Critical Value Λc Research Articles

Existence theorems for a generalized Chern–Simons equation on finite graphs

Consider G = (V, E) as a finite graph, where V and E correspond to the vertices and edges, respectively. We study a generalized Chern–Simons equation Δu=λeu(ebu−1)+4π∑j=1Nδpj on G, where λ and b are positive constants; N is a positive integer; p1, p2, …, pN are distinct vertices of V; and δpj is the Dirac delta mass at pj. We prove that there exists a critical value λc such that the equation has a solution if λ ≥ λc and the equation has no solution if λ < λc. We also prove that if λ > λc, the equation has at least two solutions that include a local minimizer for the corresponding functional and a mountain-pass type solution. Our results extend and complete those of Huang et al. [Commun. Math. Phys. 377(1), 613–621 (2020)] and Hou and Sun [Calculus Var. Partial Differ. Equations 61(4), 139 (2022)].

Multiple solutions for a generalized Chern-Simons equation on graphs

In this paper, we consider a generalized Chern-Simons equationΔu=λeu(eu−1)5+4π∑j=1Nδpj on a connected finite graph G=(V,E), where λ denotes a positive constant; N denotes a positive integer; p1,p2,⋅⋅⋅,pN denote distinct vertices of V; δpj denotes the Dirac delta mass at pj. Using the upper-lower solution method and prior estimates, we prove that there exists a critical value λc such that the generalized Chern-Simons equation admits a solution if λ≥λc. Then applying the mountain pass theorem due to Ambrosetti-Rabinowitz, we establish that the equation has at least two solutions if λ>λc.

λ-Navier-Stokes turbulence.

We investigate numerically the model proposed in Sahoo et al. (2017 Phys. Rev. Lett. 118, 164501) where a parameter λ is introduced in the Navier-Stokes equations such that the weight of homochiral to heterochiral interactions is varied while preserving all original scaling symmetries and inviscid invariants. Decreasing the value of λ leads to a change in the direction of the energy cascade at a critical value [Formula: see text]. In this work, we perform numerical simulations at varying λ in the forward energy cascade range and at changing the Reynolds number [Formula: see text]. We show that for a fixed injection rate, as [Formula: see text], the kinetic energy diverges with a scaling law [Formula: see text]. The energy spectrum is shown to display a larger bottleneck as λ is decreased. The forward heterochiral flux and the inverse homochiral flux both increase in amplitude as [Formula: see text] is approached while keeping their difference fixed and equal to the injection rate. As a result, very close to [Formula: see text] a stationary state is reached where the two opposite fluxes are of much higher amplitude than the mean flux and large fluctuations are observed. Furthermore, we show that intermittency as [Formula: see text] is approached is reduced. The possibility of obtaining a statistical description of regular Navier-Stokes turbulence as an expansion around this newly found critical point is discussed. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Giant component of the soft random geometric graph

Consider a 2-dimensional soft random geometric graph G(λ,s,ϕ), obtained by placing a Poisson(λs2) number of vertices uniformly at random in a square of side s, with edges placed between each pair x,y of vertices with probability ϕ(‖x−y‖), where ϕ: R+→[0,1] is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph G(λ,s,ϕ) in the large-s limit with (λ,ϕ) fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where λ equals the critical value λc(ϕ).

Magnetohydrodynamics (MHD) boundary layer flow of hybrid nanofluid over a moving plate with Joule heating

The proficiency of hybrid nanoparticles in augmenting the heat transfer has fascinated many researchers to further analysing the working fluid. The present paper is focused on the MHD hybrid nanofluid flow with heat transfer on a moving plate with Joule heating. The combination of metal (Cu) and metal oxide (Al2O3) nanoparticles with water (H2O) as the base fluid is used for the analysis. Similarity transformation reduces the complexity of the PDEs into a system of ODEs, which is then solved numerically using the function bvp4c from MATLAB for different values of the governing parameters. Two solutions are obtained when the plate is moved oppositely from the free stream flow. Analysis of flow stability unveils the first solution as the real physical solution, which is realizable in practice. From physical perspective, the real solution must be available for all cases of λ which affirms the finding from stability analysis. An upsurge of suction’s strength and magnetic parameter enhances the heat transfer operation and extends the critical value λc. Meanwhile, there is no change on the critical value when the Eckert number is added. This study is important in determining the thermal behavior of Cu-Al2O3/H2O when the physical parameters like magnetic field and Joule heating are embedded. The results are new and original with many practical applications in the modern industry.

