global regularity for the 2d mhd equations with mixed partial dissipation and magnetic diffusion abstract whether or not classical solutions of the 2d incompressible mhd equations without full dissipation and magnetic diffusion can develop finite time singularities is a difficult issue a major result of this paper establishes the global regularity of classical solutions for the mhd equations with mixed partial dissipation and magnetic diffusion in addition the global existence conditional regularity and uniqueness of a weak solution is obtained for the 2d mhd equations with only magnetic diffusion

necessary and sufficient conditions for nonlinear stability in the magnetic benard problem we study the magnetic benard problem with the lyapunov direct method and obtain necessary and sufficient conditions of conditional nonlinear stability by introducing a suitable change of fields and a generalized energy functional we show that whenever the magnetic prandtl number is less than the usual prandtl number the critical linear and nonlinear stability rayleigh numbers coincide for any chandrasekhar number

bmo and the regularity criterion for weak solutions to the magnetohydrodynamic equations in this paper we study the regularity criterion for weak solutions to the incompressible magnetohydrodynamic equations we derive the regularity of weak solutions in the marginal class moreover our result demonstrates that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations

on two dimensional magnetohydrodynamic equations with fractional diffusion abstract we consider the existence of the smooth solution for two dimensional magnetohydrodynamic equations with fractional diffusion in the fluid equations in particular we prove the global in time existence of fractional diffusion with power greater than 1 2

the second main theorem for holomorphic curves into semi abelian varieties we establish the second main theorem with the best truncation level one tf r l d n1 r f d tf r for an entire holomorphic curve f c a into a semi abelian variety a and an arbitrary effective reduced divisor d on a the low truncation level is important for applications we will actually prove this for the jet lifts of f finally we give some applications including the solution of a problem posed by mark green

nombre de points de hauteur bornee sur les surfaces de del pezzo de degre 5 we state manin s conjecture in the particular case of the split del pezzo s surfaces of degree 5 over q we show that for an open set u v nu q b card p u q h p b c b log b 4 b the constant c is the one conjectured by e peyre sommaire

the beale kato majda criterion for the 3d magneto hydrodynamics equations we study the blow up criterion of smooth solutions to the 3d mhd equations by means of the littlewood paley decomposition we prove a beale kato majda type blow up criterion of smooth solutions via the vorticity of velocity only namely sup j in mathbb z int 0 t delta j nabla times u infty dt where j is the frequency localization operator in the littlewood paley decomposition

global regularity for the 2d magnetic benard fluid system with mixed partial viscosity abstract inspired by the previous works the main goal of this paper is to establish the global well posedness and conditional regularity of the incompressible magnetic benard fluid system with mixed partial viscosity in two dimensions as a result our results generalize and extend the previous results

existence theorem and blow up criterion of the strong solutions to the two fluid mhd equation in r3 abstract we first give the local well posedness of strong solutions to the cauchy problem of the 3d two fluid mhd equations and then study the blow up criterion of the strong solutions by means of the fourier frequency localization and bony s paraproduct decomposition it is proved that the strong solution u b can be extended after t t if either u l t q b p 0 with 2 q 3 p 1 and b l t 1 b 0 or j l t q b p 0 with 2 q 3 p 2 where t u denotes the vorticity of the velocity and j b stands for the current density

on the distribution of points of bounded height on equivariant compactifications of vector groups we prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space

a regularity criterion for the three dimensional mhd equations in terms of one directional derivative of the pressure in this work we will establish a sufficient condition for the regularity criterion to the 3d mhd equations in terms of one directional derivative of the pressure more precisely we prove that if x 3 x t l 2 1 r 0 t m 2 3 r r 3 for 0 r 1 then the solution u b is regular we obtain some improvement of the related results in berselli and galdi 2002 cao and wu 2010 jia and zhou 2012 lin and du 2013 and yamazaki 2013

