Critical Perturbations Research Articles

On a class of elliptic equations with critical perturbations in the hyperbolic space

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a ( x ) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 ( B N ) , where B N denotes the hyperbolic space, 2 < p < 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 < p < + ∞, if N = 2, λ < ( N − 1 ) 2 4 , and 0 < a ∈ L ∞ ( B N ). We first prove the existence of a positive radially symmetric ground-state solution for a ( x ) ≡ 1. Next, we prove that for a ( x ) ⩾ 1, there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a ( x ) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.

A hybrid robust model reference adaptive controller and proportional integral controller without reference model for partially modeled systems

SummaryThis work presents the discrete‐time stability analysis of a hybrid robust model reference adaptive controller and proportional‐integral controller (MRAC‐PI) controller using Lyapunov criterion. The control structure merges the classical PI controller with a robust model reference adaptive control, resulting in an adaptive and robust control strategy, where the gains unite both control techniques in a unique structure and it does not require the design of a reference model, being able to track straightforwardly the reference signal. The adaptive gains are updated continuously by a modified gradient algorithm regarding control action, regulation error, and the system output. The stability analysis provides the controller design constraints, while also demonstrate the tendency of regulation error to a residual set, and proves the boundedness of all closed‐loop signals. Furthermore, as the controller is robust to unmodeled dynamics, it is feasible to control partially modeled systems. Simulation results in a real‐world engineering application are presented to corroborate the robustness of this controller and show its fast response in the face of critical exogenous system perturbations.

Linear stability implies nonlinear stability for Faber–Krahn-type inequalities

For a domain \Omega \subseteq \mathbb{R}^n and a small number \mathfrak{T} > 0 , let \mathcal{E}_0(\Omega) = \lambda_1(\Omega) + \mathfrak{T} \operatorname{tor}(\Omega) = \inf_{u, w \in H^1_0(\Omega)\setminus \{0\}} \frac{\int |\nabla u|^2}{\int u^2} + \mathfrak{T} \!\int \frac{1}{2} |\nabla w|^2 - w be a modification of the first Dirichlet eigenvalue of \Omega . It is well known that over all \Omega with a given volume, the only sets attaining the infimum of \mathcal{E}_0 are balls B_R ; this is the Faber–Krahn inequality. The main result of this paper is that, if for all \Omega with the same volume and barycenter as B_R whose boundaries are parametrized as small C^2 normal graphs over \partial B_R with bounded C^2 norm \int |u_\Omega - u_{B_R}|^2 + |\Omega \triangle B_R|^2 \leq C [\mathcal{E}_0(\Omega) - \mathcal{E}_0(B_R)] (i.e., the Faber–Krahn inequality is linearly stable ), then the same is true for any \Omega with the same volume and barycenter as B_R without any smoothness assumptions (i.e., it is nonlinearly stable ). Here u_\Omega stands for an L^2 normalized first Dirichlet eigenfunction of \Omega . Related results are shown for Riemannian manifolds. The proof is based on a detailed analysis of some critical perturbations of Bernoulli-type free boundary problems. The topic of when linear stability is valid, as well as some applications, are considered in a companion paper.

Critical perturbations for second-order elliptic operators, I: Square function bounds for layer potentials

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness of the Dirichlet, Neumann and Regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. For instance, this allows us to claim the first results in the setting of an unbounded domain concerning the solvability of boundary value problems for the magnetic Schr\"odinger operator $-(\nabla-i{\bf a})^2+V$ when the magnetic potential ${\bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space. In the present paper, we establish $L^2$ control of the square function via a vector-valued $Tb$ theorem and abstract layer potentials, and use these square function bounds to obtain uniform slice bounds for solutions. The existence and uniqueness of solutions, as well as bounds for the non-tangential maximal operator, are considered in the upcoming paper.

