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This work is concerned with the consistency conditions for the equations in max plus algebra. The three classes of the max plus linear equations presented in the paper Shinzawa [Ultra discrete permanent and the consistency of max plus linear equations, Linear Algebra Appl. 506 (2016) 445–477] can be equivalently converted into the corresponding system of inequalities. The equivalence relation between ultra discrete permanent and maximum cycle mean is suggested. Thus, the necessary and sufficient conditions for solvability of the equations are obtained using the maximum cycle mean in this note. [ABSTRACT FROM AUTHOR]

This paper investigates a generalized Camassa–Holm equation with cubic nonlinearities (alias the mCH-Novikov equation), which is a generalization of some special equations. The mCH-Novikov equation possesses well-known peaked solitary waves that are called peakons. The peakons were proved orbital stable by Chen et al. in [Stability of peaked solitary waves for a class of cubic quasilinear shallow-water equations. Int Math Res Not. 2022;1–33]. We mainly prove the orbital stability of the multi-peakons in the mCH-Novikov equation. In this paper, it is proved that the sum of N fully decoupled peaks is orbitally stable in the energy space by using energy argument, combining the orbital stability of single peakons and local monotonicity of the method. [ABSTRACT FROM AUTHOR]

This paper studies the asymptotic behavior of energy solutions to a non-linear generalized Hartree equation. Indeed, in the inter-critical regime, one revisits the scattering versus finite time blow-up of energy solutions with non-necessarily spherically symmetric datum. Here, one uses the new approach due to Dodson and Murphy. The novelty in this work is to express the scattering threshold in terms of some non-conserved quantities. The main result of this note seems to be stronger than the classical scattering versus finite time blow-up dichotomy given in terms of the conserved mass and energy by Holmer and Roudenko. Indeed, as an application, one investigates the scattering in three different regimes: under, at and beyond the ground state threshold. The main result given here, which can be seen as a criteria of scattering versus finite time blow-up of energy solutions, enables us to give a unified approach to deal with the above generalised Hartree equation in different regimes. This paper follows some new ideas presented recently in the classical Schrödinger equation with a local source term, by V.D. Dinh. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the uniform attractors for 3D MHD equations with nonlinear damping. Some uniform estimates for strong solutions are established. Furthermore, we prove that 3D damped equations have an $ (\mathbb {V},\mathbb {V}) $ (V , V) -uniform attractor and an $ (\mathbb {V},\mathbf{H}^{2}) $ (V , H 2) -uniform attractor, and the $ (\mathbb {V},\mathbb {V}) $ (V , V) -uniform attractor is actually the $ (\mathbb {V},\mathbf{H}^{2}) $ (V , H 2) -uniform attractor. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of ground state solution for the following Kirchhoff equation with combined nonlinearities \[ -\left(a+b\displaystyle\int_{\mathbb{R}^N} |\nabla u|^2\right)\Delta u=\lambda u+|u|^{p-2}u+\mu |u|^{q-2}u \ \hbox{in}\ \mathbb{R}^N, \quad 1\leq N\leq 3 \] − (a + b ∫ R N | ∇u | 2) Δu = λu + | u | p − 2 u + μ | u | q − 2 u in R N , 1 ≤ N ≤ 3 with prescribed mass $ \int _{\mathbb {R}^N} u^2=c^2 $ ∫ R N u 2 = c 2 , where $ a{ \gt }0,b{ \gt }0,c{ \gt }0 $ a > 0 , b > 0 , c > 0 , $ \mu { \lt }0 $ μ < 0 , $ 2{ \lt }q\leq 2+\frac {8}{N}{ \lt }p{ \lt }2^{*} $ 2 < q ≤ 2 + 8 N < p < 2 ∗ . Under certain assumptions on the parameter μ, we obtain the existence of normalized solution $ (\tilde {u},\lambda _c)\in S_c \times \mathbb {R} $ (u ~ , λ c) ∈ S c × R , where $ S_c=\{ u\in H^1(\mathbb {R}^N): \int _{\mathbb {R}^N} u^2=c^2 \} $ S c = { u ∈ H 1 (R N) : ∫ R N u 2 = c 2 }. [ABSTRACT FROM AUTHOR]