addendum to defects for ample divisors of abelian varieties schwarz lemma and hyperbolic hypersurfaces of low degrees american journal of mathematics 119 1997 1139 1172 this addendum uses bloch s original technique of ordinary differential equations to circumvent a difficulty in the proof of lemma 2 in the paper in the title this technique at the same time yields the following stronger nevanlinna s second main theorem with truncation at an order given explicitly by the chern class of the divisor for an ample divisor d in an abelian variety a of complex dimension n and for any holomorphic map f c a whose image is not contained in any translate of d the characteristic function of f for d is dominated by the counting function of f for d truncated at order k n plus an error term of logarithmic order of the characteristic function where k n is inductively given by k 0 0 k 1 1 and k l 1 k l 3 n l 1 4 k l 1 l d n for 1 l n

defect relations for holomorphic maps between spaces of different dimensions there is now a rather rich theory of defect relations for holomorphic maps between spaces of equal dimension developed by nevanlinna stoll griffiths and many other people however for the case of maps between spaces of unequal dimension essentially there is only the theory developed by weyl weyl and ahlfors when the target space is the complex projective space and their theory is built on the projective linear structure of the projective space a partial list of references for these results is given at the end of this paper in this paper we derive a defect relation for holomorphic maps between spaces of different dimensions where the target space can be an arbitrary compact complex algebraic manifold since the special case of maps from c to a compact complex manifold of complex dimension two contains already the methods and techniques needed for the general case in order to avoid unnecessarily complicated notations and to make the arguments more easily understood we give complete details only for this special case and indicate the easy modifications needed for the general case besides the usual techniques in value distribution theory of using the poincar6 lelong formula the calculus lemma of estimating the integrand by its integral and the concavity of logarithm a new key point in our argument is to introduce a meromorphic connection so that the second fundamental form of a divisor with respect to the connection is zero more precisely for the case of a meromorphic map f from c to a compact kihler surface m to consider the defect for a nonsingular zero set of a holomorphic section s of a positive holomorphic line bundle l we introduce a meromorphic connection i v form with the following two properties i for some hermitian holomorphic line bundle f over m and some holomorphic section 0 of f

hausdorff measure of the singular set in the incompressible magnetohydrodynamic equations we derive a local energy inequality for weak solutions of the three dimensional magnetohydrodynamic equations combining biot savart law interpolation inequalities and the local energy inequality we prove a partial regularity theorem for suitable weak solutions furthermore we obtain an improved estimate for the logarithmic hausdorff dimension of the singular set of suitable weak solutions

rational points on pencils of conics and quadrics with many degenerate fibres for any pencil of conics or higher dimensional quadrics over q with all degenerate bres dened over q we show that the brauer manin obstruction controls weak approximation the proof is based on the hasse principle and weak approximation for some special intersections of quadrics over q which is a consequence of recent advances in additive combinatorics

a note on the regularity criterion for 3d mhd equations in b 1 space abstract in this note we consider sufficient conditions for the regularity of leray hoph hopf solutions of the 3d incompressible magnetohydrodynamic equations via the velocity and magnetic fields in terms of b 1 spaces we prove that if u b belongs to the space l 2 0 t b 1 r 3 then the solution u b is regular this extends recent results contained in gala 2011 3

on the regularity of weak solutions to the magnetohydrodynamic equations abstract in this paper we study the regularity of weak solution to the incompressible magnetohydrodynamic equations we obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations which is similar to that of incompressible navier stokes equations moreover our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto hydrodynamic equations

global existence of smooth solutions for three dimensional magnetic benard system with mixed partial dissipation magnetic diffusion and thermal diffusivity this paper deals with the cauchy problem to the 3d system of incompressible magnetic benard fluids we prove that as the initial data satisfy u 0 h 1 r 3 2 b 0 h 1 r 3 2 0 h 1 r 3 2 e where e is a suitably small positive number then the 3d magnetic benard system with mixed partial dissipation magnetic diffusion and thermal diffusivity admit global smooth solutions