Onset and nonlinear regimes of convection in an inclined porous layer subject to a vertical temperature gradient

In this paper, we study the onset and non-linear regimes of thermal buoyancy convection in an inclined porous layer saturated with fluid. The layer is subject to a gravitational field and a strictly vertical temperature gradient. This problem is important for geological applications. The linear stability of the heat-conducting regime to two-dimensional perturbations was previously studied by Kolesnikov and Lyubimov [J. Appl. Mech. Tech. Phys. 14, 400–404 (1973)]. In the first part of our work, we numerically, using the finite difference method, investigate two-dimensional nonlinear convection regimes that arise after the loss of stability of the heat-conducting regime. In the second part of the paper, the linear stability of the heat-conducting regime to three-dimensional perturbations is investigated. It has been found that for any layer inclination angle, three-dimensional perturbations are more dangerous than two-dimensional ones, and the most dangerous perturbations have the form of longitudinal rolls. For the layer inclination angle α<45°, the wavenumber of critical perturbations is equal to zero, and for α>45°, it differs from zero. Numerical calculations by the finite volume method within the framework of the full three-dimensional nonlinear approach confirm the conclusions of the linear stability analysis.

Systematic Route to Subcritical Dynamo Branches.

We demonstrate that the nonlinear optimization of a finite-amplitude disturbance over a freely evolving and possibly even turbulent flow, can successfully identify subcritical dynamo branches as well as the structure and amplitude of their critical perturbations. As this approach does not require prior knowledge of the magnetic field amplification mechanisms, it opens a new avenue for systematically probing subcritical dynamo flows.

Structural analysis of M1AP variants associated with severely impaired spermatogenesis causing male infertility.

BackgroundImpaired meiosis can result in absence of sperm in the seminal fluid. This condition, namely non-obstructive azoospermia (NOA), is one of the reasons of male infertility. Despite the low number of studies on meiosis 1-associated protein (M1AP) in the literature, M1AP is known to be crucial for spermatogenesis. Recently, seven variants (five missense, one frameshift, one splice-site) have been reported in the M1AP gene as associated with NOA, cryptozoospermia and oligozoospermia in two separate studies. However, all missense variants were evaluated as variant of uncertain significance by these studies. Therefore, we aimed to analyze their structural impacts on the M1AP protein that could lead to NOA.MethodsWe firstly performed an evolutionary conservation analysis for the variant positions. Afterwards, a comprehensive molecular modelling study was performed for the M1AP structure. By utilizing this model, protein dynamics were sampled for the wild-type and variants by performing molecular dynamics (MD) simulations.ResultsAll variant positions are highly conserved, indicating that they are potentially important for function. In MD simulations, none of the variants led to a general misfolding or loss of stability in the protein structure, but they did cause severe modifications in the conformational dynamics of M1AP, particularly through changes in local interactions affecting flexibility, hinge and secondary structure.ConclusionsDue to critical perturbations in protein dynamics, we propose that these variants may cause NOA by affecting important interactions regulating meiosis, particularly in wild-type M1AP deficiency since the variants are reported to be homozygous or bi-allelic in the infertile individuals. Our results provided reasonable insights about the M1AP structure and the effects of the variants to the structure and dynamics, which should be further investigated by experimental studies to validate.

Experimental measurements versus linear stability analysis for primary instability of stratified two-phase flows in a square rectangular duct

The onset of the primary instability of two-phase stratified flow in a square duct was studied experimentally. Measurements were carried out by a specially designed fully non-intrusive technique. Critical values of the two superficial velocities corresponding to the primary instability of air-water stratified flows in a square duct were measured and successfully compared with the computationally predicted stability diagram. Along with the good quantitative agreement between the experimentally measured and numerically predicted critical superficial velocities, a qualitative agreement was obtained between predicted and measured frequencies of the critical perturbations.

Generalized Strichartz estimates for wave and Dirac equations in Aharonov–Bohm magnetic fields

We prove generalized Strichartz estimates for wave and massless Dirac equations in Aharonov-Bohm magnetic fields. Following a well established strategy to deal with scaling critical perturbations of dispersive PDEs, we make use of Hankel transform and rely on some precise estimates on Bessel functions. As a complementary result, we prove a local smoothing estimate for the Klein-Gordon equation in the same magnetic field.

