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This paper addresses the approximate and exact controllability issues of switched infinite dimensional linear systems. Firstly, a controllability map that is related to switching sequence and a controllability Gramian operator are proposed, based on which two necessary and sufficient conditions are established respectively for approximate and exact controllability; Secondly, it shows that the obtained conditions can degenerate into the existing controllability criteria of infinite dimensional linear systems, switched finite dimensional linear systems and single finite dimensional linear system; Finally, the new results are applied to analyze the approximate controllability of switched linear parabolic systems and the exact controllability of switched linear wave systems. Two examples are given to illustrate the effectiveness of the proposed approaches.

For stochastic quantum systems with measurement feedback, this article proposes a rapid switching control scheme based on state space partition and realizes the rapid stabilization of an eigenstate of an observable operator. Meanwhile, we apply the proposed scheme to the preparation of typical entangled states in multiqubit systems. In view of the convergence obstacle caused by the symmetric structure of the state space, especially in the case with degenerate observable operators, we first partition the state space into a subset containing the target state and its complement to distinguish the target state from its antipodal points, and then design the corresponding control laws in these two subsets, respectively, by using different Lyapunov functions. The interaction Hamiltonians are also constructed to drive the system state to the desired subset first, and further to the target state. In particular, the control law designed in the undesired subset guarantees the strictly monotonic descent of the corresponding Lyapunov function, which makes the system trajectory switch between the two subsets at most twice and has the potential to speed up the convergence process. We also prove the stability of the closed-loop system with the proposed switching control law based on the stochastic Lyapunov stability theory. By applying the proposed switching control scheme to a three-qubit system, we achieve the preparation of a GHZ state and a W state.\enlargethispage-8pt.

We propose a novel approach to realize unidirectional radiation in an asymmetric single resonator without using an external mirror, which is based on mutual coupling between two degenerate resonant modes through the background scattering of the asymmetric resonator.

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form $$L(u)=-\textrm{div} (a(x)\nabla u)$$ and a suitable nonlinearity $f$. The function $a$ vanishes on smooth 1-codimensional submanifolds of $\Omega$ where it is not allowed to be $C^{2}$. By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where $a$ vanishes.

Magnetism has revolutionized important technologies, and continues to bring forth new phenomena in emergent materials and reduced dimensions. Here, using first-principles calculations, we demonstrate that the already-synthesized two-dimensional (2D) Ni-tetracyanoquinodimethane (Ni2(TCNQ)2) lattice is a stable ferromagnetism material with multiple spin-polarized Dirac cones. The conical bands in proximity of the Fermi level can be tuned by external tensile strain and show the fourfold degenerate electronic states at the critical tensile strain of ∼2.35%, whose energy dispersion is consistent with 2D Cairo pentagonal lattice. In addition, spin-orbital coupling can open a band gap at the Dirac point of A, leading to topologically nontrivial electronic states characterized by the non-zero Chern number and the edge states of nanoribbon. Our results offer versatile platforms for the realization of massless spintronics with full-spin polarization in 2D Cairo pentagonal Ni2(TCNQ)2 Lattice.

This paper presents the properties of the minimum mean cycle-canceling algorithm for solving linear programming models. Originally designed for solving network flow problems for which it runs in strongly polynomial time, most of its properties are preserved. This is at the price of adapting the fundamental decomposition theorem of a network flow solution together with various definitions: that of a cycle and the way to calculate its cost, the residual problem, and the improvement factor at the end of a phase. We also use the primal and dual necessary and sufficient optimality conditions stated on the residual problem for establishing the pricing step giving its name to the algorithm. It turns out that the successive solutions need not be basic, there are no degenerate pivots, and the improving directions are potentially interior in addition to those on edges. For solving an m × n linear program, it requires a pseudo-polynomial number O ( n Δ ) of so-called phases, where Δ depends on the number of rows and the coefficient matrix.

