In this article, we investigate the multilinear distorted multiplier estimate (Coifman-Meyer type theorem) associated with the Schr\“{o}dinger operator $H=-\Delta + V$ in the framework of the corresponding distorted Fourier transform. Our result is the “distorted” analog of the multilinear Coifman-Meyer multiplier operator theorem in \cite{CM1}, which extends the bilinear estimates of Germain, Hani and Walsh’s in \cite{PZS} to multilinear case for all dimensions. As applications, we give the estimate of Leibniz’s law of integer order derivations for the multilinear distorted multiplier for the first time and we obtain small data scattering for a kind of generalized mass-critical NLS with good potential in low dimensions $d=1,2$.

The purpose of this work is to investigate perfect reconstruction underlying range space of operators in finite dimensional Hilbert spaces by a new matrix method. To this end, first we obtain more structures of the canonical $K$-dual. % and survey optimal $K$-dual problem under probabilistic erasures. Then, we survey the problem of recovering and robustness of signals when the erasure set satisfies the minimal redundancy condition or the $K$-frame is maximal robust. Furthermore, we show that the error rate is reduced under erasures if the $K$-frame is of uniform excess. Toward the protection of encoding frame (K-dual) against erasures, we introduce a new concept so called $(r,k)$-matrix to recover lost data and solve the perfect recovery problem via matrix equations. Moreover, we discuss the existence of such matrices by using minimal redundancy condition on decoding frames for operators. We exhibit several examples that illustrate the advantage of using the new matrix method with respect to the previous approaches in existence construction. And finally, we provide the numerical results to confirm the main results in the case noise-free and test sensitivity of the method with respect to noise.

This work is concerned with the global existence of large solutions to the three-dimensional dissipative fluid-dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the incompressible Navier–Stokes–Poisson–Nernst–Planck equations. Making full use of the algebraic structure of the system, we obtain the global existence of solutions without smallness assumptions imposed on the third component of the initial velocity field and the summation of initial densities of charged species. More precisely, we prove that there exist two positive constants c 0 , C 0 such that if the initial data satisfies ( ∥ u 0 h ∥ B _ p , 1 − 1 + 3 p + ∥ N 0 − P 0 ∥ B _ q , 1 − 2 + 3 q ) exp { C 0 ( ∥ u 0 3 ∥ B _ p , 1 − 1 + 3 p 2 + ( ∥ N 0 + P 0 ∥ B _ r , 1 − 2 + 3 r + 1 ) exp { C 0 ∥ u 0 3 ∥ B _ p , 1 − 1 + 3 p } + 1 ) } ≤ c 0 , then the incompressible Navier–Stokes–Poisson–Nernst–Planck equations admits a unique global solution.

In this article, we consider the blowup phenomenon of smooth solutions to the isentropic compressible quantum hydrodynamic model(QHD) with the initial density of compact support in arbitrary space dimensions. This result is an evolution of Xin’s work

It is comprehended that the systems without any limitation on their Zeno action are enthralled in a vast class of hybrid systems. This article is influenced by a new category of non-autonomous second order measure differential problems with state-dependent delay (SDD) and non-instantaneous impulse (NII). Some new sufficient postulates are created that guarantee solvability and approximate controllability. We employ the fixed point strategy and theory of Lebesgue–Stieltjes integral in the space of piecewise regulated functions. The measure of non-compactness is applied to establish the existence of a solution. Moreover, the measured differential equations generalize the ordinary impulsive differential equations. Thus, our findings are more prevalent than that encountered in the literature. At last, an example is comprised that exhibits the significance of the developed theory.

By constructing various Pohozaev identities, we study the uniqueness of minimizers for L 2 -subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contain x=0 as a singular point.

In this paper, the link between combinatorial structures and alternative algebras is studied, determining which configurations are associated with those algebras. Moreover, the isomorphism classes of each 2-dimensional configuration associated with these algebras is analyzed, providing a new method to classify them. In order to complement the theoretical study, two algorithmic methods are implemented: the first one constructs and draws the (pseudo)digraph associated with a given alternative algebra and the second one tests if a given combinatorial structure is associated with some alternative algebra.

In this paper, we consider the low Mach number limit problem for the compressible viscous micropolar fluid model based on the concept of dissipative measure-valued solutions. We prove that the dissipative measure-valued solutions of the compressible micropolar fluid model converge to the smooth solution of the incompressible micropolar system in the case of well-prepared initial data when the Mach number tends to zero.

In this paper, we introduce a new traveling wave with a buffer zone, which is approached by an asymmetric credit migration model with fixed migration boundary. The asymptotic behavior of the solution of the model is discussed. By constructing two sets of sub and super solutions sequences, it is proved that the solution of the credit rating migration model approaches the new traveling wave with buffer zone as time goes to infinite in a direction. Additionally, some numerical results are presented.

In this paper, we study the symmetry on a class of multiparameter Markov Processes in Khoshnevisan (2002), and obtain some important and interesting results.

