl-CoA analogues.

How does electron exchange correlation influence reactivity of metallo-β-lactamase L1 against cephalosporin antibiotics

Cephalosporin hydrolysis by the metallo-β-lactamase L1 is a reaction where electron delocalization in the substrate-protein complex affects the reaction rate: the more cephalosporin lone pairs are localized on the N atom, the higher hydrolysis rates are. We show that high Fermi holevariability is responsible for the stipulation of the electron lone pairs’ localization.One-electron potentials reveal that higher lone pairs’ localization is expressed in the dominance of the static exchange correlation (reflected in theaverage Fermi hole depth) in theN atomvicinity. However, this dominance was found to be defined by the height of the kinetic exchange potential barriers (expressing the Fermi holevariability and the response of the exchange correlation hidden in the kinetic potential to electron density variations), rather than by the features of the static exchange correlation itself. The need to account both static and kinetic exchange correlations for the reliable description of the electron delocalization explains inapplicability of the simple geometric criteria for the chemical bonding study in the considered systems.

Boundary lubrication and super lubricity, textured surfaces

Over last couple of decades, various regimes of lubrication attracted lots of attention. Among these regimes of lubrication, the least understood and studied are the regimes or boundary lubrication and lubricity. Better understanding these regimes of lubrication is necessary due to the fact that many moving joints such as, for example, bearings and gears at least part of their operating time work in these regimes. It may occur due to slow motion of counter parts or their relative overload. This paper proposes a model for analysis of these regimes of lubrication. Specifically, the model takes into account roughness of one of the contact surfaces. It is shown that for sufficiently high applied load it can be expected that contact surfaces are completely separated by a lubricant layer in spite of the presence of surface asperities. The definitions of boundary lubrication regimes and super lubricity are proposed. A method based on the method of matched asymptotic expansions is applied for analyzing such cases and an approximate analytical solution of the problem away from the contact region boundary is found. In the narrow inlet and exit zones adjacent to contact boundaries equations describing the main solution terms are derived. For the case of super lubricity, a resemblance of a superposition between the lubrication and surface roughness effects is determined and the effect of surface roughness on the lubrication film thickness is considered. Specifically, certain roughness conditions are identified for which the lubrication film thickness can be higher than for smooth surfaces while in other cases it can be lower. For the super lubricity regimes, the equations for the main pressure asymptotic terms in the inlet and exit zones are reduced to the corresponding equations for the smooth (not rough) surfaces for which a series of numerical solutions is obtained earlier. A general analysis of the effect of texturing on lubrication conditions in case of super lubricity is conducted based on a simple geometric criterion. Several models of textured surfaces and their influence on the lubrication film thickness in case of super lubricity regimes are considered.

Local $$L^p$$ norms of Schrödinger eigenfunctions on $${\mathbb {S}}^2$$

On the canonical 2-sphere and for Schrödinger eigenfunctions, we obtain a simple geometric criterion on the potential under which we can improve, near a given point and for every \(p\ne 6\), Sogge’s estimates by a power of the eigenvalue. This criterion can be formulated in terms of the critical points of the Radon transform of the potential and it is independent of the choice of eigenfunctions.

A geometric probability randomized Kaczmarz method for large scale linear systems

For solving large scale linear systems, a fast convergent randomized Kaczmarz-type method was constructed in Bai and Wu (2018) [4]. In this paper, we propose a geometric probability randomized Kaczmarz (GPRK) method by introducing a new index set Jk and three supervised probability criteria defined on Jk from a geometric point of view. Linear convergence of GPRK is proved, and the way of argument for the analysis of GPRK also leads to new sharper upper bounds for the randomized Kaczmarz (RK) method and the greedy randomized Kaczmarz (GRK) method. In practice, GPRK is implemented with a simple geometric probability criterion, i.e., the most efficient one of the aforementioned three supervised probability criteria defined on Jk. The numerical results demonstrate that GPRK is robust and efficient, and it is faster than GRK in most of the tests in the sense of computing time.