The onset of convection in liquid with the free surface cooled from above: Influence of the surface behavior on the size and formation time of critical perturbations

The onset of convection in an initially homogeneous liquid layer cooled from above is investigated by means of a non-normal approach. This method is based on the analysis of the temporal evolution of small perturbations. The most amplified perturbation and the moment, when convection can be observed, are determined. Experimental measurements are performed for a layer of distilled water cooled by evaporation. Instantaneous surface temperature fields are obtained using infrared thermography. The effect of the surface mobility on the size and onset time of convective cells is examined. The comparison of experimental results with numerical simulations for different boundary conditions reveals the presence of a film of adsorbed contaminants on the surface of distilled water, which corresponds to the no-slip condition. In the presence of the film, boundary conditions for cooling from above and heating from below become the same, and the obtained results agree with the experimental data available for a liquid layer with a heated bottom. As the layer depth increases, the onset time and the average horizontal wavenumber of convective cells become independent from the layer depth. The condition for the transition to a deep water approximation is determined by the surface heat flux and thermophysical parameters of the liquid.

Method for Predicting the Stability Limit to Acoustic Oscillations in Liquid-Propellant Rocket Engine Combustion Chambers Based on Combustion Noise

An experimental method for determining the critical pressure perturbations leading to acoustic instability in combustion chambers of liquid-propellant rocket engines has been developed and used to estimate the stability of the working process. The method involves statistical processing of recorded noise pressure pulsations near the natural resonant frequency range for all normal modes of acoustic oscillations in cylindrical combustion chambers and gas generators. The oscillation damping coefficient (decrement), characterizing the difference between the generated and dissipated energy, is used as a diagnostic criterion for predicting the stable or unstable states of a dynamic system. The method is based on the theory of self-oscillating dynamic systems and one-dimensional Markov random processes using the Fokker–Planck–Kolmogorov equation. Analysis of a nonlinear differential equation with a symmetrized stochastic right-hand side describing white noise for experimentally determined amplitudes of pressure pulsations and their statistical processing using Mera software allows the state of self-oscillating systems to be identified as stable or unstable. The method is “passive;" it is applicable without using standard external impulse disturbing devices.

Predicting Cancer Evolution Using Cell State Dynamics.

One of the biggest challenges in cancer is predicting its initiation and course of progression. In this issue of Cancer Research, Rockne and colleagues use state transition theory to predict how peripheral mononuclear blood cells in mice transition from a healthy state to acute myeloid leukemia. They found that critical transcriptomic perturbations could predict initiation and progression of the disease. This is an important step toward accurately predicting cancer evolution, which may eventually facilitate early diagnosis of cancer and disease recurrence, and which could potentially inform on timing of therapeutic interventions.See related article by Rockne et al., 3157.

Cascading failure analysis and robustness optimization of metro networks based on coupled map lattices: a case study of Nanjing, China

Cascading failure in metro networks is a dynamic chain process induced by the interaction of passenger flow and network topology. In this paper, a bi-directional coupled map lattice model is proposed to study the cascading failure of metro networks. The model considers the two-way traffic problem, and the results are closer to those of actual metro networks than the previous one-way coupling models. A $$\eta$$ -based flow redistribution method is proposed, and different passenger flow redistribution strategies after station failure can be achieved by changing the flow redistribution coefficient $$\eta$$ from 0 to 1. Moreover, the robustness of metro networks can be optimized by searching for the optimal $$\eta$$ that can maximize the critical perturbation leading to global network failure. We study the actual case of Nanjing metro. The analysis results show that the network is more vulnerable to intentional attacks than to random failures, and global network failure is triggered more easily on the largest strength station than on the stations with the largest betweenness and largest degree. The influence of coupling strengths on the critical perturbation is also investigated. The results show that larger coupling strengths correspond to smaller critical perturbations, but a change in the coupling strengths has a small impact on the optimal $$\eta$$ . Under the given traffic data, the optimal $$\eta$$ for Nanjing metro is approximately in the range (0.3, 0.4). This study provides a reference for developing strategies for dynamic safety evaluation and emergency management of passenger flow in metro networks.

