conditions imposed by tacnodes and cusps the study of linear systems of algebraic plane curves with fixed imposed singularities is a classical subject which has recently experienced important progress the horace method introduced by a hirschowitz has been successfully exploited to prove many h1 vanishing theorems even in higher dimension other specialization techniques which include degenerations of the plane are due to z ran and c ciliberto and r miranda g m greuel c lossen and e shustin use a local specialization procedure together with the horace method to give the first asymptotically proper general existence criterion for singular curves of low degree in this paper we develop a specialization method which allows us to compute the dimension of several linear systems as well as to substantially improve the bounds given by greuel lossen and shustin for curves with tacnodes and cusps

on the vanishing viscosity limit for the 3d navier stokes equations with a slip boundary condition where and below and denote the div and curl operators respectively n is the outward normal and is the unit tangential vector of the investigation of vanishing viscosity limit of solutions of the navier stokes equations both in the two and three spacial dimensional cases is a classical issue there are two related questions arising from here one is how to describe the inviscid limiting behavior of the navier stokes equation and the other is that does the euler equation can be approximated by the navier stokes equations in the case that the solution to the this research is supported in part by zheng ge ru foundation and hong kong rgc earmarked research grants cuhk 4028 04p cuhk 4040 02p and cuhk 4279 00p

on a selfadjoint realization of curl in exterior domains the first order vectorial operator curl occurs in many areas of applications e g in electrodynamics fluid mechanics hydrodynamics magnetohydrodynamics and elasticity theory it is the purpose of this note to discuss a particular instance of curl where a domain can be given to curl so that it becomes a selfadjoint operator the particular domain of selfadjointness under consideration is singled out by its relevance in applications to establish this operator we initially consider curl as an operator curl c read restricted to in l2 defined by curl c c l2 l2 0 1

helicity is the only integral invariant of volume preserving transformations we prove that any regular integral invariant of volume preserving transformations is equivalent to the helicity specifically given a functional i defined on exact divergence free vector fields of class c 1 on a compact 3 manifold that is associated with a well behaved integral kernel we prove that i is invariant under arbitrary volume preserving diffeomorphisms if and only if it is a function of the helicity

finite element methods for maxwell s transmission eigenvalues the transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory and more importantly are parts of inverse algorithms for estimating material properties in this paper we propose two finite element methods to compute a few lowest maxwell s transmission eigenvalues which are of interest in applications since the discrete matrix eigenvalue problem is large sparse and in particular non hermitian due to the fact that the problem is neither elliptic nor self adjoint we devise an adaptive method which combines the arnoldi iteration and estimation of transmission eigenvalues exact transmission eigenvalues for balls are derived and used as a benchmark numerical examples are provided to show the viability of the proposed methods and to test the accuracy of recently derived inequalities for transmission eigenvalues

existence of knotted vortex tubes in steady euler flows we prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible euler equation in mathbb r 3 more precisely given a finite collection of possibly linked and knotted disjoint thin tubes in mathbb r 3 we show that they can be transformed with a c m small diffeomorphism into a set of vortex tubes of a beltrami field that tends to zero at infinity the structure of the vortex lines in the tubes is extremely rich presenting a positive measure set of invariant tori and infinitely many periodic vortex lines the problem of the existence of steady knotted thin vortex tubes can be traced back to lord kelvin

a priori estimates for a singular limit approximation of the constitutive laws for chiral media in the time domain abstract the equations for the evolution of electromagnetic fields in chiral media in the time domain are nonlocal in time in this work we study the validity of a singular limit local in time approximation for these nonlocal in time equations by estimating the size of the difference of the fields as predicted by both models in particular we establish an a priori estimate for this difference depending on the time horizon properties of the domain spatial properties of the initial data and the source terms and the chirality measure of the approximating model

non conforming finite element methods for transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory in this paper for the helmholtz transmission eigenvalue problem we give a weak formulation which is a nonselfadjoint linear eigenvalue problem based on the weak formulation we first discuss the non conforming finite element approximation and prove the error estimates of the discrete eigenvalues obtained by the adini element the morley zienkiewicz element the modified zienkiewicz element et al and we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem

mixed methods for the helmholtz transmission eigenvalues the ciarlet raviart mixed finite element method is popular for the biharmonic equations in this paper we apply this method to the helmholtz transmission eigenvalue problem which is quadratic and

maximal rank for planar singularities of multiplicity 2 abstract we prove that general unions of singularity schemes of multiplicity two in the projective plane have maximal rank

a c0ipg method and its error estimates for the helmholtz transmission eigenvalue problem abstract the interior penalty methods using c 0 lagrange elements c 0 ipg developed in the last decade for the fourth order problems are an interesting topic in academia at present in this paper we apply the methods to the helmholtz transmission eigenvalue problem which is quadratic and non self adjoint and has important applications in the inverse scattering theory we propose a discrete scheme present a complete error analysis and first prove the c 0 ipg method can capture smooth eigenpairs efficiently using the argument in this paper we can prove that the c 0 ipg methods for the fourth order eigenvalue problems in the existing literature can also capture smooth eigenpairs efficiently numerical examples confirm our theoretical analysis