Dynamical transition theory of hexagonal pattern formations

The main goal of this paper is to understand the formation of hexagonal patterns from the dynamical transition theory point of view. We consider the transitions from a steady state of an abstract nonlinear dissipative system. To shed light on the formation of mixed mode patterns such as the hexagonal pattern, we consider the case where the linearized operator of the system has two critical real eigenvalues, at a critical value λc of a control parameter λ with associated eigenmodes having a roll and rectangular pattern. By using center manifold reduction, we obtain the reduced equations of the system near the critical transition value λc. By a through analysis of these equations, we fully characterize all possible transition scenarios when the coefficients of the quadratic part of the reduced equations do not vanish. We consider three problems, two variants of the 2D Swift-Hohenberg equation and the 3D surface tension driven convection, to demonstrate that all the main theoretical results we obtain here are indeed realizable.

Hybrid nanofluid flow and heat transfer over a permeable biaxial stretching/shrinking sheet

Purpose The purpose of this paper is to examine the axisymmetric flow and heat transfer of a hybrid nanofluid over a permeable biaxial stretching/shrinking sheet. Design/methodology/approach In this study, 0.1 solid volume fraction of alumina (Al2O3) is fixed, then consequently, various solid volume fractions of copper (Cu) are added into the mixture with water as the base fluid to form Cu-Al2O3/water hybrid nanofluid. The hybrid nanofluid equations are converted to the similarity equations by using the similarity transformation. The bvp4c solver, which is available in the Matlab software is used for solving the similarity equations numerically. The numerical results for selected values of the parameters are presented in tabular and graphical forms, and are discussed in detail. Findings It is found that dual solutions exist up to a certain value of the stretching/shrinking and suction parameters. The critical value λc < 0 for the existence of the dual solutions decreases as nanoparticle volume fractions for copper increase. The temporal stability analysis is performed to analyze the stability of the dual solutions, and it is revealed that only one of them is stable and physically reliable. Originality/value The present problem is new, original with many important results for practical problems in the modern industry.

Phase Coexistence for the Hard-Core Model on ℤ2

The hard-core model has attracted much attention across several disciplines, representing lattice gases in statistical physics and independent sets in discrete mathematics and computer science. On finite graphs, we are given a parameter λ, and an independent setIarises with probability proportional to λ|I|. On infinite graphs a Gibbs measure is defined as a suitable limit with the correct conditional probabilities, and we are interested in determining when this limit is unique and when there is phase coexistence,i.e., existence of multiple Gibbs measures.It has long been conjectured that on ℤ2this model has a critical value λc≈ 3.796 with the property that if λ < λcthen it exhibits uniqueness of phase, while if λ > λcthen there is phase coexistence. Much of the work to date on this problem has focused on the regime of uniqueness, with the state of the art being recent work of Sinclair, Srivastava, Štefankovič and Yin showing that there is a unique Gibbs measure for all λ < 2.538. Here we explore the other direction and prove that there are multiple Gibbs measures for all λ > 5.3506. We also show that with the methods we are using we cannot hope to replace 5.3506 with anything below 4.8771.Our proof begins along the lines of the standard Peierls argument, but we add two innovations. First, following ideas of Kotecký and Randall, we construct an event that distinguishes two boundary conditions and always has long contours associated with it, obviating the need to accurately enumerate short contours. Second, we obtain improved bounds on the number of contours by relating them to a new class of self-avoiding walks on an oriented version of ℤ2.