The Thomas model is a system of two reaction–diffusion equations which was originally proposed in the context of enzyme kinetics. It was subsequently realised that it offers a plausible chemical mechanism for the generation of coat markings on mammals. To that end previous investigations have focused on establishing the conditions for the Turing instability and on following the associated patterns as the bifurcation parameter increases through the instability. In this paper we use modern ideas from the theory of dynamical systems to systematically investigate the formation of localised structures in this model. The Turing instability is found to exhibit a degeneracy at which it changes from being subcritical to supercritical (or vice versa). Associated with such degenerate points is a heteroclinic connection which is a crucial requirement for the generation of localised patterns. Localised solutions containing a single ‘spike’ are associated with the so-called Belyakov-Devaney transition. Localised solutions containing either multiple ‘peaks’ or multiple ‘holes’ are associated with homoclinic snaking bifurcations. These structures are investigated using continuation software. The snaking bifurcations are found to collapse when the system crosses the Belyakov-Devaney transition. The isolated spike solutions collapse at a codimension two heteroclinic connection to create the so-called collapsed snaking. We show that the temporal stability of the localised solutions depends upon the presence of Hopf bifurcations which in turn are controlled by the diffusion ratio. For small values of the ratio only solutions in the spike region are destabilised by a Hopf bifurcation. For larger values of the diffusion ratio Hopf bifurcations destabilise most of the localised patterns.

Column generation (CG) algorithms are well known to suffer from convergence issues due, mainly, to the degenerate structure of their master problem and the instability associated with the dual variables involved in the process. In the literature, several strategies have been proposed to overcome this issue. These techniques rely either on the modification of the standard CG algorithm or on some prior information about the set of dual optimal solutions. In this paper, we propose a new stabilization framework, which relies on the dynamic generation of aggregated rows from the CG master problem. To evaluate the performance of our method and its flexibility, we consider instances of three different problems, namely, vehicle routing with time windows (VRPTW), bin packing with conflicts (BPPC), and multiperson pose estimation (MPPEP). When solving the VRPTW, the proposed stabilized CG method yields significant improvements in terms of CPU time and number of iterations with respect to a standard CG algorithm. Huge reductions in CPU time are also achieved when solving the BPPC and the MPPEP. For the latter, our method has shown to be competitive when compared with a tailored method. Summary of Contribution: Column generation (CG) algorithms are among the most important and studied solution methods in operations research. CG algorithms are suitable to cope with large-scale problems arising from several real-life applications. The present paper proposes a generic stabilization framework to address two of the main issues found in a CG method: degeneracy in the master problem and massive instability of the dual variables. The newly devised method, called dynamic separation of aggregated rows (dyn-SAR), relies on an extended master problem that contains redundant constraints obtained by aggregating constraints from the original master problem formulation. This new formulation is solved in a column/row generation fashion. The efficacy of the proposed method is tested through an extensive experimental campaign, where we solve three different problems that differ considerably in terms of their constraints and objective function. Despite being a generic framework, dyn-SAR requires the embedded CG algorithm to be tailored to the application at hand.

In this research, a novel dual-band triangular patch antenna (TPA) with linear-orthogonal-polarization is designed by utilizing degenerate modes (DMs). Current symmetries of fundamental degenerate modes of TPA are exploited to achieve dual-band antenna operation. An equilateral triangular slot is inserted along one of the edges to separate resonant frequencies of DMs and simultaneously exciting them at different resonant frequencies. Location of the slot is chosen according to the current symmetries of the horizontal mode to elongate its current paths and moving it to a lower resonant frequency, while keeping vertical mode at its fundamental resonant frequency. Prototype of proposed single probe-fed antenna is fabricated and tested. The designed antenna exhibits low frequency ratio (FR) and broadside radiation patterns with gains of 4.96dBi and 6.10dBi in the first and second band respectively. Low cross-polarization levels and linear-orthogonal-polarization are achieved in both bands.

Motivated by applications to congested traffic problems, we establish higher integrability results for the gradient of local weak solutions to the strongly degenerate or singular elliptic PDE − div ( ( | ∇ u | − 1 ) + q − 1 ∇ u | ∇ u | ) = f , in Ω , where Ω is a bounded domain in R n for n ≥ 2 , 1 q ∞ and ( ⋅ ) + stands for the positive part. We assume that the datum f belongs to a suitable Sobolev or Besov space. The main novelty here is that we deal with the case of subquadratic growth, i.e. 1 q 2 , which has so far been neglected. In the latter case, we also prove the higher fractional differentiability of the solution to a variational problem, which is characterized by the above equation. For the sake of completeness, we finally give a Besov regularity result also in the case q ≥ 2 .

