Search

We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.

In this note we show that the main result of the paper (J. Optim. Theory Appl., published online on 10 September 2011) due to A. Capata and, consequently, all its particular cases are false.

The authors of [KMM] consider a system of two functional inequalities for a function $$f : {\mathbb{R}} \rightarrow {\mathbb{R}}$$ , and they show that, if certain arithmetical conditions and inequalities for certain parameters are fulfilled, f has to be a polynomial provided that f is continuous at some point x0. This result is derived here under the weaker condition that for some x0 the limit $${\rm lim}_{x \rightarrow x_0} f(x)$$ exists. Moreover, another system of inequalities is given leading to the same result on the nature of f. The methods used also give natural explanations for the fact that Bernoulli numbers play an important role in this context.

The extrapolation theorem is an essential and important theory in signal analysis. Since the offset linear canonical transform (OLCT) has proven to be an effective method in signal processing and optics, extrapolation theorems for continuous band-limited signals associated with the OLCT have been well studied. However, the extrapolation theorem for discrete signals in the OLCT domain remains unknown. In this short paper, by using the discrete generalized prolate spheroidal sequences (DGPSSs), the extrapolation theorem for discrete signals is presented. First, the definition of DGPSSs in the OLCT domain is proposed. Subsequently, the extrapolation theorem for OLCT discrete band-limited signals is obtained using the DGPSSs. In addition, a simplified computation of the extrapolation theorem for discrete band-limited signals in the OLCT domain is also obtained. Finally, simulation results are presented to show the utility and efficacy of the derived theorems.

This paper presents a review of the elementary theories on the bending of straight and curved beams, on plates and shells, using asymptotic approximations of the basic linearized equations of elasticity in three dimensions. The maximun norm has been chosen to specify the orders of magnitude of the quantities involved. The expansions are given as usual in terms of the small geometrical parameter characterizing the thinness of the structure. Most of the ideas and results are well known. Nevertheless, in the cases where more than one small parameter may be involved, such as small curvatures (shallow structures) or the small loading parameter used to linearize the equations of elasticity, the discussion on the limits of validity of the different theories lead to some interesting newer aspects. Moreover, the main ideas presented in this paper concerning multiple parameter expansions may be applied to discuss the behaviour of the structures and to obtain valuable analytical results in more complicated situations such as moderate and strong anisotropy, dynamic problems, stability etc.

There is a folklore about the equivalence between the Melnikov method and the averaging method for studying the number of limit cycles, which are bifurcated from the period annulus of planar analytic differential systems. But there is not a published proof. In this short paper, we prove that for any positive integer k, the kth Melnikov function and the kth averaging function, modulo both Melnikov and averaging functions of order less than k, produce the same number of limit cycles of planar analytic (or $$C^\infty $$ ) near-Hamiltonian systems.

The Domain Interface Method (DIM) is extended in this contribution for the case of mixed fields as encountered in multiphysics problems. The essence of the non-conforming domain decomposition technique consists in a discretization of a fictitious zero-thickness interface as in the original methodology and continuity of the solution fields across the domains is satisfied by incorporating the corresponding Lagrange Multipliers. The multifield DIM inherits the advantages of its irreducible version in the sense that the connections between non-matching meshes, with possible geometrically non-conforming interfaces, is accounted by the automatic Delaunay interface discretization without considering master and slave surfaces or intermediate surface projections as done in many established techniques, e.g. mortar methods. The multifield enhancement identifies the Lagrange multiplier field and incorporates its contribution in the weak variational form accounting for the corresponding consistent stabilization term based on a Nitsche method. This type of constraint enforcement circumvents the appearance of instabilities when the Ladyzhenskaya---Babuska---Brezzi (LBB) condition is not fulfilled by the chosen discretization. The domain decomposition framework is assessed in a large deformation setting for mixed displacement/pressure formulations and coupled thermomechanical problems. The continuity of the mixed field is studied in well selected benchmark problems for both mixed formulations and the objectivity of the response is compared to reference monolithic solutions. Results suggest that the presented strategy shows sufficient potential to be a valuable tool in situations where the evolving physics at particular domains require the use of different spatial discretizations or field interpolations.

