Local Contraction Research Articles

An explicit Floquet-type representation of Riccati aperiodic exponential semigroups

The article presents a rather surprising Floquet-type representation of time-varying transition matrices associated with a class of nonlinear matrix differential Riccati equations. The main difference with conventional Floquet theory comes from the fact that the underlying flow of the solution matrix is aperiodic. The monodromy matrix associated with this Floquet representation coincides with the exponential (fundamental) matrix associated with the stabilising fixed point of the Riccati equation. The second part of this article is dedicated to the application of this representation to the stability of matrix differential Riccati equations. We provide refined global and local contraction inequalities for the Riccati exponential semigroup that depend linearly on the spectral norm of the initial condition. These refinements improve upon existing results and are a direct consequence of the Floquet-type representation, yielding what seems to be the first results of this type for this class of models.

On p-Robust Saturation on Quadrangulations

Abstract For the Poisson problem in two dimensions, posed on a domain partitioned into axis-aligned rectangles with up to one hanging node per edge, we envision an efficient error reduction step in an instance-optimal hp-adaptive finite element method. Central to this is the problem: Which increase in local polynomial degree ensures p-robust contraction of the error in energy norm? We reduce this problem to a small number of saturation problems on the reference square, and provide strong numerical evidence for their solution.

Fixed Points Results in G-Metric Spaces

In this paper, the concept of contraction mapping on a -metric space is extended with a consideration on local contraction. As a result, two fixed point theorems were proved for contraction on a closed ball in a complete -metric space.

Switching Regimes in Economics: The Contraction Mapping and the ω-Limit Set

This paper considers a dynamical system defined by a set of ordinary autonomous differential equations with discontinuous right-hand side. Such systems typically appear in economic modelling where there are two or more regimes with a switching between them. Switching between regimes may be a consequence of market forces or deliberately forced in form of policy implementation. Stiefenhofer and Giesl [1] introduce such a model. The purpose of this paper is to show that a metric function defined between two adjacent trajectories contracts in forward time leading to exponentially asymptotically stability of (non)smooth periodic orbits. Hence, we define a local contraction function and distribute it over the smooth and nonsmooth parts of the periodic orbits. The paper shows exponentially asymptotical stability of a periodic orbit using a contraction property of the distance function between two adjacent nonsmooth trajectories over the entire periodic orbit. Moreover it is shown that the ω-limit set of the (non)smooth periodic orbit for two adjacent initial conditions is the same.

Finite Element Analysis of the Effect of Internal Pressure on Orifice Flowmeter

The flow rate in a closed pipe is a dynamic value, the instrument for measuring the flow rate is called a flowmeter. Typical flowmeters are: differential pressure flowmeter, electromagnetic flowmeter, coriolis mass flowmeter, vortex flowmeter, ultrasonic flowmeter, etc. Among them, differential pressure flowmeter, represented by orifice flowmeter, is the mainstream flowmeter applied at home and abroad. Orifice flowmeter has the advantages of simple structure, low cost and stable performance, it is widely used in chemical industry, electric power, heating, water supply and other fields, and can be used to measure the flow of different media. When the fluid filled with pipeline flows through the orifice plate, it will produce local contraction, concentration of flow beam, increase of flow velocity, and decrease of static pressure, so there will be a static pressure difference before and after the orifice plate. The orifice flowmeter is equipped with an orifice plate on the pipeline, and the orifice plate is connected with pressure measuring tubes on both sides, and they are respectively connected with a U-type pressure differential meter. Orifice flowmeter uses the throttling effect of fluid through the sharp hole to increase the flow velocity and decrease the pressure, resulting in the pressure difference between front and back orifice plates, as the basis of measurement. In this paper, to study the influence of different pressure loads on the accuracy of the orifice flowmeter, the mathematical model is established by simulation software, and the different pressure loads of the orifice flowmeter are analyzed under the temperature load of 20°C. The results show that the location of the orifice flowmeter under the maximum stress, maximum displacement and maximum strain under the pressure load of 0.1 MPa is different from that under the pressure load of 50 MPa and 100 MPa. When the pressure loads are respectively 0.1 MPa, 50 MPa, and 100 MPa, the location of maximum stress and displacement of the orifice flowmeter are the same.