Mixed convection boundary-layer flow past a vertical flat plate with a convective boundary condition

The mixed convection boundary-layer flow on a vertical surface with an applied convective boundary condition is considered. Specific forms for the outer flow and surface heat transfer parameter are taken to reduce the problem a similarity system, which is seen to involve three parameters: m, the exponent of the outer flow; λ, the mixed convection parameter and B, the Biot number, as well as the Prandtl number. \({m = \frac{1}{5}}\) is found to be a transitional case with different behaviour depending on whether \({m > \frac{1}{5}}\) or \({m \frac{1}{5}}\), there is a critical value λc with solutions only for λ ≥ λc, a range of values of λ where there are dual solutions and the upper solution branch continuing into aiding flow. For \({0 \frac{1}{5}}\) to solutions terminating at a finite value of λ > 0 and continuing to large values of |λ|. For m = 0 (uniform outer flow) only this latter behaviour is seen to arise. When m \frac{1}{5}}\).

Noise and time delay induce critical point in a bistable system

We study relaxation time Tc of time-delayed bistable system driven by two cross-correlated Gaussian white noises that one is multiplicative and the other is additive. By means of numerical calculations, the results indicate that: (i) Combination of noise and time delay can induce two critical points about the relaxation time at some certain noise cross-correlation strength λ under the condition that the multiplicative intensity D equals to the additive noise intensity α. (ii) For each fixed D or α, there are two symmetrical critical points which locates in the regions of positive and negative correlations, respectively. Namely, as λ equals to the critical value λc, Tc is independent of the delay time and the result of Tc versus τ is a horizontal line, but as |λ|>|λc| (or |λ|<|λc|), the relaxation time Tc monotonically increases (or decreases) with the delay time increasing. (iii) In the presence of D = α, the change of λc with D is two symmetrical curves about the axis of λc = 0, and the critical value λc is close to zero for a smaller D, which approaches to +1 or -1 for a greater D.

Critical properties of the SIS model dynamics on the Apollonian network

We present an analysis of the classical SIS (susceptible–infected–susceptible) model on the Apollonian network which is scale free and displays the small word effect. Numerical simulations show a continuous absorbing-state phase transition at a finite critical value λc of the control parameter λ. Since the coordination number k of the vertices of the Apollonian network is cumulatively distributed according to a power-law P(k) ∝ 1/kη−1, with exponent η ≃ 2.585, finite size effects are large and the infinite network limit cannot be reached in practice. Consequently, our study requires the application of finite size scaling theory, allowing us to characterize the transition by a set of critical exponents β/ν⊥, γ/ν⊥, ν⊥, β. We found that the phase transition belongs to the mean-field directed percolation universality class in regular lattices but, very peculiarly, is associated with a short-range distribution whose power-law distribution of k is defined by an exponent η larger than 3.

Mixed Convection Boundary-Layer Flow Along a Vertical Cylinder Embedded in a Porous Medium Filled by a Nanofluid

The steady mixed convection boundary-layer flow on a vertical circular cylinder embedded in a porous medium filled by a nanofluid is studied for both cases of a heated and a cooled cylinder. The governing system of partial differential equations is reduced to ordinary differential equations by assuming that the surface temperature of the cylinder and the velocity of the external (inviscid) flow vary linearly with the axial distance x measured from the leading edge. Solutions of the resulting ordinary differential equations for the flow and heat transfer characteristics are evaluated numerically for various values of the governing parameters, namely the nanoparticle volume fraction \({\phi}\), the mixed convection or buoyancy parameter λ and the curvature parameter γ. Results are presented for the specific case of copper nanoparticles. A critical value λc of λ with λc λc for both aiding, λ > 0 and opposing, λ < 0, flows. Asymptotic solutions are also determined for both the free convection limit \({(\lambda \gg 1)}\) and for large curvature parameter \({(\gamma \gg 1)}\).

Characteristic time of strain induced crystallization of crosslinked natural rubber

Real time Wide-Angle X-ray Scattering (WAXS) measurements during cyclic tensile tests at high strain rates (from 8 s−1–280 s−1) and at room temperature on crosslinked Natural Rubber (NR) are performed thanks to a specific homemade device. From the observed influence of the frequency on the crystallization index at the maximum sample elongation, a characteristic crystallization time is deduced. This is done taking into account the material self-heating during such unusually high strain rates. Two regimes for the dynamic process of strain induced crystallization are evidenced. For the NR tested, the obtained characteristic time is around 20 ms when the material average elongation during the cyclic test is above a critical elongation value λc. λc is the minimum elongation needed to induce crystallization during low strain rate tensile tests. Moreover, a rapid increase of this characteristic time is found when the average elongation decreases below this critical value.