In this paper, we present four characterizations of q-Dunkl-classical orthogonal polynomials. The first one is a $ T_{\theta,q} $ T θ , q -distributional equation of Pearson type fulfilled by its associated form, where $ T_{\theta,q} $ T θ , q is the q-Dunkl operator. The second is a second order linear $ T_{\theta,q} $ T θ , q -difference equation. The third is a first order linear $ T_{\theta,q} $ T θ , q -difference equation with polynomial coefficients satisfied by the corresponding Stieltjes function. The fourth is the so-called structure relation fulfilled by q-Dunkl-classical polynomials. Moreover, we provide an example of a non-symmetric sequence of $ T_{\theta,q} $ T θ , q -classical orthogonal polynomials. [ABSTRACT FROM AUTHOR]

In this paper we propose a new parameter estimator that ensures global exponential convergence of linear regression models requiring only the necessary assumption of identifiability of the regression equation, which we show is equivalent to interval excitation of the regressor vector. An extension to – separable and monotonic – nonlinear parameterisations is also given. The estimators are shown to be robust to additive measurement noise and – not necessarily slow-parameter variations. Moreover, a version of the estimator that is robust with respect to sinusoidal disturbances with unknown internal model is given. Simulation results that illustrate the performance of the estimator compared with other algorithms are given. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is the study of second-order optimality conditions for the bilinear control of a strongly degenerate parabolic equation. The equation is degenerate at the boundary of the spatial domain. The well-posedness of the state equation, as well as weak maximum principles are established. We prove some differentiability properties of the control-to-state operator and the existence of optimal solutions. Finally, we derive first- and second-order optimality conditions for the system. [ABSTRACT FROM AUTHOR]

In this paper, we develop an integrodifference equation model that incorporates spatial memory and learning so that each year, a fraction of the population use the same dispersal kernel as the previous year, and the remaining individuals return to where they bred or were born. In temporally static environments, the equilibrium of the system corresponds to an ideal free dispersal strategy, which is evolutionarily stable. We prove local stability of this equilibrium in a special case, and we observe convergence towards this equilibrium in numerical computations. When there are periodic or stochastic temporal changes in the environment, the population is less able to match the environment, but is able to do so to some extent depending on the parameters. Overall, the mechanism proposed in this model shows a possible way for the dispersal kernel of a population to evolve towards an ideal free dispersal kernel. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonexistence of the periodic peakon and peakon for a highly nonlinear shallow-water model, which has been recently derived from the full governing equations for two dimensional flow with the Coriolis effect or with constant vorticity, under a larger scaling than the Camassa-Holm (CH) one. Note that the so obtained model not only has CH-type terms, but also exhibits cubic order nonlinearities. Thus it is interesting to study how the higher power nonlinear terms affect the existence of the (periodic) peakon. [ABSTRACT FROM AUTHOR]

In this paper, a kind of the differential equation including a time-fractional sub-diffusion equation is considered. Through this memorandum, a well-known technique, in the time direction is adopted by the p-fractional linear multistep method (p-FLMM) according to the q-fractional backward difference formula (q-FBDF) of implicit type for q = 1, 2, 3, and the spatial direction is approximated by the second-order central difference method. The stability properties of the proposed method can be investigated in combination with the Fourier technique and $ \mathcal {Z} $ Z -transformation and also its convergence is studied by using the truncated error maximum. It is shown that the method is unconditionally stable and the orders of convergence are $ \mathcal {O}(\tau ^{p}+h^2) $ O (τ p + h 2) for $ 1 \leq p \leq ~4 $ 1 ≤ p ≤ 4 , in which p is the order of accuracy in the time direction and τ and h determine temporal and spatial stepsizes, respectively. Some numerical experiments are included to demonstrate the validity and applicability of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and regularity of a capacity solution for a coupled nonlinear anisotropic parabolic-elliptic system, where the elliptic component of the parabolic equation involves thermal conductivities $ a_{i}(u) $ a i (u) that satisfy $ \lim _{s\rightarrow +\infty }a_{i}(s)=0 $ lim s → + ∞ a i (s) = 0 for all $ i=1,\ldots,N $ i = 1 , ... , N. We work with anisotropic Sobolev spaces and use Schauder's fixed-point theorem to find weak solutions to approximate problems. In addition, we validate the convergence of a subsequence of approximate solutions to a capacity solution. [ABSTRACT FROM AUTHOR]

