In this work, we are concerned with inverse nodal problems of Sturm-Liouville operators with eigenparameter dependent boundary conditions on the interval [0, 1]. It is shown that a twin dense subset of the nodal set in a subinterval [a_1,a_2]⊂[0,1] (a_1<12
Let Ω ⊂ ℝ2 be a smooth bounded domain and H 1 0 (Ω) be the standard Sobolev space consisting of functions which vanish on ∂Ω and whose gradient is in L2(Ω). In this paper, we investigate critical points of the Trudinger-Moser functional by a heat flow. Our method is based on blow-up analysis, relying on integral estimates.
An odd coloring of a graph is a proper coloring with the additional constraint that each non-isolated vertex has at least one color that appears an odd number of times in its open neighborhood. Determining if a graph is odd k-colorable is NP-complete for k ≥ 3. It is known that planar graphs are odd 8-colorable and triangle-free planar graphs are odd 7-colorable. In this paper we prove that planar graphs without 4-cycles and 5-cycles, or 4-cycles and 6-cycles, are odd 7-colorable.
In the study of the continuity of the Lyapunov exponent for the discrete quasi-periodic Schrödinger operators, there is a pioneering result by Wang-You that the authors constructed the examples whose Lyapunov exponent is discontinuous in the potential with the C0 norm for non-analytic potentials. In this paper, we consider these operators for some Gevrey potential, which is an analytic function with a Gevrey small perturbation on the multi-dimensional torus. We prove that in the large coupling regions, the Lyapunov exponent is positive and jointly continuous with the C0 norm in all parameters, such as the energy, the frequency and the potential, for two transformations, the multi-frequency shift and the skew shift. It is complementary to Wang-You’s result.
In this paper, we study the existence of standing wave solutions for the following perturbed quasilinear Schrödinger systems in ℝN {-ε^2 Δ u + V(x)u-ε^2 Δ [(1+u^2)^1/2]u/2(1+u^2)^1/2 =K(x)|u|^2^*-2u+F_u(x,u,v), -ε^2 Δ v + V(x)v-ε^2 Δ [(1+v^2)^1/2]v/2(1+v^2)^1/2 =K(x)|v|^2^*-2v+F_v(x,u,v).. Under some suitable conditions, by using the variational approach, we establish the existence of standing wave solutions for the above system for sufficiently small ε, for any m ∈ ℕ, the system admits at least m pairs of solutions for sufficiently small ε. Moreover, these solutions converge to (uε, vε) → (0, 0) in a suitable Sobolev space as ε → 0. Our results improve and supplement some existing relevant results.
We establish the global well-posedness of the three-dimensional incompressible inhomogeneous magnetohydrodynamics (MHD) system for a class of anisotropic initial data that are slowly varying in one spatial direction. The initial data may be large and are not assumed to be small in classical isotropic critical spaces. The anisotropic structure is characterized by two independent ε-dependent parameters measuring, respectively, the density perturbation and the velocitymagnetic correctors. Under explicit compatibility conditions between these parameters, we obtain global existence and uniqueness of solutions in critical anisotropic Besov spaces. The proof relies on a decomposition around a coupled two-dimensional MHD background system and on uniform anisotropic maximal regularity estimates.
In this paper, we consider potential measures of spectrally negative Lévy processes for the last exit time killed on exiting interval, some exit identities are needed for our main results. Our results are expressed in terms of scale functions.
By means of the Whitham modulation theory, this paper studies the discontinuous initial value problem of the bad Jaulent-Miodek (JM) equation, which is compared with the good JM equation. The linear stability of the good and bad JM equations is analyzed to show the difference between the two equations. Also the physical significance of the JM equations is discussed by considering the reduction of the Euler’s equation. Then the zero-, one-, two-phase periodic solutions and the corresponding Whitham equations in the framework of the bad JM equation are derived by finite-gap integration approach. The degeneration of the one-phase periodic solution along with the genus-one Whitham equation are analyzed by taking the two sides limits of the modulus m of the Jacobi elliptic functions. The basic rarefaction wave patterns and dispersive shock wave patterns are proposed analytically and graphically, and then the complete classification of solutions and the all possible waveforms evolving from discontinuous initial values in the bad JM equation are established. In the results, various patterns of emergent nonlinear waves that appear to be new and are being detected for the first time
This paper mainly presents and studies some local and parallel finite element methods for the unsteady magnetohydrodynamic equations with low electromagnetic Reynolds number. Firstly, both the semi discrete and fully discrete local algorithms are provided and investigated, and the theoretical tool crucial to the analysis of the fully discrete local algorithm is obtained. Subsequently, we generalize the fully discrete local algorithm to the fully discrete parallel algorithm. At the end, some numerical experiments are provided to validate the effectiveness and efficiency of our algorithms.
This paper considers the existence of multiple normalized solutions of the following Schrödinger-Choquard equation - Δ u = λ u + k(ε x) (I_α * |u|^q) |u|^q-2u + μ (I_α * |u|^p) |u|^p-2u, x ∈ℝ^N, ∫_ℝ^N |u|^2 dx = c^2, x ∈ℝ^N. where c, ε, μ > 0, N ≥ 3, α, ∈ (0; N), N + α N < q < 1 + α + 2 N < p ≤N + αN - 2 , λ ∈ ℝ is a Lagrange multiplier which is unknown, Iα is the Riesz potential, k:ℝN → [0; ∞) is a continuous and positive function. When ε is small enough, we prove that the numbers of normalized solutions are at least the numbers of global maximum points of k by Ekeland’s variational principle and truncated skill.