Analysis of Hypoxiamir-Gene Regulatory Network Identifies Critical MiRNAs Influencing Cell-Cycle Regulation Under Hypoxic Conditions

Hypoxia is a pathophysiological condition which arises due to low oxygen concentration in conditions like cardiovascular diseases, inflammation, ascent to higher altitude, malignancies, deep sea diving, prenatal birth, etc. A number of microRNAs (miRNAs), Transcription Factors (TFs) and genes have been studied separately for their role in hypoxic adaptation and controlling cell-cycle progression and apoptosis during this stress. We hypothesize that miRNAs and TFs may act in conjunction to regulate a multitude of genes and play a crucial and combinatorial role during hypoxia-stress-responses and associated cellcycle control mechanisms. We collected a comprehensive and non-redundant list of human hypoxia-responsive miRNAs (also known as hypoxiamiRs). Their experimentally validated gene-targets were retrieved from various databases and a comprehensive hypoxiamiR-gene regulatory network was built. Functional characterization and pathway enrichment of genes identified phospho-proteins as enriched nodes. The phospho-proteins which were localized both in the nucleus and cytoplasm and could potentially play important role as signaling molecules were selected; and further pathway enrichment revealed that most of them were involved in NFkB signaling. Topological analysis identified several critical hypoxiamiRs and network perturbations confirmed their importance in the network. Feed Forward Loops (FFLs) were identified in the subnetwork of enriched genes, miRNAs and TFs. Statistically significant FFLs consisted of four miRNAs (hsa-miR-182-5p, hsa- miR-146b-5p, hsa-miR-96, hsa-miR-20a) and three TFs (SMAD4, FOXO1, HIF1A) both regulating two genes (NFkB1A and CDKN1A). Detailed BioCarta pathway analysis identified that these miRNAs and TFs together play a critical and combinatorial role in regulating cell-cycle under hypoxia, by controlling mechanisms that activate cell-cycle checkpoint protein, CDKN1A. These modules work synergistically to regulate cell-proliferation, cell-growth, cell-differentiation and apoptosis during hypoxia. A detailed mechanistic molecular model of how these co-regulatory FFLs may regulate the cell-cycle transitions during hypoxic stress conditions is also put forth. These biomolecules may play a crucial and deterministic role in deciding the fate of the cell under hypoxic-stress.

The features of circulating and tumor-infiltrating γδ T cells in melanoma patients display critical perturbations with prognostic impact on clinical outcome

ABSTRACT γδT cells hold a pivotal role in tumor immunosurveillance through their prompt activation and cytokine secretion, their ability to kill tumor cells in an Human Leukocyte Antigen (HLA)-unrestricted manner, and their combination of features of both innate and adaptive immunity. These unique properties and functional plasticity render them very attractive both as targets and vectors for cancer immunotherapy. Yet, these potent and fascinating antitumor effectors have not been extensively explored in melanoma. We provided here a detailed investigation of the phenotypic and functional properties of circulating and tumor-infiltrating γδT cells in melanoma patients, and their impact on clinical evolution. High proportions of circulating- and tumor-infiltrating γδT and δ2+ subset were associated with better clinical outcome. We reported however that circulating and tumor-infiltrating γδT cells from melanoma patients displayed an altered expression of NCR, KIR, and immune checkpoints, and identified NKp44, PD1, 41BB/41BBL, TIM3, and LAG3 as crucial checkpoints allowing immune escape and tumor progression. Notably, melanoma drastically impaired the ability of γδT cells to exhibit activation molecules, secrete cytokines, and display cytotoxicity toward melanoma in response to stimulation with phosphoantigens. It drove them toward regulatory and Th17 profiles associated with poor clinical outcomes. Our study highlights that melanoma hijacked γδT cells to escape from immune control, and revealed that circulating and tumor-infiltrating γδT cell features are promising potential biomarkers of clinical evolution. Such understanding of the physiopathology of γδT cells may help designing new therapeutic approaches exploiting the antitumor potential of γδT cells while counteracting their skewing by tumors to improve patient outcomes.