Lifting weighted blow-ups

Let f\colon X \to Z be a local, projective, divisorial contraction between normal varieties of dimension n with \mathbb Q -factorial singularities. Let Y \subset X be a f -ample Cartier divisor and assume that f_{|Y}\colon Y \to W has a structure of a weighted blow-up. We prove that f\colon X \to Z , as well, has a structure of weighted blow-up. As an application we consider a local projective contraction f\colon X \to Z from a variety X with terminal \mathbb Q -factorial singularities, which contracts a prime divisor E to an isolated \mathbb Q -factorial singularity P\in Z , such that -(K_X + (n-3)L) is f -ample, for a f -ample Cartier divisor L on X . We prove that (Z,P) is a hyperquotient singularity and f is a weighted blow-up.

Deformation-based Morphometry MRI Reveals Brain Structural Modifications in Living Mu Opioid Receptor Knockout Mice.

Mu opioid receptor (MOR) activation facilitates reward processing and reduces pain, and brain networks underlying these effects are under intense investigation. Mice lacking the MOR gene (MOR KO mice) show lower drug and social reward, enhanced pain sensitivity and altered emotional responses. Our previous neuroimaging analysis using Resting-state (Rs) functional Magnetic Resonance Imaging (fMRI) showed significant alterations of functional connectivity (FC) within reward/aversion networks in these mice, in agreement with their behavioral deficits. Here we further used a structural MRI approach to determine whether volumetric alterations also occur in MOR KO mice. We acquired anatomical images using a 7-Tesla MRI scanner and measured deformation-based morphometry (DBM) for each voxel in subjects from MOR KO and control groups. Our analysis shows marked anatomical differences in mutant animals. We observed both local volumetric contraction (striatum, nucleus accumbens, bed nucleus of the stria terminalis, hippocampus, hypothalamus and periacqueducal gray) and expansion (prefrontal cortex, amygdala, habenula, and periacqueducal gray) at voxel level. Volumetric modifications occurred mainly in MOR-enriched regions and across reward/aversion centers, consistent with our prior FC findings. Specifically, several regions with volume differences corresponded to components showing highest FC changes in our previous Rs-fMRI study, suggesting a possible function-structure relationship in MOR KO-related brain differences. In conclusion, both Rs-fMRI and volumetric MRI in live MOR KO mice concur to disclose functional and structural whole-brain level mechanisms that likely drive MOR-controlled behaviors in animals, and may translate to MOR-associated endophenotypes or disease in humans.

New classes of local almost contractions

Abstract Contractions represents the foundation stone of nonlinear analysis. That is the reason why we propose to unify two different type of contractions: almost contractions, introduced by V. Berinde in [2] and local contractions (Martins da Rocha and Filipe Vailakis in [7]). These two types of contractions operate in different space settings: in metric spaces (almost contractions) and semimetric spaces (for local contractions). That new type of contraction was built up in a new space setting, which is the pseudometric space. The main results of this paper represent the extension for various type of operators on pseudometric spaces, such as: generalized ALC, Ćirić-typeALC, quasi ALC, Ćirić-Reich-Rustype ALC. We propose to study the existence and uniqueness of their fixed points, and also the continuity in their fixed points, with a large number of examples for ALC-s.

Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction

In this paper we analyze local structure of several chaotic attractors recently suggested in literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the well known finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finite time Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finite time Lyapunov exponents we demonstrate that specific to the pseudohyperbolicity average expanding and contracting properties are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace.

Directional Stiffness Control Through Geometric Patterning and Localized Heating of Field’s Metal Lattice Embedded in Silicone

This research explores a new realm of soft robotic materials where the stiffness magnitude, directionality, and spatial resolution may be precisely controlled. These materials mimic biological systems where localized muscle contractions and adjustment of tissue stiffness enables meticulous, intelligent movement. Here we propose the use of a low-melting-point (LMP) metal lattice structure as a rigid frame using localized heating to allow compliance about selectable axes along the lattice. The resulting shape of the lattice is modeled using product of exponentials kinematics to describe the serial chain of tunably compliant axes; this model is found to match the behavior of the physical test piece consisting of a Field’s metal (FM) lattice encased in silicone rubber. This concept could enable highly maneuverable robotic structures with significantly improved dexterity.