In this paper, we have analysed the influence of viscoelastic and frictional damping on the decay rate of solutions for a Kirchhoff-type viscoelastic wave equation with a distributed delay acting on nonlinear internal damping. Taking the relaxation function of a fairly large class and using the method of energy in which we introduce an adapted Lyapunov functional and by exploiting certain properties of convex functions, under certain assumptions on the constants of system, we obtain the optimal decay rate of energy in the sense that it is compatible with the decay rate of the relaxation function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the well-posedness and large time behavior of solutions to the 3D incompressible magneto-micropolar equations. By virtue of the $ L_p-L_q $ L p − L q estimate obtained through the spectral decomposition of the linearized magneto-micropolar equations, we show the existence and uniqueness of small $ L_3 $ L 3 -strong solutions of the equations with small initial data. Then basing on this result, we derive sharp time decay estimates of the $ L_3 $ L 3 -strong solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a singular Adams' inequality with logarithmic weights in the unit ball of $ \mathbb {R}^4 $ R 4 . Our results extend the results of Zhu and Wang [Adams' inequality with logarithmic weights in $ \mathbb {R}^4 $ R 4 . Proc Amer Math Soc. 2021;149(8):3463–3472] on Adams' inequality with logarithmic weights to singular case. Then, we study the existence of solutions for some weighted mean field equations, relying on variational methods and the singular Adams' inequality with logarithmic weights we previously established. [ABSTRACT FROM AUTHOR]

This paper deals with the following critical Choquard equation with a Kirchhoff type perturbation in bounded domains, \[ \begin{cases} -(1+b\|u\|^{2})\Delta u=\left(\int_{\Omega}\frac{u^{2}(y)}{|x-y|^{4}}\,{\rm d}y\right)u +\lambda u & {\rm in}\ \Omega,\\ u=0 & {\rm on}\ \partial\Omega, \end{cases} \] { − (1 + b ‖ u ‖ 2) Δu = (∫ Ω u 2 (y) | x − y | 4 d y) u + λu in Ω , u = 0 on ∂Ω , where $ \Omega \subset \mathbb {R}^{N}(N\geq 5) $ Ω ⊂ R N (N ≥ 5) is a smooth bounded domain and $ \|\cdot \| $ ‖ ⋅ ‖ is the standard norm of $ H_{0}^{1}(\Omega) $ H 0 1 (Ω). Under the suitable assumptions on the constant $ b\geq 0 $ b ≥ 0 , we prove the existence of solutions for $ 0 \lt \lambda \leq \lambda _{1} $ 0 < λ ≤ λ 1 , where $ \lambda _{1} \gt 0 $ λ 1 > 0 is the first eigenvalue of $ -\Delta $ − Δ on Ω. Moreover, we prove the multiplicity of solutions for $ \lambda \gt \lambda _{1} $ λ > λ 1 and b>0 in suitable intervals. [ABSTRACT FROM AUTHOR]

This paper takes an objective look at how papers using structural equation models are received in the review process of academic marketing research journals. The focus is examining whether or not papers with structural equation modeling (SEM) applications enjoy an advantage in the review process and what the nature of such an advantage may be. A sample of submission records from a top marketing research journal is used to examine this issue. Results suggest that papers with SEM applications enjoy a modest advantage in the review process but that this difference is partially due to the fact that papers with SEM applications are rated more highly on multiple paper characteristics. For example, authors using SEM tend to have their papers rated more highly than others on the theoretical and conceptual development attribute. Thus, learning and applying SEM may have advantages not directly related to the application of the statistical approach. [ABSTRACT FROM AUTHOR]

