PERSISTENT PATH LAPLACIAN.

Persistent homology enables fast and computable comparison of topological objects. We give some instances of a recent extension of the theory of persistence, guaranteeing robustness and computability for relevant data types, like simple graphs and digraphs. We focus on categorical persistence functions that allow us to study in full generality strong kinds of connectedness—clique communities, k-vertex, and k-edge connectedness—directly on simple graphs and strong connectedness in digraphs.

Results from a series of two‐phase fluid flow experiments in Leopard, Berea, and Bentheimer sandstones are presented. Fluid configurations are characterized using laboratory‐based and synchrotron based 3‐D X‐ray computed tomography. All flow experiments are conducted under capillary‐dominated conditions. We conduct geometry‐topology analysis via persistent homology and compare this to standard topological and watershed‐partition‐based pore‐network statistics. Metrics identified as predictors of nonwetting fluid trapping are calculated from the different analytical methods and are compared to levels of trapping measured during drainage‐imbibition cycles in the experiments. Metrics calculated from pore networks (i.e., pore body‐throat aspect ratio and coordination number) and topological analysis (Euler characteristic) do not correlate well with trapping in these samples. In contrast, a new metric derived from the persistent homology analysis, which incorporates counts of topological features as well as their length scale and spatial distribution, correlates very well (R2 = 0.97) to trapping for all systems. This correlation encompasses a wide range of porous media and initial fluid configurations, and also applies to data sets of different imaging and image processing protocols.

While protein engineering, which iteratively optimizes protein fitness by screening the gigantic mutational space, is constrained by experimental capacity, various machine learning models have substantially expedited protein engineering. Three-dimensional protein structures promise further advantages, but their intricate geometric complexity hinders their applications in deep mutational screening. Persistent homology, an established algebraic topology tool for protein structural complexity reduction, fails to capture the homotopic shape evolution during the filtration of a given data. This work introduces a Topology-offered protein Fitness (TopFit) framework to complement protein sequence and structure embeddings. Equipped with an ensemble regression strategy, TopFit integrates the persistent spectral theory, a new topological Laplacian, and two auxiliary sequence embeddings to capture mutation-induced topological invariant, shape evolution, and sequence disparity in the protein fitness landscape. The performance of TopFit is assessed by 34 benchmark datasets with 128,634 variants, involving a vast variety of protein structure acquisition modalities and training set size variations.

Investigating the longterm behaviour of cells in the lymph nodes releasing an activating substance into an intercellular space is a demanding task for mathematical modelling and numerical simulations of reaction-diffusion systems coupled with ordinary differential equations on unstructred 3D grids. To explore the vast amount of computed spatio-temporal data the visualization has to be fast and concise. Persistent homology in discrete Morse theory is a mighty tool for meaningful data reduction. Isosurfaces at characteristic thresholds can be extracted from the data without missing the crucial changes in the topology of the manifold. Visualization results are presented for single and multiple cells with details on the topological concept and computational efforts of the preprocessing steps.

Snow aggregates evolve into a variety of observed shapes and densities. Despite this diversity, models and observational studies employ fractal or Euclidean geometric measures that are assumed universal for all aggregates. This work therefore seeks to improve understanding and representation of snow aggregate geometry and its evolution by characterizing distributions of both observed and Monte Carlo–generated aggregates. Two separate datasets of best-fit ellipsoid estimates derived from Multi-Angle Snowflake Camera (MASC) observations suggest the use of a bivariate beta distribution model for capturing aggregate shapes. Product moments of this model capture shape effects to within 4% of observations. This mathematical model is used along with Monte Carlo simulated aggregates to study how combinations of monomer properties affect aggregate shape evolution. Plate aggregates of any aspect ratio produce a consistent ellipsoid shape evolution whereas thin column aggregates evolve to become more spherical. Thin column aggregates yield fractal dimensions much less than the often-assumed value of 2.0. Ellipsoid densities and fractal analogs of density (lacunarity) are much more variable depending on combinations of monomer size and shape. Simple mathematical scaling relationships can explain the persistent triaxial ellipsoid shapes that appear in both observed and modeled aggregates. Overall, both simulations and observations prove aggregates are rarely oblate. Therefore, the use of the proposed bivariate ellipsoid distribution in models will allow for similar-sized aggregates to exhibit a realistic dispersion of masses and fall speeds.

A closed loop series parallel resonant converter with LCL-T configuration has been simulated and presented in this paper. The fuzzy/PID controller has been used for closed loop operation and the performance of proposed converter has been estimated with the closed loop and the open loop condition. The mathematical model using state space approach has been developed and simulated using MATLAB. The performance of the converter has been found to be better when the fuzzy controller has been considered. The Harmonic Spectrum and steady state error for various load condition have been obtained to validate the role of controllers. The proposed approach is expected to provide better voltage regulation for dynamic load conditions.

The purpose of the work is to improve the maintenance technique of power asynchronous motors with short-circuited rotor.Scientific novelty of the obtained results is improving the technique of automated diagnostics of power asynchronous electric motors without their extraction from operation with the help of spectral analysis of a stator current. The mathematical model of a three-phase asynchronous motor was improved, which was allowed to define all primary parameters of the asynchronous motor scheme by the limited passport data. It was allowed to calculate the voltages and currents in windings of stator and rotor, their longitudinal and transverse components. An example of calculations at fundamental harmonic of 50 Hz frequency was given for operating mode of the electric motor MEUL355L4 (under load).The adequacy of the mathematical model was verified using Fisher's criterion. Relative error between measured and calculated data did not exceed 5 %.By comparing the primary parameters of the three-phase asynchronous motor scheme, calculated by the experimental results, and the currents (voltages) of stator and rotor windings with the nominal ones, it became possible to determine the type and the failure causes for investigated object.The spectrum and amplitudes of harmonics are analyzed, which are characteristic for various types of faults of asynchronous motor. The application of the proposed monitoring system will be allowed to detect the failures at an early stage of appearance and in time, to minimize the costs of the damage elimination, as well as to modern maintenance and to change the planned preventive repair to the repair of the status of an object in further.