In this short paper, We establish the weighted Logarithmic Sobolev inequalities for sub-Gaussian measure of high dimension with explicit constants via phase transition and the well-known Bakry-Emery criterion.

In this short paper, we will prove the following Theorem: Let \(\mathcal {F}\) be a codimension-one foliation of a complete and connected Riemannian manifold \(M^{n+1}\) with constant sectional curvature \(c {\leq} 0\). Suppose that \(\mathcal {F} \) is transversely orientable, i.e., there exists an unit vector field \(N \epsilon {\mathfrak{X}}(M)\) such that N is normal to the leaves of \(\mathcal {F} \). Suppose that N is geodesic. Then, if the foliation \(\mathcal {F} \) has an umbilical leaf, every leaf of foliation \(\mathcal {F} \) must be umbilical.

In this short paper, we introduce a new difference approximation for singular perturbation problem and prove the necessary condition of uniform convergence. Selecting apposite weight factor, we obtain the same difference schemes as in the case of Ilin's method.

A recent note of Ih-Ching Hsu poses an unsolved problem, to wit, the general solution of the functional equation g(x1, x2) + g(φ1(x1), φ2(x2)) = g(x1, φ2(x2)) + g(φ1(x1),x2), where the φi are given functions. This short paper obtains the general solution. It gives conditions which imply that anycontinuous solution has form g1(x1) + g2(x2).

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.

Fractional calculus has been widely used to study the flow of viscoelastic fluids recently, and fractional differential equations have attracted a lot of attention. However, the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena. In this paper, the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically. Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow. The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed. Based on the L1-approximation formula of Caputo fractional derivatives, the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented, and the numerical solutions are derived. In order to test the correctness and availability of numerical schemes, two numerical examples are established to give the exact solutions. The comparisons between the numerical solutions and the exact solutions have been made, and their high consistency indicates that the present numerical methods are effective. Then, this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions, and discusses the effects of the weight coefficient ϖ(α) in distributed order time fractional derivatives, the order α(r, t) in variable fractional order derivatives, the relaxation time λ, and the frequency ω of the periodic pressure gradient on the fluid flow velocity. Finally, the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.

Incorporating the arc-length constraint, the dynamic relaxation strategy has been widely used to trace full equilibrium path in the post-buckling analysis of structures. This combined numerical scheme has been shown to be successful for solving snap-through problems, but its applicability to snap-back problems has been rarely investigated and remains unclear. This paper proposes a direct and more general finite-difference equation to investigate the numerical stability of this combined numerical scheme, which is dominated by the spectral radius of amplification matrix. And a key discovery of this paper is that a first minor of the tangent stiffness matrix is always negative once snap back occurs. Due to this negative minor stiffness, the spectral radius is invariably greater than one, resulting in unconditional instability, which demonstrates the invalidity of dynamic relaxation arc-length method for snap-back problems. These important conclusions are corroborated by the numerical results of three representative examples in one-, two- and three-dimensional spaces.

In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.

This paper is devoted to study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. The properties of solutions, such as vacuum isolating phenomena, global existence, extinction, exponentially decay, exponentially growth, and finite time blow-up were studied by potential well method and energy estimate method. The results of this paper extend and complete the recent studies on this model.

The well posedness of the two-phase Stefan problem with convection is established in $$L^{1}$$ . First we consider the case with a singular enthalpy and we fix the convection velocity. In the second part of the paper we study the case of a smoothed enthalpy, but the convection velocity is the solution to a Navier-Stokes equation. In the last section we give some numerical illustrations of a physical case simulated using the models studied in the paper.

