Symmetric Case Research Articles

On the large solutions to a class of k-Hessian problems

Abstract In this paper, we consider the k-Hessian problem S k (D 2 u) = b(x)f(u) in Ω, u = +∞ on ∂Ω, where Ω is a C ∞-smooth bounded strictly (k − 1)-convex domain in R N ${\mathbb{R}}^{N}$ with N ≥ 2, b ∈ C∞(Ω) is positive in Ω and may be singular or vanish on ∂Ω, f ∈ C[0, ∞) ∩ C 1(0, ∞) (or f ∈ C 1 ( R ) $f\in {C}^{1}\left(\mathbb{R}\right)$ ) is a positive and increasing function. We establish the first expansions (equalities) of k-convex solutions to the above problem when f is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of f and principal curvatures of ∂Ω on the first expansion of solutions. For the latter, we find the principal curvatures of ∂Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including k = N). Moreover, we know the existence of k-convex solutions to the above problem (including k = N) is still an open problem when b possesses high singularity on ∂Ω and f satisfies Keller–Osserman type condition. For the radially symmetric case in the ball, we give a positive answer to this open problem, and then we further show the global estimates for all radial large solutions.

Coalescence-Induced Self-Propelled Particle Transport with Asymmetry Arrangement.

We experimentally investigated the coalescence-induced droplet-particle jumping phenomenon on a submillimeter scale in symmetric and asymmetric particle arrangements with poly(methyl methacrylate) (PMMA) particles and stainless steel (SS) particles. Coalescence-induced droplet-particle jumping exhibited excellent capability and interesting behavior for both droplet jumping enhancement and particle transport. The particle increased the normalized droplet jumping velocity from 0.250 for no particle case to 0.315 and 0.320 for symmetric and asymmetric particle cases. Compared with similar-sized macrostructures fixed between droplets, better jumping performance with particles may be attributed to avoiding the work of adhesion during droplet-macrostructure separation. Besides, all particles always sunk at the bottom in the symmetric cases, while the stick mode for PMMA particles and sink, wander, and jet modes for SS particles appeared in the asymmetry cases. We revealed that the asymmetric particle arrangement induces an unbalanced surface tension force, which may provide a driving force in the vertical direction. Additionally, a small enough resistive force caused by hydrophobic particles is another necessary condition for the wonder and jet mode. Finally, we realized a significantly superior particle transport in the asymmetric SS particle cases with maximum particle height reaching ∼2.1 mm, ∼12.4 times the particle radius, the most significant vertical self-propelled transport distance currently.

Models of Protocells Undergoing Asymmetrical Division

The conditions that allow for the sustained growth of a protocell population are investigated in the case of asymmetrical division. The results are compared to those of previous studies concerning models of symmetrical division, where synchronization (between duplication of the genetic material and fission of the lipid container) was found under a variety of different assumptions about the kinetic equations and about the place where molecular replication takes place. Such synchronization allows a sustained proliferation of the protocell population. In the asymmetrical case, there can be no true synchronization, since the time to duplication may depend upon the initial size, but we introduce a notion of homogeneous growth that actually allows for the sustained reproduction of a population of protocells. We first analyze Surface Reaction Models, defined in the text, and we show that in many cases they undergo homogeneous growth under the same kinetic laws that lead to synchronization in the symmetrical case. This is the case also for Internal Reaction Models (IRMs), which, however, require a deeper understanding of what homogeneous growth actually means, as discussed below.

Dynamic Analysis of a Simple Cournot Duopoly Model Based on a Computed Cost

The paper is organized to study some mathematical properties and dynamics of a simple Cournot duopoly game based on a computed quadratic cost. The time evolution of this game is described by a two-dimensional noninvertible discrete time map using the bounded rationality mechanism. For this map, some dynamic characteristics such as multistability and synchronization are investigated. Its equilibrium points are obtained for the asymmetric case, and their conditions of stability are obtained. Our results investigate that the Nash equilibrium point may be unstable due to flip bifurcation and under certain parameter values, and Neimark–Sacker bifurcation is born after the period-4 cycle. Through some restrictions, the coordinate axes of the map construct an invariant manifold, and therefore, their dynamics can be analyzed by using a one-dimensional map. In the symmetric case, both firms behave identically, and this implies that the diagonal set forms an invariant manifold, and hence the synchronization phenomena take place. Furthermore, the global bifurcation of the map is confirmed through contact between critical curves and the boundaries of infeasible domains.