Lattice expansion and local lattice distortion in Nb- and La-doped SrTiO3 single crystals investigated by x-ray diffraction and first-principles calculations

Electron-doped $\mathrm{SrTi}{\mathrm{O}}_{3}$ (where the dopant can be Nb or La) has been widely investigated for both its fundamental interest in condensed matter physics and for industrial applications. Its electronic properties are closely related to the Ti-O bonding states in the $\mathrm{SrTi}{\mathrm{O}}_{3}$ crystal. To further develop and control these properties, it is crucial to understand the factors controlling the change in lattice parameters and local lattice distortion upon doping with various atoms. Herein, we report the changes in lattice parameters and local lattice distortion in Nb- and La-doped $\mathrm{SrTi}{\mathrm{O}}_{3}$ single crystals, investigated by in-plane x-ray diffraction and first-principles calculations. The lattice parameter of Nb- and La-doped $\mathrm{SrTi}{\mathrm{O}}_{3}$ single crystals increased with dopant concentration. The broad intensities around the Bragg peak observed in the in-plane x-ray-diffraction experiments indicated that the local lattice expansion and contraction, or local lattice distortions, in the crystal were caused by the dopant atoms. First-principles calculations similarly showed that the lattice expansion and local lattice distortions in the $\mathrm{SrTi}{\mathrm{O}}_{3}$ crystals were caused by doped Nb and La atoms. Atoms surrounding Schottky pairs of O and Sr vacancies were displaced both away from and towards the vacancies, resulting in a reduction in the lattice expansion of donor-doped $\mathrm{SrTi}{\mathrm{O}}_{3}$. Donor atoms and Schottky pairs thus play an important role in determining the lattice parameters of $\mathrm{SrTi}{\mathrm{O}}_{3}$ crystals. These fundamental structural analyses provide a useful basis to further investigate the electronic conductivity of electron-doped $\mathrm{SrTi}{\mathrm{O}}_{3}$.

In Situ Imprinting of Topographic Landscapes at the Cell-Substrate Interface.

In their native environments, adherent cells encounter dynamic topographical cues involved in promoting differentiation, orientation, and migration. Ideally, such processes would be amenable to study in cell culture using tools capable of imposing dynamic, arbitrary, and reversible topographic features without perturbing environmental conditions or causing chemical and/or structural disruptions to the substrate surface. To address this need, we report here development of an in vitro strategy for challenging cells with dynamic topographical experiences in which protein-based hydrogel substrate surfaces are modified in real time by positioning a pulsed, near-infrared laser focus within the hydrogel, promoting chemical cross-linking which results in local contraction of the protein matrix. Scanning the laser focus through arbitrary patterns directed by a dynamic reflective mask creates an internal contraction pattern that is projected onto the hydrogel surface as features such as rings, pegs, and grooves. By subjecting substrates to a sequence of scan patterns, we show that topographic features can be created, then eliminated or even reversed. Because laser-induced shrinkage can be confined to 3D voxels isolated from the cell-substrate interface, hydrogel modifications are made without damaging cells or disrupting the chemical or structural integrity of the surface.

Semi-solid deformation of Al-Cu alloys: A quantitative comparison between real-time imaging and coupled LBM-DEM simulations

Semi-solid alloys are deformed in a wide range of casting processes; an improved understanding and modelling capability is required to minimise defect formation and optimise productivity. Here we combine thin-sample in-situ X-ray radiography of semisolid Al-Cu alloy deformation at 40–70% solid with 2D coupled lattice Boltzmann method - discrete element method (LBM-DEM) simulations. The simulations quantitatively capture the key features of the in-situ experiments, including (i) the local contraction and dilation of the grain assembly during shear deformation; (ii) the heterogeneous strain fields and localisation features; (iii) increases in local liquid pressure in regions where liquid was expelled from the free surface in the experiment; and (iv) decreases in liquid pressure in regions where surface menisci are sucked-in in experiments. The verified DEM simulations provide new insights into the role of initial solid fraction on the stress-deformation response and support the hypothesis that the behaviour of semi-solid alloys can be described using critical state soil mechanics.

