solutions to the master equations governing fractional vortices abstract by means of variational methods in this paper we establish sharp existence results for solutions of the master equations governing fractional multiple vortices in the doubly periodic situation the conditions for existence are both necessary and sufficient and give the upper bounds for the vortex numbers in terms of the size of the periodic cell domain in the planar situation there is no restriction on the vortex numbers in both situations the solutions are uniquely determined by the prescribed locations and the local winding numbers of the vortices

inversive congruential pseudorandom numbers distribution of triples this paper deals with the inversive congruential method with power of two modulus m for generating uniform pseudorandom numbers statistical independence properties of the generated sequences are studied based on the distribution of triples of successive pseudorandom numbers it is shown that on the average over the parameters in the inversive congruential method the discrepancy of the corresponding point sets in the unit cube is of an order of magnitude between m 1 2 and m 1 2 logm 3 the method of proof relies on a detailed discussion of the properties of certain exponential sums

on the global well posedness theory for a class of pde models for criminal activity abstract we study a class of reaction advection diffusion system of partial differential equations which can be taken as basic models for criminal activity this class of models are based on routine activity theory and other theories such as the repeat and near repeat victimization effect and were first introduced in short et al 2008 11 in these models the criminal density is advected by a velocity field that depends on a scalar field which measures the appeal to commit a crime we refer to this scalar field as the attractiveness field we prove local well posedness of solutions for the general class of models furthermore we prove global well posedness of solutions to a fully parabolic system with a velocity field that depends logarithmically on the attractiveness field our final result is the global well posedness of solutions the fully parabolic system with velocity field that depends linearly on the attractiveness field for small initial mass

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

a singular sphere covering inequality uniqueness and symmetry of solutions to singular liouville type equations we derive a singular version of the sphere covering inequality which was recently introduced in gui and moradifam invent math https doi org 10 1007 s00222 018 0820 2 2018 suitable for treating singular liouville type problems with superharmonic weights as an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains as well as new self contained proofs of previously known results such as the uniqueness of spherical convex polytopes first established in luo and tian proc am math soc 116 4 1119 1129 1992 furthermore we derive new symmetry results for the spherical onsager vortex equation

the sphere covering inequality and its applications in this paper we show that the total area of two distinct surfaces with gaussian curvature equal to 1 which are also conformal to the euclidean unit disk with the same conformal factor on the boundary must be at least 4 pi in other words the areas of these surfaces must cover the whole unit sphere after a proper rearrangement we refer to this lower bound of total area as the sphere covering inequality the inequality and its generalizations are applied to a number of open problems related to moser trudinger type inequalities mean field equations and onsager vortices etc and yield optimal results in particular we prove a conjecture proposed by chang and yang acta math 159 3 4 215 259 1987 in the study of nirenberg problem in conformal geometry

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

the nirenberg problem and its generalizations a unified approach making use of integral representations we develop a unified approach to establish blow up profiles compactness and existence of positive solutions of the conformally invariant equations p sigma v kv frac n 2 sigma n 2 sigma on the standard unit sphere mathbb s n for all sigma in 0 n 2 where p sigma is the intertwining operator of order 2 sigma finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures when sigma 1 it is the prescribing scalar curvature problem or the nirenberg problem and when sigma 2 it is the prescribing q curvature problem

lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus the inversive congruential method with modulus m 2w for the generation of uniform pseudorandom numbers has recently been introduced the discrepancy z l of a tuples of consecutive pseudorandom numbers generated by such a generator with maximal period length m 2 is the cru cial quantity for the analysis of the statistical independence properties of these pseudorandom numbers by means of the serial test it is proved that for a pos itive proportion of the inversive congruential generators with maximal period length the discrepancy ij2 is at least of the order of magnitude m xl2 for all k 2 this shows that the bound d 2 2 0 m 1 2 logw 2 established by the second author is essentially best possible