Dynamics of a bubble-pair between two parallel rigid walls

The dynamics of two gas bubbles placed side-by-side in liquid between parallel rigid plates is studied numerically using the multiphase volume of fluid (VOF) model in OpenFOAM. This model solves a single set of Navier–Stokes equations for both the gas in the bubbles and the surrounding liquid. Being initially stationary, the bubbles first expand to their maximum volume due to the high gas pressure inside them and then collapse. Both phases are treated as viscous and immiscible without any phase change. The model is validated against available experimental data of a single bubble oscillation between parallel plates and a bubble-pair oscillation near a single rigid wall. Parametric studies are then performed, with the main variables being the initial sizes of the bubbles and their standoff distances from the rigid plates. The bubbles exhibit symmetric or asymmetric behavior depending on these two variables. Three distinct scenarios are identified in the symmetric case based on the relative influence of bubble-bubble and bubble-wall interactions. The jet strength and direction, bubble oscillation period, coalescence, and early asymmetric bubble collapse are characterized by initial distances and radii. This study provides valuable insights into intricate hydrodynamic interactions and behaviors of bubble pairs bounded by rigid surfaces.

Exact closed forms for the transfer matrix of free oscillations in finite periodic Timoshenko–Ehrenfest beams

In this work, we obtain closed-form expressions for the transfer matrix of free oscillations in finite periodic Timoshenko–Ehrenfest beams with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for free oscillations of a beam composed of N cells, we obtain a fourth-order recursive relation for the matrix coefficients, which defines the so-called Tetranacci Polynomials. Such recursive relation provides an algorithm to compute the Nth power of the transfer matrix, avoiding the matrix product of the N matrices. Furthermore, in the symmetric case of free oscillations in finite periodic Timoshenko-Ehrenfest beams, closed-form expressions for the solutions to the recursive relation have recently been derived, which we use to write the transfer matrix of a finite beam composed of N cells in a closed form. We find a good agreement between the natural frequencies calculated with the obtained expressions and finite element simulations. These expressions are very useful for studying the interaction of evanescent oscillations and find application, for instance, in the development of phononic topological insulators. Our formalism can be applied to waves propagating in finite periodic layers described by a 4 × 4-transfer matrix, such as electromagnetic waves in anisotropic optical media.

The steady state of the boundary-driven multiparticle asymmetric diffusion model

We consider the multiparticle asymmetric diffusion model (MADM) introduced by Sasamoto and Wadati with integrability preserving reservoirs at the boundaries. In contrast to the open asymmetric simple exclusion process the number of particles allowed per site is unbounded in the MADM. Taking inspiration from the stationary measure in the symmetric case, i.e. the rational limit, we first obtain the length 1 solution and then show that the steady state can be expressed as an iterated product of Jackson q-integrals. In the proof of the stationarity condition, we observe a cancellation mechanism that closely resembles the one of the matrix product ansatz. To our knowledge, the occupation probabilities in the steady state of the boundary-driven MADM were not available before.

Single-quasiparticle eigenstate thermalization.

Quadratic Hamiltonians that exhibit single-particle quantum chaos are called quantum-chaotic quadratic Hamiltonians. One of their hallmarks is single-particle eigenstate thermalization introduced in Łydżba etal. [Phys. Rev. B 104, 214203 (2021)2469-995010.1103/PhysRevB.104.214203], which describes statistical properties of matrix elements of observables in single-particle eigenstates. However, the latter has been studied only in quantum-chaotic quadratic Hamiltonians that obey the U(1) symmetry. Here, we focus on quantum-chaotic quadratic Hamiltonians that break the U(1) symmetry and, hence, their "single-particle" eigenstates are actually single-quasiparticle excitations introduced on the top of a many-body state. We study their wave functions and matrix elements of one-body observables, for which we introduce the notion of single-quasiparticle eigenstate thermalization. Focusing on spinless fermion Hamiltonians in three dimensions with local hopping, pairing, and on-site disorder, we also study the properties of disorder-induced near zero modes, which give rise to a sharp peak in the density of states at zero energy. Finally, we numerically show equilibration of observables in many-body eigenstates after a quantum quench. We argue that the latter is a consequence of single-quasiparticle eigenstate thermalization, in analogy to the U(1) symmetric case from Łydżba etal. [Phys. Rev. Lett. 131, 060401 (2023)0031-900710.1103/PhysRevLett.131.060401].

