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In 2002, F. Luca and P. G. Walsh studied the Diophantine equations of the form ( ak - 1)( bk - 1) = x2, for all ( a, b) in the range 2 ≤≤ b < a ≤≤ 100 with sixty-nine exceptions. In this paper, we solve two of the exceptions. In fact, we consider the equations of the form ( ak - 1)( bk - 1) = x2, with ( a, b) = (13, 4), (28, 13). [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (− Δ) s u + μ u = ( I α * F (u)) F ′ (u) in R N , (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ R N u 2 = m > 0 prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, "Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities," Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, "Nonlinear scalar field equations II: existence of infinitely many solutions," Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C1-regularity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the general dual fractional parabolic problem ∂ t α u (x , t) + L u (x , t) = f (t , u (x , t)) in R n × R. We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons' conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L and the Marchaud time derivative ∂ t α , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons' conjecture. It is worth noting that our results are novel even for a special case of L , the fractional Laplacian (−Δ)s, and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators. [ABSTRACT FROM AUTHOR]

This paper studies equable parallelograms whose vertices lie on the Eisenstein lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the set of these parallelograms forms naturally an infinite tree, all of whose vertices have degree 4, bar the root which has degree 3. This study naturally complements the authors' previous study of equable parallelograms whose vertices lie on the integer lattice. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the following nonlinear elliptic problem A (u) = ρ (u) | ∇ φ | 2 in Ω , div (ρ (u) ∇ φ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω. where A(u) = −div a(x, u, ∇u) is a Leray-Lions operator of order p. The second member of the first equation is only in L1(Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique. [ABSTRACT FROM AUTHOR]

The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine p-Laplace nonlocal operator Δ p 풜 , which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math. 386 2021, Article ID 107808] driven by the affine L p energy ℰ p , Ω from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine L p Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive C 1 solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ℰ p , Ω and by the comparison ℰ p , Ω ⁢ (u) ≤ ∥ u ∥ W 0 1 , p ⁢ (Ω) generally strict. [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

In this paper, we study a type of nonlinear hybrid Δ-difference equations of fractional-order. The main objective is to establish some stability criteria including the Ulam–Hyers stability, generalized Ulam–Hyers stability together with the Mittag-Leffler–Ulam–Hyers stability for the addressed problem. Prior to the stabilization processes, solvability criteria for the existence and uniqueness of solutions are considered. For this purpose, a hybrid fixed point theorem for triple operators and the Banach contraction mapping principle are applied, respectively. For the sake of illustrating the practical impact of the proposed theoretical criteria, we finish the paper with particular examples. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the parallel surfaces of the nonlightlike solution of vortex filament equations, namely, the parallel surfaces to Hasimoto surfaces in E 1 3 . We examine the relation between the geometric structures of Hasimoto surfaces and parallel surfaces to them. Moreover, we give concluding remarks about the parameter curves of parallel surface to Hasimoto surfaces. [ABSTRACT FROM AUTHOR]

Let (Pm)m≥0 be the sequence of Pell numbers given by P0 = 0, P1 = 1, and Pm+2 = 2Pm+1 + Pm for all m ≥ 0. In this paper, for an integer d ≥ 2 which is square free, we show that there is at most one value of the positive integer x participating in the Pell equation x2 − dy2 = ± 1, which is a product of two Pell numbers. [ABSTRACT FROM AUTHOR]

In this paper, the generalized inverses and the solution of the generalized inverse equation in a specific set are used to characterize the SEP element in a ring with involution. [ABSTRACT FROM AUTHOR]

In this paper, we apply a Landweber iterative regularization method to determine a space-dependent source for a time-space fractional diffusion-wave equation from the final measurement. In general, this problem is ill-posed, and a Landweber iterative regularization method is used to obtain the regularization solution. Under the a priori parameter choice rule and the a posteriori parameter choice rule, we give the error estimates between the regularization solution and the exact solution, respectively. Some numerical results in the one-dimensional and two-dimensional cases show the utility of the used regularization method. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and interior W 2 , p -regularity of the solution, and the regularity of the free boundary ∂ ⁡ { u > ϕ } to the obstacle problem of the porous medium equation, u t = Δ ⁢ u m ( m > 1 ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ ⁡ { u > ϕ } and ∂ ⁡ { u > 0 } , we consider two cases on the initial data which make the free boundary ∂ ⁡ { u > ϕ } separate from the free boundary ∂ ⁡ { u > 0 } . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 -regularity of the free boundary ∂ ⁡ { u > ϕ } is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator. [ABSTRACT FROM AUTHOR]