In order to construct global solutions to two-dimensional (2D for short) Riemann problems for nonlinear hyperbolic systems of conservation laws, it is important to study various types of wave interactions. This paper deals with two types of wave interactions for a 2D nonlinear wave system with a nonconvex equation of state: Rarefaction wave interaction and shock-rarefaction composite wave interaction. In order to construct solutions to these wave interactions, the authors consider two types of Goursat problems, including standard Goursat problem and discontinuous Goursat problem, for a 2D self-similar nonlinear wave system. Global classical solutions to these Goursat problems are obtained by the method of characteristics. The solutions constructed in the paper may be used as building blocks of solutions of 2D Riemann problems.

Running is the basic mode of fast locomotion for legged animals. One of the most successful mathematical descriptions of this gait is the so-called spring–mass model constructed upon an inverted elastic pendulum. In the description of the grounded phase of the step, an interesting boundary value problem arises where one has to determine the leg stiffness. In this paper, we find asymptotic expansions of the stiffness. These are conducted perturbatively: once with respect to small angles of attack, and once for large velocities. Our findings are in agreement with previous results and numerical simulations. In particular, we show that the leg stiffness is inversely proportional to the square of the attack angle for its small values, and proportional to the velocity for large speeds. We give exact asymptotic formulas to several orders and conclude the paper with a numerical verification.

This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if the Fokker–Planck–Kolmogorov equation has non-trivial Lie symmetry, then the boundary crossing identity exists and depends only on parameters of process and symmetry. For time-homogeneous diffusion processes we found the necessary and sufficient conditions of the symmetries’ existence. This paper shows that if a drift function satisfies one of a family of Riccati equations, then the problem has nontrivial Lie symmetries. For each case we present symmetries in explicit form. Based on obtained results, we derive two-parametric boundary crossing identities and prove its uniqueness. Further, we present boundary crossing identities between different process. We show, that if the problem has 6 or 4 group of symmetries then the first passage time density to any boundary can be explicitly represented in terms of the first passage time by a Brownian motion or a Bessel process. Many examples are presented to illustrate the method.

This paper deals with the water waves problem for uneven bottom under the influence of surface tension. We consider here an asymptotic limit for the Green–Naghdi equations in KdV scale, that is the Boussinesq system. The derivation of the KdV equation with uneven bottom under the influence of surface tension has been established. Indeed, this derivation is obtained in a formal way by using the Whitham technique, then the analytic solution to this equation has been obtained in case of flat bottom. However, in case of uneven bottom an $$H^s$$ -consistent solution has been obtained. Also, an $$H^s$$ -consistent solution for the Boussinesq system has been established, taking into consideration the influence of surface tension and uneven bottom. Finally, we confirmed the obtained theoretical results of this paper numerically, by devoting the last section to make a numerical validation. Moreover, analytic solutions to the KdV $$\sigma $$ and Boussinesq system were established in case of several bottom parametrization including the linear one.

In this paper, we propose a novel difference finite element (DFE) method based on the P1-element for the 3D heat equation on a 3D bounded domain. One of the novel ideas of this paper is to use the second-order backward difference formula (BDF) combining DFE method to overcome the computational complexity of conventional finite element (FE) method for the high-dimensional parabolic problem. First, we design a fully discrete difference FE solution ${u^{n}_{h}}$ by the second-order backward difference formula in the temporal t-direction, the center difference scheme in the spatial z-direction, and the P1-element on a almost-uniform mesh Jh in the spatial (x, y)-direction. Next, the H1-stability of ${u_{h}^{n}}$ and the second-order H1-convergence of the interpolation post-processing function on ${u_{h}^{n}}$ with respect to u(tn) are provided. Finally, numerical tests are presented to show the second-order H1-convergence results of the proposed DFE method for the heat equation in a 3D spatial domain.

The main motive of the paper is to look on rectifying and osculating curves on a smooth surface. In this paper we find the normal and geodesic curvature for a rectifying curve on a smooth surface and we also prove that geodesic curvature is invariant under the isometry of surfaces such that rectifying curves remain. We find a sufficient condition for which an osculating curve on a smooth surface remains invariant under isometry of surfaces and also we prove that the component of the position vector of an osculating curve α(s) on a smooth surface along any tangent vector to the surface at α(s) is invariant under such isometry.

