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We prove optimal radially weighted L-2-norm inequalities for the Fourier extension operator associated to the unit sphere in R-n. Such inequalities valid at all scales are well understood. The purpose of this short paper is to establish certain more delicate single-scale versions of these.

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given., MTM2017-89677-P, MINECO/ AEI/FEDER, UE.

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

In the present paper we consider a boundary homogenization problem for the Poisson’s equation in a bounded domain and with a part of the boundary conditions of highly oscillating type (alternating between homogeneous Neumman condition and a nonlinear Robin type condition involving a small parameter). Our main goal in this paper is to investigate the asymptotic behavior as ε → 0 of the solution to such a problem in the case when the length of the boundary part, on which the Robin condition is specified, and the coefficient, contained in this condition, take so-called critical values. We show that in this case the character of the nonlinearity changes in the limit problem. The boundary homogenization problems were investigate for example in [1, 2, 4]. For the first time the effect of the nonlinearity character change via homogenization was noted for the first time in [5]. In that paper an effective model was constructed for the boundary value problem for the Poisson’s equation in the bounded domain that is perforated by the balls of critical radius, when the space dimension equals to 3. In the last decade a lot of works appeared, e.g., [6–10], in which this effect was studied for different geometries of perforated domains and for different differential operators. We note that in [6–10] only perforations by balls were considered. In papers [11, 12] the case of domains perforated by an arbitrary shape sets in the critical case was studied.

To achieve numerical simulation of large deformation evolution processes in underground engineering, the barycentric interpolation test function is established in this paper based on the manifold cover idea. A large-deformation numerical simulation method is proposed by the double discrete method with the fixed Euler background mesh and moving material points, with discontinuous damage processes implemented by continuous simulation. The material particles are also the integration points. This method is called the manifold cover Lagrangian integral point method based on barycentric interpolation. The method uses the Euler mesh as the background integral mesh and describes the deformation behavior of macroscopic objects through the motion of particles between meshes. Therefore, this method can avoid the problem of computation termination caused by the distortion of the mesh in the calculation process. In addition, this method can keep material particles moving without limits in the set region, which makes it suitable for simulating large deformation and collapse problems in geotechnical engineering. Taking a typical slope as an example, the results of a slope slip surface obtained using the manifold cover Lagrangian integral point method based on barycentric interpolation proposed in this paper were basically consistent with the theoretical analytical method. Hence, the correctness of the method was verified. The method was then applied for simulating the collapse process of the side slope, thereby confirming the feasibility of the method for computing large deformations.

In this paper, our main objective is to establish certain new fractional integral by applying the Saigo hypergeometric fractional integral operators and by employing some integral transforms on the resulting formulas, we presented their image formulas involving the product of the generalized k -Mittag-Leffler function. Furthermore, We develop a new and further generalized form of the fractional kinetic equation involving the product of the generalized k -Mittag-Leffler function. The manifold generality of the generalized k -Mittag-Leffler function is discussed in terms of the solution of the fractional kinetic equation and their graphical interpretation is interpreted in the present paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably) new results. Keywords: k-Pochhammer symbol, k-gamma function, Generalized k-Mittag-Leffler function, Laplace transform, Fractional kinetic equations, MATLAB

The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT.

This paper is concerned with a non-conserved phase field system of Caginalp type in which the main operators are fractional versions of two fixed linear operators $A$ and $B$. The operators $A$ and $B$ are supposed to be densely defined, unbounded, self-adjoint, monotone in the Hilbert space $L^2(\Omega)$, for some bounded and smooth domain $\Omega$, and have compact resolvents. Our definition of the fractional powers of operators uses the approach via spectral theory. A nonlinearity of double-well type occurs in the phase equation and either a regular or logarithmic potential, as well as a non-differentiable potential involving an indicator function, is admitted in our approach. We show general well-posedness and regularity results, extending the corresponding results that are known for the non-fractional elliptic operators with zero Neumann conditions or other boundary conditions like Dirichlet or Robin ones. Then, we investigate the longtime behavior of the system, by fully characterizing every element of the $\omega$-limit as a stationary solution. In the final part of the paper we study the asymptotic behavior of the system as the parameter $\sigma$ appearing in the operator $B^{2\sigma}$ that plays in the phase equation decreasingly tends to zero. We can prove convergence to a phase relaxation problem at the limit, in which an additional term containing the projection of the phase variable on the kernel of $B$ appears., Comment: This paper is dedicated to Prof. Dr. Juergen Sprekels on the occasion of his 70th birthday, with best wishes from the authors. Key words: Fractional operators, Allen-Cahn equations, phase field system, well-posedness, regularity, asymptotics