We give explicit isomorphisms between simple modules of degenerate cyclotomic Hecke algebras defined via various cellular bases. A special case gives a generalized Mullineux involution in the degenerate case., Comment: 21 pages

The current paper is devoted to stochastic fractional Boussinesq equations (FBEs) on the Torus T 2 = [ 0 , 2 π ] × [ 0 , 2 π ] with degenerate random forcing. We first show the existence and pathwise uniqueness of strong solutions by virtue of the Yamada-Watanabe Theorem, and then we investigate the asymptotic properties of invariant measures in two aspects: the first is, focusing on the stochastic systems, to prove the existence of an invariant measure in the case of multiplicative noise, and then prove that the invariant measure can be unique and exponentially mixing if we are restricted to degenerate additive noise. The second is, from the perspective of stochastic stability, to prove that the zero-noise limits of stationary measures for the perturbed stochastic systems are invariant for the deterministic system and their supports are concentrated on the Birkhoff center.

In this paper we mainly investigate the Cauchy problem of the Doi-Edwards polymer model with dimension d ≥ 2 . The model was derived in the late 1970s to describe the dynamics of polymers in melts. The system contains a Navier-Stokes equation with an additional stress tensor which depend on the deformation gradient tensor and the memory function. The deformation gradient tensor satisfies a transport equation and the memory function satisfies a degenerate parabolic equation. We first proved the local well-posedness for the Doi-Edwards polymer model in Besov spaces by using the Littlewood-Paley theory. Moreover, if the initial velocity and the initial memory is small enough, we obtain a global existence result.

A dispersive decay estimate is proven for the linear propagator of the three-scale rotating stratified Boussinesq equations in the stratification-dominant regime. Because the phase function of that propagator is both singular and degenerate, obtaining the estimate requires novel techniques involving cutting phase-space shells into several pieces and using different methods in each region. Using the decay estimate, we prove the long-time existence of solutions to the initial-value problem for the nonlinear equations and obtain a rate of convergence to the limit system and intermediate asymptotics.

In this paper we develop a potential theory for strongly degenerate parabolic operators of the form L : = ∇ X ⋅ ( A ( X , Y , t ) ∇ X ) + X ⋅ ∇ Y − ∂ t , in unbounded domains of the form Ω = { ( X , Y , t ) = ( x , x m , y , y m , t ) ∈ R m − 1 × R × R m − 1 × R × R | x m > ψ ( x , y , y m , t ) } , where ψ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator L . Concerning A = A ( X , Y , t ) we assume that A is bounded, measurable, symmetric and uniformly elliptic (as a matrix in R m ). Beyond the solvability of the Dirichlet problem and other fundamental properties our results include scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures. All of our estimates are translation- and scale-invariant with constants only depending on the constants defining the boundedness and ellipticity of A and the Lipschitz constant of ψ. Our results represent a version, for operators of Kolmogorov type with bounded, measurable coefficients, of the by now classical results of Fabes and Safonov, and several others, concerning boundary estimates for uniformly parabolic equations in (time-dependent) Lipschitz type domains.

We consider asymptotic behavior of a solution to the drift-diffusion equation for a fast-diffusion case. In the degenerate drift-diffusion equation, it is known that large time behavior of solutions converges to the Zel'dovich–Kompaneetz–Barenblatt (ZKB) function. For a fast-diffusion case, we show that the asymptotic profile for a solution is the generalized ZKB function such as the Talenti function. We use the entropy dissipation method combining the logarithmic Sobolev and the Shannon inequalities for the Renyi entropy that is known as an extension of the Boltzmann–Shannon entropy.