In this paper, we address a long-standing open problem in stochastic simulation: construction of a random walk on spheres (RWS) algorithm for solving a system of elasticity equations, known as the Lamé equation. Many attempts to generalize the classic probabilistic representations like the Kac formula for parabolic and scalar elliptic equations failed. A different approach based on a branching random walk on spheres (BRWS) introduced in our paper of 1995 [K. K. Sabelfeld and D. Talay, Integral formulation of the boundary value problems and the method of random walk on spheres, Monte Carlo Methods Appl. 1 1995, 1, 1–34] made little progress in solving this problem. In the present study, we further improve the BRWS algorithm by a special implementation of a branching anisotropic random walk on spheres process. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with positive solutions for the Dirichlet boundary value problem for equations and systems of Kirchhoff type. We obtain existence and localization results of positive solutions using Krasnosel'skiĭ's fixed point theorem in cones and a weak Harnack-type inequality. The localization is given in terms of energy norm, being of interest from a physical point of view. In the case of systems, the results on the localization are established componentwise using the vector version of Krasnosel'skiĭ's theorem, which allows some of the equations of the system to satisfy the compression condition and others the expansion one. [ABSTRACT FROM AUTHOR]

In this paper we consider the Navier–Stokes–Brinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spaces-based approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor are introduced as auxiliary unknowns, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation becomes a nonlinear perturbation of, in turn, a perturbed saddle point linear system, which is then rewritten as an equivalent fixed-point equation whose operator involved maps the velocity space into itself. The well-posedness of it is then analyzed by applying the classical Banach fixed point theorem, along with a smallness assumption on the data, the Babuška–Brezzi theory in Banach spaces, and a slight variant of a recently obtained solvability result for perturbed saddle point formulations in Banach spaces as well. The resulting Galerkin scheme is momentum-conservative. Its unique solvability is analyzed, under suitable hypotheses on the finite element subspaces, using a similar fixed-point strategy as in the continuous problem. A priori error estimates are rigorously derived, including also that for the pressure. We show that PEERS and AFW elements for the stress, the velocity, and the rotation, together with piecewise polynomials of a proper degree for the strain rate tensor, yield stable discrete schemes. Then, the approximation properties of these subspaces and the Céa estimate imply the respective rates of convergence. Finally, we include two and three dimensional numerical experiments that serve to corroborate the theoretical findings, and these tests illustrate the performance of the proposed mixed finite element methods. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the singular k-Hessian equations S k ⁢ (λ ⁢ (D 2 ⁢ u)) = h ⁢ (x) ⁢ f ⁢ (- u) + g ⁢ (| D ⁢ u |) in ⁢ Ω and S k ⁢ (λ ⁢ (D 2 ⁢ u)) = h ⁢ (x) ⁢ f ⁢ (- u) ⁢ (1 + g ⁢ (| D ⁢ u |) ) in ⁢ Ω with the Dirichlet boundary condition u = 0 on ∂ ⁡ Ω , where Ω ⊂ ℝ N ( N ≥ 2 ) is a strictly convex, bounded smooth domain. Using the method of upper and lower solutions and the Karamata regular variation theory, we get new criteria of the existence and asymptotic behavior of strictly convex solutions under different conditions imposed on h, f and g. This problem is more difficult to solve than the k-Hessian problem without gradient terms, and requires additional new conditions in the proof process. [ABSTRACT FROM AUTHOR]

In this paper, we consider Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. In part I, we establish Carleman estimates for the coupled quantitative thermoacoustic equations by assuming that the coefficients satisfy suitable conditions and taking the usual weight function φ ⁢ (x , t) = e λ ⁢ ψ ⁢ (x , t) , ψ ⁢ (x , t) = | x - x 0 | 2 - β ⁢ | t - t 0 | 2 + β ⁢ t 0 2 for x in a bounded domain in ℝ n with C 3 -boundary and t ∈ (0 , T) , where t 0 = T 2 . We will discuss applications of the Carleman estimates to some inverse problems for the coupled quantitative thermoacoustic equations in the succeeding part II paper [M. Cristofol, S. Li and Y. Shang, Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. Part II: Inverse problems, preprint 2020, https://hal.archives-ouvertes.fr/hal-02863385]. [ABSTRACT FROM AUTHOR]