This paper deals with the initial-value problem for the linearized equations of coupled sound and heat flow, in a bounded domain Ω in RN, with homogeneous Dirichlet boundary conditions. Existence and uniqueness of solutions to the problem are established by using the Hille-Yosida theorem. This paper gives a simpler proof than one by Carasso (1975). Moreover, regularity of solutions is established.

We establish the general form of a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$ which holds for arbitrary $n$ and $d$. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in a companion paper., Comment: 17 pages, v2: updated reference to companion paper

In this paper, we generalize the concept of gamma bridge in the sense that the length will be random, that is, the time to reach the given level is random. The main objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. We show that the gamma bridge with random length is a pure jump process and that its jumping times are countable and dense in the random interval bounded by 0 and the random length. Moreover, we prove that this process is a Markov process with respect to its completed natural filtration as well as with respect to the usual augmentation of this filtration, which leads us to conclude that its completed natural filtration is right continuous. Finally, we give its canonical decomposition with respect to the usual augmentation of its natural filtration.

This paper is a follow-up of a previous paper of the author published in Mathematics journal in 2015, which treats the so-called continuous fractional orthogonal derivative. In this paper, we treat the discrete case using the fractional orthogonal difference. The theory is illustrated with an application of a fractional differentiating filter. In particular, graphs are presented of the absolutel value of the modulus of the frequency response. These make clear that for a good insight into the behavior of a fractional differentiating filter, one has to look for the modulus of its frequency response in a log-log plot, rather than for plots in the time domain.

In this paper, we consider the final state problem for the nonlinear Schr\"odinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [10], the first and the second authors consider one- and two-dimensional cases and gave a sufficient condition on the nonlinearity for that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converges to a given asymptotic profile in a faster rate than in the lower dimensional cases. To obtain the necessary convergence rate, we employ the end-point Strichartz estimate and modify a time-dependent regularizing operator, introduced in [10]. Moreover, we present a candidate of the second asymptotic profile to the solution., Comment: 23 pages

In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of the relative volume inequality $(1.9)$ on conformally compact Einstein manifolds. Therefore we obtain the complete proof of the rigidity theorem for conformally compact Einstein manifolds in general dimensions with no spin structure assumption (cf. $[29, 15]$) as well as the new curvature pinch estimates for conformally compact Einstein manifolds with conformal infinities of very large Yamabe constant. We also derive the curvature estimates for conformally compact Einstein manifolds with conformal infinities of large Yamabe constant., Comment: 28 pages, 1 figure(with one sentence added)

This paper deals with the classic problem of the synthesis of planar linkages for path generation. Based on the Fourier theory, the task curve and the synthesized four-bar coupler curve are regarded as the same curve if their Fourier descriptors match. Using Fourier analysis, a curve must be given as a function of time, termed a parametrization. In practical applications, different parametrizations can be associated with the same task and coupler curve, respectively; however, these parametrizations are Fourier analyzed to different Fourier descriptors, thus resulting in the mismatch of the task and coupler curve. In this paper, we present a parametrization-invariant method to eliminate the influence of parametrization on the values of Fourier descriptors by unifying given parametrizations to the arc length parametrization; meanwhile, a new design space decoupling scheme is introduced to separate the shape, size, orientation, and location matching of the task and four-bar curve, which leads naturally to an efficient synthesis approach.

In this paper we investigate a thermoelastic theory obtained from the Taylor approximation for the heat flux vector proposed by Choudhuri. This new thermoelastic theory gives rise to interesting mathematical questions. We here prove a uniqueness theorem and instability of solutions under the relaxed assumption that the elasticity tensor can be negative. Later we consider the one-dimensional and homogeneous case and we prove the existence of solutions. We finish the paper by proving the slow decay of the solutions. That means that the solutions do not decay in a uniform exponential way. This last result is relevant if it is compared with other thermoelastic theories where the decay of solutions for the one-dimensional case is of exponential way.