Balanced dielectric resonator (DR) filters with multiple reconfigurable passbands are proposed in this article. Based on the field distributions of two pairs of orthogonal degenerate modes, quad-mode differential-mode (DM) operations are achieved by the specific probe feeding structure. More importantly, the four modes are flexibly controlled by different inserted screws and the detailed principle is illustrated for reconfiguration. For demonstration, a dual-band balanced filter with independently frequency-tunable passbands and a quad-band one with selectable number of passbands are fabricated and measured. Good agreement between the simulated and measured results is observed. The exhibited low DM loss, high common-mode (CM) rejection, and independent tuning abilities verify the proposed method in miniaturized reconfigurable filter designs.

In this work, we consider a coupled nonlinear viscoelastic Kirchhoff equations with degenerate damping, dispersion and source terms. Under suitable hypothesis, we will prove that when the initial data are large enough (in the energy point of view), the energy grows exponentially and thus so the $ L^{2(p+2)} $-norm.

We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscosity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper.

A new approach based on a strong degeneracy state to design oscillators that exhibit low phase noise and low power consumption is presented. A 100-MHz oscillator is designed using a periodic structure based on a degenerate band edge that is augmented with an NMOS cross-coupled pair. The analysis and simulation results show 16 dB phase noise improvement compared to oscillators based on a conventional single-ladder periodic structure. It is also shown that this oscillator dissipates less power than a conventional LC oscillator that requires a buffer to drive a 50Ω load.

In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $ p $-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $ {p(\cdot)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.

The paper deals with stochastic homogenization of a system modeling immiscible compressible two-phase, such as water and gas, flow in random porous media. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. We consider the behavior of compressible two-phase flow in heterogeneous reservoirs with permeability and porosity being realizations of given statistically homogeneous random fields. We derive the effective (macroscopic) problem and prove the convergence of solutions. Our approach relies on stochastic two-scale convergence techniques, the realization-wise notion of stochastic two-scale convergence being used. Also, we exploit various a priori estimates as well as monotonicity and compactness arguments. To our best knowledge, this is the first stochastic homogenization result in the case of compressible two-phase flow in random porous media.

Degenerate points/lines in the bulk band structures of crystals have become a staple of the growing number of topological materials. The bulk-boundary correspondence provides a relation between bulk topology and surface states. While line degeneracies of bulk excitations have been extensively characterized, line degeneracies of surface states are not well understood. We show that SnIP, a quasi-one-dimensional van der Waals material with a double helix crystal structure, exhibits topological nodal rings/lines in both the bulk phonon modes and their corresponding surface states. Using a combination of first-principles calculations, symmetry-based indicator theories and Zak phase analysis, we find that two neighbouring bulk nodal rings form doubly degenerate lines in their drumhead-like surface states, which are protected by the combination of time-reversal and glide mirror symmetries $\mathcal{T}\bar{M}_y$. Our results indicate that surface degeneracies can be generically protected by symmetries such as $\mathcal{T}\bar{M}_y$, and phonons provide an ideal platform to explore such degeneracies., Comment: 13 pages, 5 figures

In this paper, a Gause type predator–prey system with constant-yield prey harvesting and monotone ascending functional response is proposed and investigated. We focus on the influence of the harvesting rate on the predator–prey system. First, equilibria corresponding to different situations are investigated, as well as the stability analysis. Then bifurcations are explored at nonhyperbolic equilibria, and we give the conditions for the occurrence of two saddle–node bifurcations by analyzing the emergence, coincidence and annihilation of equilibria. We calculate the Lyapunov number and focal values to determine the stability and the quantity of limit cycles generated by supercritical, subcritical and degenerate Hopf bifurcations. Furthermore, the system is unfolded to explore the repelling and attracting Bogdanov–Takens bifurcations by perturbing two bifurcation parameters near the cusp. It is shown that there exists one limit cycle, or one homoclinic loop, or two limit cycles for different parameter values. Therefore, the system is susceptible to both the constant-yield prey harvesting and initial values of the species. Finally, we run numerical simulations to verify the theoretical analysis.