Flow and heat transfer performance of asymmetric fractal tree network in fractal porous media

This study aims to investigate the fluid transport and heat transfer characteristics in fractal porous media, introduce asymmetric factors to derive a generalized optimization principle for asymmetric branching flow and heat transfer, and obtain the optimal radius ratio for the superior flow resistance/heat resistance model; and the accurate models of permeability and thermal conductivity of asymmetric tree-fractal networks are developed and validated against the traditional Murray's law and symmetric tree-fractal network models. The results show that (a) the symmetric case can be regarded as a special case of the asymmetric fractal network model, and Murray's law is correct only for the symmetric bifurcation (flow percentage Ψi = 0.5), and the errors predicted by Murray's law for the asymmetric case with a flow percentage of 10% (Ψi = 0.1, n = 1) are 23.5% and 33.1% with respect to the optimal radius ratio of flow and heat transfer, respectively. (b) The symmetric case has the largest flow resistance and the smallest thermal resistance. The asymmetric length factor and radius ratio have significant effects on the dimensionless flow resistance/thermal resistance of the asymmetric fractal network, and there is a critical radius ratio (βm = 0.84), where a larger asymmetric length factor is detrimental to the flow of the tree-like branching network when β < βm, while the opposite effect is observed when β > βm. (c) The asymmetric radius factor affects the optimal radius ratio for thermal conductivity, but does not change the maximum value of thermal conductivity. (d) The pressure gradient and heat transfer coefficient in the fractal microchannel are related to the variation of the volume flow rate and the increase in the heat flux will weaken heat transfer.

Aerodynamic drag and noise reduction of a pantograph of high-speed trains with a novel cavity structure

The design of the cavity structure is one of the effective means to reduce the resistance and noise of the pantograph installed on the roof of a high-speed train. This research first investigated the flow and acoustic characteristics of a pantograph with four different cavity structures, namely the rectangular cavity (original), the rounded edge cavity (case 1), and the other two rounded edge cavities with asymmetric (case 2) and symmetric (case 3) connecting tubes. The results show that the three cavity treatment methods all improve the aerodynamic performance, and the cavity model of case 2 is determined to be the optimal structure. In case 2, the tube installed at the front of the cavity destroys the separated shear layer and reduces the unstable airflow, reducing cavity resistance and noise by 9.64% and 5.2 dBA (A-weighted decibels), respectively. The pantograph is placed inside the previously determined improved cavity, which reduces the airflow velocity and the recirculation region upstream of the pantograph, decreases the impingement on the components in the middle and lower regions of the pantograph and the generation of highly intense vortices, and improves the wake structure and flow separation at the rear surface of the cavity. Thus, the aerodynamic drag for the pantograph and the whole system is reduced by 3.82% and 3.25%, respectively, and the aerodynamic noise is also decreased by 1.4 and 1.9 dBA, respectively. This study provides a novel structural design for the pantograph cavity region.

Leader-Following Consensus of Discrete-Time Nonlinear Multi-Agent Systems with Asymmetric Saturation Impulsive Control

Impulsive control is an effective approach for coordinating multi-agent systems in practical environments due to its high robustness and low cost. However, impulsive control exhibits characteristics such as high amplitude and rapid variation, potentially presenting threats to the equipment. Additionally, multi-agent systems are constrained by input saturation due to limitations in physical controller structures and information-processing capabilities. These saturation constraints may be asymmetrical. Therefore, it is necessary to consider the saturation constraint when implementing impulsive control, as it can also mitigate the threats posed by the impulse to agents. This paper investigates the leader-following consensus for a class of discrete-time nonlinear multi-agent systems, proposing an asymmetric saturation impulsive control protocol to reduce the energy consumption and damage to the equipment. Regarding the handle of asymmetric saturation, an approach is proposed that eliminates the need for transformation from the asymmetric case to the symmetric case, which retains the saturation function and directly introduces the sector condition to deal with saturation nonlinearity. Furthermore, based on Lyapunov stability theory and matrix theory, sufficient conditions for leader-following consensus in discrete-time nonlinear multi-agent systems under asymmetric saturation impulsive control are established, and the admissible region of the system is estimated. Finally, numerical simulations are provided to verify the validity of the theoretical results.

AROFRAG─A Systematic Approach for Fragmentation of Aromatic Molecules.