In this paper, we study persistent piecewise linear multidimensional random motions. Their velocities, switching at Poisson times, are uniformly distributed on a sphere. The changes of direction are accompanied with subsequent jumps of random length and of uniformly distributed orientation. In this paper, we obtain some useful properties and formulae of distributions of these processes. In particular, we get these distributions in the cases of jumps with Gaussian and exponential distributions of jump magnitudes. © 2017 John Wiley and Sons, Ltd.

The aim of this paper is to extend the modified Fourier-Ritz approach to evaluate the free vibration of four-parameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary conditions. The first-order shear deformation theory is employed to formulate the theoretical model. In the modified Fourier-Ritz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solve the natural frequencies by means of the Ritz method. As one merit of this paper, the functionally graded cylindrical, conical, spherical shells are, respectively, regarded as a special functionally graded cylindrical, conical, spherical panels, and the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian. The excellent accuracy and reliability of the unified computational model are compared with the results found in the literatures.

We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.

27 pages. This is the third and final part of the series of 3 publications stemming from the preprint arXiv:1312.3161; International audience; In the third paper of the series we complete the proof of our main result: a description of the ergodic decomposition of infinite Pickrell measures. We first prove that the scaling limit of the determinantal measures corresponding to the radial parts of Pickrell measures is precisely the infinite Bessel process introduced in the first paper of the series. We prove that the `Gaussian parameter' for ergodic components vanishes almost surely. To do this, we associate a finite measure with each configuration and establish convergence to the scaling limit in the space of finite measures on the space of finite measures. We finally prove that the Pickrell measures corresponding to different values of the parameter are mutually singular.

One of the fundamental problems in communications is finding the energy distribution of signals in time and frequency domains. It should, therefore, be of great interest to find the most energy concentration hypercomplex signal. The present paper finds a new kind of hypercomplex signals whose energy concentration is maximal in both time and frequency under quaternionic Fourier transform. The new signals are a generalization of the prolate spheroidal wave functions (also known as Slepian functions) to quaternionic space, which are called quaternionic prolate spheroidal wave functions. The purpose of this paper is to present the definition and properties of the quaternionic prolate spheroidal wave functions and to show that they can reach the extreme case in energy concentration problem both from the theoretical and experimental description. In particular, these functions are shown as an effective method for bandlimited signals extrapolation problem., 36 pages, 4 figures

This paper introduces a family of transportation costs between non-negative measures. This family is used to obtain parabolic and reaction-diffusion equations with drift, subject to Dirichlet boundary condition, as the gradient flow of the entropy functional $\int_{\Omega}\rho\log\rho+V\rho+1\hspace{1mm}dx$. In 2010, Alessio Figalli and Nicola Gigli introduced a transportation cost that can be used to obtain parabolic equations with drift subject to Dirichlet boundary condition. However, the drift and the boundary condition are coupled in their work. The costs in this paper allow the drift and the boundary condition to be detached.

We show that the Teichmuller space of a surface without boundary and with punctures, equipped with the Thurston metric, is the limit in an appropriate sense of Teichmuller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the translation distances of mapping classes for their actions on Teichmuller spaces equipped with the Thurston metric. In this paper, we show that the arc metrics on the Teichmuller space of surfaces with boundary limit to the Thurston metric on the Teichmuller space of a surface without boundary, by making the boundary lengths tend to zero. We use this to prove a result on the translation distances for mapping classes. We introduce some notation before stating precisely the results. In all this paper, S = Sg,p,n is a connected orientable surface of finite type, of genus g with p punctures and n boundary components. We assume that S has negative Euler characteristic, i.e., χ(S) = 2 − 2g − p − n 0, we denote by ∂S the boundary of S. A hyperbolic structure on S is a complete metric of constant curvature −1 such that (i) each puncture has a neighborhood isometric to a cusp, i.e., to the quotient {z = x + iy ∈ H 2 | y > a}/hz 7→z + 1i

In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a natural continuation of our critical reading of A. Stern's results for Dirichlet eigenfunctions in the square, see arXiv:14026054., Comment: Accepted for publication in "Monatshefte f{\"u}r Mathematik"

This paper presents an analytical solution of the simply supported horizontally composite curved I-beam by trigonometric series considering the effect of partial interaction in the tangential direction. Governing equations and boundary conditions are obtained by using the Vlasov curved beam theory and the principle of minimum potential energy. The deflection functions and the Lagrange multiplier functions are expressed as trigonometric series to satisfy the governing equations and the simply supported constraints at both ends. The numerical results of deflections and forces which are obtained by this method are compared with both FEM results and experimental results, and the inaccuracy between the analytical solutions in this paper and the FEM results is small and reasonable.