influence of spatiotemporally correlated noise on structure formation in excitable media we discuss the influence of additive spatiotemporally correlated i e colored noise on pattern formation in a two dimensional network of excitable systems the signature of spatiotemporal stochastic resonance stsr is analyzed using cross correlation and information theoretic measures it is found that the stsr behavior is affected by both the spatial and temporal correlations of the noise due to an interplay with the length scales of the deterministic network increasing the spatiotemporal noise correlation shifts the occurrence of stsr to smaller values of the noise variance additionally if the spatial correlation of the noise exceeds that of the network the excitation patterns disappear in favor of cloudy structures directly rendering the underlying spatial noise field

correlated noise induced spatiotemporal coherence resonance in a square lattice network abstract the effects of additive correlated noise which is composed of common gaussian white noise and local gaussian colored noise on a square lattice network locally modelled by the rulkov map are studied we focus on the ability of noise to induce pattern formation in a resonant manner it is shown that local gaussian colored noise is able to induce pattern formation which is more coherent at some noise intensity or correlation time so it is able to induce spatiotemporal coherence resonance in the network when common gaussian white noise is applied in addition it is seen that the correlated noise can induce coherent spatial structures at some intermediate noise correlation while this is not the case for smaller and larger noise intensities the mechanism of the observed spatiotemporal coherence resonance is discussed it is also found that the correlation time of local colored noise has no evident effect on the optimal value of the noise strength for spatiotemporal coherence resonance in the network

invariant conformal metrics on ⁿ in this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its schouten tensor also we study conformal metrics on the sphere which are invariant by a k parameter subgroup of conformal diffeomorphisms of the sphere giving a bound on its maximum dimension moreover we classify conformal metrics on the sphere whose eigenvalues of the shouten tensor are all constant we call them isoparametric conformal metrics and we use a clas

extremals for moser inequalities this paper shows that extremals exist for four moser inequalities involving exponential integrals the results are for sobolev spaces defined on the spheres s2 and s3 on the disk d2 and on rp2 the method that carleson and chang used for the n ball with boundary value zero can be modified and applied to functions with mean value zero on these domains this approach also provides an elementary proof of the moser inequality for sn n 2 which is a special case of a result of fontana

the serial test for congruential pseudorandom numbers generated by inversions two types of congruential pseudorandom number generators based on inversions were introduced recently we analyze the statistical independence properties of these pseudorandom numbers by means of the serial test the results show that these pseudorandom numbers perform satisfactorily under the serial test the methods of proof rely heavily on bounds for character sums such as the weil stepanov bound for character sums over finite fields

on the solutions of liouville systems for some prescribed m 0 together with the dirichlet boundary condition u 0 on 0 the liouville equation finds applications in various fields of physics and mathematics to name a few it represents the newtonian potential of a cluster of self gravitating mass distribution where a 0 1 6 36 37 the electric potential induced by charge carriers in the theory of electrolytes where a 0 31 and maximal entropy solutions of the incompressible euler equation 23 9 in the field of differential geometry it stands for the problem of finding a metric corresponding to a prescribed gaussian curvature 3 10 the system analog of l takes the form

existence and non existence results for the su 3 singular toda system on compact surfaces we consider the su 3 singular toda system on a compact surface g u1 2 1 h1eu1 h1eu1dvg 1 2 h2eu2 h2eu2dvg 1 4 m 1m 1m pm 1 u2 2 2 h2eu2 h2eu2dvg 1 1 h1eu1 h1eu1dvg 1 4 m 1m 2m pm 1 where hi are smooth positive functions on i r pm and im 1 we give both existence and non existence results under some conditions on the parameters i and im existence results are obtained using variational methods which involve a geometric inequality of new type non existence results are obtained using blow up analysis and localized poho aev type identities

sharp moser trudinger inequalities for the laplacian without boundary conditions we derive a sharp moser trudinger inequality for the borderline sobolev imbedding of w2 n 2 bn into the exponential class where bn is the unit ball of rn the corresponding sharp results for the spaces w0d n d are well known for general domains and are due to moser and adams when the zero boundary condition is removed the only known results are for d 1 and are due to chang yang cianchi and leckband the proof of our result is based on a new integral representation formula for the canonical solution of the poisson equation on the ball that is the unique solution of the equation u f which is orthogonal to the harmonic functions on the ball the main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation expressed in terms of its distribution function