Keywords: Equivalence; double-phase equation; viscosity solution; distributional solution; A H (-) {\mathcal{A}{H(\,{\cdot}\,)}}-harmonic function; comparison principle; 35J92; 35D40; 35C45; 46E30 EN Equivalence double-phase equation viscosity solution distributional solution A H (-) {\mathcal{A}{H(\,{\cdot}\,)}}-harmonic function comparison principle 35J92 35D40 35C45 46E30 811 829 19 10/03/22 20221001 NES 221001 1 Introduction Let be a bounded domain in811. Partial Differential Equations 57 (2018), no. 4, Paper No. 95. 43 V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Partial Differential Equations 56 (2017), no. 2, Paper No. 46. 10 I. Chlebicka and C. De Filippis, Removable sets in non-uniformly elliptic problems, Ann. [Extracted from the article]

In this paper, we focused on a one-dimensional elastic system with a double porosity structure and frictional damping acting on both porous equations. We introduce two stability numbers χ 0 {\chi_{0}} and χ 1 {\chi_{1}} and prove that the solution of the system decays exponentially provided that χ 0 = 0 {\chi_{0}=0} and χ 1 ≠ 0 {\chi_{1}\neq 0} . Otherwise, we prove the absence of exponential decay. Our results improve the results of [N. Bazarra, J. R. Fernández, M. C. Leseduarte, A. Magaña and R. Quintanilla, On the thermoelasticity with two porosities: Asymptotic behaviour, Math. Mech. Solids 24 2019, 9, 2713–2725] and [A. Nemsi and A. Fareh, Exponential decay of the solution of a double porous elastic system, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 83 2021, 1, 41–50]. [ABSTRACT FROM AUTHOR]

The study of thermoelasticity problems holds significant importance in the field of engineering. When analyzing non-Fourier thermoelastic problems, it was found that as the thermal relaxation time increases, the finite element solution will face convergence difficulties. Therefore, it is necessary to use alternative methods to solve. This paper proposes a physics-informed neural network (PINN) based on the DeepXDE deep learning library to analyze thermoelastic problems, including classical thermoelastic problems, thermoelastic coupling problems, and generalized thermoelastic problems. The loss function is constructed based on equations, initial conditions, and boundary conditions. Unlike traditional data-driven methods, this approach does not rely on known solutions. By comparing with analytical and finite element solutions, the applicability and accuracy of the deep learning method have been validated, providing new insights for the study of thermoelastic problems. [ABSTRACT FROM AUTHOR]

Generalized Abel equations have been employed in the recent literature to invert Radon transforms which arise in a number of important imaging applications, including Compton Scatter Tomography (CST), Ultrasound Reflection Tomography (URT), and X-ray CT. In this paper, we present novel injectivity results and inversion methods for generalized Abel operators. We apply our theory to a new Radon transform, R j , of interest in URT, which integrates a square integrable function of compact support, 푓, over ellipsoid and hyperboloid surfaces with centers on a plane. Using our newly established theory on generalized Abel equations, we show that R j is injective and provide an inversion method based on Neumann series. In addition, using algebraic methods, we present image phantom reconstructions from R j ⁢ f data with added pseudo-random noise. [ABSTRACT FROM AUTHOR]

Identification of physical properties of materials is very important because they are in general unknown. Furthermore, their direct experimental measurement could be costly and inaccurate. In such a situation, a cheap and efficient alternative is to mathematically formulate an inverse, but difficult, problem that can be solved, in general, numerically; the challenge being that the problem is, in general, nonlinear and ill-posed. In this paper, the reconstruction of a lower-order unknown time-dependent coefficient in a Cahn–Hilliard-type fourth-order equation from an additional integral observation, which has application to characterizing the nonlinear saturation of the collisional trapped-ion mode in a tokamak, is investigated. The local existence and uniqueness of the solution to such inverse problem is established by utilizing the Rothe method. Moreover, the continuous dependence of the unknown coefficient upon the measured data is derived. Next, the Tikhonov regularization method is applied to recover the unknown coefficient from noisy measurements. The stability estimate of the minimizer is derived by investigating an auxiliary linear fourth-order inverse source problem. Henceforth, the variational source condition can be verified. Then the convergence rate is obtained under such source condition. [ABSTRACT FROM AUTHOR]

In this paper, we study the analytic properties of solutions to the Vafa–Witten equation over a compact Kähler manifold. Simple obstructions to the existence of nontrivial solutions are identified. The gauge theoretical compactness for the C ∗ \mathbb{C}^{*} invariant locus of the moduli space is shown to behave similarly to the Hermitian Yang–Mills connections. More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions. When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers. This gives a simpler proof for Taubes’ results on rank two solutions over Kähler surfaces together with a new complex geometric interpretation. The moduli space of SU ⁢ ( 2 ) \mathsf{SU}(2) monopoles and some related examples are also discussed in the final section. [ABSTRACT FROM AUTHOR]