We present a new systematic fragmentation scheme of polycyclic aromatic hydrocarbons (PAHs), including fullerenes and nanotubes, based on an idea to treat a sextet ring as a single unbreakable unit so that the basic unit of aromaticity remains preserved upon fragmentation. In the approach, denoted as AROFRAG (from aromatic fragmentation), a set of predefined elementary subsystems, such as naphthalene and biphenyl in the first model and larger PAHs in the second and third models, is generated with appropriate weights with the aim of reproducing the structure of the original molecule. The energies of the molecules are approximated as weighted sums of the energies of these subsystems. For symmetric cases, e.g., fullerenes, the point-group symmetry is preserved during the decomposition. The AROFRAG is used in conjunction with the molecule-in-molecule (MIM) technique to obtain an accurate description of the electronic energies. The new approach has been applied for selected graphene structures and fullerene doped with boron and nitrogen atoms, for which isomerization energies were calculated, as well as for several nanotubes and regular fullerene molecules. The combination of the third AROFRAG model and the MIM approach leads to the reproduction of electronic energies with a few milli-hartree accuracy at a fraction of the computational cost of the original method.

Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons

The Brownian separable permutons are a one-parameter family – indexed by p∈(0,1) – of universal limits of random constrained permutations. We show that for each p∈(0,1), there are explicit constants 1/2<α⁎(p)≤β⁎(p)<1 such that the length of the longest increasing subsequence in a random permutation of size n sampled from the Brownian separable permuton is between nα⁎(p)−o(1) and nβ⁎(p)+o(1) with probability tending to 1 as n→∞. In the symmetric case p=1/2, we have α⁎(p)≈0.812 and β⁎(p)≈0.975. We present numerical simulations which suggest that the lower bound α⁎(p) is close to optimal in the whole range p∈(0,1).Our results work equally well for the closely related Brownian cographons. In this setting, we show that for each p∈(0,1), the size of the largest clique (resp. independent set) in a random graph on n vertices sampled from the Brownian cographon is between nα⁎(p)−o(1) and nβ⁎(p)+o(1) (resp. nα⁎(1−p)−o(1) and nβ⁎(1−p)+o(1)) with probability tending to 1 as n→∞.Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002). We expect that our techniques can be extended to prove similar bounds for uniform separable permutations and uniform cographs.

Symmetry analysis for nonlinear fractional terminal system under $ w $-Hilfer fractional derivative in different weighted Banach spaces

&lt;abstract&gt;&lt;p&gt;Our objective in this study is to investigate the behavior of a nonlinear terminal fractional system under $ w $-Hilfer fractional derivative in different weighted Banach spaces. We examine the system's dynamics and understand the effects of different weighted Banach spaces on the properties of solutions, including existence, uniqueness, stability, and symmetry. We derive the equivalent integral equations and employ the Schauder and Banach fixed point theorems. Additionally, we discuss three symmetric cases of the system to show how the choice of the weighted function $ w(\iota) $ impacts the solutions and their symmetry properties. We study the stability of the solutions in the Ulam sense to assess the robustness and reliability of these solutions under various conditions. Finally, to understand the system's behavior, we present an illustrative example with graphs of the symmetric cases.&lt;/p&gt;&lt;/abstract&gt;

Asymmetric contact-induced selective doping of CVD-grown bilayer WS2 and its application in high-performance photodetection with an ultralow dark current.

Two-dimensional (2D) transition metal dichalcogenides (TMDs) are excellent candidates for high-performance optoelectronics due to their high carrier mobility, air stability and strong optical absorption. However, photodetectors made with monolayer TMDs often exhibit a high dark current, and thus, there is a scope for further improvement. Herein, we developed a 2D bilayer tungsten disulfide (WS2) based photodetector (PD) with asymmetric contacts that exhibits an exceptionally low dark current and high specific detectivity. High-quality and large-area monolayer and bilayer WS2 flakes were synthesized using a thermal chemical vapor deposition system. Compared to conventional symmetric contact electrodes, utilizing metal electrodes with higher and lower work functions relative to bilayer WS2 aids in achieving asymmetric lateral doping in the WS2 flakes. This doping asymmetry was confirmed through the photoluminescence spectral profile and Raman mapping analysis. With the asymmetric contacts on bilayer WS2, we find evidence of selective doping of electrons and holes near the Ti and Au contacts, respectively, while the WS2 region away from the contacts remains intrinsic. When compared with the symmetric contact case, the dark current in the WS2 PD with asymmetric (Au, Ti) contact decreases by an order of magnitude under reverse bias with a concomitant increase in the photocurrent, resulting in an improved on/off ratio of ∼105 and overall improved device performance under identical illumination conditions. We explained this improved performance based on the energy band alignment showing a unidirectional charge flow under light illumination. Our results indicate that the planar device structure and compatibility with current nanofabrication technologies can facilitate its integration into advanced chips for futuristic low-power optoelectronic and nanophotonic applications.