explicit inversive congruential pseudorandom numbers with power of two modulus an explicit version of the inversive congruential method with power of two modulus for generating uniform pseudorandom numbers is introduced statistical independence properties of the generated sequences are studied by means of the serial test the method of proof relies on a detailed analysis of certain exponential sums

the scalar curvature flow on s n perturbation theorem revisited for the problem of finding a geometry on s n for n 3 with a prescribed scalar curvature there is a well known result which is called the perturbation theorem it is due to chang and yang duke math j 64 27 69 1991 their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm it is important to know how large that difference in sup norm can possibly be here we consider prescribing scalar curvature problem using the scalar curvature flow

application of an adams type inequality to a two chemical substances chemotaxis system abstract this paper deals with positive solutions of the fully parabolic system u t u u v in 0 1 v t v v w in 0 2 w t w w u in 0 under homogeneous neumann boundary conditions or mixed boundary conditions no flux and dirichlet conditions in a smooth bounded domain r n n 4 with positive parameters 1 2 0 and nonnegative smooth initial data u 0 v 0 w 0 in the lower dimensional case n 3 it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded in the case n 4 it is shown that in the radially symmetric setting solutions to the neumann boundary value problem of the system exist globally in time and remain bounded if u 0 l 1 8 2 as to the mixed boundary value problem we will establish global existence and boundedness of solutions if u 0 l 1 8 2 without radial symmetry the key ingredients are a lyapunov functional and an adams type inequality a lyapunov functional of the above problems will be constructed and the constant 8 2 is deduced from the critical constant in the adams type inequality this result is regarded as a generalization of the well known 8 problem in the keller segel system to higher dimensions

construction of inversive congruential pseudorandom number generators with maximal period length abstract the inversive congruential method for generating uniform pseudorandom numbers is a particularly attractive alternative to linear congruential generators with their well known inherent deficiencies like the unfavourable coarse lattice structure in higher dimensions in the present paper the modulus in the inversive congruential method is chosen as a power of an arbitrary odd prime the existence of inversive congruential generators with maximal period length is proved by a new constructive characterization of these generators

method for detecting the signature of noise induced structures in spatiotemporal data sets spatiotemporal stochastic resonance stsr is a phenomenon where the stability of spatial patterns in an extended dynamical system displays a resonance type dependence on the noise amplitude with the patterns being optimal at intermediate noise level this dynamical behavior has been found in theoretical systems as well as in biochemical processes where the noise level has been controlled externally however it is an open question how to identify the signature of a spatiotemporal stochastic resonance in a natural system e g in ecology when the noise amplitude is not known this question is addressed in the present paper we provide analysis tools which allow to reconstruct the noise intensity in a spatiotemporal data set from the data alone these tools are based on nearest neighbor considerations inspired by cellular automata and are an appropriate method for detecting stsr when combined with some measure of spatial order as a test of our analysis tools we apply them to sample data generated by four theoretical model systems we show explicitly that without knowledge of the theoretical value of the noise amplitude for those systems displaying stsr the corresponding resonance curve can be reconstructed from the data alone in addition the other nonresonant cases are properly identified by our method with no resonance curve being found

non degeneracy and uniqueness of solutions to singular mean field equations on bounded domains abstract the aim of this paper is to complete the program initiated in 51 23 and then carried out by several authors concerning non degeneracy and uniqueness of solutions to mean field equations in particular we consider mean field equations with general singular data on non smooth domains the argument is based on the alexandrov bol inequality and on the eigenvalues analysis of linearized singular liouville type problems

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

statistical independence of a new class of inversive congruential pseudorandom numbers linear congruential pseudorandom numbers show several undesirable regularities which can render them useless for certain stochastic simulations this was the motiviation for important recent developments in nonlinear congruential methods for generating uniform pseudorandom numbers it is particularly promising to achieve nonlinearity by employing the operation of multiplicative inversion with respect to a prime modulus in the present paper a new class of such inversive congruential generators is introduced and analyzed it is shown that they have excellent statistical independence properties and model true random numbers very closely the methods of proof rely heavily on weil stepanov bounds for rational exponential sums 39 refs