In this paper, we consider the blow-up curve of semilinear wave equations. Merle and Zaag (Am J Math 134:581–648, 2012) considered the blow-up curve for $$\partial _t^2 u- \partial _x^2 u = |u|^{p-1}u$$ and showed that there is the case that the blow-up curve is not differentiable at some points when the initial value changes its sign. Their analysis depends on the variational structure of the problem. In this paper, we consider the blow-up curve for $$\partial _t^2 u- \partial _x^2 u = |\partial _t u|^{p-1}\partial _t u$$ which does not have the variational structure. Nevertheless, we prove that the blow-up curve is not differentiable if the initial data changes its sign and satisfies some conditions.

The paper deals with the identification of three-dimensional anisotropic diffusion properties of polymer-matrix composite materials with complex texture, based on the exploitation of short-time gravimetric tests. According to the Thermodynamics of Irreversible Processes, the diffusion behavior can be isotropic or orthotropic: for many materials, due to the complexity of the microscopic texture, the principal directions of orthotropy are not known a priori and enter the identification issue. After reviewing some identification methods (proper generalized decomposition) for isotropic and orthotropic material whose orthotropy directions are known, the paper proposes an experimental protocol and an identification algorithm for the full three-dimensional diffusion case, aiming at establishing the 3 coefficients of diffusion along the principal directions of orthotropy and the orientation of the orthotropic reference frame with respect to the sample frame. The identification of the physical properties is done through the minimization of a distance in the space of the physical parameters. The problem being non-convex, the numerical strategy used for the search of the global minimum is a particle swarm optimization, the code adaptive local evolution-particle swarm optimization with adaptive coefficients.

In this paper, we study the component configuration issue of the line-shaped wave networks which is made of two viscoelastic components and an elastic component and the viscoelastic parts produce the infinite memory and damping and distributed delay. The structural memory of viscoelastic component results in energy dissipative and the damping memory arouses the instability, and the elastic component is energy conservation, such a hybrid effects lead to complex dynamic behaviour of network. Our purpose of the present paper is to find out stability condition of such a network, in particular, the configuration condition of the wave network under which the network is exponentially stable. At first, using a resolvent family approach, we prove the well-posed of the wave network systems under suitable assumptions on the memory kernel $g(s)$ , the damping coefficient $\mu _{1}$ and delay distributed kernel $\mu _{2}(s)$ . Next, using the Lyapunov function method, we seek for a structural condition of the wave networks under which the wave networks are exponentially stable. By constructing new functions we obtain the sufficient conditions for the exponential stability of the wave networks, the structural conditions are given as inequalities.

A nonlinear parabolic equation from a two-phase problem is considered in this paper. The existence of weak solutions is proved by the standard parabolically regularized method. Different from the related papers, one of diffusion coefficients in the equation, $b(x)$ b ( x ) , is degenerate on the boundary. Then the Dirichlet boundary value condition may be overdetermined. In order to study the stability of weak solution, how to find a suitable partial boundary value condition is the foremost work. Once such a partial boundary value condition is found, the stability of weak solutions will naturally follow.

This paper extends approach of our recent paper together with Kahler to building special classes of exact solutions of the static Maxwell system in inhomogeneous isotropic media by means of different generalizations of the Cauchy–Riemann system with variable coefficients. A new class of three-dimensional solutions of the static Maxwell system in some special cylindrically layered media is obtained using class of exact solutions of the elliptic Euler–Poisson–Darboux equation in cylindrical coordinates. The principal invariants of the electric field gradient tensor within a wide range of meridional fields are described using a family of Vekua type systems in cylindrical coordinates. Analytic models of meridional electrostatic fields in accordance with different generalizations of the Cauchy–Riemann system with variable coefficients allow us to introduce the concept of $$\alpha $$ -meridional mappings of the first and second kind depending on the values of a real parameter $$\alpha $$ . In particular, in case $$\alpha =0$$ , geometric properties of harmonic meridional mappings of the second kind are demonstrated explicitly within meridional fields in homogeneous media.