Numerical investigations of lift suppression by feedback rotary oscillation of circular cylinder at low Reynolds number

This article describes a strategy of active flow control for lift force reduction of circular cylinder subjected to uniform flow at low Reynolds numbers. The flow control is realized by rotationally oscillating the circular cylinder about its axis with ω(t)=−λCL(t), where ω(t) is the dimensionless angular speed of rotation cylinder, λ is the control parameter and CL(t) is the feedback signal of lift coefficient. The study focuses on seeking optimum λ for the low Reynolds numbers of 60, 80, 100, 150, and 200. The effectiveness of the proposed flow control in suppressing lift force is examined comprehensively by a numerical model based on the finite element solution of two-dimensional Navier–Stokes equations. The dependence of lift reduction on the control parameter λ is investigated. The threshold of λ, denoted by λc, is identified for the Reynolds numbers considered in this work. The numerical results show that the present active rotary oscillation of circular cylinder is able to reduce the amplitude of lift force significantly as long as λ≤λc, at least 50% for the laminar flow regime. Meanwhile, the present active flow control does not result in the undesirable increase in the drag force. The Strouhal number is observed to decrease slightly with the increase of λ. As for a specific Reynolds number, the larger λ gives rise to the larger amount of lift reduction. The lift reduction reaches the maximum at λ=λc. The mechanism behind the present lift reduction method is revealed by comparing the flow patterns and pressure distributions near the active rotationally oscillating circular cylinder and the stationary circular cylinder. It is found that the critical value λc generally increases with Reynolds number. Two types of lift shift are observed in the numerical results for the cases with λ&amp;gt;λc. The first is characterized by the regular fluctuation of lift coefficient but with nonzero mean value, while the second is associated with the sustaining increase of lift coefficient. The phenomenon of lift shift is found to be related closely to the evolution of vortex pattern in the near wake of circular cylinder.

A self-regulating and patch subdivided population

We consider an interacting particle system on a graph which, from a macroscopic point of view, looks like ℤ d and, at a microscopic level, is a complete graph of degree N (called a patch). There are two birth rates: an inter-patch birth rate λ and an intra-patch birth rate ϕ. Once a site is occupied, there is no breeding from outside the patch and the probability c(i) of success of an intra-patch breeding decreases with the size i of the population in the site. We prove the existence of a critical value λc(ϕ, c, N) and a critical value ϕc(λ, c, N). We consider a sequence of processes generated by the families of control functions {c n } n∈ℕ and degrees {N n } n∈ℕ; we prove, under mild assumptions, the existence of a critical value n c(λ, ϕ, c). Roughly speaking, we show that, in the limit, these processes behave as the branching random walk on ℤ d with inter-neighbor birth rate λ and on-site birth rate ϕ. Some examples of models that can be seen as particular cases are given.

A self-regulating and patch subdivided population

We consider an interacting particle system on a graph which, from a macroscopic point of view, looks like ℤd and, at a microscopic level, is a complete graph of degree N (called a patch). There are two birth rates: an inter-patch birth rate λ and an intra-patch birth rate ϕ. Once a site is occupied, there is no breeding from outside the patch and the probability c(i) of success of an intra-patch breeding decreases with the size i of the population in the site. We prove the existence of a critical value λc(ϕ, c, N) and a critical value ϕc(λ, c, N). We consider a sequence of processes generated by the families of control functions {cn}n∈ℕ and degrees {Nn}n∈ℕ; we prove, under mild assumptions, the existence of a critical value nc(λ, ϕ, c). Roughly speaking, we show that, in the limit, these processes behave as the branching random walk on ℤd with inter-neighbor birth rate λ and on-site birth rate ϕ. Some examples of models that can be seen as particular cases are given.