This paper provides a modification of a recently developed common weight model for technology selection. The discrimination power of the existing model depends on a discriminating parameter. Although larger values of this parameter can give more discrimination, it should also be small enough to guarantee the existence of at least one efficient DMU. Hence, different values of this parameter should be examined to determine the proper value. In the model proposed here, the largest possible value of the parameter is utilised to maximise the discrimination power, and the existence of an efficient DMU is guaranteed by adding a few constraints. In addition, the paper presents comments on existing models, and the properties and the advantages of the model are explained. The contents of the paper are illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

The first goal of this paper is to establish a result on the existence and uniqueness of solution to an initial-boundary value problem for parabolic equations of Kirchhoff type involving the 1-Laplace operator. The second goal is to discuss some qualitative properties, such as asymptotic behaviour and the extinction of solutions for the considered problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to the uniqueness of global-in-time conservative solutions and generic regularity for the shallow water waves of moderate amplitude equation (Constantin-Lannes equation). The Constantin-Lannes equation possible development of singularities in finite time, and beyond the occurrence of wave breaking, it exists global conservative solutions. In the present paper, we will prove the uniqueness of global-in-time conservative solutions for the Constantin-Lannes equation with general initial data u 0 ∈ H 1 (R) by analyzing the evolution of the quantities u and v = 2 arctan ⁡ u x along each characteristic. Moreover, we consider that piecewise smooth solutions with only generic singularities are dense in the whole solution set by Thom's transversality Lemma. [ABSTRACT FROM AUTHOR]

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This paper is an overview from a personal perspective on the various ways Lucas has shaped today's economics in general and in terms of 'Chicago economics' in particular. In honor of the 50th anniversary of its publication, much focus is given to his 1972 neutrality paper and its impact. I discuss how the paper was a trigger of the subsequent emergence of rational expectations macroeconomics. Further, I touch upon his fundamental contributions to growth theory, asset pricing and the characteristic use of the Bellman equations. After covering these topics, the paper concludes with a portrayal of the Money and Banking Workshop to describe the environment that Lucas established at the Chicago department, and to illustrate his enduring influence on the culture of teaching and discussing macroeconomics at the University of Chicago. [ABSTRACT FROM AUTHOR]

In this paper, analytic solutions and dynamic properties of a variety of Boussinesq-type equations are established via the complete discrimination system for polynomial method. All the existing single traveling wave solutions to these equations as well as some new solutions are shown, and the Hamiltonian and topological properties to these equations are also presented. Considering the significance of the Boussinesq-type equations, our results would have wide applications in ocean engineering and fluid mechanics, like describing and predicting the solitary and periodic waves in various shallow water models. [ABSTRACT FROM AUTHOR]

In this paper, we study the following critical nonlinear Schrödinger–Kirchhoff equation: ($P$) $$\begin{align*} \left \{ \begin{array}{@{}l@{}} \displaystyle -\left(a+b\int_{R^{N}}|\nabla u|^{2}\,{\rm d}x\right)\Delta u + V(x)u =P(x)|u|^{2^*-2}u+\mu|u|^{q-2}u, \ {\rm in}\ \mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N) \end{array} \right. \end{align*}$$ { − (a + b ∫ R N | ∇u | 2 d x) Δu + V (x) u = P (x) | u | 2 ∗ − 2 u + μ | u | q − 2 u , in R N , u ∈ H 1 (R N) where $ a, b, \mu \gt 0 $ a , b , μ > 0 , $ N\geq 3 $ N ≥ 3 , $ \max \{2^*-1, 2\} \lt q \lt 2^*=\frac {2N}{N-2} $ max { 2 ∗ − 1 , 2 } < q < 2 ∗ = 2 N N − 2 , $ V(x) \gt 0 $ V (x) > 0 and $ P(x)\geq 0 $ P (x) ≥ 0 are two continuous functions. By using the variational method and truncation technique, we prove the multiplicity of solutions for Equation (P). [ABSTRACT FROM AUTHOR]