Excitation of Convection in a System of Layers of a Binary Solution and an Inhomogeneous Porous Medium in a High-Frequency Vibration Field

The onset of convection in a system of horizontal layers of a binary solution and an inhomogeneous porous medium saturated with the solution is studied. The system is subjected to high-frequency transverse vibrations in the gravity field. It is assumed that the porosity of the medium depends linearly on the vertical coordinate. The permeability is estimated by the Karman–Kozeny formula for different values of the dimensionless gradient of the porosity, mz. The convection of the fluid under the action of high-frequency vibrations in the gravity field is described using the averaging method. The linear problem of the stability of the mechanical equilibrium of the fluid is solved numerically by the shooting method. The values of the critical parameters corresponding to the convection initiation threshold are determined for the system heated from below or from above. The heating from below is distinguished by a sharp change in the character of the instability with a variation in the porosity gradient or the vibration intensity. It is shown that, when the porosity increases with depth, at mz =–0.2, the instability is caused by the development of long-wave perturbations involving the fluid and the porous layers. When the porosity decreases with depth, at mz = 0.2, the most dangerous perturbations are short-wave perturbations localized in the fluid layer. For intermediate values of the porosity gradient,–0.2< mz < 0.2, the values of the minimum Rayleigh–Darcy critical numbers, which determine the equilibrium stability threshold with respect to short-wave and long-wave disturbances, approach one another. The neutral curves are bimodal. Upon heating from below, vertical vibrations effectively suppress convection in the fluid layer; therefore, with an increase in their intensity, the transition from short-wave vibrations, which are most dangerous, to long-wave perturbations is observed. A noticeable increase in the stability threshold is observed when the porosity decreases with depth. Upon heating from above, vibrations destabilize the equilibrium in the system and lead to a reduction in the wavelength of the critical perturbations. The wavelength decreases monotonically. Its maximum change is detected in layers whose porosity increases with depth.

Instantons in Self-Organizing Logic Gates

Self-organizing logic is a recently-suggested framework that allows the solution of Boolean truth tables "in reverse," i.e., it is able to satisfy the logical proposition of gates regardless to which terminal(s) the truth value is assigned ("terminal-agnostic logic"). It can be realized if time non-locality (memory) is present. A practical realization of self-organizing logic gates (SOLGs) can be done by combining circuit elements with and without memory. By employing one such realization, we show, numerically, that SOLGs exploit elementary instantons to reach equilibrium points. Instantons are classical trajectories of the non-linear equations of motion describing SOLGs, and connect topologically distinct critical points in the phase space. By linear analysis at those points, we show that these instantons connect the initial critical point of the dynamics, with at least one unstable direction, directly to the final fixed point. We also show that the memory content of these gates only affects the relaxation time to reach the logically consistent solution. Finally, we demonstrate, by solving the corresponding stochastic differential equations, that since instantons connect critical points, noise and perturbations may change the instanton trajectory in the phase space, but not the initial and final critical points. Therefore, even for extremely large noise levels, the gates self-organize to the correct solution. Our work provides a physical understanding of, and can serve as an inspiration for, new models of bi-directional logic gates that are emerging as important tools in physics-inspired, unconventional computing.

The Onset of Double-Diffusive Convection in a Superposed Fluid and Porous Layer Under High-Frequency and Small-Amplitude Vibrations