Let $M$ be a Riemannian manifold and $\Omega$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $\Omega$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the normal derivative of the solution (heat flow) is a constant function on the boundary. We express this fact by saying that such domains have the constant flow property. In constant curvature spaces known examples of such domains are given by geodesic balls and, more generally, by domains whose boundary is connected and isoparametric. The question is: are they all like that? In this paper we give an affirmative answer to this question: in fact we prove more generally that, if a domain in an analytic Riemannian manifold has the constant flow property, then every component of its boundary is an isoparametric hypersurface. For space forms, we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the $r$-mean curvatures of the boundary and with the isoparametric property. Finally, we discuss the constant flow property in relation to other well-known overdetermined problems involving the Laplace operator, like the Serrin problem or the Schiffer problem., Comment: 41 pages

In this paper, we characterize the pointwise slant submersions from cosymplectic manifolds onto Riemannian manifolds and give several examples. In this paper, we characterize the pointwise slant submersions from cosymplectic manifolds onto Riemannian manifolds and give several examples.

The theorems on the estimates of solutions for nonlinear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for suitable implicit finite difference functional schemes are proved. The proofs are based on the com- parison technique. The convergent and stable difference method is considered without the assumption of the global generalized Perron condition posed on the functional variable but with the local one only. It is a consequence of our estimates theorems. In particular, these results cover quasi-linear equations. However, such equations are also treated separately. The functional dependence is of the Volterra type. The aim of the paper is to prove theorems on the estimates of solutions for non- linear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for generated by them implicit finite difference functional schemes. We also give the applications of the results. More precisely, we prove the theorem on the convergence of a difference method to a classical solution for the differential functional problem, which by the given estimates, may be treated in the subspace C (,R) ⊂ C (,R), where R ⊂ R is an interval. It is a new idea in area of nonlinear implicit difference methods which was studied for explicit methods by K. Kropielnicka and L. Sapa (14). This considerably extends the class of problems which are solvable by the described method. Therefore, the Lipschitz, Perron or generalized Perron conditions posed onf with respect toz need not be global, inC (,R), as in the papers due to M. Malec, Cz. Mączka, W. Voigt, M. Rosati and L. Sapa (15-19,24,25)

Conservation laws of the form ∂tu + ∂xf(x;u) = 0 with space-discontinuous flux f(x;⋅) = fl(⋅)1x + fr(⋅)1x>0 were deeply investigated in the past ten years, with a particular emphasis in the case where the fluxes are "bell-shaped". In this paper, we introduce and exploit the idea of transmission maps for the interface condition at the discontinuity, leading to the well-posedness for the Cauchy problem with general shape of fl,r. The design and the convergence of monotone Finite Volume schemes based on one-sided approximate Riemann solvers are then assessed. We conclude the paper by illustrating our approach by several examples coming from real-life applications.

It is well known that the standard Lipschitz space in Euclidean space, with exponent α ∈ (0, 1), can be characterized by means of the inequality , where is the Poisson integral of the function f. There are two cases: one can either assume that the functions in the space are bounded, or one can not make such an assumption. In the setting of the Ornstein–Uhlenbeck semigroup in ℝn, Gatto and Urbina defined a Lipschitz space by means of a similar inequality for the Ornstein–Uhlenbeck Poisson integral, considering bounded functions. In a preceding paper, the authors characterized that space by means of a Lipschitz-type continuity condition. The present paper defines a Lipschitz space in the same setting in a similar way, but now without the boundedness condition. Our main result says that this space can also be described by a continuity condition. The functions in this space turn out to have at most logarithmic growth at infinity.