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

on a nonlinear congruential pseudorandom number generator a nonlinear congruential pseudorandom number generator with modulus m 2 w is proposed which may be viewed to comprise both linear as well as inversive congruential generators the condition for it to generate sequences of maximal period length is obtained it is akin to the inversive one and bears a remarkable resemblance to the latter

sharp exponential integrability for critical riesz potentials and fractional laplacians on rn abstract we derive sharp adams inequalities for the riesz and more general riesz like potentials on the whole of r n as a consequence we obtain sharp moser trudinger inequalities for the critical sobolev spaces w n r n 0 n these inequalities involve fractional laplacians higher order gradients general homogeneous elliptic operators with constant coefficients and general trace type borel measures

conformal metrics with prescribed gaussian curvature on s2 we consider on s 2 the problem of which functions k can be the scalar curvature of a metric conformal to the standard metric on s 2 we assume that k is a function of one variable and we obtain a necessary and sufficient condition for the problem to be solvable we also obtain several new sufficient conditions on k which are easy to check in order that the problem be solvable

delay sustained pattern formation in subexcitable media the influence of time delayed feedback on pattern formation in subexcitable media represented by a net of fitzhugh nagumo elements a minimal model of neuronal dynamics is studied without feedback wave fronts die out after a short propagation length subexcitable net dynamics applying time delayed feedback with appropriate feedback parameters pattern formation is sustained and the wave fronts may propagate through the whole net signature of excitable behavior the coherence of noise induced patterns is significantly enhanced if feedback with appropriately chosen parameters is applied and shows a resonancelike dependency on the delay time in a next step the transition to the excitable regime is investigated in dependence on the quota of elements which get the feedback signal it is sufficient to control approximately half of the elements to achieve excitable behavior regarding a medical application where the external control of a neural tissue would affect not single neurons but clusters of neurons the spatial correlation of the controlled elements is of importance the selection of the elements which get the feedback signal is based on a spatially correlated random distribution it is shown that the correlation length of this distribution affects the pattern formation

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

supercritical mean field equations on convex domains and the onsager s statistical description of two dimensional turbulence we are motivated by the study of the microcanonical variational principle within onsager s description of two dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails we obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding mean field equation on convex and thin enough domains in the supercritical with respect to the moser trudinger inequality regime this is a brand new achievement since existence results in the supercritical region were previously known only on multiply connected domains we then study the structure of these solutions by the analysis of their linearized problems and we also obtain a new uniqueness result for solutions of the mean field equation on thin domains whose energy is uniformly bounded from above finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and combining it with all the results obtained so far we solve the microcanonical variational principle in a small range of supercritical energies where the entropy is shown to be concave

a nonlinear congruential pseudorandom number generator with power of two modulus a nonlinear congruential pseudorandom number generator is studied where the modulus is a power of two investigation of this generator was suggested by knuth 7 a simple necessary and sufficient condition is given for this generator to have the maximal period length

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

on steady distributions of self attracting clusters under friction and fluctuations we consider the vlasov fokker planek equation with a newtonian attracting potential and study its stationary solutions given by the generalized lane emden equation in a two dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse for a one dimensional model we prove some results on existence uniqueness stability and symmetry breaking of stationary solutions

on a sharp sobolev type inequality on two dimensional compact manifolds motivated by the asymptotic analysis of double vortex condensates in the chern simons higgs theory we construct a suitable minimizing sequence for a sharp sobolev inequality a la moser for two dimensional compact manifolds as a consequence we first obtain a direct proof of the sharp character of such an inequality secondly and more interestingly we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum

on prescribing geodesic curvature on d2 abstract some new results are obtained for the problem of prescribing geodesic curvature k on d when k possesses some kinds of symmetries

lower bounds for the discrepancy of inversive congruential pseudorandom numbers the inversive congruential method is a uniform pseudorandom number generator which was introduced recently for a prime modulus p the discrepancy d k of k tuples of successive pseudorandom numbers generated by this method determines the statistical independence properties of these pseudorandom numbers it was shown earlier by the author that d k o p 1 2 logp k for 2 2

