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This paper mainly gives some sufficient and necessary conditions for an element in a ring to be EP , partial isometry and strongly EP by some equalities, using the solutions of certain equations and constructing the invertible elements in a ring. [ABSTRACT FROM AUTHOR]

In this paper, we study the Lipschitz stability in inverse source problems for degenerate/singular parabolic equations in the case of a boundary observation. First, we establish new global Carleman estimates, which improve that derived by Vancostenoble (2011). Then, following the general lines of the approach introduced by Imanuvilov and Yamamoto (1998), we prove the Lipschitz stability in the inverse source problems. The results obtained are natural extensions of many previous results for parabolic equations without/with degeneracy or singularity. [ABSTRACT FROM AUTHOR]

Let d(n) denotes the number of positive integers dividing the positive integer n, and let φ(n) denotes Euler's function representing the number of numbers less than and prime to n. In this paper, we present some notes on the equation d(n) = d(φ(n)). Several results on the related inequalities are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we construct stochastic simulation algorithms for solving an elastostatics problem governed by the Lamé equation. Two different stochastic simulation methods are suggested: (1) a method based on a random walk on spheres, which is iteratively applied to anisotropic diffusion equations that are related through the mixed second-order derivatives (this method is meshless and can be applied to boundary value problems for complicated domains); (2) a randomized algorithm for solving large systems of linear algebraic equations that is the core of this method. It needs a mesh formation, but even for very fine grids, the algorithm shows a high efficiency. Both methods are scalable and can be easily parallelized. [ABSTRACT FROM AUTHOR]

This paper extends the Realized-GARCH framework, by allowing the conditional variance equation to incorporate exogenous variables related to intra-day realized measures. The choice of these measures is motivated by the so-called heterogeneous auto-regressive (HAR) class of models. Our augmented model is found to outperform both the Realized-GARCH and the various HAR models in terms of in-sample fitting and out-of-sample forecasting accuracy. The new model specification is examined under alternative parametric density assumptions for the return innovations. Non-normality seems to be very important for filtering the return innovations to which variance responds and helps significantly upon the prediction performance of the suggested model. [ABSTRACT FROM AUTHOR]

In the present paper, we address the global solvability and finite-time attractivity of strong solutions for an abstract Rayleigh–Stokes-type equation involving nonlinear perturbations in Hilbert spaces. Based on regularity estimates of resolvent operator, local estimates on Hilbert scales and fixed point arguments, we obtain some results on global existence and finite-time attractivity of strong solutions to our problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the blowup, as well as global existence, and uniqueness of mild solutions for the initial-boundary value problem to a class of fractional extended Fisher–Kolmogorov equations with a general nonlinear term. We establish a general framework to find the global mild solutions for fractional extended Fisher–Kolmogorov equations with general nonlinear function, which will provide an effective way to deal with such problems. The results obtained in this paper can be considered as a contribution to this nascent field. [ABSTRACT FROM AUTHOR]

This paper relates to a recent trend in complex differential equations which studies solutions in a given domain. The classical settings in complex equations were widely studied for meromorphic solutions in the complex plane. For functions in the complex plane, we have a lot of results of general nature, in particular, the classical value distributions theory describing numbers of a-points. Many of these results do not work for functions in a given domain. A recent principle of derivatives permits us to study the numbers of Ahlfors simple islands for functions in a given domain; the islands play, to some extend, a role similar to that of the numbers of simple a-points. In this paper, we consider a large class of higher order differential equations admitting meromorphic solutions in a given domain. Applying the principle of derivatives, we get the upper bounds for the numbers of Ahlfors simple islands of similar solutions. [ABSTRACT FROM AUTHOR]