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

This paper addresses the exponential synchronization problem of a class of master-slave distributed parameter systems (DPSs) with spatially variable coefficients and spatiotemporally variable nonlinear perturbation, modeled by a couple of semilinear parabolic partial differential equations (PDEs). With a locally Lipschitz constraint, the perturbation is a continuous function of time, space, and system state. Firstly, a sufficient condition for the robust exponential synchronization of the unforced semilinear master-slave PDE systems is investigated for all admissible nonlinear perturbations. Secondly, a robust distributed proportional-spatial derivative (P-sD) state feedback controller is desired such that the closed-loop master-slave PDE systems achieve exponential synchronization. Using Lyapunov’s direct method and the technique of integration by parts, the main results of this paper are presented in terms of spatial differential linear matrix inequalities (SDLMIs). Finally, two numerical examples are provided to show the effectiveness of the proposed methods applied to the robust exponential synchronization problem of master-slave PDE systems with nonlinear perturbation.

This paper presents a new method to give an analytical solution in Laplace domain directly that is used to describe pressure transient behavior of partially penetrating hydraulic fractures in a box-shaped reservoir with closed boundaries. The basic building block of the method is to solve diffusivity equation with the integration of Dirac function over the distance that is presented for the first time. Different from the traditional method of using the source solution and Green’s function presented by Gringarten and Ramey, this paper uses Laplace transform and Fourier transform to solve the diffusivity equation and the analytical solution obtained is accurate and simple. The effects of parameters including fracture height, fracture length, the position of the fracture, and reservoir width on the pressure and pressure derivative are fully investigated. The advantage of the analytical solution is easy to incorporate storage coefficient and skin factor. It can also reduce the amount of computation and compute efficiently and quickly.

In this paper, we describe generalized fractional integral operator and its inverse with generalized Bessel-Maitland function (BMF-Ⅴ) as its kernel. We discuss its convergence, boundedness, its relation with other well known fractional operators (Saigo fractional integral operator, Riemann-Liouville fractional operator), and establish its integral transform. Moreover, we have given the relationship of BMF-Ⅴ with Mittag-Leffler functions.

In this paper, the Poincare (or hyperbolic) metric and the associated distance are investigated for a plane domain based on the detailed properties of those for the particular domain Open image in new window In particular, another proof of a recent result of Gardiner and Lakic [7] is given with explicit constant. This and some other constants in this paper involve particular values of complete elliptic integrals and related special functions. A concrete estimate for the hyperbolic distance near a boundary point is also given, from which refinements of Littlewood’s theorem are derived.

Using the one-to-one correspondence between Kahler-Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics were established in my previous paper [19]. In the presented paper, we prove that there is a strict relation between the holomorphic Weyl and Bochner conformal curvature tensors and similarly their covariant derivatives are strictly related. Especially, we find necessary and sufficient conditions for the holomorphic Weyl conformal curvature tensor of a Kahler-Norden manifold to be holomorphically recurrent.

Whether, in the presence of a boundary, solutions of the Navier-Stokes equations converge to a solution to the Euler equations in the vanishing viscosity limit is unknown. In a seminal 1983 paper, Tosio Kato showed that the vanishing viscosity limit is equivalent to having sufficient control of the gradient of the Navier-Stokes velocity in a boundary layer of width proportional to the viscosity. In a 2008 paper, the present author showed that the vanishing viscosity limit is equivalent to the formation of a vortex sheet on the boundary. We present here several observations that follow on from these two papers. Compiled on Sunday 20 March 2016 1. Definitions and past results 3 2. A 3D version of vorticity accumulation on the boundary 6 3. L-norms of the vorticity blow up for p > 1 7 4. Improved convergence when vorticity bounded in L 8 5. Some kind of convergence always happens 9 6. Width of the boundary layer: 2D 10 7. Optimal convergence rate: 2D 12 8. A condition on the boundary equivalent to (V V ): 2D 14 9. Examples where condition on the boundary holds: 2D 17 10. On a result of Bardos and Titi: 2D 22 Appendix A. A Trace Lemma 23 Acknowledgements 25 References 26 The Navier-Stokes equations for a viscous incompressible fluid in a domain Ω ⊆ Rd, d ≥ 2, with no-slip boundary conditions can be written, (NS)  ∂tu+ u · ∇u+∇p = ν∆u+ f in Ω, div u = 0 in Ω, u = 0 on Γ := ∂Ω. Date: (compiled on Sunday 20 March 2016). 2010 Mathematics Subject Classification. Primary 76D05, 76B99, 76D10.