optimal control in prescribing webster scalar curvatures on 3 dimensional pseudo hermitian manifolds abstract in this work we give new existence and multiplicity results for the solutions of the prescription problem for the webster scalar curvature on a 3 dimensional pseudo hermitian manifold the critical points of prescribed functions verify mixed conditions we establish some morse inequalities at infinity and a poincare hopf type formula to give a lower bound on the number of solutions as well as an upper bound for the morse index of such solutions

on the best constant in the moser onofri aubin inequality let s2 be the 2 dimensional unit sphere and let j denote the nonlinear functional on the sobolev space h1 s2 defined by j alpha u frac alpha 16 pi int s 2 nabla u 2 d mu 0 frac 1 4 pi int s 2 u d mu 0 rm ln int s 2 e u frac d mu 0 4 pi where d mu 0 sin theta d theta wedge d phi onofri had established that j is non negative on h1 s2 provided 1 in this note we show that if j is restricted to those u in h 1 s 2 that satisfies the aubin condition int s 2 e u x j d mu 0 0 quad rm for all 1 leq j leq 3 then the same inequality continues to hold i e j u 0 whenever alpha geq frac 2 3 epsilon 0 for some epsilon 0 0 the question of chang yang on whether this remains true for all alpha geq frac 1 2 remains open

a quantization property for blow up solutions of singular liouville type equations abstract motivated by the study of self dual vortices in gauge field theory cf vortices and monopoles birkhauser boston 1980 solitons in field theory and nonlinear analysis monographs in mathematics springer new york 2001 we consider a concentrating solution sequence u k satisfying u k z 2 k v k z e u k in b 1 z 2 k v k z e u k z 0 weakly in the sense of measures in b 1 where i k r k ii v k c 0 1 b 1 0 a v k b and v k a in b 1 we prove that necessarily 8 n 8 1 8 z the result is sharp as shown by explicit examples and should be compared with that obtained by li shafrir ind univ math j 43 4 1994 1255 concerning the situation where k 0 k n

the serial test for a nonlinear pseudorandom number generator let m 2 and g m 1 3 m 1 a sequence y n y n g m is obtained by the formula y n 1 ay n b cy n mod m the sequence x n x n y n m is a sequence of pseudorandom numbers of the maximal period length m 2 if and only if a c 1 mod 4 b 2 mod 4 in this note the uniformity is investigated by the 2 dimensional serial test for the sequence we follow closely the method of papers by eichenauer herrmann and niederreiter

a remark on the discrepancy of quadratic congruential pseudorandom numbers abstract the linear congruential method for generating uniform pseudorandom numbers has several known deficiencies which can render the generated sequences useless for certain simulation purposes one of the alternatives is knuth s quadratic congruential method recently it was proved that pairs of quadratic congruential pseudorandom numbers have uniformly excellent statistical independence properties in the present paper it is shown that unfortunately a similar result cannot be obtained for k tuples with k 3 it is proved that for a positive percentage of the possible generators the discrepancy of these k tuples is too large the method of proof relies on the evaluation of certain exponential sums in view of the present result it becomes doubtful whether the quadratic congruential method is absolutely suitable for generating uniform pseudorandom numbers

adams inequalities for bi laplacian and extremal functions in dimension four let be a smooth oriented bounded domain be the sobolev space and be the first eigenvalue of the bi laplacian operator 2 then for any a 0a o we have and the above supremum is infinity when a o this strengthens adams inequality in dimension 4 d adams a sharp inequality of j moser for high order derivatives ann of math 128 1988 365 398 where he proved the above inequality holds for a 0 moreover we prove that for sufficiently small a an extremal function for the above inequality exists as a special case of our results we thus show that there exists with such that this establishes the existence of an extremal function of the original adams inequality in dimension 4

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

remarks on prescribing gauss curvature we study the nonlinear partial differential equation for the problem of prescribing gauss curvature k on s 2 we give an example of a rotationally symmetric k for which the kazdan warner obstruction is satisfied but the equation has no rotationally symmetric solution on the other hand we give a simple sufficient condition for solvability of the equation when k is rotationally symmetric finally we give a sufficient condition for solvability when k is not necessarily rotationally symmetric