THE COSMOLOGICAL CONSTANT PROBLEM AND INFLATION IN THE STRING LANDSCAPE

An earlier paper points out that a quantum treatment of the string landscape is necessary. It suggests that the wave function of the universe is mobile in the landscape until the universe reaches a meta-stable site with its cosmological constant Λ0 smaller than the critical value Λc, where Λc is estimated to be exponentially small compared to the Planck scale. Since this site has an exponentially long lifetime, it may well be today's universe. We investigate specific scenarios based on this quantum diffusion property of the cosmic landscape and find a plausible scenario for the early universe. In the last fast tunneling to the Λ0(&lt;Λc) site in this scenario, all energies are stored in the nucleation bubble walls, which are released to radiation only after bubble collisions and thermalization. So the Λ0 site is chosen even if Λ0 plus radiation is larger than Λc, as long as the radiation does not destabilize the Λ0 vacuum. A consequence is that inflation must happen before this last fast tunneling, so the inflationary scenario that emerges naturally is extended brane inflation, where the brane motion includes a combination of rolling, fast tunnelings, slow-roll, hopping and percolation in the landscape. We point out that, in the brane world, radiation during nucleosynthesis are mostly on the standard model branes (brane radiation, as opposed to radiation in the bulk). This distinction may lead to interesting dynamics. We consider this paper as a road map for future investigations.

Mixed convection flow near an axisymmetric stagnation point on a vertical cylinder

The mixed convection flow near an axisymmetric stagnation point on a vertical cylinder is considered. The equations for the fluid flow and temperature fields reduce to similarity form that involves a Reynolds number R and a mixed convection parameter λ, as well as the Prandtl number σ. Numerical solutions are obtained for representative values of these parameters, which show the existence of a critical value λ c = λ c (R, σ) for the existence of solutions in the opposing (λ < 0) case. The variation of λ c with R is considered. In the aiding (λ > 0) case solutions are possible for all λ and the asymptotic limit λ → ∞ is obtained. The limits of large and small R are also treated and the nature of the solution in the asymptotic limit of large Prandtl number is briefly discussed.

Steady mixed convection boundary-layer flow over a vertical flat surface in a porous medium filled with water at 4°C: variable surface heat flux

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water at 4°C (maximum density) when the surface heat flux varies as xm and the velocity outside the boundary layer varies as x(1+2m)/2, where x measures the distance from the leading edge, is discussed. Assisting and opposing flows are considered with numerical solutions of the governing equations being obtained for general values of the flow parameters. For opposing flows, there are dual solutions when the mixed convection parameter λ is greater than some critical value λc (dependent on the power-law index m). For assisting flows, solutions are possible for all values of λ. A lower bound on m is found, m > −1 being required for solutions. The nature of the critical point λc is considered as well as various limiting forms; the forced convection limit (λ = 0), the free convection limit (λ → ∞) and the limits as m → ∞ and as m → −1.

完全層状組織 TiAl 合金における &amp;alpha;2/&amp;gamma; 界面への界面転位の導入と格子ミスフィット変化に及ぼすNb および Zr 添加の影響

Changes in microstructures of α2/γ lamellar boundaries was studied on fully lamellar Ti-38Al-xM (at%, x=0, 3, M=Zr, Nb) alloys over a wide range of average lamellar thickness from 290 to 20 nm. The lamellar structure was formed by isothermal aging and lamellar thickness was controlled by changing aging temperature. The addition of Nb or Zr alters the lattice mismatch between α2 and γ phases. At lamellar boundaries the mismatch is accommodated by elastic deformation and/or by introducing misfit dislocations. The elastic deformation totally accommodates the lattice mismatch (coherent boundary) when γ lamellae are thin enough. Misfit dislocations are introduced to accommodate partly the mismatch when γ lamellar thickness exceeds a critical value λc. Density of the misfit dislocations saturates to an upper limit ds in thick lamellar structures. The critical value of λc decreases and the saturation level ds increases with increasing lattice mismatch. Zr increases and Nb decreases the lattice mismatch.