Immediate and substantial evolution of T-cell repertoire in peripheral blood and tumor microenvironment of patients with esophageal squamous cell carcinoma treated with preoperative chemotherapy.

Preoperative chemotherapy could decrease tumor size and improve overall survival for esophageal squamous cell carcinoma (ESCC), and moreover, rational combination of chemotherapy and immunotherapy could increase likelihood of inducing an effective antitumor immune response. However, the immunologic impact of chemotherapeutic drugs originally chosen for cancer treatment due to the direct toxicity is poorly understood. We assess the effect of a combination of clinically approved chemotherapeutic drugs [paclitaxel-nedaplatin (PTX-NDP)] on T-cell receptor (TCR) repertoires of peripheral T cells and tumor-infiltrating lymphocytes (TILs) from five patients with primary ESCC. We found that PTX-NDP therapy induced immediate and substantial changes in clonotype frequencies of peripheral T cells and TILs, and moreover, compared with clonal amplification, clonal contraction was more likely to occur in more abundant clones in patients with ESCC. Significant increases in TCR diversity were observed in peripheral T cells but not in TILs after PTX-NDP therapy. Reconstruction of posttreatment TILs was not merely a result of local expansion or contraction of pretreatment TILs, but also-at least in part-a consequence of the migration of peripheral T cells into the chronically inflamed tumor microenvironment. Our findings uncover further insight into T-cell immune response modulated with chemotherapy, providing for theoretical bases for rational combination strategy of chemotherapy and immunotherapy.

Finite element modelling of complex movements during self-sealing of ring incisions in leaves of Delosperma cooperi

A numerical computer model was developed in order to describe the complex self-sealing mechanism of injured Delosperma cooperi leaves. For this purpose, the leaf anatomy was simplified to a model consisting of five concentric tissue layers. Specific parameters (modulus of elasticity, permeability, porosity, etc.) were assigned to each tissue type for modelling its physical properties. These parameters were either determined experimentally from living plant material or taken from literature. The developed computer model considers the leaf as a liquid-filled porous body within a continuum approach in order to determine the governing equations. The modelling of the wound accounts for both the injury of peripheral tissues and the free surfaces caused by the incision. The loss of water through these free surfaces initiates the self-sealing process. It is further shown that the tissue permeability and the reflection coefficient (relative permeability of a cell membrane for solutes) are the determining parameters of the self-sealing process, whereas the modulus of elasticity has a negligible influence. Thus, the self-sealing mechanism is a hydraulically driven process which leads to a local (incision region) and global (total leaf) contraction of the leaf. The accuracy of the modelled self-sealing process was validated by comparing simulation results with experiments conducted on natural plant leaves. The results will serve as valuable input for developing novel, bio-inspired technical products with self-sealing function.

Time deformations of master equations

Convolutionless and convolution master equations are the two mostly used physical descriptions of open quantum systems dynamics. We subject these equations to time deformations: local dilations and contractions of time scale. We prove that the convolutionless equation remains legitimate under any time deformation (results in a completely positive dynamical map) if and only if the original dynamics is completely positive divisible. Similarly, for a specific class of convolution master equations we show that uniform time dilations preserve positivity of the deformed map if the original map is positive divisible. These results allow witnessing different types of non-Markovian behavior: the absence of complete positivity for a deformed convolutionless master equation clearly indicates that the original dynamics is at least weakly non-Markovian; the absence of positivity for a class of time-dilated convolution master equations is a witness of essentially non-Markovian original dynamics.