counting algebraic numbers with large height ii let denote the field of rational numbers open image in new window an algebraic closure of and h open image in new window the absolute multiplicative weil height for each positive integer d and real number mathcal h geqslant 1 it is well known that the number open image in new window of points in open image in new window having degree d over and satisfying h left alpha right leqslant mathcal h is finite this is the one dimensional case of northcott s theorem 8 see also 5 page 59 the systematic study of the counting function open image in new window and that of related functions in higher dimensions was begun by schmidt 10 it is relatively easy to prove the existence of a positive constant c c d such that open image in new window 1 and also the existence of positive constants c c d and mathcal h 0 mathcal h 0 left d right such that open image in new window 2

two regularity criteria for the 3d mhd equations this work establishes two regularity criteria for the 3d incompressible mhd equations the first one is in terms of the derivative of the velocity field in one direction while the second one requires suitable boundedness of the derivative of the pressure in one direction

global regularity to the 3d mhd equations with large initial data in bounded domains this paper considers the global regularity to the 3d incompressible mhd equations with large initial data in bounded domains let u and b denote the viscosity coefficient magnetic diffusivity velocity field and magnetic field respectively we construct new systems for u b and u b to overcome the difficulties caused by the large initial data it is shown that u b h1 is globally bounded as long as u0 b0 h1 1 or u0 b0 h1 1 is sufficiently small which indicates that the navier stokes equations can be regularized by the magnetic field

global existence of three dimensional incompressible magneto micropolar system with mixed partial dissipation magnetic diffusion and angular viscosity abstract the magneto micropolar fluid flows describe the motion of electrically conducting micropolar fluids in the presence of a magnetic field the issue of whether the strong solution of magneto micropolar equations in three dimensional can exist globally in time with large initial data is still unknown in this paper we deal with the cauchy problem of the three dimensional magneto micropolar system with mixed partial dissipation magnetic diffusion and angular viscosity more precisely the global existence of smooth solutions to the three dimensional incompressible magneto micropolar fluid equations with mixed partial dissipation magnetic diffusion and angular viscosity are obtained by energy method under the assumption that h 1 norm of the initial data u 0 b 0 w 0 sufficiently small namely u 0 b 0 0 h 1 r 3 2 e 2 where e is a sufficiently small positive number this work follows the techniques in the paper of cao and wu 2011

a generalized bloch s theorem and the hyperbolicity of the complement of an ample divisor in an abelian variety bloch s theorem on the hyperbolicity of nonlinear subvarieties of abelian varieties states that the zariski closure of the image of a holomorphic map from c to an abelian variety is precisely the translate of an abelian subvariety b26 ks0 o77 w80 gg79 for a survey on the proofs of bloch s theorem see e g si95 chapter 7 the purpose of this note is to prove the following generalization of bloch s theorem if the image of a holomorphic map f from c to an abelian variety is zariski dense then the zariski closure of the image of the differential of any order of the map f is invariant under any translation of the abelian variety theorem 2 2 below this generalization has as a corollary the conjecture of lang la91 that the complement of an ample divisor in an abelian variety is hyperbolic the following slightly more general statement which together with bloch s theorem implies lang s conjecture see corollary 3 2 is a corollary of our generalized bloch s theorem the image from c to an abelian variety with zariski dense image must intersect any complex hypersurface of the abelian variety theorem 3

regularity criteria for a lagrangian averaged magnetohydrodynamic α model abstract in this paper we study the n dimensional lagrangian averaged magnetohydrodynamic lamhd model with 5 n 7 throughout the whole space various regularity criteria are established for both the cases with and those without viscosity

regularity criterion for 3d mhd fluid passing through the porous medium in terms of gradient pressure abstract the aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in r 3 our results show that if p l with 2 3 3 1 then the weak solution is regular and unique if p l with 2 3 2 3 2 then the weak solution is regular and unique

partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations in this paper we study the local behavior of the solutions to the three dimensional magnetohydrodynamic equations we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions it is shown that in some neighborhood of x0 t0 the gradients of the velocity field u and the magnetic field b are locally uniformly bounded in l norm as long as that either the scaled local l2 norm of the gradient or the scaled local total energy of the velocity field is small and the scaled local total energy of the magnetic field is uniformly bounded these estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory as an immediately corollary we can derive an estimates of hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid various partial regularity results are obtained as consequences of our blow up estimates

an improved regularity criterion for the 3d hall mhd equations via the vorticity abstract in this paper we consider the three dimensional incompressible hall magnetohydrodynamic equations and establish an improved regularity criterion of local in time classical solutions involving only the vorticity field consequently this improves the previous result