The relevance of the stated subject is conditioned upon the presence of a real possibility to simulate vibrations of elastic membranes in space according to the Hamilton principle using degenerate three-dimensional hyperbolic equations, which is of particular practical importance from the standpoint of the prospects for mathematical modelling of the heat propagation process in oscillating elastic membranes. The purpose of this paper is to study the sequence of the procedure for mathematical modelling of heat propagation in oscillating elastic membranes which is leading to degenerate three-dimensional hyperbolic equations. The methodological approach of this study is based on a combination of theoretical study of the possibilities of constructing mathematical models of heat propagation in oscillating elastic membranes with the practical application of methods for constructing three-dimensional hyperbolic equations with type and order degeneracy to find a single solution to a mixed problem. In the course of this study, the results were presented in the form of a mathematical proof of the possibility of obtaining a single solution to a mixed problem for three-dimensional hyperbolic equations with type and order degeneracy. The results obtained in this study and the conclusions formulated on their basis are of significant practical importance for developers of methods of mathematical modelling of heat propagation processes in oscillating artificial membranes, which is of key importance from the standpoint of prospects for improving methods of mathematical modelling of processes occurring in technical devices used in various fields of modern industries. [ABSTRACT FROM AUTHOR]

This paper presents the research data on the study of the basic equation and the length of the perfect hydraulic jump, which lasted for a total of more than 500 years. The perfect hydraulic jump is a unique natural phenomenon, which is expressed in a sharp increase in the depth of the flow from h < hcr to h > hcr (where hcr – critical depth) on a small stretch of land with the formation of a surface whirlpool. Having experimental data of conjugate depths, the length of the hydraulic jump was found according to known empirical formulas. This study applied the theoretical equation of Professor Abduramanov to determine the length of a perfect hydraulic jump. As a result, according to the comparison of experimental data, the equations of Professor Abduramanov turned out to be more suitable, giving the most accurate values. However, since the equation of the length of a perfect hydraulic jump is theoretical, the study suggests an empirical formula obtained experimentally based on this equation. [ABSTRACT FROM AUTHOR]

This paper demonstrates a novel form of substrate integrated waveguide SIW antenna array based on Fourier Series Equations. An optimized SIW power divider feeds the proposed structure where a micro-strip transition is used for impedance matching at 50 Ohm. The proposed antenna is traveling wave antenna type, in conventional methods the curve of antenna is of a linear or exponential form. In our method, we used the FSE to modify the contour of the antenna. The method is based on optimization of the Fourier series coefficients in order to maximize the gain for φ = π/2 and θ = π/2. The results of the return loss, the field's distribution, the radiation pattern and the realized gain were analysed. Furthermore, the antenna is designed in HFSS and CST with a comparison between them. The simulated return loss was compared with experimental measurements. The return loss of the proposed 1 × 4 antenna array is less than −10 dB at 5.5–6.8 GHz with an average gain about of 6 dBi and a peak gain about of 7.2 dBi. The proposed 1 × 4 antenna array is optimized with HFSS and designed using FR4 substrate, and considered in C-band for centimeters waves applications. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the stability and error estimates of the fully discrete schemes for parabolic equations, in which local discontinuous Galerkin methods with generalized alternating numerical fluxes and a novel spectral deferred correction method based on second-order time integration methods are adopted. With the energy techniques, we obtain both the second- and fourth-order spectral deferred correction time-marching with local discontinuous Galerkin spatial discretization are unconditional stable. The optimal error estimates for the corresponding fully discrete scheme are derived by the aid of the generalized Gauss–Radau projection. We extend the analysis to problems with higher even-order derivatives. Numerical examples are displayed to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of singular integrals which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation ∂ t ⁡ θ + u ⋅ ∇ ⁡ θ + κ ⁢ Λ 2 ⁢ β ⁢ θ = 0 , (x , t) ∈ ℝ 2 × ℝ + , κ > 0 , where u = - ∇ ⊥ ⁡ Λ - 2 + 2 ⁢ α ⁢ θ , α ∈ [ 0 , 1 2 ] and β ∈ (0 , 1 ] . First, we give a relationship between this kind of singular integrals and Calderón–Zygmund singular integral operators and obtain a uniform Besov estimates. As an application, we give the well-posedness of the generalized 2D dissipative quasi-geostrophic (QG) in the critical Besov space. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of weak solutions to the boundary value problem - Δ γ ⁢ u = f ⁢ (x , u) + g ⁢ (x , u) in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , where Ω is a bounded domain with smooth boundary in ℝ N (N ≥ 2) , f ⁢ (x , ξ) is odd in ξ, g ⁢ (x , ξ) is a perturbation term and Δ γ is a subelliptic operator of the type Δ γ := ∑ j = 1 N ∂ x j ⁡ (γ j 2 ⁢ ∂ x j ) , ∂ x j := ∂ ∂ ⁡ x j , γ := (γ 1 , γ 2 , ... , γ N) . By using the variant of Rabinowitz's perturbation method, under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special approach to the investigation of a mixed boundary value problem (BVP) for the Laplace equation in the case of a three-dimensional bounded domain Ω ⊂ ℝ 3 , when the boundary surface S = ∂ ⁡ Ω is divided into two disjoint parts S D and S N where the Dirichlet—Neumann-type boundary conditions are prescribed, respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W 2 1 ⁢ (Ω) . Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the L p -based Besov spaces, which under appropriate boundary data implies C α -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ¯ with α = 1 2 - ε , where ε > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the limit theory of stochastic McKean–Vlasov equations. First, we prove the optimal L p ( p ⩾ 2 ) strong convergence rate of the Wong–Zakai approximation for stochastic McKean–Vlasov equations. Then we show the support theorem for stochastic McKean–Vlasov equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the ground states for the following fractional Choquard equation with magnetic fields and critical exponents: (- Δ) A s ⁢ u + V ⁢ (x) ⁢ u = λ ⁢ f ⁢ (x , u) + [ | x | - α ∗ | u | 2 α , s * ] ⁢ | u | 2 α , s * - 2 ⁢ u in ⁢ ℝ N , where λ > 0 , α ∈ (0 , 2 ⁢ s) , N > 2 ⁢ s , u : ℝ N → ℂ is a complex-valued function, 2 α , s * = (2 ⁢ N - α) / (N - 2 ⁢ s) is the fractional Hardy–Littlewood–Sobolev critical exponent, V ∈ (ℝ N , ℝ) is an electric potential, V and f are asymptotically periodic in x, A ∈ (ℝ N , ℝ N) is a magnetic potential, and (- Δ) A s is a fractional magnetic Laplacian operator with s ∈ (0 , 1) . We prove that the equation has a ground state solution for large λ by using the Nehari method and the concentration-compactness principle. [ABSTRACT FROM AUTHOR]