The main work of this paper is to construct the Wronskian solution and investigate the integrability characteristics of the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation. Firstly, the Wronskian technique is used to acquire a sufficient condition of the Wronskian solution. According to the Wronskian form, the soliton solution is obtained by selecting the elements in the determinant that satisfy the linear partial differential systems. Secondly, the bilinear Bäcklund transformation and Bell-polynomial-typed Bäcklund transformation are derived directly via the Hirota bilinear method and the Bell polynomial theory, respectively. Finally, Painlevé analysis proves that this equation possesses the Painlevé property, and a Painlevé-typed Bäcklund transformation is constructed to solve a family of exact solutions by selecting appropriate seed solution. It shows that the Wronskian technique, Bäcklund transformation, Bell polynomial and Painlevé analysis are applicable to obtain the exact solutions of the nonlinear evolution equations, e.g., soliton solution, single-wave solution and two-wave solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

The article studies quantitative relations (error identities) that characterize distances between exact solutions of nonlinear evolutionary problems and functions considered as approximations. The restrictions imposed on such a function are minimal and actually come down to the condition that it belongs to the same functional class as the generalized solution of the problem under consideration. Functional identities of this type reflect the most general relations between deviations from exact solutions of parabolic initial boundary value problems and those data that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class regardless of the method by which it was constructed. Therefore, they can serve as basic tools for deriving fully reliable a posteriori estimates of approximation errors as well as for analysis of modeling errors. The corresponding examples are discussed in the paper. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of modelling a system of aggregating particles, that are being transported with stationary velocities dependent on masses of the particles in one-dimensional case. A numerical method based on the ideas of POD (Proper Orthogonal Decomposition) is constructed, and its capacity to speed up the solution up to 40 times is demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the role of the damping term μ ⁢ u t in unique determination of unknown spatial load F ⁢ (x) in a damped Euler–Bernoulli beam equation u t ⁢ t + μ ⁢ u t + (r ⁢ (x) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ (x) ⁢ G ⁢ (t) from the measured final time displacement u T ⁢ (x) := u ⁢ (x , T) or velocity ν T ⁢ (x) := u t ⁢ (x , T) . In [A. Hasanov Hasanoğlu and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017] it was shown that the unique determination of the spatial load F ⁢ (x) in the undamped wave equation u t ⁢ t - (k ⁢ (x) ⁢ u x) x = F ⁢ (x) ⁢ G ⁢ (t) from final data (displacement or velocity) is not possible. This result is also valid for the undamped beam equation u t ⁢ t + (r ⁢ (x) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ (x) ⁢ G ⁢ (t) . We show that in the presence of the damping term, the spatial load can be uniquely determined from the final time output under some acceptable conditions with respect to the final time T > 0 and the damping coefficient μ > 0 . The cases of pure spatial load G ⁢ (t) ≡ 1 , and the exponentially decaying load G ⁢ (t) = e - η ⁢ t are analyzed separately, leading also to sufficient conditions for the uniqueness of the inverse problem solution. The results presented in this paper clearly demonstrate the key role of the damping term in the unique determination of the unknown spatial load F ⁢ (x) in the damped Euler–Bernoulli beam equation. [ABSTRACT FROM AUTHOR]

The paper deals with the existence of nontrivial solutions for (p , Q) equations in the Heisenberg group H n with critical exponential growth at infinity and a singular behavior at the origin. The main features and novelty of the paper are the above generality on the right-hand side of the equation, the (p , Q) growth of the elliptic operator and the fact that the equation is studied in the entire Heisenberg group. [ABSTRACT FROM AUTHOR]

In recent years, many papers have been devoted to the regularity of doubly nonlinear singular evolution equations. Many of the proofs are unnecessarily complicated, rely on superfluous assumptions or follow an inappropriate approximation procedure. This makes the theory unclear and quite chaotic to a nonspecialist. The aim of this paper is to fix all the misprints, to follow correct procedures, to exhibit, possibly, the shortest and most elegant proofs and to give a complete and self-contained overview of the theory. [ABSTRACT FROM AUTHOR]