On cyclic strong exceptional collections of line bundles on surfaces

We study exceptional collections of line bundles on surfaces. We prove that any full cyclic strong exceptional collection of line bundles on a rational surface is an augmentation in the sense of Lutz Hille and Markus Perling. We find simple geometric criteria of exceptionality (strong exceptionality, cyclic strong exceptionality) for collections of line bundles on weak del Pezzo surfaces. As a result, we classify smooth projective surfaces admitting a full cyclic strong exceptional collection of line bundles. Also, we provide an example of a weak del Pezzo surface of degree 2 and a full strong exceptional collection of line bundles on it which does not come from augmentations, thus answering a question by Hille and Perling.

A geometrical criterion for nonexistence of constant-sign solutions for some third-order two-point boundary value problems

We give a simple geometrical criterion for the nonexistence of constant-sign solutions for a certain type of third-order two-point boundary value problem in terms of the behavior of nonlinearity in the equation. We also provide examples to illustrate the applicability of our results.

Support optimization in additive manufacturing for geometric and thermo-mechanical constraints

Supports are often required to safely complete the building of complicated structures by additive manufacturing technologies. In particular, supports are used as scaffoldings to reinforce overhanging regions of the structure and/or are necessary to mitigate the thermal deformations and residual stresses created by the intense heat flux produced by the source term (typically a laser beam). However, including supports increase the fabrication cost and their removal is not an easy matter. Therefore, it is crucial to minimize their volume while maintaining their efficiency. Based on earlier works, we propose here some new optimization criteria. First, simple geometric criteria are considered like the projected area and the volume of supports required for overhangs: they are minimized by varying the structure orientation with respect to the baseplate. In addition, an accessibility criterion is suggested for the removal of supports, which can be used to forbid some parts of the structure to be supported. Second, shape and topology optimization of supports for compliance minimization is performed. The novelty comes from the applied surface loads which are coming either from pseudo gravity loads on overhanging parts or from equivalent thermal loads arising from the layer-by-layer building process. Here, only the supports are optimized, with a given non-optimizable structure, but of course many generalizations are possible, including optimizing both the structure and its supports. Our optimization algorithm relies on the level set method and shape derivatives computed by the Hadamard method. Numerical examples are given in 2-d and 3-d.

Pattern localization to a domain edge.

The formation of protein patterns inside cells is generically described by reaction-diffusion models. The study of such systems goes back to Turing, who showed how patterns can emerge from a homogenous steady state when two reactive components have different diffusivities (e.g., membrane-bound and cytosolic states). However, in nature, systems typically develop in a heterogeneous environment, where upstream protein patterns affect the formation of protein patterns downstream. Examples for this are the polarization of Cdc42 adjacent to the previous bud site in budding yeast and the formation of an actin-recruiter ring that forms around a PIP3 domain in macropinocytosis. This suggests that previously established protein patterns can serve as a template for downstream proteins and that these downstream proteins can "sense" the edge of the template. A mechanism for how this edge sensing may work remains elusive. Here we demonstrate and analyze a generic and robust edge-sensing mechanism, based on a two-component mass-conserving reaction-diffusion (McRD) model. Our analysis is rooted in a recently developed theoretical framework for McRD systems, termed local equilibria theory. We extend this framework to capture the spatially heterogeneous reaction kinetics due to the template. This enables us to graphically construct the stationary patterns in the phase space of the reaction kinetics. Furthermore, we show that the protein template can trigger a regional mass-redistribution instability near the template edge, leading to the accumulation of protein mass, which eventually results in a stationary peak at the template edge. We show that simple geometric criteria on the reactive nullcline's shape predict when this edge-sensing mechanism is operational. Thus, our results provide guidance for future studies of biological systems and for the design of synthetic pattern forming systems.