In this paper, the local existence and uniqueness of weak solutions to a d-dimensional non-resistive MHD equations in homogeneous Besov spaces are studied. Specifically we obtain the local existence of a weak solution $ (u,b) $ (u , b) of the non-resistive MHD equations for the initial data $ u_0\in \dot B^{\frac {d}{p}-1}_{p,1}(\mathbb {R}^{d}) $ u 0 ∈ B ˙ p , 1 d p − 1 (R d) and $ b_0\in \dot B^{\frac {d}{p}}_{p,1}(\mathbb {R}^{d}) $ b 0 ∈ B ˙ p , 1 d p (R d) with $ 1\le p \le \infty $ 1 ≤ p ≤ ∞ , and the uniqueness of the weak solution when $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d. Compared with the previous results for the non-resistive MHD equations, in the local existence part, the range of p extends to $ 1\le p \le \infty $ 1 ≤ p ≤ ∞ from $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d , but the uniqueness of the solution requires $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d yet. [ABSTRACT FROM AUTHOR]

In this paper, we propose a sparse least squares (SLS) optimization model for solving multilinear equations, in which the sparsity constraint on the solutions can effectively reduce storage and computation costs. By employing variational properties of the sparsity set, along with differentiation properties of the objective function in the SLS model, the first-order optimality conditions are analysed in terms of the stationary points. Based on the equivalent characterization of the stationary points, we propose the Newton Hard-Threshold Pursuit (NHTP) algorithm and establish its locally quadratic convergence under some regularity conditions. Numerical experiments conducted on simulated datasets including cases of Completely Positive(CP)-tensors and symmetric strong M-tensors illustrate the efficiency of our proposed NHTP method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the iterative solutions to a class of generalized Sylvester quaternion tensor equations. From the Kronecker product and the properties of quaternion tensors, we propose the biconjugate A-orthogonal residual stabilized and the generalized product-type biconjugate A-orthogonal residual methods in their tensor forms for solving the original tensor equation. As an application, the proposed methods are tested with some colour video restoration problems. Numerical results show the effectiveness of our methods compared with some existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we shall investigate the large-time behavior of solutions to the Cauchy problem for the three-dimensional bipolar isentropic Euler–Poisson equations with time-dependent damping effects $ -\frac {\mu }{(1+t)^{\lambda }}n_iu_i(i=1, 2) $ − μ (1 + t) λ n i u i (i = 1 , 2) for $ -1 − 1 < λ < 1 , μ > 0. Here, we consider a more general case that the two pressure functions are different and the doping profile is non-zero. Under the assumption that the initial data are close to the constant equilibrium states, we show that the smooth solutions to the Cauchy problem exist uniquely and globally. The algebraic time-decay rates of the solutions toward the constant equilibrium states are also obtained. The main ingredient of the proof is the time-weighted energy method with artfully chosen time-weighted functions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the incompressible limit of the non-isentropic ideal magnetohydrodynamic equations with periodic boundary conditions and general initial data. Weak convergence results of the smooth solutions are established by filtering method in the torus. [ABSTRACT FROM AUTHOR]

This paper investigates explicit power series solutions and conservation laws of variable coefficient time fractional Kawahara equation with Riemann–Liouville derivative which describs oscillatory solitary wave propagation in dispersive media. The similarity reductions and explicit solutions are derived by using Lie symmetry method. Further, using Ibragimov's theorem conservation laws for this equation are discussed. [ABSTRACT FROM AUTHOR]

This paper studies the resolution of the max-min compositional fuzzy relation equation with the constraint of The solvability and the unique solvability of this constrained fuzzy relation equation are characterized. Furthermore, this paper presents a resolution of it and designs a corresponding tabular method. Finally, a numerical example is provided to illustrate the resolution procedure. [ABSTRACT FROM AUTHOR]