We define the fractional powers L s = ( − a i j ( x ) ∂ i j ) s , 0 s 1 , of nondivergence form elliptic operators L = − a i j ( x ) ∂ i j in bounded domains Ω ⊂ R n , under minimal regularity assumptions on the coefficients a i j ( x ) and on the boundary ∂Ω. We show that these fractional operators appear in several applications such as fractional Monge–Ampere equations, elasticity, and finance. The solution u to the nonlocal Poisson problem { ( − a i j ( x ) ∂ i j ) s u = f in Ω u = 0 on ∂ Ω is characterized by a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge–Ampere geometry and prove the interior Harnack inequality and Holder estimates for solutions to the extension problem when the coefficients a i j ( x ) are bounded, measurable functions. This in turn implies the interior Harnack inequality and Holder estimates for solutions u to the fractional problem.

In this paper, we find that using either the singular boundary integral equation (BIE) alone or the hypersingular BIE alone may solve the symmetric and anti-symmetric problems containing a degenerate boundary subject to the remote anti-plane shear, respectively. Although the rank of the influence matrix is deficient due to the degenerate boundary, it is still solvable. The mechanism was analytically demonstrated by using the degenerate kernel and numerically performed by using the singular value decomposition (SVD) technique. Besides, solvability was also discussed by using the Fredholm alternative theorem in both the boundary integral equation and boundary element method (BEM). We can analytically prove and numerically explain that the boundary density solved by using the singular BIE alone is different from the exact solution but the field solution is acceptable. The solvability of the linear algebraic system was discussed from the mapping table of the vector space of influence matrix where the domain is composed by the right singular vectors and the vector space of the range is composed of left singular vectors. The corresponding null space and the left null space due to the zero singular value indicates the homogeneous solution and the range deficiency, respectively. Several approaches, the singular or hypersingular BIE alone, adding symmetric or anti-symmetric boundary constraints, the self-regularized approach provided us alternative ways to solve the degenerate-boundary problem without using the dual BEM.

It is commonly stated that the Poisson’s ratio associated with bond-based peridynamics is $$\tfrac{1}{4}$$ for three-dimensional isotropic elasticity. This manuscript critically revisits this statement from a variational perspective for both two-dimensional and three-dimensional problems. To do so, a purely geometrical description of Poisson’s ratio is considered. Unlike the commonly established treatment of the problem, the Poisson’s ratio here is calculated via minimizing the internal energy density, rather than quantifying it and comparing it to its counterpart in classical linear elasticity. The advantage of the proposed approach is threefold. Firstly, elements of Cauchy linear elasticity such as “strain”, “stress” and “elastic parameters” are entirely absent throughout the derivations here. This is particularly important since peridynamics is a non-local formulation, and therefore, using local notions such as “strain” and “stress” implies locality and is misleading. Secondly, unbound by linear elasticity, the proposed approach unlocks the limitation of the analysis to small deformations. Hence, it can be immediately applied to large deformations, resulting in a nonlinear Poisson’s ratio that is no longer constant. Thirdly, the two-dimensional analysis here is purely two-dimensional, corresponding to a two-dimensional manifold in a three-dimensional space. That is, the two-dimensional formulation is neither plane stress nor plane strain that are rather degenerate three-dimensional cases. This contribution introduces the notion of nonlinear Poisson’s ratio in peridynamics for the first time and proves that the nonlinear Poisson’s ratio at the reference configuration coincides with $$\tfrac{1}{4}$$ for three-dimensional and $$\tfrac{1}{3}$$ for two-dimensional problems.

The stability of linear conservative gyroscopic systems with a degenerate potential matrix is considered. First, using a Lyapunov-type approach, a simple necessary and sufficient condition for stability of the systems with positive semi-definite potential matrices is derived. After that, a generalization of the famous Kelvin–Tait–Chetaev theorem is given, which covers the degenerate case.