A model for seagrass species competition: Dynamics of the symmetric case

We propose a general population dynamics model for two seagrass species growing and interacting in two spatial dimensions. The model includes spatial terms accounting for the clonal growth characteristics of seagrasses, and coupling between species through the net mortality rate. We consider both intraspecies and interspecies facilitative and competitive interactions, allowing density-dependent interaction mechanisms. Here we study the case of very similar species with reciprocal interactions, which allows reducing the number of the model parameters to just four, and whose bifurcation structure can be considered the backbone of the general case. We find that the parameter space can be divided into ten regions with qualitatively different bifurcation diagrams. These regions can be further grouped into just five regimes with different ecological interpretations. Our analysis allows the classification of all possible density distributions and dynamical behaviors of meadows with two coexisting species.

Hamiltonian Hopf bifurcations near a chaotic Hamiltonian resonance

The 1:2:3 Hamiltonian resonance is one of the four genuine first order resonances which is non-integrable. For this resonance, chaotic behavior of the normal form has been shown due to the existence of a transverse homoclinic orbit on the energy manifold. Considering the detuning parameters, in the mirror symmetric cases of the Poisson manifold, by a reduction theory and computing the classical normal form of non-degenerate Hamiltonian Hopf bifurcation, we show there are just non-degenerate Hamiltonian Hopf bifurcations. However in the general case by considering some case studies and specially of FPU (Fermi–Pasta–Ulam) chains for a fiber passaging 1:2:3 resonance, we may deal with complex dynamics and specially Hamiltonian Hopf bifurcation in a complicated region. Actually, we can see some special Krein collisions in a complex region.

A flexible Clayton-like spatial copula with application to bounded support data

The Gaussian copula is a powerful tool that has been widely used to model spatial and/or temporal correlated data with arbitrary marginal distributions. However, this kind of model can potentially be too restrictive since it expresses a reflection symmetric dependence. In this paper, we propose a new spatial copula model that makes it possible to obtain random fields with arbitrary marginal distributions with a type of dependence that can be reflection symmetric or not.Particularly, we propose a new random field with uniform marginal distributions that can be viewed as a spatial generalization of the classical Clayton copula model. It is obtained through a power transformation of a specific instance of a beta random field which in turn is obtained using a transformation of two independent Gamma random fields.For the proposed random field, we study the second-order properties and we provide analytic expressions for the bivariate distribution and its correlation. Finally, in the reflection symmetric case, we study the associated geometrical properties. As an application of the proposed model we focus on spatial modeling of data with bounded support. Specifically, we focus on spatial regression models with marginal distribution of the beta type. In a simulation study, we investigate the use of the weighted pairwise composite likelihood method for the estimation of this model. Finally, the effectiveness of our methodology is illustrated by analyzing point-referenced vegetation index data using the Gaussian copula as benchmark. Our developments have been implemented in an open-source package for the R statistical environment.

Effect of double quantum dot asymmetry on electron localization

In this paper, we study the localization of an electron in a binary quantum system formed by a pair of quantum dots (QDs). The traditional theoretical consideration of such systems is limited to the symmetrical case when QDs in such double quantum dot (DQD) are assumed identical in all respects. In this paper, we model the effects of breaking QD similarities in a DQD by studying two-dimensional (2D) DQDs as a double quantum well (DQW). This is done by solving the Schrödinger equation, with parameters chosen to describe an InAs/GaAs heterostructure. We calculate the energy spectrum of the electron confinement and the spectral distribution of localized/delocalized spatial states. Both symmetric and asymmetric QW shapes are considered and their effects are compared. The effects of symmetry breaking are explained within the framework of the two-level system theory. We delineate the QW weak and strong coupling cases in DQW. In particular, we show that the coherence in ideal DQW is unstable in the case of a weak QW coupling. Within the framework of the proposed approach, a charge qubit realized on a DQD is discussed and, as an example, a qubit based on an almost ideal DQD is proposed.

Analytical solutions to the isentropic Euler system with the logarithmic equation of state

In this paper, we study the analytical solutions to the isentropic Euler system with the logarithmic equation of state, which has been introduced for the dark energy fluid to investigate the dynamical evolution of a late-time universe. By using some ansatz, we construct some analytical solutions for one-dimensional case, [Formula: see text]-dimensional radially symmetric case and three-dimensional cylindrically symmetric case, respectively. The concentration and cavitation phenomena are investigated and some global analytical solutions are obtained under some conditions.