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

big rational surfaces we prove that the cox ring of a smooth rational surface with big anticanonical class is finitely generated we classify surfaces of this type that are blow ups of mathbb p 2 at distinct points lying on a possibly reducible cubic

regularity criteria and small data global existence to the generalized viscous hall magnetohydrodynamics in this paper we establish the regularity criteria of solutions to the three space dimensions generalized viscous hall magnetohydrodynamics moreover we also prove that there exist classical solutions globally in time to the generalized viscous hall mhd if the initial data is small enough or the fractional laplacian dissipation powers and are suitably large these results are some improvements of very recent work chae and lee 2014 established by chae and lee

a new regularity criterion for weak solutions to the viscous mhd equations in terms of the vorticity field abstract in this paper we consider regularity criterion for solutions to the 3d viscous incompressible mhd equations in morrey campanato spaces it is proved that if the vorticity field u belongs to m 2 3 r for 0 r 1 on 0 t then the solution remains smooth on 0 t

remarks on singularities dimension and energy dissipation for ideal hydrodynamics and mhd for weak solutions of the incompressible euler equations there is energy conservation if the velocity is in the besov space b 3 s with s greater than 1 3 b 3 s consists of functions that are lip s i e holder continuous with exponent s measured in the l p norm here this result is applied to a velocity field that is lip 0 except on a set of co dimension on which it is lip agr 1 with uniformity that will be made precise below we show that the frisch parisi multifractal formalism is valid at least in one direction for such a function and that there is energy conservation if analogous conservation results are derived for the equations of incompressible ideal mhd i e zero viscosity and resistivity for both energy and helicity in addition a necessary condition is derived for singularity development in ideal mhd generalizing the beale kato majda condition for ideal hydrodynamics

global well posedness for the 3 d incompressible inhomogeneous mhd system in the critical besov spaces abstract in this paper we study the cauchy problem for the incompressible inhomogeneous magnetohydrodynamic mhd system in r 3 we obtain the global existence of a unique solution for this system without any small conditions imposed on the third component of the initial velocity field and magnetic field in the critical besov spaces our methods mainly rely upon the fourier frequency localization and bony s paraproduct decomposition

rational points of bounded height on general conic bundle surfaces a conjecture of manin predicts the asymptotic distribution of rational points of bounded height on fano varieties in this paper we use conic bundles to obtain correct lower bounds for a wide class of surfaces over number fields for which the conjecture is still far from being proved for example we obtain the conjectured lower bound of manin s conjecture for any del pezzo surface whose picard rank is sufficiently large or for arbitrary del pezzo surfaces after possibly an extension of the ground field of small degree

regularity criteria in terms of the pressure for the three dimensional mhd equations in this paper we consider the cauchy problem for the 3d incompressible mhd equations a regularity criterion by providing sufficient condition in terms of the pressure is established more precisely we show that if p l 0 t l r 3 with 3 2 7 4 and 12 5 4 then any weak solutions u b of the 3d mhd equations is regular on 0 t

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

manin s conjecture for quartic del pezzo surfaces with a conic fibration an asymptotic formula is established for the number of q rational points of bounded height on a non singular quartic del pezzo surface with a conic bundle structure

a new regularity criterion for the 3d incompressible mhd equations in terms of one component of the gradient of pressure abstract in this paper we study the global regularity issue for the 3d incompressible mhd equations it is proved that if the pressure p satisfies 3 p l 0 t l r 3 with 2 3 2 3 and 1 then the corresponding weak solution u b is regular on 0 t

tamagawa numbers of diagonal cubic surfaces numerical evidence a refined version of manin sconjecture aboutthe asymptotics of points of bounded height on fano varieties has been developped by batyrev and the authors we test numerically this refined conjecture for some diagonal cubic surfaces