This paper examines serial production lines with unreliable non-identical parallel machines at each workstation and intermediate buffers with finite capacities. All machines are assumed to have exponential service times, times to failure and repair times. An efficient decomposition technique is introduced for the performance evaluation of such lines. Rather than replacing each parallel-machine workstation with an equivalent single-server workstation, the main contribution of this paper is the presentation of a direct approach to derive and apply decomposition equations directly for every parallel machine at each workstation. Experimental results indicate that such a method can provide a computationally efficient algorithm to analyse large serial unreliable multi-server production lines with a good accuracy compared against simulation and other available methods. [ABSTRACT FROM AUTHOR]

This paper is concerned with identification of a spatial source function from final time observation in a bi-parabolic equation, where the full source function is assumed to be a product of time dependent and a space dependent function. Due to the ill-posedness of the problem, recently some authors have employed different regularization method and analysed the convergence rates. But, to the best of our knowledge, the quasi-reversibility method is not explored yet, and thus we study that in this paper. As an important implication, the Hölder rates for the apriori and aposteriori error estimates obtained in this paper improve upon the rates obtained in earlier works. Also, in some cases we show that the rates obtained are of optimal order. Further, this work seems to be the first one that has broaden the applicability of the problem by allowing the time dependent component of the source function to change sign. To the best of our knowledge, the earlier known work assumed the fixed sign of the time dependent component by assuming some bounded below condition. [ABSTRACT FROM AUTHOR]

The stability and large-time behavior problem on the magneto-micropolar equations has evoked a considerable interest in recent years. In this paper, we study the stability and exponential decay near magnetic hydrostatic equilibrium to the two-dimensional magneto-micropolar equations with partial dissipation in the domain $ \Omega =\mathbb {T}\times \mathbb {R} $ Ω = T × R . In particular, we takes advantage of the geometry of the domain $ \mathbb {T}\times \mathbb {R} $ T × R to divide u into zeroth mode and the nonzero modes, and obey a strong version of the Poincaré's inequality, which plays a crucial role in controlling the nonlinearity. Moreover, we find that the oscillation part of the solution decays exponentially to zero. Finally, our result mathematically verifies that the stabilization effect of a background magnetic field on magneto-micropolar fluids. [ABSTRACT FROM AUTHOR]

In this paper, we studied the quantum dynamics of the TiC, NiC, and CuLi diatomic molecules interacting with the Varshni-Eckart potential (VEP) model in topological defects geometry associated with cosmic string under the influence of magnetic and Aharonov-Bohm (AB) flux fields. We employ the Greene-Aldrich approximation scheme in the centrifugal term and determine the approximate eigenvalue solution using the Nikiforov-Uvarov-Functional Analysis (NUFA) method. The numerical energy spectra for topological defects geometry and external fields with this potential are determined. Subsequently, we applied the eigenvalue solution to various cases of the potential model and obtained their analytical solutions. When compared to Minkowski flat space results, the presence of topological defects and external fields eliminates degeneracy and shifts the energy spectra of the diatomic molecules. [ABSTRACT FROM AUTHOR]

In this paper, the following conformable fractional sub-elliptic equation is investigated: \[ \begin{cases} -\Delta_H^{\alpha} u = f(\xi,u,\nabla_H^\alpha u), & {\rm in} \ \Omega,\\ u > 0, & {\rm in} \ \Omega,\\ u=0, & {\rm on} \ \partial \Omega. \end{cases} \] { − Δ H α u = f (ξ , u , ∇ H α u) , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω. Ω represents the bounded domain in space $ \mathbb {H}^n $ H n , $ \alpha \in (0,1) $ α ∈ (0 , 1) , and f>0 is a bounded function. This paper is organized into three sections. In the first section, we present some of the features of the Heisenberg Laplacian. To prove the solution to the above equation, we use the Arzelà–Ascoli theorem and Bolzano–Weierstrass theorem in the second section. We then prove the uniqueness and cylindrical symmetry of the solution in the third section. [ABSTRACT FROM AUTHOR]