In this paper, we construct virtual cycles on moduli spaces of solutions to the perturbed gauged Witten equation over a fixed smooth r-spin curve, under the framework of [G. Tian and G. Xu, Analysis of gauged Witten equation, J. reine angew. Math. 740 (2018), 187–274]. Together with the wall-crossing formula proved in the companion paper [G. Tian and G. Xu, A wall-crossing formula for the correlation function of gauged linear σ-model, preprint], this paper completes the construction of the correlation function for the gauged linear σ-model announced in [G. Tian and G. Xu, Correlation functions in gauged linear σ-model, Sci. China Math. 59 (2016), 823–838] as well as the proof of its invariance. [ABSTRACT FROM AUTHOR]

The paper is concerned with the reliable numerical solution of a class of nonlinear interface problems governed by the Poisson–Boltzmann equation. Arising in electrostatic biomolecular models these problems typically contain measure-type source terms and their solution often exposes drastically different behaviour in different subdomains. The interface conditions reflect the requirement that the potential and its normal derivative must be continuous. In the first part of the paper, we discuss an appropriate weak formulation of the problem that guarantees existence and uniqueness of the generalized solution. In the context of the considered class of nonlinear equations, this question is not trivial and requires additional analysis, which is based on a special splitting of the problem into simpler subproblems whose weak solutions can be defined in standard Sobolev spaces. This splitting also suggests a rational numerical solution strategy and a way of deriving fully guaranteed error bounds. These bounds (error majorants) are derived for each subproblem separately and, finally, yield a fully computable majorant of the difference between the exact solution of the original problem and any energy-type approximation of it. The efficiency of the suggested computational method is verified in a series of numerical tests related to real-life biophysical systems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new diophantine equation with prime numbers. Let [ ⋅ ] be the floor function. We prove that, when 1 < c < 23 21 and θ > 1 is fixed, then every sufficiently large positive integer 푁 can be represented in the form N = [ p 1 c ⁢ tan θ ⁡ (log ⁡ p 1) ] + [ p 2 c ⁢ tan θ ⁡ (log ⁡ p 2) ] + [ p 3 c ⁢ tan θ ⁡ (log ⁡ p 3) ] , where p 1 , p 2 , p 3 are prime numbers. We also establish an asymptotic formula for the number of such representations. [ABSTRACT FROM AUTHOR]