defects for ample divisors of abelian varieties schwarz lemma and hyperbolic hypersurfaces of low degrees we prove that the defect vanishes for a holomorphic map f from the affine complex line to an abelian variety a and for an ample divisor d in a the proof uses the translational invariance of the zariski closure of the k jet space of the image of f and the theorem of riemann roch to construct a nonidentically zero meromorphic k jet differential whose pole divisor is dominated by a divisor equivalent to pd and which vanishes along the k jet space of d to order q with p q smaller than a prescribed small positive number then estimates involving the theta function with divisor d and the logarithmic derivative lemma are used we also prove a pointwise schwarz lemma which gives the vanishing of the pullback by a holomorphic map from the affine complex line to a compact complex manifold of a holomorphic jet differential vanishing on an ample divisor this pointwise schwarz lemma is a slight modification of a statement whose proof green and griffiths sketched in their alternative treatment of bloch s theorem on entire curves in abelian varieties the log pole case of the pointwise schwarz lemma is also given we construct examples of hyperbolic hypersurface whose degree is only 16 times the square of its dimension

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

global regularity of the two dimensional incompressible generalized magnetohydrodynamics system abstract in this paper we consider the global regularity of the two dimensional generalized incompressible magnetohydrodynamics system with velocity dissipation exponent 1 4 and magnetic diffusion exponent 1 based on the improved integrability in terms of vorticity this result is a further improvement of very recent works jiu and zhao 2014 and yamazaki 2013 which require 1 3 and 1 therefore our result is more closer to the resolution of the well known global regularity issue on the 2d magnetohydrodynamics system 0 1

the second main theorem in the hyperbolic case we develop nevanlinna s theory for a class of holomorphic maps when the source is a disc such maps appear in the theory of foliations by riemann surfaces

on the uniqueness of weak solutions for the 3d viscous magneto hydrodynamic equations abstract a uniqueness result of weak solutions for the 3d viscous magneto hydrodynamic equations is proved by means of the fourier localization technique and bony s paraproduct decomposition

regularity criterion of the 2d benard equations with critical and supercritical dissipation abstract in this paper we investigate the cauchy problem for the two dimensional 2d incompressible benard equations on the one hand we prove global in time existence of smooth solutions to the 2d benard equations with critical dissipation on the other hand we establish several regularity criteria involving temperature for 2d benard equations with supercritical dissipation

remarks on the regularity criteria of generalized mhd and navier stokes systems we study the regularity criteria of the three dimensional generalized magnetohydrodynamics mhd and navier stokes systems in particular we show that the regularity criteria of the generalized mhd system may be reduced to depend only on two diagonal entries of the jacobian matrix of the velocity vector field or one vorticity component and one entry of the jacobian matrix of the velocity vector field

a toric extension of faltings diophantine approximation on abelian varieties for divisors on abelian varieties faltings established an optimal bound on the proximity of rational points to the same we extend this both to the quasiprojective category where the role of abelian varieties is played by their toroidal extensions and to holomorphic maps from the line proving along the way some wholly general dynamic intersection estimates in value distribution theory of independent interest

a regularity criterion for the 3d magneto micropolar fluid equations in triebel lizorkin spaces abstract we consider the regularity criterion for the 3d magneto micropolar fluid equations in triebel lizorkin spaces it is proved that if u l p 0 t f q 2 q 3 0 with 2 p 3 q 2 3 2 q then the solution remains smooth in 0 t as a corollary we obtain the classical beal kato majda criterion that is the condition u l 1 0 t b 0 ensures the smoothness of the solution

steady states of hall mhd system abstract this paper concerns steady states of the resistive incompressible hall magnetohydrodynamics hall mhd system with emphasis on the effect of the hall term on the solutions existence of h 1 weak solutions is proved for a general domain and an infinite number of weak solutions exist if the domain has holes existence of h 2 weak solutions is proved for either small data or small hall parameter the asymptotic behavior of the solutions in the vanishing hall parameter limit is examined and the asymptotic stability of the steady states is also investigated