We numerically simulate the initiation of an average convective flow in a system composed of a horizontal binary fluid layer overlying a homogeneous porous layer saturated with the same fluid under gravitational field and vibration. In the layers, fixed equilibrium temperature and concentration gradients are set. The layers execute high-frequency oscillations in the vertical direction. The vibration period is small compared with characteristic timescales of the problem. The averaging method is applied to obtain vibrational convection equations. Using for computation the shooting method, a numerical investigation is carried out for an aqueous ammonium chloride solution and packed glass spheres saturated with the solution. The instability threshold is determined under two heating conditions—on heating from below and from above. When the solution is heated from below, the instability character changes abruptly with increasing solutal Rayleigh number, i.e., there is a jump-wise transition from the most dangerous shortwave perturbations localized in the fluid layer to the long-wave perturbations covering both layers. The perturbation wavelength increases by almost 10 times. Vibrations significantly stabilize the fluid equilibrium state and lead to an increase in the wavelength of its perturbations. When the fluid with the stabilizing concentration gradient is heated from below, convection can occur not only in a monotonous manner but also in an oscillatory manner. The frequency of critical oscillatory perturbations decreases by 10 times, when the long-wave instability replaces the shortwave instability. When the fluid is heated from above, only stationary convection is excited over the entire range of the examined parameters. A lower monotonic instability level is associated with the development of perturbations with longer wavelength even at a relatively large fluid layer thickness. Vibrations speed up the stationary convection onset and lead to a decrease in the wavelength of most dangerous perturbations of the motionless equilibrium state. In this case, high enough amplitudes of vibration are needed for a remarkable change in the stability threshold. The results of numerical simulation show good agreement with the data of earlier works in the limiting case of zero fluid layer thickness.

Influence of Sedimentation Length on the Convective Stability of a Colloidal Suspension

The сonvective stability of a colloidal suspension is studied in the case when the vertical dimension of the cavity is less than or comparable with the sedimentation length of nanoparticles. The analysis is carried out within the Boussinesq approximation on the basis of a modified model that takes into account the dependence of a thermodiffusion flow on the local value of impurity concentration. A new parameter of the problem is the ratio of the sedimentation length to the vertical dimension of the cavity. For a quiescent colloidal suspension, exact and approximate (in the case of small concentrations) solutions are obtained that describe the distributions of nanoparticles. A transformation is obtained that allows one to investigate the convective stability of a colloidal suspension stratified in the gravitational field by the Galerkin method with a set of simple trial functions. Instability boundaries and the characteristics of critical perturbations are determined. It is shown that, in the case of negative thermodiffusion, a decrease in the sedimentation length leads to a decrease in the convection threshold and the frequency of neutral oscillations.

Vibration effect on double-diffusive instability in an inhomogeneous porous layer underlying a binary fluid layer

A linear stability problem for mechanical equilibrium in an inhomogeneous fluid-saturated porous layer underlying a horizontal binary fluid layer is numerically simulated. The layers are subjected to high-frequency vertical vibration in a gravity field. Different values of temperature and concentration are fixed at the outer impermeable boundaries of the two-layer system. Porosity of a porous layer linearly depends on a transverse z-coordinate. Permeability is given by the Carman-Kozeny formula. To describe a fluid flow in layers, we use the vibrational convection equations written in the Boussinesq approximation and obtained by the averaging method. It is shown that when the fluid is heated from below, vibration effectively increases the equilibrium stability threshold and wavelength of its most dangerous perturbations. If a temperature gradient coincides with the direction of a concentration gradient for the heavier component of binary fluid, average convection is excited in an oscillatory manner. In the opposite case, there is a monotonic instability of equilibrium. Effects of the dimensionless porosity gradient mz and vibrational parameter pv on the instability threshold are studied. For pv<0.0134 and a fixed solutal Rayleigh number (-5⩽Rmc⩽15) an abrupt jump-like transition from the long-wave to short-wave most dangerous perturbations occurs as mz grows in the range of 0.030–0.162. Long-wave perturbations penetrate both layers. Short-wave perturbations mainly locate in the fluid layer. The transition is smoothed for pv>0.0134. When the fluid is heated from above, wavelength of critical perturbations smoothly varies with a change in pv and mz. A convective fluid flow arises monotonously and mostly in the form of long wavelength rolls (a wave number is of k<3.3). Vibration weakly lowers the equilibrium stability threshold in layers. The effect is most pronounced at high enough Rayleigh numbers (Rm=-40) for a porous medium with the porosity increasing with depth at mz=-0.2.