The photodissociation of methyl hydroperoxide (CH3OOH) at 193 nm has been studied using a direct dynamics trajectory surface-hopping (TSH) method. The potential energies, energy gradients, and nonadiabatic couplings are calculated on the fly at the MRCIS(6,7)/aug-cc-pVDZ level of theory. The hopping of a trajectory from one electronic state to another is decided on the basis of Tully's fewest switches algorithm. An analysis of the trajectories reveals that the cleavage of the weakest O-O bond leads to major products CH3O(2E) + OH(2Π), contributing about 72.7% of the overall product formation. This OH elimination was completed in the ground degenerate product state where both the ground singlet (S0) and first excited singlet (S1) states become degenerate. The O-H bond dissociation of CH3OOH is a minor channel contributing about 27.3% to product formation, resulting in products CH3OO + H. An inspection of the trajectories indicates that unlike the major channel OH elimination, the H-atom elimination channel makes a significant contribution (∼3% of the overall product formation) through the nonadiabatic pathway via conical intersection S1/S0 leading to ground-state products CH3OO(X 2A″) + H(2S) in addition to adiabatic dissociation in the first excited singlet state, S1, correlating to products CH3OO(1 2A') + H(2S). The computed translational energy of the majority of the OH products is found to be high, distributed in the range of 70 to 100 kcal/mol, indicating that the dissociation takes place on a strong repulsive potential energy surface. This finding is consistent with the nature of the experimentally derived translational energy distribution of OH with an average translational energy of 67 kcal/mol after the excitation of CH3OOH at 193 nm.

The Pauli exclusion principle forbids indistinguishable fermions to occupy the same quantum mechanical state. Its implications are profound and it for example accounts for the electronic shell structure of atoms. Here we perform measurements on the scattering of off-resonant light from ultracold gasses of fermionic atoms. For Fermi gases in the quantum degenerate regime, we observe a marked suppression in light scattering as compared to a similarly prepared thermal Bose gas. We attribute the observed increased transmission of light through the quantum degenerate Fermi gas to Pauli blocking, where Fermi-Dirac statistics causes atoms to occupy a large region of the momentum space limiting the number of accessible states for the scattered atom. Our work confirms a longstanding fundamental result in the theory of the optical response of quantum gases and is an important step towards novel cooling and thermometry mechanisms for degenerate Fermi gases.

The basic characteristics of cylindrical as well as spherical solitary and shock waves in degenerate electron-nucleus plasmas are theoretically investigated. The electron species is assumed to be cold, ultra-relativistically degenerate, negatively charged gas, whereas the nucleus species is considered a cold, non-degenerate, positively charged, viscous fluid. The reductive perturbation technique is utilized in order to reduce the basic equations (governing the degenerate electron-nucleus plasmas under consideration) to the modified Korteweg-de Vries and Burgers equations. The latter are numerically solved and analyzed to detect the basic characteristics of solitary and shock waves in such electron-nucleus plasmas. The nonlinear nucleus-acoustic waves are found to be propagated in the form of solitary as well as shock waves in such degenerate electron-nucleus plasmas. Their basic properties as well as their time evolution are significantly modified by the effects of cylindrical as well as spherical geometries. The results of this study is expected to be applicable not only to astrophysical compact objects, but also to ultra-cold dense plasmas produced in laboratory.

In this paper we develop some KAM techniques to prove the persistence of lower dimensional elliptic-type degenerate invariant tori with prescribed frequencies in Hamiltonian systems. The proof is based on a formal KAM theorem, which allows us to solve the equation of equilibrium points and choose the parameter of small divisors after the KAM iteration, instead of in each KAM step. The proof is also based on the Leray–Schauder continuation theorem, which insures the existence of a path of real roots of an approximating odd-order real polynomial which depends continuously on parameters. This result is very important for us to tackle the Melnikov condition in the elliptic-type degenerate case.

Cyclic loading-induced consolidation behavior of soft soil is of great interest for the analysis of offshore and onshore structures. In this study, an analytical solution for one-dimensional (1D) nonlinear consolidation of saturated double-layered soil under various types of cyclic loadings such as trapezoidal cyclic loading, rectangular cyclic loading, and triangular cyclic loading was derived. The proposed solution was subsequently degenerated into solutions for special cases and compared to the existing solutions. The degenerate solutions show good agreement with the existing results, which proves that the proposed solutions are more general ones for 1D nonlinear consolidation of saturated soils under time-dependent loading. Finally, a comprehensive parametric study was conducted to investigate the influences of different layer parameters, drainage conditions, and loading parameters on nonlinear consolidation of saturated double-layered soil under cyclic loadings.