Let $H_c$ be a $(2n)\times(2n)$ symmetric tridiagonal matrix with diagonal elements $c \in \mathbb{R}$ and off-diagonal elements one, and $S$ be a $(2n)\times(2n)$ diagonal matrix with the first $n$ diagonal elements being plus ones and the last $n$ being minus ones. Davies and Levitin studied the eigenvalues of a linear pencil $\mathcal{A}_c=H_c-\lambda S$ as $2n$ approaches to infinity. It was conjectured by DL that for any $n \in \mathbb{N}$ the non-real eigenvalues $\lambda$ of $\mathcal{A}_c$ satisfy both $|\lambda + c|<2$ and $|\lambda - c|<2$. The conjecture has been verified numerically for a wide range of $n$ and $c$, but so far the full proof is missing. The purpose of the paper is to support this conjecture with a partial proof and several numerical experiments which allow to get some insight in the behaviour of the non-real eigenvalues of $\mathcal{A}_c$. We provide a proof of the conjecture for $n \leq 3$, and also in the case where $|\lambda + c|=|\lambda - c|$. In addition, numerics indicate that some phenomena may occur for more general linear pencils.

In this paper, we consider the Ricker model with delay and constant or periodic stocking. The impact of delay and stocking on stability is known to reflect opposing effect, which motivates investigating the interaction between these factors and their influence on overall stability. We found that the high stocking density tends to neutralize the delay effect. Conditions are established on the parameters in order to guarantee the global stability of the equilibrium solution in the case of constant stocking, as well as the global stability of the 2-periodic solution in the case of 2-periodic stocking. Our approach extensively relies on the utilization of the embedding technique. Whether constant stocking or periodic stocking, the mode has the potential to undergo a Neimark-Sacker bifurcation in both cases. However, the Neimark-Sacker bifurcation in the 2-periodic case results in the emergence of two invariant curves that collectively function as a single attractor. Finally, we pose open questions in the form of conjectures about global stability for certain choices of the parameters.

The inappropriate use of insecticides may lead to disastrous pest outbreaks. Aiming at avoiding the outbreak of pests by optimizing the control strategies, we propose a novel mathematical model base on the idea of pulse in this study, namely a discrete-time model of pest-natural enemies, which considers the implementation of spraying insecticides within the time interval between two generations of pests. We first investigates the existence and stability of fixed points, and then using the central manifold theory, we proved that the system can exist period-doubling bifurcation. The main theoretical results are verified by numerical simulations. The experiments show that if the insecticidal time is within a certain range, the lower insecticidal rate stimulates the growth of pests and natural enemies, while the larger insecticidal rate can inhibit the growth of pests and natural enemies. In addition, the effects of pest growth rate, timing of pesticide spraying, and pesticide intensity on the system are comprehensively discussed. The main research provides good reference significance to choose an appropriate time to prevent the pest outbreaks.

In the present paper, we study a class of quasilinear Choquard equations involving N-Laplacian and the nonlinearity with the critical exponential growth. We discuss the existence of positive solutions of such equations.

Abstract: In this paper, the nonlinear Schrödinger-type equation −(∇ + iA) ^2 u + u + λ[I_α*(K|u|^2)]Ku=af(|u|)u/|u| in R ^3 is considered in the presence of magnetic field, where A ∈ C ^1 (R ^3 ,R^ 3 ), α ∈ (0,3), I_α denotes the Riesz potential, K ∈ L^ p (R ^3 ,(0,∞)) for some p ∈ (6/(1 + α),∞], a ∈ L^ q (R 3 ,[0,∞)) \ {0} for some q ∈ (3/2,∞], and f ∈ C(R,[0,∞)) is assumed to be asymptotically linear at infinity. Under suitable assumptions regarding A, K, a, and f, variational methods are used to establish the existence of ground-state solutions of the above equation for sufficiently small values of the parameter λ.

This article proposes an adaptive neural output feedback control scheme in combination with state and disturbance observers for uncertain fractional-order nonlinear systems containing unknown external disturbance, input saturation and immeasurable state. The radial basis function neural network (RBFNN) approximation is used to estimate unknown nonlinear function, and a state observer as well as a fractional-order disturbance observer is developed simultaneously by using the approximation output of the RBFNN to estimate immeasurable states and unknown compounded disturbances, respectively. Then, a fractional-order auxiliary system is constructed to compensate the effects caused by the saturated input. In addition, by introducing a dynamic surface control strategy, the tedious analytic computation of time derivatives of virtual control laws in the conventional backstepping method is avoided. The proposed method guarantees that the boundness of all signals in the closed loop system and the tracking errors converge to a small neighbourhood around the origin. Finally, two examples are provided to verify the effectiveness of the proposed control method.

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure - measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples - classes of homogeneous integral operators on various metric spaces, and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article - various specific cases - illustrate general statements and results given in the paper and at the same time are of interest in their own way.

We study the Helmholtz equation with periodic coefficients in a closed wave-guide. A functional analytic approach is used to formulate and to solve the radiation problem in a self-contained exposition. In this context, we simplify the non-degeneracy assumption on the frequency. Limiting absorption principles (LAPs) are studied and the radiation condition corresponding to the chosen LAP is derived; we include an example to show different LAPs lead, in general, to different solutions of the radiation problem. Finally, we characterize the set of all bounded solutions to the homogeneous problem.