We consider a class of iterative methods based on block splitting (BBS) to solve absolute value equations A ⁢ x - | x | = b . Recently, several works were devoted to deriving sufficient conditions for the convergence of iterative methods of this type under certain assumptions including ν := ∥ A - 1 ∥ < 1 . However, the BBS-type iterative methods tend to converge slowly when 휈 is very close to one (i.e., ν ≈ 1 ). In this paper, using an auxiliary matrix, we develop a new approach by first rewriting the main problem into a new equivalent block system having shifted (1 , 1) -block and then constructing a fixed point iteration. The exploited strategy can significantly improve the convergence speed of the BBS-type iterative methods when ν ≈ 1 . Numerical experiments are reported to demonstrate the superiority of the new modified iterative scheme over the existing original form of BBS-type methods in the literature. [ABSTRACT FROM AUTHOR]

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The previous paper was concerned with systems of equations over a certain family 𝓢 of quasigroups. In that work a method of elimination of an outermost variable from the system of equations was suggested and it was shown that further elimination of variables requires that the family 𝓢 of quasigroups satisfy the generalized distributive law (GDL). In this paper we describe families 𝓢 that satisfy GDL. The results are applied to construct classes of easily solvable systems of equations. [ABSTRACT FROM AUTHOR]

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider control systems for which the underlying semigroup is analytic, and the resolvent of its generator is compact. In that case we give a characterization of the stabilizability of such control systems. When the stabilizability condition is satisfied the system is also stabilizable by finite-dimensional controls. We end the paper by giving an application of this result to the stabilizability of the Oseen equations with mixed boundary conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the inverse problem of determining the spatially dependent source in a time fractional diffusion-wave equation, with the aid of extra measurement data at a subboundary. A uniqueness result is obtained by using the analyticity and the newly established unique continuation principle provided that the coefficients are all temporally independent. We also derive a Lipschitz stability of our inverse source problem under a suitable topology whose norm is given via the adjoint system of the fractional diffusion-wave equation. [ABSTRACT FROM AUTHOR]

In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg–Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Let [ ⋅ ] be the floor function. In this paper, we prove that if 1 < c < 16559 15276 , then every sufficiently large positive integer N can be represented in the form N = [ p 1 c ] + [ p 2 c ] + [ p 3 c ] , where p 1 , p 2 , p 3 are primes, such that p 1 = x 2 + y 2 + 1 . [ABSTRACT FROM AUTHOR]

The prime intention of this paper is to study the conditions under which certain non-linear generalized delay-differential equations possess a solution. In this respect, by extending and improving recent results of [5, 15], we characterize the nature of solutions. By providing relevant examples in a remark, we also show that for the uniqueness of solutions the conditions on the Borel exceptional value can not be removed. Finally, to determine explicitly all forms of the solutions of a traditional generalized delay-differential equation, we deal with the situation under the aegis of generalized c-delay-differential equations. We exhibit some examples to show that the conclusions of the theorems actually occur. [ABSTRACT FROM AUTHOR]

In the paper, similarity solutions are constructed for a model equation with multiple characteristics of an arbitrary integer order. It is shown that the structure of similarity solutions depends on the mutual simplicity of the orders of derivatives with respect to the variable x and y, respectively. Frequent cases are considered in which they are shown as fundamental solutions of well-known equations, expressed in a linear way through the constructed similarity solutions. [ABSTRACT FROM AUTHOR]