The developers of the 2021 CKDEPI equation also introduced a race-free equation that combines creatinine and cystatin C, and preliminary data has demonstrated strong performance of this equation in a non-Western population [[15]]. In their Opinion Paper, Delanaye et al. argue that European laboratories should continue using the 2009 CKDEPI equation (without applying the race-correction factor) or switch to the recently developed European Kidney Function Consortium (EKFC) equation (which does not contain a race-correction factor) [[5]]. CI, confidence interval; CKDEPI, Chronic Kidney Disease Epidemiology Collaboration; eGFR, estimated glomerular filtration rate; EKFC, European Kidney Function Consortium. Keywords: Chronic Kidney Disease Epidemiology Collaboration (CKDEPI); creatinine; estimated glomerular filtration rate (eGFR); Europe; European Kidney Function Consortium (EKFC) EN Chronic Kidney Disease Epidemiology Collaboration (CKDEPI) creatinine estimated glomerular filtration rate (eGFR) Europe European Kidney Function Consortium (EKFC) e192 e195 4 08/22/23 20230901 NES 230901 To the Editor, The 2009 Chronic Kidney Disease Epidemiology Collaboration (CKDEPI) equation for estimated glomerular filtration rate (GFR) has been the standard equation in most European clinical settings for the last decade. [Extracted from the article]

In this paper, the perturbed Gerdjikov–Ivanov equation with spatio-temporal dispersion is investigated. The trial equation method, the complex envelope travelling wave transformation, and the complete discriminant system for polynomial method, respectively, are all utilized. As a consequence, twenty-eight exact chirped solutions are obtained, and some representative two-dimensional patterns under concrete parameters are presented. The results acquired are conducive to studying the dynamic behavior of solitons. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be a fixed integer. The k-generalized Lucas sequence { L n (k) } n ≥ 0 starts with the positive integer initial values k, 1, 3, ..., 2k−1 – 1, and each term afterward is the sum of the k consecutive preceding elements. An integer n is said to be close to a positive integer m if n satisfies | n − m | < m . In this paper, we combine these two concepts. We solve completely the diophantine inequality | L n (k) − 2 m | < 2 m / 2 in the non-negative integers k, n, and m. This problem is equivalent to the resolution of the equation L n (k) = 2 m + t with the condition |t| < 2m/2, t ∈ ℤ . We also discovered a new formula for L n (k) which was very useful in the investigation of one particular case of the problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the hypercritical deformed Hermitian-Yang–Mills equation on compact Kähler manifolds and resolve two conjectures of Collins–Yau [Moment maps, nonlinear PDE, and stability in mirror symmetry, preprint (2018), https://arxiv.org/abs/1811.04824]. [ABSTRACT FROM AUTHOR]

In the present paper, the authors rigorously derive Richards' multicontinuum model using the multicontinuum homogenization approach. This approach is based on formulating constraint cell problems and a homogenization-like expansion. We present numerical results for the two continua case with separable coefficients. First, we explore the relationships between the effective coefficients and the hydraulic conductivity. Then, we solve test problems with different contrasts to study the developed multicontinuum model. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of stochastic plate equations with memory in unbounded domains. More specifically, we obtain the uniform in time estimates for solutions of the problem. Based on the estimates above, we prove the existence and uniqueness of random attractors in unbounded domains. Finally, we show the upper semicontinuity of the attractors when stochastic perturbations approaches zero. [ABSTRACT FROM AUTHOR]

In this paper, the analytic solution of the delay fractional model is derived by the method of steps. The theoretical result implies that the regularity of the solution at s + is better than that at 0 + , where s is a constant time delay. The behavior of derivative discontinuity is also discussed. Then, improved regularity solution is obtained by the decomposition technique and a fitted L ⁢ 1 numerical scheme is designed for it. For the case of initial singularity, the optimal convergence order is reached on uniform meshes when α ∈ [ 2 3 , 1) , α is the order of fractional derivative. Furthermore, an improved fitted L ⁢ 1 method is proposed and the region of optimal convergence order is larger. For the case t > s , stability and min ⁡ { 2 ⁢ α , 1 } order convergence of the fitted L ⁢ 1 scheme are deduced. At last, the numerical tests are carried out and confirm the theoretical result. [ABSTRACT FROM AUTHOR]

The paper demonstrates two approaches to constructing monotonic difference schemes for the transfer equation in divergent form from the family of semi-Lagrangian methods: Eulerian–Lagrangian and Lagrangian–Eulerian. Within each approach, a monotonic conservative difference scheme is proposed. It is shown that within the framework of the Lagrangian–Eulerian approach, based on the use of curvilinear grids formed by the characteristics of the approximated transfer operator, it is possible to construct monotonic difference schemes of second order accuracy. [ABSTRACT FROM AUTHOR]