In this work, we investigate a class of nonlinear boundary value problems within the framework of Sobolev spaces with variable exponents. After establishing precise formulations of these problems, we reformulate them as nonlinear hyperbolic-type equations. For each of the six problems considered, we establish both existence and uniqueness results. Furthermore, we address the associated stationary problems and prove the existence of solutions by applying a suitable variant of Brouwer’s fixed-point theorem.
In this paper, we investigate some asymptotic properties of the least squares estimator in nonlinear regression model with independent and identically distributed random errors under sub-linear expectations. The large deviation results for the estimator are established under some general conditions. As applications, the results on weak consistency and strong consistency are obtained under the meaning of capacity. A simulation study is also presented to verify the validity of the theoretical results.
This paper considers the customer’s equilibrium joining-balking behavior in a retrial queue with two-phase service and priority-purchasing. If the arriving customer finds the server busy, the customer can make three choices (joining the orbit, purchasing the priority or balking) based on different levels of the system information. We consider two cases: the partially observable case and the fully unobservable case. The customer’s equilibrium strategy for the partially observable case and the fully unobservable case are derived. Then, an algorithm is developed to solve the optimal fee that maximizes the service provider’s profit. Finally, the influence of system parameters on performance measures, equilibrium joining strategies and the service provider’s profits is illustrated by numerical examples.
In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary incompressible Magnetohydrodynamic equations in the whole space, the half-space, or a periodic slab, and presenting the solution must vanish under the condition that for some 0 ≤ δ ≤ 1 < L and q = 6(3 - δ)6 - δ,lim inf_R →∞1 R‖(u,h)‖_R < | x | < LR^3 - δ = 0 . We also deduce sufficient conditions by allowing shrinking ratio L = 1 + R−α. When in slab with zero boundary condition, stronger decay rate is needed. We do not assume the global bound of the velocity field u and the magnetic field h and investigate the Liouville type theorems by the conditions lim inf rather than lim.
In this paper, we address exponential stabilization of transmission problem of the wave equation with dynamical boundary conditions. We consider waves passing from a medium in which the speed is a1 into a medium in which the speed a2 in the case of total internal reflection and show that such a system can be controlled by introducing both dynamical boundary control along the exterior boundary and distributed control near the transmission boundary. We obtain that the system is exponential stabilization without any restriction on a1, a2 and the transmission boundary.
Sparse signal recovery is one of the key problems in the field of compressive sensing. Restricted isometry property (RIP) is an important metric for the recoverability of sparse signals. It can provide explicit and simple sufficient conditions for the convergences of many reconstruction methods such as orthogonal matching pursuit, basis pursuit (BP), and hard thresholding pursuit. However, RIP has several drawbacks. One drawback is that RIP can not be preserved for the scalar rescaling. In order to overcome this drawback, a new metric named combinatorial condition number is defined in this paper. It is invariant for the scalar rescaling. Subsequently, the Wielandt inequality and robust null space property are utilized to present a sufficient condition for the sparse signal recovery by BP in terms of the new metric.
In this paper, an efficient algorithm is proposed for Toeplitz matrix recovery via hybrid thresholding operator. The algorithm is based on the mean-value augmented Lagrangian multiplier algorithm and the singular values are processed by hybrid singular value threshold operator. The new algorithm ensures that the matrix generated by the iteration has a Toeplitz structure, which reduces the calculation time and obtains a more accurate Toeplitz matrix. The convergence of the new algorithm is discussed under certain assumptions. Numerical experiments show that the new algorithm achieves lower CPU time than the mean-value augmented Lagrangian multiplier algorithm, smooth augmented Lagrangian multiplier algorithm, and augmented Lagrangian multiplier algorithm.
In this paper, we consider the Cauchy problem and the vacuum free boundary problem of the three-dimensional non-isentropic compressible Euler equations with spherical symmetry and cylindrical symmetry, respectively. By using some ansatzes, we construct some self-similar analytical solutions with the initial specific entropy of the form S0(r) = b0r, where b0 is a constant. Moreover, we investigate the global existence and blowup of the constructed solutions and study the spreading rate of the free boundary for the vacuum free boundary problem by using the averaged quantities method. Finally, we provide some analytical solutions for the non-isentropic compressible Euler equations with Coriolis force. We find that the rotation can suppress the formation of singularity for the Cauchy problem and prevent the free boundary from spreading out infinitely for the free boundary problem.
In this article, we develop a jackknife model averaging (JMA) method for modal linear regression with rightly censored responses. The weights in model averaging are obtained by maximizing the leave-one-out cross-validation criterion function involving the kernel density estimation, inverse-probability-of-censoring weighting and Kaplan-Meier estimation techniques. Under rather mild conditions, we establish the asymptotic optimality of the JMA estimator in the sense of minimizing out-of-sample prediction risk asymptotically. Besides, we also obtain some theoretical properties of regression coefficient within each misspecified candidate model, including the convergence rate and asymptotic normality. Some simulations are conducted to evaluate the performance of the proposed JMA approach against some competing model selection and model averaging methods, which show that the suggested JMA method is superior to its competitors. The proposed JMA method is also applied to a real data analysis as an illustration.
In Lauritzen-Wermuth-Frydenberg (LWF) chain graphical models (CGMs), conditional independence collapsibility means that the structure of a marginal model over a subset of variables can be exactly captured by the corresponding induced subgraph. Collapsibility helps reduce model complexity and improves interpretability. To describe collapsibility using graphical criteria, prior work has explored various interaction models. In this paper, we study conditions and criteria for collapsibility in LWF chain graphs. We introduce the concept of CI-removable node, which offers an equivalent condition for collapsibility when marginalizing over one variable. Extending this idea, we define a sequentially CI-removable set as a sufficient condition for collapsibility over multiple variables. We also use this condition to find minimal I-maps of marginal models in LWF chain graphs. This approach allows the efficient recovery of structural information from marginal models while preserving the original conditional independence relationships.