In the present paper we utilize the Proper Orthogonal Decomposition (POD) method for model order reduction in application to Smoluchowski aggregation equations with source and sink terms. In particular, we show in practice that there exists a low-dimensional space allowing to approximate the solutions of aggregation equations. We also demonstrate that it is possible to model the aggregation process with the complexity depending only on dimension of such a space but not on the original problem size. In addition, we propose a method for reconstruction of the necessary space without solving of the full evolutionary problem, which can lead to significant acceleration of computations, examples of which are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of radially symmetric k-admissible solutions for the k-Hessian equation with 0-Dirichlet boundary condition { S k (D 2 u) = f (− u) in B , u = 0 on ∂ B , and the corresponding one-parameter problem, where B is a unit ball in ℝn with n ≥ 1, k ∈ {1,..., n}, f: [0, +∞) → [0, +∞) is continuous. We show that the k-admissible solutions are not convex, so we construct a new cone and obtain the existence of triple and arbitrarily many k-admissible solutions via the Leggett-Williams' fixed point theorem. [ABSTRACT FROM AUTHOR]

Let 퐺 be a discrete group. In 2001, Rosenblatt and Willis proved that 퐺 is amenable if and only if every possible system of configuration equations admits a normalized solution. In this paper, we show independently that 퐺 is locally finite if and only if every possible system of configuration equations admits a strictly positive solution. Also, we give a procedure to get equidecomposable subsets 퐴 and 퐵 of an infinite finitely generated or a locally finite group 퐺 such that A ⊊ B A\subsetneq B , directly from a system of configuration equations not having a strictly positive solution. [ABSTRACT FROM AUTHOR]

The paper is concerned with existence of nonnegative solutions of a Schrödinger–Choquard–Kirchhoff-type fractional p-equation. As a consequence, the results can be applied to the special case (a + b ⁢ ∥ u ∥ s p ⁢ (θ - 1) ) ⁢ [ (- Δ) p s ⁢ u + V ⁢ (x) ⁢ | u | p - 2 ⁢ u ] = λ ⁢ f ⁢ (x , u) + (∫ ℝ N | u | p μ , s * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | p μ , s * - 2 ⁢ u in ⁢ ℝ N , (a+b\|u\|_{s}^{p(\theta-1)})[(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u]=\lambda f(x,u)% +\Bigg{(}\int_{\mathbb{R}^{N}}\frac{|u|^{p_{\mu,s}^{*}}}{|x-y|^{\mu}}\,dy% \Biggr{)}|u|^{p_{\mu,s}^{*}-2}u\quad\text{in }\mathbb{R}^{N}, where ∥ u ∥ s = ( ∬ ℝ 2 ⁢ N | u ⁢ (x) - u ⁢ (y) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ℝ N V ⁢ (x) ⁢ | u | p ⁢ 𝑑 x ) 1 p , \|u\|_{s}=\Bigg{(}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}% \,dx\,dy+\int_{\mathbb{R}^{N}}V(x)|u|^{p}\,dx\Biggr{)}^{\frac{1}{p}}, a , b ∈ ℝ 0 + {a,b\in\mathbb{R}^{+}_{0}} , with a + b > 0 {a+b>0} , λ > 0 {\lambda>0} is a parameter, s ∈ (0 , 1) {s\in(0,1)} , N > p ⁢ s {N>ps} , θ ∈ [ 1 , N / (N - p ⁢ s)) {\theta\in[1,N/(N-ps))} , (- Δ) p s {(-\Delta)^{s}_{p}} is the fractional p-Laplacian, V : ℝ N → ℝ + {V:\mathbb{R}^{N}\rightarrow\mathbb{R}^{+}} is a potential function, 0 < μ < N {0<\mu

In this paper, we consider the solvability of the Pellian equation x 2 − (d 2 + 1) y 2 = − m , $$\begin{array}{} \displaystyle x^2-(d^2+1)y^2 = -m, \end{array} $$ in cases d = nk, m = n2l−1, where k, l are positive integers, n is a composite positive integer and d = pq, m = pq2, p, q are primes. We use the obtained results to prove results on the extendibility of some D(−1)-pairs to quadruples in the ring Z [ − t ] $\begin{array}{} {\mathbb{Z}}[\sqrt{-t}] \end{array} $ , with t > 0. [ABSTRACT FROM AUTHOR]