limiting case for the regularity criterion to the 3 d magneto hydrodynamics equations abstract we prove that any weak solution u b of three dimensional incompressible magneto hydrodynamics equations is regular if u l 0 t l 3 r 3 and b l 0 t vmo 1 r 3 the proof is based on the blow up analysis and backward uniqueness for the parabolic operator developed by escauriaza seregin sverak

integral points on subvarieties of semiabelian varieties ii this paper proves a finiteness result for families of integral points on a semiabelian variety minus a divisor generalizing the corresponding result of faltings for abelian varieties combined with the main theorem of the first part of this paper this gives a finiteness statement for integral points on a closed subvariety of a semiabelian variety minus a divisor in addition the last two sections generalize some standard results on closed subvarieties of semiabelian varieties to the context of closed subvarieties minus divisors

quadratic families of elliptic curves and unirationality of degree 1 conic bundles we study families of elliptic curves whose coefficients are degree 2 polynomials in a variable t all such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers we prove that such conic bundles are unirational as a consequence one obtains that for infinitely many values of t the resulting elliptic curve has rank at least 1

remarks on partial regularity for suitable weak solutions of the incompressible magnetohydrodynamic equations abstract in this paper we are concerned with the partial regularity for suitable weak solutions of the tri dimensional magnetohydrodynamic equations with the help of the de giorgi iteration method we obtain the results proved by he and xin c he z xin partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations j funct anal 227 2005 113 152 namely the one dimensional parabolic hausdorff measure of the possible singular points of the velocity field and the magnetic field is zero

on surfaces of general type with maximal albanese dimension abstractgiven a minimal surface s equipped with a generically nite map to an abelianvariety and c s a rational or an elliptic curve we show that the canonical degreeof c is bounded by four times the self intersection of the canonical divisor of s asa corollary we obtain the niteness of rational and elliptic curves with an optimaluniform bound on their canonical degrees on any surface of general type with twolinearly independent regular one forms 1 1 introduction the object of this paper is to give an e ective bound on the canonical degrees of rationaland elliptic curves on a minimal surface of general type with generically nite albanese maplinearly in terms of the canonical volume of the surface here the canonical degree is thedegree with respect to the canonical polarization of the surface and the self intersection ofthis polarization is the canonical volume we will work in the complex analytic setting so that all varieties are complex analytic wewill assume the rudiments from the the theory of classi cation of complex projective surfacesas found for example in 3 we call a smooth projective surface to be of maximal albanesedimension if its albanese map is generically nite n b this terminology is di erent fromthat introduced in 21 we recall that any smooth projective surface s of maximal albanesedimension admits a morphism to a minimal one whose canonical divisor pulls back to thepositive part p of the zariski decomposition of the canonical divisor k of s and the canonicalvolume of s is by de nition vol k p

a new unicity theorem and erdos problem for polarized semi abelian varieties in 1988 erdos asked if the prime divisors of xn 1 for all n 1 2 determine the given integer x the problem was affirmatively answered by corrales rodriganez and schoof j number theory 64 276 290 1997 but a solution could also be deduced from an earlier result of schinzel bull acad polon sci 8 307 309 2007 together with its elliptic version analogously yamanoi forum math 16 749 788 2004 proved that the support of the pulled back divisor f d of an ample divisor on an abelian variety a by an algebraically non degenerate entire holomorphic curve f c a essentially determines the pair a d by making use of the main theorem of noguchi forum math 20 469 503 2008 we here deal with this problem for semi abelian varieties namely given two polarized semi abelian varieties a1 d1 a2 d2 and algebraically non degenerate entire holomorphic curves fi c ai i 1 2 we classify the cases when the inclusion rm supp f 1 d 1 subset rm supp f 2 d 2 holds we shall remark in 5 that these methods yield an affirmative answer to a question of lang formulated in 1966 our answer is more general and more geometric than the original question finally we interpret the main result of corvaja and zannier invent math 149 431 451 2002 to provide an arithmetic counterpart in the toric case