We study the spectrum of the Laplace–Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids sufficiently close to the sphere, the first $L^2$ eigenvalues have multiplicity at most two, and characterize those that are simple. For the triaxial ellipsoids sufficiently close to the sphere that are not biaxial, we show that at least the first 16 eigenvalues are all simple. We also give the results of various numerical experiments, including comparisons to our results from the analytic perturbation theory, and approximations for the eigenvalues of ellipsoids that degenerate into infinite cylinders or two-dimensional disks. We propose a conjecture on the exact number of nodal domains of near-sphere ellipsoids.

Two-dimensional (2D) transition-metal dichalcogenide (TMDC)-based semiconducting van der Waals (vdW) heterostructures are considered as potential candidates for next-generation nanoelectronics due to their unique and tunable properties. Controlling the carrier type and band alignment in 2D TMDCs and their vdW heterostructures is critical for realizing heterojunctions with the desired performances and functionalities. In this report, controlling the carrier type and band alignment in a vertical MoTe2/MoS2 heterojunction is presented via thickness engineering and surface charge transfer doping. A highly rectifying p-n diode and a nonrectifying n-n junction are obtained with different MoTe2 thicknesses due to their different doping conditions. A vertical tunnel diode is subsequently achieved with a controlled oxygen plasma treatment, which selectively induces degenerate p-type doping to MoTe2, whereas the intrinsic n-type characteristic of MoS2 is maintained during the treatment. These techniques to realize multifunctional diodes are universal and applicable to emerging nanoelectronics based on 2D materials.

We prove higher integrability of the spatial gradient of weak solutions to parabolic systems with $\phi$-growth, where $\varphi=\varphi(t)$ is a general Orlicz function. The parabolic systems need be neither degenerate nor singular. Our result is a generalized version of the one of J. Kinnunen and J. Lewis [Duke Math. J. 102 (2000), no. 2, 253--271] for the parabolic $p$-Laplace systems.

Let M be a smooth pseudoconvex hypersurface in ℂn+1 whose Levi form has at most one degenerate eigenvalue. For any tangent vector field L of type (1, 0), we prove the equality of the commutator type and the Levi form type associated to L. We also show that the regular contact type, the commutator type and the Levi form type of the real hypersurface are the same.

An accurate and efficient method is proposed to characterize waveguides with arbitrary cross sections exhibiting discrete rotational symmetries, known in the literature as $C_{N}$ symmetries. This new method is based on a novel tailored 2-D finite element method (FEM) formulation exploiting rotationally periodic boundary conditions involved in the minimum sector that defines the waveguide cross section. While retaining the proven accuracy and convergence rate of FEM, it allows for a reduction in dimensionality of the problem and classifies the modes with respect to their rotational symmetry. Most importantly, it discriminates reliably and unambiguously degenerate modes (no matter how high their order is) with the exact same computed cutoff frequency value. This is especially interesting in modal analysis of devices involving circular polarization, since the formulation inherently provides different mode sets for clockwise and counterclockwise circularly polarized modes. The method has been tested on a wide variety of waveguides and computed higher order modes have been compared against analytical solutions and general-purpose FEM.

By employing a two-fluid quantum magnetohydrodynamic (QMHD) model, linear and nonlinear aspects of magnetoacoustic wave propagation in dense magnetized electron–positron (e-p) plasma are examined in the presence of exchange-correlation potential as well. The degenerate plasma pressure and the Bohm potential effects are already taken in the model. The reductive perturbation method (RPM) is used, and the modified Korteweg–de Vries (modified KdV) equation (or KdV equation containing linear drift term) is derived for nonlinear magnetoacoustic wave propagation in degenerate pair plasma under periodic boundary conditions. The well-known Sagdeev potential approach is opted to obtain the analytical solution of magnetoacoustic cnoidal wave and soliton in dense pair plasmas. The numerical plots of magnetoacoustic cnoidal wave and soliton structures are also presented, and the condition of their formation in dense e-p plasma is discussed in detail. The parametric analysis in the absence and the presence of exchange-correlation potential effects of electrons and positrons is discussed. Moreover, the parametric analysis of the variations of plasma density and magnetic field intensity on the formation and propagation of nonlinear structures in dense e-p plasma in the presence of exchange-correlation force effects in the model is also done using the astrophysical plasma parameters mentioned in the literature for pulsars (or white dwarfs).