In this paper, we consider singularly perturbed differential equations having delay on the convection and reaction terms. The considered problem exhibits boundary layer on the left or right side of the domain depending on the sign of the coefficient of the convection term. We approximated the terms containing the delay using Taylor series approximation. The resulting singularly perturbed boundary value problem is treated using exponentially fitted operator mid-point upwind finite difference method. The stability of the scheme is analysed and investigated using maximum principle and by constructing barrier functions for solution bound. We formulated the uniform converges of the scheme. The scheme converges uniformly with linear order of convergence. To validate the theoretical analysis and finding, we consider three examples exhibiting boundary layer on the left and right side of the domain. The obtained numerical result in this paper is accurate and parameter uniformly convergent.

The main purpose of this paper is to analyze the Hopf-like bifurcations in 3D piecewise linear systems. Such bifurcations are characterized by the birth of a piecewise smooth limit cycle that bifurcates from a singular point located at the discontinuity manifold. In particular, this paper concerns systems of the form $${\dot{x}}=Ax+b^{\pm }$$ which are ubiquitous in control theory. For this class of systems, we show the occurrence of two distinct types of Hopf-like bifurcations, each of which gives rise to a crossing limit cycle (CLC). Conditions on the system parameters for the coexistence of two CLCs and the occurrence of a saddle-node bifurcation of these CLCs are provided. Furthermore, the local asymptotic stability of the pseudo-equilibrium point is analyzed and applications in discontinuous control systems are presented.

This paper is devoted to the study of existence and uniqueness of a mild solution for a parabolic equation with conformable derivative. The nonlocal problem for parabolic equations appears in many various applications, such as physics, biology. The first part of this paper is to consider the well-posedness and regularity of the mild solution. The second one is to investigate the existence by using Banach fixed point theory.

This paper is devoted to consider the existence and bifurcation of subharmonic solutions of two types of 2n-dimensional nonlinear systems with time-dependent perturbations. When the unperturbed system is a Hamiltonian system, we obtain the extended Melnikov function by means of performing the curvilinear coordinate frame and constructing a Poincare map. Then some conditions of the bifurcation of subharmonic solutions are obtained. The results obtained in this paper contain and improve the existing results for $$n=2,3$$ . When the unperturbed system contains an isolated invariant torus, we investigate the bifurcation of subharmonic solutions by analyzing the Poincare map. We apply the extended Melnikov method to study the bifurcation and number of subharmonic solutions of the ice-covered suspension system. The maximum number of subharmonic solutions of this system is 2, and the relative parameter control condition is obtained.

In earlier studies of circular data, the corresponding probability distributions considered were mostly assumed to be symmetric. However, the assumption of symmetry may not be meaningful for some data. Thus there has been increased interest, more recently, in developing skewed circular distributions. In this article we introduce three skewed circular models based on inverse stereographic projection (ISP), originally introduced by Minh and Farnum (Comput. Stat.–Theory Methods, 32, 1–9, 2003), by considering three different versions of skewed-t distribution on real line considered in the literature, namely skewed-t by Azzalini (Scand. J. Stat., 12, 171–178, 1985), two-piece skewed-t, (seemingly first considered in Gibbons and Mylroie Appl. Phys. Lett., 22, 568–569, 1973 and later by Fernandez and Steel J. Amer. Statist. Assoc., 93, 359–371 1998) and skewed-t by Jones and Faddy (J. R. Stat. Soc. Ser. B (Stat. Methodol.), 65, 159–174, 2003). Unimodality and skewness of the resulting distributions are addressed in this paper. Further, real data sets are used to illustrate the application of the new models. It is found that under certain condition on the original scaling parameter, the resulting distributions may be unimodal. Furthermore, the study in this paper concludes that ISP circular distributions obtained from skewed distributions on the real line may provide an attractive alternative to other asymmetric unimodal circular distributions, especially when combined with a mixture of uniform circular distribution.