global strong solution to the 2d nonhomogeneous incompressible mhd system abstract in this paper we first prove the unique global strong solution with vacuum to the two dimensional nonhomogeneous incompressible mhd system as long as the initial data satisfies some compatibility condition as a corollary the global existence of strong solution with vacuum to the 2d nonhomogeneous incompressible navier stokes equations is also established our main result improves all the previous results where the initial density need to be strictly positive the key idea is to use some critical sobolev inequality of logarithmic type which is originally due to brezis and wainger 1980 7

global regularity for the 212d incompressible hall mhd system with partial dissipation abstract in this paper we investigate the cauchy problem for the 2 1 2 dimensional incompressible hall magnetohydrodynamic equations with partial dissipation by using energy estimate we obtain the local existence of strong solution and the global in time existence of the strong solution

geometric measure type regularity criteria for the 3d magnetohydrodynamical system abstract several formulations of a local geometric measure type condition are imposed on super level sets of mild solutions to the homogeneous incompressible 3d magnetohydrodynamical system with bounded initial data to prevent finite time singularity formation supporting this results regarding the existence uniqueness and real analyticity of mild solutions are established as is a sharp lower bound on the radius of analyticity

remarks on the blow up criteria for three dimensional ideal magnetohydrodynamics equations in this paper we establish some improved regularity conditions for the three dimensional 3d magnetohydrodynamics equations we proved the logarithmically improved caflisch klapper steele s criteria remarks on sigularities dimension and energy dissipation for ideal hydrodynamics and mhd commun math phys 184 443 1997 as well as the logarithmically improved he xin s on the regularity of weak solutions to the magnetohydrodynamic equations j differ equ 23 235 2005 and wu s bounds and new approach for the 3d mhd equations j nonlinear sci 12 395 2002 results for 3d ideal mhd equations

global well posedness for the generalized magneto hydrodynamic equations in the critical fourier herz spaces abstract this paper concerns the cauchy problem of the n dimensional generalized incompressible magneto hydrodynamic gmhd equations with 1 2 1 by using the fourier localization argument and the littlewood paley theory we get the global well posedness of the gmhd equations with small initial data u 0 b 0 belongs to the critical fourier herz spaces b q 2 1 with q 1 2 in addition for 2 q ill posedness for the case 1 in b q 1 is also established

initial boundary value problem to 2d boussinesq equations for mhd convection with stratification effects abstract this paper is concerned with the initial boundary value problem to 2d magnetohydrodynamics boussinesq system with the temperature dependent viscosity thermal diffusivity and electrical conductivity first we establish the global weak solutions under the minimal initial assumption then by imposing higher regularity assumption on the initial data we obtain the global strong solution with uniqueness moreover the exponential decay rates of weak solutions and strong solution are obtained respectively

boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations abstract we present some new regularity criteria for suitable weak solutions of magnetohydrodynamic equations near boundary in dimension three we prove that suitable weak solutions are holder continuous near boundary provided that either the scaled l x t p q norm of the velocity with 3 p 2 q 2 2 q or the scaled l x t p q norm of the vorticity with 3 p 2 q 3 2 q are sufficiently small near the boundary

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

on oka s extra zero problem and examples after the solution of cousin ii problem by oka iii in 1939 he thought an extra zero problem in 1945 his posthumous paper asking if it is possible to solve an arbitrarily given cousin ii problem adding some extra zeros whose support is disjoint from the given one by the secondly named author some special case was affirmatively confirmed in dimension two and a counter example in dimension three or more was given the purpose of the present paper is to give a complete solution of this problem with examples and some new questions

the 2d mhd equations with horizontal dissipation and horizontal magnetic diffusion this paper studies the global regularity of classical solutions to the 2d incompressible magnetohydrodynamic mhd equations with horizontal dissipation and horizontal magnetic diffusion it is shown here that the horizontal component of any solution admits a global in time bound in any lebesgue space l2r with 1 r and the bound grows no faster than the order of rlogr as r increases in addition we establish a conditional global regularity in terms of the lt2lx norm of the horizontal component and the global regularity of a slightly regularized version of the aforementioned mhd equations