In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior Holder estimates are obtained when the order of the fractional diffusion is less or equal than one, and Lipschitz estimates when it is bigger than one. In the latter case, the estimates are robust enough to conclude interior C 1 , α regularity by an improvement of the flatness procedure, which is possible when the nonlocal term is close enough to a second-order diffusion.

In this paper, we employ the boundary integral equation (BIE) to analytically derive the solution of potential flow across a circular cylinder, an elliptical cylinder and a thin airfoil. It is found that the symmetric or antisymmetric potential flow problem can be solved by using the UT or the LM equation alone. Both analytical and numerical approaches were considered. The key tool is using the degenerate kernel instead of the closed-form fundamental solution. A closed-form fundamental solution is expressed in terms of degenerate kernel by using polar and elliptical coordinates for a circular cylinder and an elliptical cylinder, respectively. The role of the dual BEM is also examined by showing the singular vectors of influence matrix. Besides, either the singular or the hypersingular BIE are respectively employed alone to solve the problem of symmetric and anti-symmetric flow field. Analytical derivation as well as the BEM is demonstrated. The orientation of the ellipse, as well as the angle of attack for the thin airfoil, is considered. The result shows that the degenerate kernel can be an alternative tool for solving some BVPs, although the complex variable is always employed. Once the degenerate kernel for the closed-form fundamental solution is available, the BIE is nothing more than the linear algebra and an analytical study is possible. The extension to 3-D object is promising once the degenerate kernel is available.

We investigate the roles of symmetry and bulk-boundary correspondence in characterizing topological edge states in generalized Jackiw–Rebbi (JR) models. We show that time-reversal (T), charge-conjugation (C), parity (P), and discrete internal field rotation ($$Z_n$$ Z n ) symmetries protect and characterize the various types of edge states such as chiral and nonchiral solitons via bulk-boundary correspondence in the presence of the multiple vacua. As two representative models, we consider the JR model composed of a single fermion field having a complex mass and the generalized JR model with two massless but interacting fermion fields. The JR model shows nonchiral solitons with the $$Z_2$$ Z 2 rotation symmetry, whereas it shows chiral solitons with the broken $$Z_2$$ Z 2 rotation symmetry. In the generalized JR model, only nonchiral solitons can emerge with only $$Z_2$$ Z 2 rotation symmetry, whereas both chiral and nonchiral solitons can exist with enhanced $$Z_4$$ Z 4 rotation symmetry. Moreover, we find that the nonchiral solitons have C, P symmetries while the chiral solitons do not, which can be explained by the symmetry-invariant lines connecting degenerate vacua. Finally, we find the symmetry correspondence between multiply-degenerate global vacua and solitons such that T, C, P symmetries of a soliton inherit from global minima that are connected by the soliton, which provides a novel tool for the characterization of topological solitons.

We consider the fully nonlinear equation with variable-exponent double phase type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$ Under some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques we obtain the sharp local $C^{1,\alpha}$ regularity of viscosity solutions to such equations.

The Richards equation is a nonlinear degenerate advection diffusion equation that models flow in saturated/unsaturated porous media, it's crucially important for prediction of disasters when heavy ...

In the paper, we first define a degenerate Pochhammer symbol by using the degenerate gamma function and investigate its properties. By using the degenerate Pochhammer symbol, we introduce and investigate a degenerate hypergeometric function. We also define a degenerate Sumudu transform and investigate its properties by using degenerate exponential function. Finally, we give certain the integral representations, derivative formulas, integral transforms, factional calculus applications, and generating functions of the degenerate hypergeometric function.