Regularization and Geometry of Piecewise Smooth Systems with Intersecting Discontinuity Sets

In this work, we study the dynamics of piecewise smooth systems on a codimension-2 transverse intersection of two codimension-1 discontinuity sets. The Filippov convention can be extended to such intersections, but this approach does not provide a unique sliding vector and, as opposed to the classical sliding vector-field on codimension-1 discontinuity manifolds, there is no agreed notion of stability in the codimension-2 context. In this paper, we perform a regularization of the piecewise smooth system, introducing two regularization functions and a small perturbation parameter. Then, based on singular perturbation theory, we define sliding and stability of sliding through a critical manifold of the singularly perturbed, regularized system. We show that this notion of sliding vector-field coincides with the Filippov one. The regularized system gives a parameterized surface (the canopy) independent of the regularization functions. This surface serves as our natural basis to derive new and simple geometric criteria on the existence, multiplicity and stability of the sliding flow, depending only on the smooth vector fields around the intersection. Interestingly, we are able to show that if there exist two sliding vector-fields then one is a saddle and the other is of focus/node/center type. This means that there is at most one stable sliding vector-field. We then investigate the effect of the choice of the regularization functions, and, using a blowup approach, we demonstrate the mechanisms through which sliding behavior can appear or disappear on the intersection and describe what consequences this has on the dynamics on the adjacent codimension-1 discontinuity sets. Finally, we show the existence of canard explosions of regularizations of PWS systems in $\mathbb R^3$ that depend on a single unfolding parameter.

The regular reflection→Mach reflection transition in unsteady flow over convex surfaces

The non-stationary transition from regular reflection (RR) to Mach reflection (MR) over convex segments has been the focus of many recent studies. Until recently, the problem was thought to be very complicated because it was believed that many parameters such as the radius of curvature, initial angle and geometrical shape of the reflecting surface influenced this process. In this study, experiments and inviscid numerical computations were performed in air ($\unicode[STIX]{x1D6FE}=1.4$) at an incident shock-wave Mach number of 1.3. The incident shock waves were reflected over cylindrical and elliptical convex surfaces. The computations were validated by high-resolution experiments, which enabled the detection of features in the flow having characteristic lengths as small as 0.06 mm. Therefore, the RR →MR transition and Mach stem growth were successfully validated in the early stages of the Mach stem formation and closer to the surface than ever before. The evolution of the RR, the transition to MR and the Mach stem growth were found to depend only on the radius of the reflecting surface. The reflected shock wave adjusts itself to the changing angles of the reflecting surface. This feature, which was demonstrated at Mach numbers 1.3 and 1.5, distinguishes the unsteady case from the self-similar pseudo-steady case and requires the formulation of the conservation equations. A modification of the standard two-shock theory (2ST) is presented to predict the flow properties behind a shock wave that propagates over convex surfaces. Until recently, the determination of the time-dependent flow properties was possible solely by numerical computations. Moreover, this derivation explains the controversial issue on the delay in the transition from the RR to the MR that was observed by many researchers. It turns out that the entire RR evolution and the particular moment of transition to MR, are based on the essential ‘no-penetration’ condition of the flow. Therefore, we proposed a simple geometrical criterion for the RR →MR transition.

Combustion in vortex chambers with a fluidized particle bed

This paper is devoted to methods of estimating the combustion regimes of gaseous fuel in vortex chambers of various designs in the presence of a bed of inert particles in the chamber, which rotate with the gas flow. A simple geometric criterion for determining the possibility of fuel combustion in the particle bed without the formation of a flame above the bed is proposed.

Squeezing xenon into phenylether bis-urea nanochannels

One-dimensional nanochannels, hundreds of microns in persistence length but with elliptical cross-sectional dimensions of only ∼3.7 Å × 4.8 Å, are formed by the columnar assembly of phenylether bis-urea macrocycles. Hyperpolarized Xe-129 NMR is utilized to investigate the Xe atom packing and Xe diffusion inside the needle shaped crystals. The elliptical channel structure produces a Xe-129 powder pattern characteristic of an asymmetric chemical shift tensor extending to well over 300 ppm with respect to the gas phase, reflecting the highly anisotropic electronic environment and extreme confinement of the atom. Consistent with the simple geometrical criterion, hyperpolarized tracer exchange NMR data reveals single-file diffusion in the bis-urea nanochannels.

Geometric approach to string analysis for biosequence classification

Summary Tools that effectively analyze and compare sequences are of great importance in various areas of applied computational research, especially in the framework of molecular biology. In the present paper, we introduce simple geometric criteria based on the notion of string linearity and use them to compare DNA sequences of various organisms, as well as to distinguish them from random sequences. Several other theoretical and statistical results are outlined as well. Our experiments reveal a substantial difference between biosequences and random sequences – the former having much higher deviation from linearity than the latter – as well as a general trend of increasing deviation from linearity between primitive and biologically complex organisms.