In this paper, some exact solutions of the stochastic generalized nonlinear shallow water wave equation are investigated. This equation is important in fluid mechanics' fields since it can model the propagation of disturbances in water and other incompressible fluids. Opposite to what is usually considered in the literature, the two dispersion coefficients of the nonlinear terms are considered dependent random quantities as a more realistic case. The modified extended-tanh function (METF) method is combined with the random variable transformation (RVT) technique to get full probabilistic solutions of the problem via computing the probability density functions (PDFs) of the solution processes. Based on the probability density function, any statistical moment of the solution can be evaluated. Through two different applications for the input random variables (dispersion coefficients), my findings are applied efficiently. Finally, numerical results are presented graphically along the spatial dimension at a certain wave speed and time. The obtained results ratify that the proposed technique is efficient and powerful for obtaining analytical probabilistic solutions for the problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long-term behaviour of 3D double time-delayed magnetohydrodynamics (MHD)-Voight equations with non-autonomous forcing term. First, we establish the existence, forward compactness and forward stability of pullback attractors by the forward equi-continuity and forward pullback asymptotic compactness of solutions. We then study the upper semi-convergence of pullback attractors as the time parameter tends to positive infinity. Since the solution of this equation has no high regularity, we use the spectrum decomposition technique to prove the forward pointwise pullback asymptotic compactness of solutions. [ABSTRACT FROM AUTHOR]

This paper explores the development of dynamic modelling of macroeconomic fluctuations at the Cowles Commission from Roos, Dynamic Economics (Cowles Monograph No. 1, 1934) and Davis, Analysis of Economic Time Series (Cowles Monograph No. 6, 1941) to Koopmans, ed., Statistical Inference in Dynamic Economic Models (Cowles Monograph No. 10, 1950) and Klein's Economic Fluctuations in the United States, 1921–1941 (Cowles Monograph No. 11, 1950), emphasising the emergence of a distinctive Cowles Commission approach to structural modelling of macroeconomic fluctuations influenced by Cowles Commission work on structural estimation of simulation equations models, as advanced by Haavelmo ("A Probability Approach to Econometrics," Cowles Commission Paper No. 4, 1944) and in Cowles Monographs Nos. 10 and 14. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalized Riemann-type hydrodynamical equation and establish the existence of a weak solution to the equation in a lower order Sobolev space (Hs(ℝ))N with 1 < s ≤ 3/2 by using a modified pseudo-parabolic regularization method. Although the absence of some important conservation laws leads to the failure of global existence of the approximate solutions, it has been addressed by making good use of the structure of the system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the prediction of the helium concentrations as function of a spatially variable source term perturbed by fractional Brownian motion. For the direct problem, we show that it is well-posed and has a unique mild solution under some conditions. For the inverse problem, the uniqueness and the instability are given. In the meanwhile, we determine the statistical properties of the source from the expectation and covariance of the final-time data u (r , T). Finally, numerical implements are given to verify the effectiveness of the proposed reconstruction. [ABSTRACT FROM AUTHOR]

In present paper, we mainly consider the wave-breaking phenomena and persistence properties of a nonlinear dissipative Camassa–Holm equation. The precise blow-up criteria of the Cauchy problem are established and wave-breaking phenomenon is investigated based on the method of characteristics and the Riccati-type differential inequality. Due to the presence of high order nonlinear term, the conservation law E = ∫ R u 2 + u x 2 d x is losed. This difficulty has been addressed by establishing the energy inequality. Finally, we establish the persistence properties and some unique continuation properties of the solutions in the weighted L p -spaces L ϕ p (R) := L p (R , ϕ p d x). [ABSTRACT FROM AUTHOR]