Preliminary investigation of limits of shock train jumps in a hypersonic inlet-isolator

During the acceleration process of an air-breathing propulsion vehicle, when the Mach number is closed to the transition point of the combustion mode, the drag force increases and thrust-to-drag reduces. Thus, equivalence ratio and higher values of heat release are demanded. A shock train will form in an isolator and will climb upstream as the backpressure increasing. In now days, a jumping characteristic was found as the shock train climbs upstream, while the deep mechanism is still not clear. Few works focus on the limits that trigger a jumping of the shock train. This paper aims to investigate the boundary of the trigger condition for it, which is named as the limits of a shock train jumps. Numerical study was carried out in a Mach 5 inlet-isolator model, and the shock train jumping movements were captured. The results indicate a local throat-like shape was formed by the separated flow at the head of the shock train, and the contraction ratio of the local throat-like shape was reduced to a small value when shock train jumping occurred. Based on the numerical data, the minimum value of the contraction ratio was discussed with flow physics. The reason that the head of a shock train jumps upstream with separation zone was discussed by a theoretical analysis, which assumed that the core flow is compressed through an oblique shock, normal shock, and returns to sonic at the local throat that formed by the throat-like shape at the head of the shock train. Then, a coefficient of correction was set for the minimum value of the local contraction ratio, which is used to improve the robustness of this analytical method. At last, the analysis method for limits of shock train jumps was employed for the experimental results. The phenomenon that a varied contraction ratio of a local throat-like shape was reproduced in the experiment and the analysis method in this paper was validated.

Spatially modulated ephrinA1:EphA2 signaling increases local contractility and global focal adhesion dynamics to promote cell motility

Recent studies have revealed pronounced effects of the spatial distribution of EphA2 receptors on cellular response to receptor activation. However, little is known about molecular mechanisms underlying this spatial sensitivity, in part due to lack of experimental systems. Here, we introduce a hybrid live-cell patterned supported lipid bilayer experimental platform in which the sites of EphA2 activation and integrin adhesion are spatially controlled. Using a series of live-cell imaging and single-molecule tracking experiments, we map the transmission of signals from ephrinA1:EphA2 complexes. Results show that ligand-dependent EphA2 activation induces localized myosin-dependent contractions while simultaneously increasing focal adhesion dynamics throughout the cell. Mechanistically, Src kinase is activated at sites of ephrinA1:EphA2 clustering and subsequently diffuses on the membrane to focal adhesions, where it up-regulates FAK and paxillin tyrosine phosphorylation. EphrinA1:EphA2 signaling triggers multiple cellular responses with differing spatial dependencies to enable a directed migratory response to spatially resolved contact with ephrinA1 ligands.

Fixed Point Theorems for Maps With Local and Pointwise Contraction Properties

AbstractThis paper constitutes a comprehensive study of ten classes of self-maps on metric spaces ⟨X, d⟩ with the pointwise (i.e., local radial) and local contraction properties. Each of these classes appeared previously in the literature in the context of fixed point theorems.We begin with an overview of these fixed point results, including concise self contained sketches of their proofs. Then we proceed with a discussion of the relations among the ten classes of self-maps with domains ⟨X, d⟩ having various topological properties that often appear in the theory of fixed point theorems: completeness, compactness, (path) connectedness, rectifiable-path connectedness, and d-convexity. The bulk of the results presented in this part consists of examples of maps that show non-reversibility of the previously established inclusions between these classes. Among these examples, the most striking is a differentiable auto-homeomorphism f of a compact perfect subset X of ℝ with f′ ≡ 0, which constitutes also a minimal dynamical system. We finish by discussing a few remaining open problems on whether the maps with specific pointwise contraction properties must have the fixed points.

Theoretical basis of the method for measuring direct muscular excitability (experimental research)

During “in vivo experiments” there was compared the amplitude and shape of induced contractions mechanograms (MGS) with changes in muscle contractions impedanceograms (IGS). It was made with the help of an atraumatic needle electrode for direct myostimulation and registration of muscle contraction impedance. So the impedance decrease was registered at threshold (local) and maximum IGS. The absolute amplitude values of MGS and IGS were changed proportionally, however, the configuration of the MGS and IGS were different. In a way the action potential and local muscle contraction were preceded the beginning of the "mechanical" response. According to the results of the study, the IGS parameters are proportional to the number of open ion channels Ca2+ and can be used to assess the degree of muscle tissue damage based on changes in the parameters of direct muscle excitability (PMV).