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black–Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]

This paper concerns the scalar field equation - Δ ⁢ u = λ ⁢ u + | u | q - 2 ⁢ u + a ⁢ | u | 4 ⁢ u + b ⁢ (I 2 ∗ | u | 5) ⁢ | u | 3 ⁢ u in ⁢ ℝ 3 under the normalized constraint ∫ ℝ 3 u 2 ⁢ 푑 x = c 2 , where a , b , c > 0 , 2 < q < 10 3 and I 2 is the Riesz potential. We prove that for small prescribed mass c the above equation has a positive ground state solution and an infinite sequence of normalized solutions with negative energies tending to zero. Asymptotic properties of ground state solutions as a → 0 + and as b → 0 + are also studied. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following fractional (N/s)-Laplacian Choquard equation: ε N ⁢ (- Δ) N / s s ⁢ u + V ⁢ (x) ⁢ | u | N s - 2 ⁢ u = ε μ ⁢ ( 1 | x | N - μ ∗ F ⁢ (u)) ⁢ f ⁢ (u) , x ∈ ℝ N , where (- Δ) N / s s denotes the (N/s)-Laplacian operator, 0 < μ < N , and V and f are continuous real functions satisfying some mild assumptions. Applying the weak growth conditions on the exponential critical nonlinearity f and without using the strictly monotone condition, we use some refined analysis and develop the arguments in the existing results to establish the existence of the ground state solution of the fractional (N/s)-Laplacian Choquard equation. Moreover, we also study the concentration phenomenon of the ground state solutions. As far as we know, our results seem to be new concerning the fractional (N/s)-Laplacian equation with the Choquard reaction. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence and finite time blow-up of solution for the Cauchy problem of one-dimensional fifth-order Boussinesq equation with logarithmic nonlinearity. Fist we prove the existence and uniqueness of local mild solutions in the energy space by means of the contraction mapping principle. Further under some restriction on the initial data, we establish the results on existence and uniqueness of global solutions and finite time blow-up of solutions by using the potential well method. Moreover, the sufficient and necessary conditions of global existence and finite time blow-up of solutions are given. [ABSTRACT FROM AUTHOR]

In this work, the dynamical behaviors of the Jimbo–Miwa equation that describes certain interesting (3 + 1)-dimensional waves in physics but does not pass any of the conventional integrability tests are studied. One-, two-, and three-M-lump waves are constructed successfully. Interactions between one-M-lump and one-soliton wave, between one-M-lump and two-soliton wave as well as between two-M-lump and one-soliton solution are reported. Also, complex multi-soliton, solutions are offered. The simplified Hirota's method and a long-wave method are used to construct these types of solutions. The velocity of a one-M-lump wave is studied. Straight Lines of travel for M-lump waves are also reported. To our knowledge, all gained solutions in this research paper are novel and not reported beforehand. Moreover, the gained solutions are presented graphically in three dimensions to better understand the physical phenomena of the suggested equation. [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 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 a time-explicit weak Galerkin finite element method is introduced and analyzed for parabolic equations. The main idea relies on the inclusion of a stabilization term in the temporal direction in addition to the usual static stabilization in the weak Galerkin framework. Both semi-discrete and fully-discrete schemes in time are presented, as well as their stability and error analysis. Numerical results are reported for this new explicit weak Galerkin finite element method. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate the reverse-space–time nonlocal nonlinear Fokas–Lenells equation by the idea of Ablowitz and Musslimani. The reverse-space–time Fokas–Lenells equation, associated with a 2 × 2 matrix Lax pair, is the important integrable system, which can be reduced to the nonlocal Fokas–Lenells equation. Based on its Lax pair, we construct nonlocal version of N-fold Darboux transformation (DT) for the Fokas–Lenells equation, and obtain two kinds of soliton solutions from vanishing and plane wave backgrounds. Further some novel one-soliton and two-soliton are derived with the zero and nonzero seed solutions through complex computations, including the bright soliton, kink soliton and breather wave soliton. Moreover, various graphical analyses on the presented solutions are made to reveal the dynamic behaviors, which display the elastic interactions between two solitons and their amplitudes keeping unchanged after the interactions except for the phase shifts. It is clearly shown that these solutions have new properties which differ from ones of the classical Fokas–Lenells equation. [ABSTRACT FROM AUTHOR]