The concern of this paper is in expanding and computing initial-value problems of the form y ′= f ( y )+ h ω ( t ), where the function h ω oscillates rapidly for ω ≫1. Asymptotic expansions for such equations are well understood in the case of modulated Fourier oscillators and they can be used as an organizing principle for very accurate and affordable numerical solvers. However, there is no similar theory for more general oscillators, and there are sound reasons to believe that approximations of this kind are unsuitable in that setting. We follow in this paper an alternative route, demonstrating that, for a much more general family of oscillators, e.g. linear combinations of functions of the form e i ωg k ( t ) , it is possible to expand y ( t ) in a different manner. Each r th term in the expansion is for some ς >0 and it can be represented as an r -dimensional highly oscillatory integral. Because computation of multivariate highly oscillatory integrals is fairly well understood, this provides a powerful method for an effective discretization of a numerical solution for a large family of highly oscillatory ordinary differential equations.

Based on the moving least-squares (MLS) approximation, an improved interpolating moving least-squares (IIMLS) method based on nonsingular weight functions is presented in this paper. Then combining the IIMLS method and the Galerkin weak form, an improved interpolating element-free Galerkin (IIEFG) method is presented for two-dimensional potential problems. In the IIMLS method, the shape function of the IIMLS method satisfies the property of Kroneckerδfunction, and there is no difficulty caused by singularity of the weight function. Then in the IIEFG method presented in this paper, the essential boundary conditions are applied naturally and directly. Moreover, the number of unknown coefficients in the trial function of the IIMLS method is less than that of the MLS approximation; then under the same node distribution, the IIEFG method has higher computational precision than element-free Galerkin (EFG) method and interpolating element-free Galerkin (IEFG) method. Four selected numerical examples are presented to show the advantages of the IIMLS and IIEFG methods.

WOS: 000340737800002, In this paper, we write Seiberg-Witten-like equations on contact metric manifolds of dimension 5. Since any contact metric manifold has a Spin(e)-structure, we use the generalized Tanaka-Webster connection on a Spin(e) spinor bundle of a contact metric manifold to define the Dirac-type operators and write the Dirac equation. The self-duality of 2-forms needed for the curvature equation is defined by using the contact structure. These equations admit a nontrivial solution on 5-dimensional strictly pseudoconvex CR manifolds whose contact distribution has a negative constant scalar curvature., Scientific Research Foundation of Anadolu University [1105F099], This paper was supported by Project 1105F099 of the Scientific Research Foundation of Anadolu University.

Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces. As a corollary we state precise asymptotic formulas for the beta-invariant introduced around 2000 by R. Hain and D. Reed. These formulas are a refinement of a result Hain and Reed prove in their paper. We illustrate our results with some explicit calculations on degenerating genus two surfaces., 17 pages

This paper presents the nonlinear free vibration analysis of axisymmetric polar orthotropic circular membrane, based on the large deflection theory of membrane and the principle of virtual displacement. We have derived the governing equations of nonlinear free vibration of circular membrane and solved them by the Galerkin method and the Bessel function to obtain the generally exact formula of nonlinear vibration frequency of circular membrane with outer edges fixed. The formula could be degraded into the solution from small deflection vibration; thus, its correctness has been verified. Finally, the paper gives the computational examples and comparative analysis with the other solution. The frequency is enlarged with the increase of the initial displacement, and the larger the initial displacement is, the larger the effect on the frequency is, and vice versa. When the initial displacement approaches zero, the result is consistent with that obtained on the basis of the small deflection theory. Results obtained from this paper provide the accurate theory for the measurement of the pretension of polar orthotropic composite materials by frequency method and some theoretical basis for the research of the dynamic response of membrane structure.

In this paper, we study the long time decay of global solution to the 3D incompressible Navier-Stokes equations. We prove that if u ∈ C ( R + , X − 1 , σ ( R 3 ) ) u\in {\mathcal{C}}\left({{\mathbb{R}}}^{+},{{\mathcal{X}}}^{-1,\sigma }\left({{\mathbb{R}}}^{3})) is a global solution to the considered equation, where X i , σ ( R 3 ) {{\mathcal{X}}}^{i,\sigma }\left({{\mathbb{R}}}^{3}) is the Fourier-Lei-Lin space with parameters i = − 1 i=-1 and σ ≥ − 1 \sigma \ge -1 , then ‖ u ( t ) ‖ X − 1 , σ \Vert u\left(t){\Vert }_{{{\mathcal{X}}}^{-1,\sigma }} decays to zero as time goes to infinity. The used techniques are based on Fourier analysis.

We consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term $B$ and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in the class of global solutions. This paper extends our previous paper (Da Prato, Flandoli, Priola and M. Rockner, Annals of Prob., published online in 2012) which generalized Veretennikov's fundamental result to infinite dimensions assuming boundedness of the drift term. As in our previous paper pathwise uniqueness holds for a large class, but not for every initial condition. We also include an application of our result to prove existence of strong solutions when the drift $B$ is only measurable, locally bounded and grows more than linearly., The paper will be published in Journal of Theoretical Probability. arXiv admin note: text overlap with arXiv:1109.0363

In this paper we prove convergence results for the homogenization of the Dirichlet problem with rapidly oscillating boundary data in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as $L^p$ convergence results. For larger exponents $p$ we prove that the $L^p$ convergence rate is close to optimal. We shall also suggest several directions of possible generalization of the result in this paper.

This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is $H^s$ {with $s>\frac52$.} Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the {lifespan $T_\epsilon$} of the solutions is bounded below by $\log\log\log\frac1\epsilon$, where $\epsilon$ denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system, when the parameter $\epsilon$ goes to zero. In the second part of the paper we address the same problem but for the Besov critical regularity $B_{2,1}^{\frac52}$. This case turns out to be more subtle at least due to two facts. The first one is related to the Beale-Kato-Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate $L^1_TL^\infty$ for the acoustic parts $(\nabla\Delta^{-1}\diver\vepsilon,\cepsilon)$: it scales in the space variables like the space of the initial data., Comment: 47 pages

In this paper we prove an integral representation formula for the inverse Fueter mapping theorem for monogenic functions defined on axially symmetric open sets U ⊆ ℝ n+1, i.e. on open sets U invariant under the action of SO(n), where n is an odd number. Every monogenic function on such an open set U can be written as a series of axially monogenic functions of degree k, i.e. functions of type $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x) = \left[ {A\left( {x_{0,\rho } } \right) + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } {\rm B}\left( {x_{0,\rho } } \right)} \right]\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ , where A(x 0, ρ) and B(x 0, ρ) satisfy a suitable Vekua-type system and $$\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ is a homogeneous monogenic polynomial of degree k. The Fueter mapping theorem says that given a holomorphic function f of a paravector variable defined on U, then the function $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ given by $$\Delta ^{k + \tfrac{{n - 1}} {2}} \left( {f(x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )} \right) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ is a monogenic function. The aim of this paper is to invert the Fueter mapping theorem by determining a holomorphic function f of a paravector variable in terms of $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ . This result allows one to invert the Fueter mapping theorem for any monogenic function defined on an axially symmetric open set.

We continue our previous study of improved Hardy, Rellich and uncertainty principle inequalities on a Riemannian manifold M, started in our earlier paper from 2009. In the present paper we prove new weighted Hardy-Poincaré, Rellich type inequalities as well as an improved version of our uncertainty principle inequalities on a Riemannian manifold M. In particular, we obtain sharp constants for these inequalities on the hyperbolic space ?n. © 2013 American Mathematical Society.

In this paper, we consider an inverse boundary value problem for a mixed type partial differential equation with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The differential equation depends from another positive parameter in mixed derivatives. With respect to first variable this equation is a fractional-order nonhomogeneous differential equation in the positive part of the considering segment, and with respect to second variable is a second-order differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of direct and inverse boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. It is proved the stability of the solution with respect to redefinition functions, and with respect to parameter given in mixed derivatives. For irregular values of the spectral parameter, an infinite number of solutions in the form of a Fourier series are constructed.

In this paper we investigate bifurcation phenomena in a single-species reaction-diffusion model with spatiotemporal delay under the conditions of the weak and strong kernel functions. We have found that when the weak kernel function is introduced there is Hopf bifurcation but no Turing bifurcation and wave bifurcation to occur, but when the strong kernel function is introduced there exist Hopf bifurcation and wave bifurcation but no Turing bifurcation to occur. Especially, taking the inverse of the average time delay as a bifurcation parameter, we investigate influences of the time delay on the formation of spatiotemporal patterns through the numerical method. Some spatiotemporal patterns induced by Hopf bifurcation and wave bifurcation are respectively shown to illustrate the mechanism of the complexity of spatiotemporal dynamics.