In the current paper, we consider a system of N-coupled weakly dissipative fractional nonlinear Schrödinger equations. The well-posedness of the initial value problem is established by a refined analysis based on a limiting argument as well as the study of the asymptotic dynamics of the solutions. This asymptotic behavior is described by the existence of a compact global attractor in the appropriate energy space. [ABSTRACT FROM AUTHOR]

In this paper we study the existence as well as uniform decay rates of the energy associated with the nonlinear damped Schrödinger equation, i ⁢ u t + Δ ⁢ u + | u | α ⁢ u - g ⁢ (u t) = 0 in ⁢ Ω × (0 , ∞) , iu_{t}+\Delta u+|u|^{\alpha}u-g(u_{t})=0\quad\text{in }\Omega\times(0,\infty), subject to Dirichlet boundary conditions, where Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , n ≤ 3 {n\leq 3} , is a bounded domain with smooth boundary ∂ ⁡ Ω = Γ {\partial\Omega=\Gamma} and α = 2 , 3 {\alpha=2,3}. Our goal is to consider a different approach than the one used in [B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z. 254 2006, 4, 729–749], so instead than using the properties of pseudo-differential operators introduced by cited authors, we consider a nonlinear damping, so that we remark that no growth assumptions on g ⁢ (z) {g(z)} are made near the origin. [ABSTRACT FROM AUTHOR]

In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space R n - 1 × R \mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems. [ABSTRACT FROM AUTHOR]

In this paper, based on the shift splitting technique, a shift splitting (SS) iteration method is presented to solve the generalized absolute value equations. Convergence conditions of the SS method are discussed in detail when the involved matrices are some special matrices. Finally, numerical experiments show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of reconstructing a space-dependent coefficient in a linear Benjamin–Bona–Mahony (BBM)-type equation from a single measurement of its solution at a given time. We analyze the well-posedness of the forward initial-boundary value problem and characterize the inverse problem as a constrained optimization one. Our objective consists on reconstructing the variable coefficient in the BBM equation by minimizing an appropriate regularized Tikhonov-type functional constrained by the BBM equation. The well-posedness of the forward problem is studied and approximated numerically by combining a finite-element strategy for spatial discretization using the Python-FEniCS package, together with a second-order implicit scheme for time stepping. The minimization process of the Tikhonov-regularization adopted is performed by using an iterative L-BFGS-B quasi-Newton algorithm as described for instance by Byrd et al. (1995) and Zhu et al. (1997). Numerical simulations are presented to demonstrate the robustness of the proposed method with noisy data. The local stability and uniqueness of the solution to the constrained optimization problem for a fixed value of the regularization parameter are also proved and illustrated numerically. [ABSTRACT FROM AUTHOR]

The stability of implicit semi-Lagrangian schemes for time-integration of the non-hydrostatic atmosphere dynamics equations is analyzed in the present paper. The main reason for the instability of the considered class of schemes is the semi-Lagrangian advection of stratified thermodynamic variables coupled to the fixed point iteration method used to solve the implicit in time upstream trajectory computation problem. We identify two types of unstable modes and obtain stability conditions in terms of the scheme parameters. Stabilization of sound modes requires the use of a pressure reference profile and time off-centering. Gravity waves are stable only for an even number of fixed point method iterations. The maximum time step is determined by inverse buoyancy frequency in the case when the reference profile of the potential temperature is not used. Generally, applying time off-centering and reference profile to pressure variable is necessary for stability. Using reference profile for potential temperature and an even number of the iterations allows one to significantly increase the maximum time-step value. [ABSTRACT FROM AUTHOR]

This paper tries to examine the recovery of the time-dependent implied volatility coefficient from market prices of options for the time fractional Black–Scholes equation (TFBSM) with double barriers option. We apply the linearization technique and transform the direct problem into an inverse source problem. Resultantly, we get a Volterra integral equation for the unknown linear functional, which is then solved by the regularization method. We use L1-forward difference implicit approximation for the forward problem. Numerical results using L1-forward difference implicit approximation (L1-FDIA) for the inverse problem are also discussed briefly. [ABSTRACT FROM AUTHOR]