The extended Fisher–Kolmogorov (EFK) system is a strong nonlinear fourth-order reaction diffusion evolution equation. So far, the numerical methods for solving this problem are only a few. This paper deals with the finite difference solution to the two-dimensional EFK equation. The scheme is proved to be uniquely solvable and second-order convergent in both time and space in maximum norm. Finally, some numerical examples have been illustrated to verify the efficiency and simplicity of the proposed technique. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces B p , r s with s > 1 + 1 p and s = 1 + 1 p , r = 1 , p ∈ [ 1 , ∞). Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, a semilinear fourth-order Petrovsky equation with nonlinear weak damping term and linear strong damping term is considered in the frame of the potential well. We prove the existence and uniqueness of the global strong solution with sub-critical initial energy. In addition, we obtain the global existence, asymptotic behaviour and finite time blowup for the weak (non-steady state) solution with a critical initial energy level. [ABSTRACT FROM AUTHOR]

The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function F D (N) (a ; b , c ; z) , depending on N complex variables (z 1 , ... , z N) =: z and having complex parameters (a 1 , ... , a N) =: a , b, c. For an arbitrary number N of variables, we have obtained explicit representations for such basis functions in the vicinity of points (0 , ... , 0) and (∞ , ... , ∞) in terms of the Horn type hypergeometric series in N variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons. [ABSTRACT FROM AUTHOR]

Developing suitable equivalent circuit models for multiple winding transformers provides great convenience in obtaining system equations. In general, circuits models are available for two-winding transformers. The paper proposes an equivalent circuit model based on ideal transformer representations consisting of dependent sources for multiple winding transformers. The use of equivalent circuit simplifies analysis and makes solutions more understandable. Since the model has a systematic structure, the increase in the number of windings does not affect the obtaining of the system equations. To validate the proposed equivalent circuit model, an application example, whose theoretical results can be easily seen, has been added to the study. [ABSTRACT FROM AUTHOR]

This paper focuses on the exponential stabilisation problem of inertial memristive neural networks (IMNNs) with unbounded discrete time-varying delays. The considered IMNNs are modelled by second-order derivatives equations by introducing the inertial terms. By using nonsmooth analysis, Lyapunov stability theory, inequality techniques and integral-differential of Lyapunov functional method, a feedback controller is designed to guarantee pth moment exponential stabilisation of the addressed IMNNs under the framework of Filippov solutions. It is worth noticing that the considered time delays of IMNNs can be unbounded. Finally, a numerical example is presented to illustrate the effectiveness of the main theoretical result. [ABSTRACT FROM AUTHOR]

This paper is devoted to a class of general nonlocal fourth-order elliptic equation Δ 2 u − (a + b ∫ R N | ∇ u | 2 d x) Δ u + V (x) u = f (x , u) , x ∈ R N , where Δ 2 = Δ (Δ) is the biharmonic operator, a > 0 , b ≥ 0 are constants and the potential V is allowed to be sign-changing. Under certain assumptions on f, infinitely many high energy solutions are established. Our results enrich that obtained in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in a class of critical nonlocal problems with variable exponents of the form: { M (T p (⋅ , ⋅) (u)) [ (− Δ) p (x , y) s u + | u | p ~ (x) − 2 u ] = λ f (x , u) + | u | q (x) − 2 u , in R N , u ∈ W s , p (⋅ , ⋅) (R N) , p ~ (x) = p (x , x) , where T p (⋅ , ⋅) (u) := ∬ R 2 N | u (x) − u (y) | p (x , y) p (x , y) | x − y | N + s p (x , y) d x d y + ∫ R N 1 p ~ (x) | u | p ~ (x) d x , M is the Kirchhoff function, λ is a real parameter and f is a continuous function. We also assume that { x ∈ R N : q (x) = p s ∗ (x) } ≠ ∅ , where p s ∗ (x) = N p ~ (x) / (N − s p ~ (x)) is the critical Sobolev exponent for variable exponents. The strategy of the proof for these results is to approach the problem variationally by using the mountain pass theorem and the concentration-compactness principles for fractional Sobolev spaces with variable exponents. In addition, we obtain the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases. [ABSTRACT FROM AUTHOR]