This paper shows the reliability of the symmetrical columns with eccentric loading about one and two axes due to the maximum intensity stress and minimum intensity stress. In this paper, a new lifetime distribution is introduced which is obtained by compounding exponential and gamma distributions (named as Lindley distribution). Hazard rates, mean time to failure and estimation of single parameter Lindley distribution by maximum likelihood estimator have been discussed. It is observed that when the load and the area of the cross section increase, failure of the column also increases at two intensity stresses. It is observed from the results that reliability decreases when scale parameter increases.

A family of planar nilpotent reversible systems with an equilibrium point located at the origin has been studied in the recent paper Algaba et al. (Nonlinear Dyn 87:835–849, 2017). The authors investigate the candidate for an universal unfolding of a codimension-three degenerate case which exhibits a rich bifurcation scenario. However, a codimension-two point is missed in one of the two cases considered. In this paper, we complete the bifurcation set demonstrating the existence of this new organizing center and analyzing the dynamics generated in this case. Moreover, by means of the Melnikov theory, we study analytically four different global connections present in the system under consideration. Numerical continuation of the bifurcation curves illustrates that the first-order analytical approximation is valid in a large region of the parameter space.

The waveguide considered coincides with a strip having two narrows of width e. An electron wave function satisfies the Dirichlet boundary value problem for the Helmholtz equation. The part of the waveguide between the narrows serves as a resonator, and conditions for the electron resonant tunneling may occur. In the paper, asymptotic formulas as e → 0 for characteristics of the resonant tunneling are used. The asymptotic results are compared with the numerical ones obtained by approximate calculation of the scattering matrix for energies in the interval between the second and third thresholds. The comparison allows us to state an interval of e, where the asymptotic and numerical approaches agree. The suggested methods can be applied to more complicated models than that considered in the paper. In particular, the same approach can be used for asymptotic and numerical analysis of the tunneling in three-dimensional quantum waveguides of variable cross-sections. Bibliography: 3 titles.

The biharmonic operator plays a central role in a wide array of physical models, such as elasticity theory and the streamfunction formulation of the Navier–Stokes equations. Its spectral theory has been extensively studied. In particular the one-dimensional case (over an interval) constitutes the basic model of a high order Sturm–Liouville problem. The need for corresponding numerical simulations has led to numerous works. The present paper relies on a discrete biharmonic calculus. The primary object of this calculus is a high-order compact discrete biharmonic operator (DBO). The DBO is constructed in terms of the discrete Hermitian derivative. However, the underlying reason for its accuracy remained unclear. This paper is a contribution in this direction, expounding the strong connection between cubic spline functions (on an interval) and the DBO. The first observation is that the (scaled) fourth-order distributional derivative of the cubic spline is identical to the action of the DBO on grid functions. It is shown that the kernel of the inverse of the discrete operator is (up to scaling) equal to the grid evaluation of the kernel of $$\left[ \left( \frac{d}{dx}\right) ^4\right] ^{-1}$$ , and explicit expressions are presented for both kernels. As an important application, the relation between the (infinite) set of eigenvalues of the fourth-order Sturm–Liouville problem and the finite set of eigenvalues of the discrete biharmonic operator is studied. The discrete eigenvalues are proved to converge (at an “optimal” $$O(h^4)$$ rate) to the continuous ones. Another consequence is the validity of a comparison principle. It is well known that there is no maximum principle for the fourth-order equation. However, a positivity result is derived, both for the continuous and the discrete biharmonic equation, showing that in both cases the kernels are order preserving.

The purpose of this paper is to investigate a more general rotation property of Gaussian processes on the function space $$C_{a,b}[0,T]$$ . The function space $$C_{a,b}[0,T]$$ can be induced by a generalized Brownian motion process. The Gaussian processes used in this paper are neither centered nor stationary.

In this paper the authors investigate a class of $p$ -Laplace equations with logarithmic nonlinearity, which were considered in Le and Le (Acta Appl. Math. 151:149–169, 2017), where, among other things, global existence and finite time blow-up of solutions were proved when the initial energy is subcritical and critical, that is, initial energy smaller than or equal to the depth of the potential well. Their results are complemented in this paper in the sense that an abstract criterion is given for the existence of global solutions that vanish at infinity or solutions that blow up in finite time, when the initial energy is supercritical. As a byproduct it is shown that the problem admits a finite time blow-up solution for arbitrarily high initial energy.