on the representation of inhomogeneous linear force free fields it is shown that there is a false assumption hidden in the description of a relaxed state with inhomogeneous boundary conditions as the vector sum of a potential field satisfying the boundary conditions and a sum of eigenfunctions of the associated eigenvalue problem expanded by certain coefficients in particular although the jensen and chu formula 1984 can provide the correct expansion coefficients it contains an implicit paradox in its derivation according to a general vector theorem the same paradox led chu et al 1999 to be concerned about a contradiction obtained by taking the curl of their magnetic field expansion which if permitted becomes inconsistent with a current normal to the surface the assumption that the curl can be commuted across an infinite sum of terms is the mechanism leading to these apparently paradoxical conclusions two mechanisms for resolving this apparent paradox are possible one of which will be described in some detail below and the other discussed further in a forthcoming more theoretical paper laurence et al 2000 the decomposition of the magnetic field above is valid with convergence in the mean squared sense but a decomposition of the current needs to be reinterpreted in terms of negative sobolev spaces to avoid this and remain in a more easily managable and familiar setting we derive the expansion coefficients in a way that involves the commuting of the inverse curl as opposed to the curl and the series the resulting series converges in a mean square sense when this is done the calculation can conform to the general vector theorem and a new gauge invariant expression for the coefficients is obtained however the consequence of the non commutability is nullified in the jensen and chu formula in both simply and multiply connected domains by the important extra requirement of a boundary condition on the vector potential eigenfunctions this excludes magnetic field eigenfunctions that carry flux but there remains a complete set for the expansion and all flux is carried by the potential field the two formulas are then identical on a different issue it is shown that if the general expansion is taken over a half space by combining positive and negative eigenvalue terms then the coefficients are anisotropic that is they are tensors except when evaluated at the first eigenvalue a specific example is presented to illustrate the situation and to validate the new method of deriving the coefficients

local representation and construction of beltrami fields ii solenoidal beltrami fields and ideal mhd equilibria abstract object of the present paper is the local theory of solution for steady ideal euler flows and ideal mhd equilibria the present analysis relies on the lie darboux theorem of differential geometry and the local theory of representation and construction of beltrami fields developed in sato and yamada 2019 a theorem for the construction of harmonic orthogonal coordinates is proved using such coordinates families of solenoidal beltrami fields with different topologies are obtained in analytic form existence of global solenoidal beltrami fields satisfying prescribed boundary conditions while preserving the local representation is considered it is shown that only singular solutions are admissible an explicit example is given in a spherical domain and a theorem on existence of singular solutions is proven local conditions for existence of solutions and local representation theorems are derived for generalized beltrami fields ideal mhd equilibria and general steady ideal euler flows the theory is applied to construct analytic examples

helical bifurcation and tearing mode in a plasma a description based on casimir foliation the relation between the helical bifurcation of a taylor relaxed state a beltrami equilibrium and a tearing mode is analyzed in a hamiltonian framework invoking an eulerian representation of the hamiltonian the symplectic operator defining a poisson bracket becomes non canonical i e the symplectic operator has a nontrivial cokernel dual to its nullspace foliating the phase space into level sets of casimir invariants a taylor relaxed state is an equilibrium point on a casimir helicity leaf changing the helicity equilibrium points may bifurcate to produce helical relaxed states a necessary and sufficient condition for bifurcation is derived tearing yields a helical perturbation on an unstable equilibrium producing a helical structure approximately similar to a helical relaxed state a slight discrepancy found between the helically bifurcated relaxed state and the linear tearing mode viewed as a perturbed singular equilibrium state is attributed to a casimir element named helical flux pertinent to a resonance singularity of the non canonical symplectic operator while the helical bifurcation can occur at discrete eigenvalues of the beltrami parameter the tearing mode being a singular eigenfunction exists for an arbitrary beltrami parameter bifurcated beltrami equilibria appearing on the same helicity leaf are isolated by the helical flux casimir foliation the obstacle preventing the tearing mode to develop in the ideal limit turns out to be the shielding current sheet on the resonant surface preventing the release of the potential energy when this current is dissipated by resistivity reconnection is allowed and tearing instability occurs the criterion for linear tearing instability of beltrami equilibria is shown to be directly related to the spectrum of the curl operator

a new error analysis for discontinuous finite element methods for linear elliptic problems the standard a priori error analysis of discontinuous galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm in this paper a new error analysis of discontinuous galerkin methods is developed using only the h k weak formulation of a boundary value problem of order 2k this is accomplished by replacing the galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in h k

a special class in vanishing helicity ideal flows the derivation of a lagrangian invariant so called spirality is reviewed through the lagrangian coordinates the value of the spirality is fixed up to a gauge transformation the helicity conservation follows directly from this invariant among all ideal flows with zero helicity in a domain of flow frozen into the fluid motion a special class is introduced this special topological class has the possibility to transform its spirality to be identically zero everywhere in the domain for those simply connected domains of motion a necessary and a sufficient condition is presented for these zero spirality flows