Severe burn injuries often lead to skin contraction, leading to stresses in and around the damaged skin region. If this contraction leads to impaired joint mobility, one speaks of contracture. To optimize treatment, a mathematical model, that is based on finite element methods, is developed. Since the finite element-based simulation of skin contraction can be expensive from a computational point of view, we use machine learning to replace these simulations such that we have a cheap alternative. The current study deals with a feed-forward neural network that we trained with 2D finite element simulations based on morphoelasticity. We focus on the evolution of the scar shape, wound area, and total strain energy, a measure of discomfort, over time. The results show average goodness of fit (R2) of 0.9979 and a tremendous speedup of 1815000X. Further, we illustrate the applicability of the neural network in an online medical app that takes the patient's age into account.

In the present paper we propose a geometrically motivated mathematical model, which reveals the key features of natural coastal and fluvial fragment shape evolution from the earliest stages of the abrasion. Our collisional polygon model governs the evolution through an ordinary differential equation (ODE), which determines the rounding rate of initially sharp corners in the function of the size reduction of the fragment. As an approximation, the basic structure of our model adopts the concept of Bloore’s partial differential equation (PDE) in terms of the curvature-dependent local collisional frequency. We tested our model under various conditions and made comparisons with the predictions of Bloore’s PDE. Moreover, we applied the model to discover and quantify the mathematical conditions corresponding to typical and special shape evolution. By further extending our model to investigate the self-dual and mixed cases, we outline a possible explanation of the long-term preservation of initial pebble shape characteristics.

A few basic motifs, such as hexagonal lattices, are prevalent in a wide variety of pattern-forming biological and other chemical and physical systems. These include phyllotaxis (the arrangement of elements such as leaves or seeds on plants) [1], biomineralization processes [2], and nanostructures formed by the bombardment of binary materials by ions [3]. The chemical and physical mechanisms behind the formation of these similar patterns are, however, very different. It is a challenge to compare mathematical models with observations or experiments since very disparte mechanisms can produce similar steady-state patterns. Over the last decade, a powerful new tool of computational topology called persistent homology has been developed to characterize the topological properties of a set of points or a surface at all length scales. Applying variations on this technique to the surfaces produced by simulations of pattern-forming systems, we find that, although the topological properties of the steady-state solutions are similar, the evolution of the topological properties to that steady state can differ significantly. This provides a potential way of distinguishing between mechanisms and for determining parameters. ...

A mathematical model based on continuity and Navier-Stokes equations, considering laminar flow in the gap between the disks, is presented to estimate the drag torque in open multidisks wet clutches. By taking into account the effects of Poiseuille and centrifugal forces, the flow pressure and velocity fields are investigated. The model quantifies the volume fraction of fluids and predicts the evolution of film shape. The drag torque estimated by the model is the sum of drag torque due to shearing of automatic transmission fluid (ATF) and the mist (suspension of ATF in air) film. In order to validate the model, experiments are performed on SAE# 2 test-setup under actual operating conditions of clutches. The model is capable of predicting the drag torque under conditions of variable flow rate and different disks rotational state for higher clutch speed range.

Alterations in nuclear morphology are closely associated with essential cell functions, such as cell motility and polarization, and correlate with a wide range of human diseases, including cancer, muscular dystrophy, dilated cardiomyopathy and progeria. However, the mechanics and forces that shape the nucleus are not well understood. Here, we demonstrate that when an adherent cell is detached from its substratum, the nucleus undergoes a large volumetric reduction accompanied by a morphological transition from an almost smooth to a heavily folded surface. We develop a mathematical model that systematically analyzes the evolution of nuclear shape and volume. The analysis suggests that the pressure difference across the nuclear envelope, which is influenced by changes in cell volume and regulated by microtubules and actin filaments, is a major factor determining nuclear morphology. Our results show that physical and chemical properties of the extracellular microenvironment directly influence nuclear morphology and suggest that there is a direct link between the environment and gene regulation.

The growth and the fluid dynamics of protein crystals and soft organic tissues: models and simulations, similarities and differences

Theoretical and experimental study of nanosecond pulse amplification in a CW CO2 amplifier

The transient process of growing axisymmetric crystals by the liquid-encapsulated Czochralski method is considered. The mathematical model accounts for heat transfer in the crucible-crystal-melt-encapsulant, formation of the melt/encapsulant meniscus, crystallization interface movement, and changes in crystal radius. A new algorithm was developed to determine the lateral surface shape of the crystal during the process. The proposed numerical approach utilizes a geometrically conservative difference scheme that guarantees the fulfillment of conservation laws of energy and mass. The special splitting technique is used to solve the corresponding set of finite difference equations. The proposed approach ensures the consistency of crystal shape evolution with conservation laws. The designed numerical procedure is used to evaluate the impact of external thermal regime on the shape of the growing crystal. When the heater temperature is maintained constant, the crystal radius gradually decreases over time. To study the influence of the external temperature field on the shape of the lateral surface, the mathematical model is supplemented with a proportional-integral temperature controller equation that links the change in heater temperature to the radius of the growing crystal. In a general case, the application of an integral temperature controller leads to fluctuations in the crystal radius around a set value, with both the frequency and amplitude of fluctuations increasing progressively. Based on the results of the numerical experiments, the parameter values for a proportional-integral temperature controller that ensure the growth of crystals with a nearly constant radius are determined. The results of transient numerical simulations are compared with the results obtained using the quasi-steady-state model of crystal growth.