Geometric approach to string analysis for biosequence classification.

Tools that effectively analyze and compare sequences are of great importance in various areas of applied computational research, especially in the framework of molecular biology. In the present paper, we introduce simple geometric criteria based on the notion of string linearity and use them to compare DNA sequences of various organisms, as well as to distinguish them from random sequences. Several other theoretical and statistical results are outlined as well. Our experiments reveal a substantial difference between biosequences and random sequences – the former having much higher deviation from linearity than the latter – as well as a general trend of increasing deviation from linearity between primitive and biologically complex organisms.

Self-assembly of mesoscale isomers: the role of pathways and degrees of freedom.

The spontaneous self-organization of conformational isomers from identical precursors is of fundamental importance in chemistry. Since the precursors are identical, it is the multi-unit interactions, characteristics of the intermediates, and assembly pathways that determine the final conformation. Here, we use geometric path sampling and a mesoscale experimental model to investigate the self-assembly of a model polyhedral system, an octahedron, that forms two isomers. We compute the set of all possible assembly pathways and analyze the degrees of freedom or rigidity of intermediates. Consequently, by manipulating the degrees of freedom of a precursor, we were able to experimentally enrich the formation of one isomer over the other. Our results suggest a new approach to direct pathways in both natural and synthetic self-assembly using simple geometric criteria. We also compare the process of folding and unfolding in this model with a geometric model for cyclohexane, a well-known molecule with chair and boat conformations.

Isogeometric segmentation: The case of contractible solids without non-convex edges

We present a novel technique for segmenting a three-dimensional solid with a 3-vertex-connected edge graph consisting of only convex edges into a collection of topological hexahedra. Our method is based on the edge graph, which is defined by the sharp edges between the boundary surfaces of the solid. We repeatedly decompose the solid into smaller solids until all of them belong to a certain class of predefined base solids. The splitting step of the algorithm is based on simple combinatorial and geometric criteria. The segmentation technique described in the paper is part of a process pipeline for solving the isogeometric segmentation problem that we outline in the paper.

Fatigue degradation and electric recovery in Silicon solar cells embedded in photovoltaic modules

Cracking in Silicon solar cells is an important factor for the electrical power-loss of photovoltaic modules. Simple geometrical criteria identifying the amount of inactive cell areas depending on the position of cracks with respect to the main electric conductors have been proposed in the literature to predict worst case scenarios. Here we present an experimental study based on the electroluminescence (EL) technique showing that crack propagation in monocrystalline Silicon cells embedded in photovoltaic (PV) modules is a much more complex phenomenon. In spite of the very brittle nature of Silicon, due to the action of the encapsulating polymer and residual thermo-elastic stresses, cracked regions can recover the electric conductivity during mechanical unloading due to crack closure. During cyclic bending, fatigue degradation is reported. This pinpoints the importance of reducing cyclic stresses caused by vibrations due to transportation and use, in order to limit the effect of cracking in Silicon cells.

Increasing the stability of T-shape tube hydroforming process under stochastic framework

Metal forming processes present several sources of uncertainties coming from material properties, geometric characteristics, and loading paths. During the manufacturing phase, such parameters may vary affecting the process stability and increasing the defect parts. Stochastic framework seems more pertinent than classical deterministic approaches to treat such problems since it is intended to include variabilities at the early design stage. In the present work, tube hydroforming process widely used in various industry applications is investigated. To ensure the process stability, loading paths should be optimized with taking into account randomness associated to the input parameters. To control the potential failure modes, the Forming Limit Stress Diagram is implemented in the finite element code to avoid necking while a simple geometrical criterion is defined for wrinkling. A global sensitivity analysis using the variance-based method is done which shows that the selected random parameters impact considerably the variance of failure indicators. Then, a numerical example of T-shape tube hydroforming process is proposed to show the efficiency of the stochastic framework. Statistical and probabilistic observations of the optimum solution show that the stochastic approach yields to an optimum less sensitive to such fluctuations which improves the process stability and minimizes considerably the percentage of defect parts in a mass production environment.