discontinuous galerkin methods of the non selfadjoint steklov eigenvalue problem in inverse scattering abstract in this paper we apply discontinuous galerkin methods to the non selfadjoint steklov eigenvalue problem arising in inverse scattering the variational formulation of the problem is non selfadjoint and does not satisfy h1 elliptic condition by using the spectral approximation theory of compact operators we prove the spectral approximation and optimal convergence order for the eigenvalues finally some numerical experiments are reported to show that the proposed numerical schemes are efficient for real and complex steklov eigenvalues

on the secant varieties to the tangent developable of a veronese variety abstract let v n d p n for n n d n 1 be the order d veronese embedding of p n x n d t v n d p n the tangent developable of v n d and s s 1 x n d p n the s secant variety of x n d i e the closure in p n of the union of all s 1 linear spaces spanned by s points of x n d s s 1 x n d has expected dimension min n 2 n 1 s 1 catalisano geramita and gimigliano conjectured that s s 1 x n d has always the expected dimension except when d 2 n 2 s or d 3 and n 2 3 4 in this paper we prove their conjecture when n 2 and n 3

a c0 discontinuous galerkin formulation for kirchhoff plates abstract a particular discontinuous galerkin finite element formulation for the simulation of kirchhoff plates is presented it is rotation free and utilises standard c 0 lagrange finite element basis functions with the required continuity imposed in a weak sense across element boundaries the implications of the scheme in terms of coercivity and convergence of the galerkin problem in various norms are studied with the formulation shown to be stable for any positive value of a penalty parameter a priori error estimates are supported by a range of numerical examples properties of the approach for the important eigenvalue problems of plate buckling and vibration are also examined through numerical examples based on the results of the analysis and numerical examples it is concluded that the formulation is robust accurate and relatively simple

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

recursive integral method for transmission eigenvalues the transmission eigenvalue problem arose in the inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing the problem is numerically challenging because of the nonself adjointness and complicated spectrum in this talk we introduce a novel recursive contour integral method using the spectrum projection the method can compute both real and complex eigenvalues and does not require any a priori information numerical examples show that the method is effective and robust

iterative methods for transmission eigenvalues transmission eigenvalues have important applications in inverse scattering theory they can be used to obtain useful information of the physical properties such as the index of refraction of the scattering target despite considerable effort devoted to the existence and estimation for the transmission eigenvalues the numerical treatment is limited since the problem is nonstandard classical finite element methods result in non hermitian matrix eigenvalue problems in this paper we focus on the computation of a few lowest transmission eigenvalues which are of practical importance instead of a non hermitian problem we work on a series of generalized hermitian problems we first use a fourth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem the roots of these functions are the transmission eigenvalues then we apply iterative methods to compute the transmission eigenvalues we show the convergence of the numerical schemes the effectiveness of the methods is demonstrated using various numerical examples

on partial polynomial interpolation thealexander hirschowitztheoremsaysthatageneralcollectionof k double points in p n imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions we generalize this theorem to arbitrary zero dimensional schemes contained in a general union of double points we work in the polynomial interpolation setting in this framework our main result says that the affine space of polynomials of degree d in n variables with assigned values of any number of general linear combinations of first partial derivatives has the expected dimension if d 2w ith only five exceptional cases if d 2 the exceptional cases are fully described

spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem we first develop an efficient spectral galerkin method and a rigorous error analysis for the generalized eigenvalue problems associated to a transmission eigenvalue problem then we present an iterative scheme based on computation of the first transmission eigenvalue to estimate the index of refraction of an inhomogeneous medium we present ample numerical results to demonstrate the effectiveness and accuracy of our approach

on the determination of higher order terms of singular elastic stress fields near corners in two dimensional elasticity stresses at reentrant corners exhibit singular behavior the stress field is of the form sigma sigma rm k iota r lambda 2 1 f i theta lambda i where r are polar coordinates centered at the tip of the corner andf i i are smooth functions for practical use of this series the eigenvalues i which are generally complex numbers are required in order of ascending real part the problem then is to find the roots of a transcendental equation eigenequation in the complex plane and arranged in order of ascending real part a theorem is proved on the number location and nature of the roots of this equation with the real part in fixed intervals of length excellent initial estimates of the real part of the complex roots become available and so are bounds within which single real roots exist this enables the determination of any number of roots in ascending order of real part the critical angles at which the eigenvalues change nature are also determined it is shown that for certain cases and for the symmetric mode of deformation the eigenvalue 1 does not represent a rigid body rotation therefore it has to be included in the series representation of the stresses the coefficientsk i can be determined by recently developed extraction techniques thus allowing complete determination of the elastic field and enabling its correlation with experimental data on brittle fracture crack initiation plastic zone estimation etc