This paper extends a conventional, general framework for online adaptive estimation problems for systems governed by unknown nonlinear ordinary differential equations. The central feature of the theory introduced in this paper represents the unknown function as a member of a reproducing kernel Hilbert space (RKHS) and defines a distributed parameter system (DPS) that governs state estimates and estimates of the unknown function. This paper 1) derives sufficient conditions for the existence and stability of the infinite dimensional online estimation problem, 2) derives existence and stability of finite dimensional approximations of the infinite dimensional approximations, and 3) determines sufficient conditions for the convergence of finite dimensional approximations to the infinite dimensional online estimates. A new condition for persistency of excitation in a RKHS in terms of its evaluation functionals is introduced in the paper that enables proof of convergence of the finite dimensional approximations of the unknown function in the RKHS. This paper studies two particular choices of the RKHS, those that are generated by exponential functions and those that are generated by multiscale kernels defined from a multiresolution analysis., Comment: 24 pages, Submitted to CMAME

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.

The load distribution of simultaneously meshed gear teeth pairs has a significant influence in determining the relevant stress for the working capacity calculation of cylindrical gear pairs with respect to surface and volume strength. According to conventional calculation procedures, the influence of the boundary-ideally uniform load distribution, when all simultaneously meshed teeth pairs are equally engaged in the load transfer, on the teeth flanks stress state is taken into account by approximate expressions for contact ratio factor $$Z_{\varepsilon }$$ . In addition, this factor is not defined, no assumptions were given on the basis of which approximate expressions were determined, and no procedure for determining these expressions was disclosed. This paper defines a factor on the basis of which an analysis of the boundary-ideally uniform load distribution can be carried out. By observing the contact lines in the meshing area, when one and two or two and three meshed gear teeth pairs are alternating during the meshing period, a set of analytical expressions is developed to describe the boundary-ideally uniform load distribution of simultaneously meshed gear teeth pairs. Based on the developed analytical expressions, the limiting values of the gear teeth flank stresses can be determined more precisely and the degree of accuracy of the approximate expressions for the contact ratio factor which are given in conventional calculation procedures can be tested. The analysis carried out in this paper can be used for more accurate considerations of all cylindrical gear pairs occurrences which depend on load distribution, such as: load capacity, mesh stiffness, vibration, noise, energy efficiency, etc.

This paper is concerned with entire solutions of a two species time periodic bistable Lotka–Volterra competition-diffusion systems in $${\mathbb {R}}^N$$ . Here an entire solution refers to a solution that is defined for all time and in the whole space. It is known that “annihilating-front” type entire solutions have been obtained recently in $${\mathbb {R}}$$ . In the present paper, we first prove that there is a new type of entire solution which behaves as three time periodic moving planar traveling fronts as time goes to $$-\infty $$ and as a time periodic V-shaped traveling front as time goes to $$+\infty $$ in $${\mathbb {R}}^N$$ . Furthermore, we show that the propagating speed of such entire solutions coincides with the unique speed of the time periodic planar front, regardless of the shape of the level sets of the fronts.

The single particle reconstruction (SPR) in cryogenic electron microscopy is considered in this paper. This is an emerging technique for determining the three-dimensional (3D) structure of biological specimens from a limited number of the micrographs. Because the micrographs are modulated by contrast transfer functions and corrupted by heavy noise, the number of micrographs might be limited, in general it is a serious ill-posed problem to reconstruct the original particle. In this paper, we propose a constrained total variation (TV) model for single particle reconstruction. The TV norm is represented by the dual formulation that changes the SPR problem into a minimax one. The primal-dual method is applied to find the saddle point of the minimax problem, and the convergence condition is given. Numerical results show that the proposed model is very effective in reconstructing the particle.