Let F be the finite field of order q2. In this paper we continue the study in [24], [23], [22] of F-maximal curves defined by equations of type y n = x ℓ (x m + 1). ${{y}^{n}}={{x}^{\ell }}\left({{x}^{m}}+1 \right).$ New results are obtained via certain subcovers of the nonsingular model of v N = u t 2 − u ${{v}^{N}}={{u}^{{{t}^{2}}}}-u$ where q = tα, α ≥ 3 is odd and N = (tα + 1)/(t + 1). We observe that the case α = 3 is closely related to the Giulietti–Korchmáros curve. [ABSTRACT FROM AUTHOR]

Analytical solutions of both the nonlinear Schrödinger equation (NLSE) and NLSE with a perturbation term have been attained. Besides, analytical solutions of nonlocal NLSE have also been obtained. In this paper, the nonlocal NLSE with a perturbation term is discussed. By virtue of the dependent variable substitution, trilinear forms of this equation is attained. Lax pairs and Darboux transformation of this equation are obtained. Via the Darboux transformation, two kinds solutions of this equation with the different seed solutions are attained. [ABSTRACT FROM AUTHOR]

In this paper, we propose the concept of a weighted pseudo δ-almost automorphic function under the matched space for time scales and we present some properties. Also, we obtain sufficient conditions for the existence of weighted pseudo δ-almost automorphic mild solutions to a class of semilinear dynamic equations under the matched spaces for time scales. [ABSTRACT FROM AUTHOR]

Prescribed mean curvature problems on the torus have been considered in one dimension. In this paper, we prove the existence of a graph on the n-dimensional torus 𝕋 n {\mathbb{T}^{n}} , the mean curvature vector of which equals the normal component of a given vector field satisfying suitable conditions for a Sobolev norm, the integrated value, and monotonicity. [ABSTRACT FROM AUTHOR]

In this paper, the inverse scattering approach is applied to the Kundu-Eckhaus equation with two cases of zero boundary condition (ZBC) and nonzero boundary conditions (NZBCs) at infinity. Firstly, we obtain the exact formulae of soliton solutions of three cases of N simple poles, one higher-order pole and multiple higher-order poles via the associated Riemann-Hilbert problem (RHP). Moreover, given the initial data that allow for the presence of discrete spectrum, the higher-order rogue waves of the equation are presented. For the case of NZBCs, we can construct the infinite order rogue waves through developing a suitable RHP. Finally, by choosing different parameters, we aim to show some prominent characteristics of this solution and express them graphically in detail. Our results should be helpful to further explore and enrich the related nonlinear wave phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we study two inverse problems for stochastic parabolic equations with additive noise. One is to determinate the history of a stochastic heat process and the random heat source simultaneously by the observation at the final time 𝑇. For this inverse problem, we obtain a conditional stability result. The other one is an inverse source problem to determine two kinds of sources simultaneously by the observation at the final time and on the lateral boundary. The main tool for solving the inverse problems is a new global Carleman estimate for the stochastic parabolic equation. [ABSTRACT FROM AUTHOR]

In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in ℝn, whose level set {u = 0} is contained in a half-space. If n ≤ 3, we prove that the solution must be one-dimensional. In dimension n ≥ 4, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study following Kirchhoff type equation: {− (a + b ∫Ω|∇u|2dx)Δu = ƒ(u) + h in Ω, u = 0 on ∂Ω. We consider first the case that Ω ⊂ ℝ3 is a bounded domain. Existence of at least one or two positive solutions for above equation is obtained by using the monotonicity trick. Nonexistence criterion is also established by virtue of the corresponding Pohožaev identity. In particular, we show nonexistence properties for the 3-sublinear case as well as the critical case. Under general assumption on the nonlinearity, existence result is also established for the whole space case that Ω = ℝ3 by using property of the Pohožaev identity and some delicate analysis. [ABSTRACT FROM AUTHOR]