holomorphic curves and integral points off divisors we deal with the distributions of holomorphic curves and integral points off divisors we will simultaneously prove an optimal dimension estimate from above of a subvariety w off a divisor d which contains a zariski dense entire holomorphic curve or a zariski dense d integral point set provided that in the latter case everything is defined over a number field then if the number of components of d is large the estimate leads to the constancy of such a holomorphic curve or the finiteness of such an integral point set at the beginning we extend logarithmic bloch ochiai s theorem to the kahler case

remarks on the regularity criteria of three dimensional magnetohydrodynamics system in terms of two velocity field components we study the three dimensional magnetohydrodynamics system and obtain its regularity criteria in terms of only two velocity vector field components eliminating the condition on the third component completely the proof consists of a new decomposition of the four nonlinear terms of the system and estimating a component of the magnetic vector field in terms of the same component of the velocity vector field this result may be seen as a component reduction result of many previous works c he and z xin on the regularity of weak solutions to the magnetohydrodynamic equations j differ equ 213 2 234 254 2005 y zhou remarks on regularities for the 3d mhd equations discrete contin dyn syst 12 5 881 886 2005

on manin s conjecture for a family of châtelet surfaces the manin conjecture is established for ch telet surfaces over q arising as minimal proper smooth models of the surface y 2 z 2 f x where f is a totally reducible polynomial of degree 3 without repeated roots these surfaces do not satisfy weak approximation

strong solutions to the incompressible magnetohydrodynamic equations with vacuum we consider the incompressible magnetohydrodynamic mhd equations with the coefficients depending on the density and temperature we prove the existence of unique local strong solutions for all initial data satisfying a natural compatibility condition the initial density need not be positive and may vanish in an open set

continuous data assimilation for the 2d benard convection through velocity measurements alone 2015 elsevier b v an algorithm for continuous data assimilation for the two dimensional benard convection problem is introduced and analyzed it is inspired by the data assimilation algorithm developed for the navier stokes equations which allows for the implementation of variety of observables low fourier modes nodal values finite volume averages and finite elements the novelty here is that the observed data is obtained for the velocity field alone i e no temperature measurements are needed for this algorithm we provide conditions on the spatial resolution of the observed data under the assumption that the observed data is free of noise which are sufficient to show that the solution of the algorithm approaches at an exponential rate the unique exact unknown solution of the benard convection problem associated with the observed finite dimensional projection of velocity

descent on fibrations over the projective line we formulate a general set up for the descent method of j l colliot thelene and j j sansuc applied to varieties fibred over the projective line over a number field this makes it possible to prove that the manin obstruction to the hasse principle and weak approximation is the only one provided there are only few degenerate fibres usually two three in some cases and that sufficiently many smooth fc fibres satisfy the hasse principle and weak approximation we introduce a new concept of what should be called a degenerate fibre the so called split fibres the property which depends only on the generic fibre and not on the choice of a particular model this yields a simpler and more general approach to the previous results of that kind 0 introduction motivated by hasse s theorem on quadratic forms one says that the hasse principle holds for a family of geometrically integral smooth and proper varieties over a number field k9 if for every member of this family the existence of rational points over all the completions of k implies the existence of a rational point the fact that the hasse principle holds for some family signif icantly simplifies the study of its arithmetic because the problem of determining whether or not there is a rational point is then reduced to a finite number of

counting rational points on quartic del pezzo surfaces with a rational conic upper and lower bounds of the expected order of magnitude are obtained for the number of rational points of bounded height on any quartic del pezzo surface over mathbb q that contains a conic defined over mathbb q

on the regularity of generalized mhd equations abstract this paper studies the regularity of generalized magneto hydrodynamics equation on the condition 0 3 2 it will show if u l p q on 0 t with 2 p 3 q 2 3 2 q then the solution can be extended past t

a note on jets of entire curves in semi abelian varieties we prove a product decomposition of the zariski closure of every jet lift of an entire curve f c a into a semi abelian variety a provided that f is of finite order on the other hand by giving an example of f into a three dimensional abelian variety we show that this product decomposition does not hold in general there was a gap in the proofs of 2 proposition 1 8 ii and of 6 theorem 2 2