lattice polytopes cut out by root systems and the koszul property we show that lattice polytopes cut out by root systems of classical type are normal and koszul generalizing a well known result of bruns gubeladze and trung in type a we prove similar results for cayley sums of collections of polytopes whose minkowski sums are cut out by root systems the proofs are based on a combinatorial characterization of diagonally split toric varieties

coverage probability and achievable rate analysis of ffr aided multi user ofdm based mimo and simo systems expressions are derived for the coverage probability and average rate of both multi user multiple input multiple output mu mimo and single input multiple output simo systems in the context of a fractional frequency reuse ffr scheme in particular given a reuse region of frac 1 3 fr3 and a reuse region of 1 fr1 as well as a signal to interference plus noise ratio sinr threshold s th which decides the user assignment to either the fr1 or fr3 regions we theoretically show that 1 the optimal choice of s th which maximizes the coverage probability is s th t where t is the target sinr required for ensuring adequate coverage and 2 the optimal choice of s th which maximizes the average rate is given by s th t prime where t prime is a function of the path loss exponent the number of antennas and of the fading parameters the impact of frequency domain correlation amongst the ofdm sub bands allocated to the fr1 and fr3 cell regions is analysed and it is shown that the presence of correlation reduces both the coverage probability and the average throughput of the ffr network furthermore the performance of our ffr aided mu mimo and simo systems is compared our analysis shows that the 2 times 2 mu mimo system achieves 22 5 higher rate than the 1 times 3 simo system and for lower target sinrs the coverage probability of a 2 times 2 mu mimo system is comparable to a 1 times 3 simo system hence the former one may be preferred over the latter our simulation results closely match the analytical results

the existence of solitary wave solutions of delayed camassa holm equation via a geometric approach abstract this paper is concerned with the camassa holm equation which is a model for shallow water waves we first establish the existence of solitary wave solutions for the equation without delay and then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system especially the geometric singular perturbation theory and invariant manifold theory according to the relationship between solitary wave and homoclinic orbit the camassa holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution it is proved that the equation with disturbance also possesses homoclinic orbit and there exists solitary wave solution of the delayed camassa holm equation

continuity and minimization of spectrum related with the periodic camassa holm equation abstract an important point in looking for period solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y m t y the first aim of this paper is to study the dependence of eigenvalues for the periodic camassa holm equation on potentials as an infinitely dimensional parameter to be precise we prove that as nonlinear functionals of potentials eigenvalues for the periodic camassa holm equation are continuous in potentials with respect to the weak topologies in the l p lebesgue spaces the second aim of this paper is to find the optimal lower bound of the lowest eigenvalue for the periodic camassa holm equation when the l 1 norm of potentials are given in order to make our results more applicable we will find the optimal lower bound for the lowest eigenvalue in the more general setting of measure differential equations

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

multigraded regularity and the koszul property abstract we give a criterion for the section ring of an ample line bundle to be koszul in terms of multigraded regularity we discuss applications to adjoint bundles on toric varieties as well as to polytopal semigroup rings

continuous dependence and estimates of eigenvalues for periodic generalized camassa holm equations abstract in this paper we are concerned with the spectral problem y 1 y 2 x 1 2 1 2 m 1 2 m 1 2 y 1 y 2 for the periodic generalized camassa holm equations the first aim is to study the structure of eigenvalues and prove the continuous dependence of eigenvalues on potentials we prove that as nonlinear functionals of potentials eigenvalues are continuous in potentials with respect to the weak topologies in the lebesgue space l 1 0 t the second aim is to find the estimates of the eigenvalues when the l 1 norm of potentials is given based on an application of trace formulas

integrable evolution equations on spaces of tensor densities and their peakon solutions we study a family of equations defined on the space of tensor densities of weight on the circle and introduce two integrable pde one of the equations turns out to be closely related to the inviscid burgers equation while the other has not been identified in any form before we present their lax pair formulations and describe their bihamiltonian structures we prove local wellposedness of the corresponding cauchy problem and include results on blow up as well as global existence of solutions moreover we construct peakon and multi peakon solutions for all 0 1 and shock peakons for 3 we argue that there is a natural geometric framework for these equations that includes other well known integrable equations and which is based on v arnold s approach to euler equations on lie groups

the cauchy problem of a periodic 2 component μ hunter saxton system in besov spaces abstract this paper is concerned with the local well posedness and the precise blow up scenario for a periodic 2 component hunter saxton system in besov spaces moreover we state a new global existence result to the system our obtained results for the system improve considerably earlier results

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

generalized hunter saxton equation and the geometry of the group of circle diffeomorphisms we study an equation lying mid way between the periodic hunter saxton and camassa holm equations and which describes evolution of rotators in liquid crystals with external magnetic field and self interaction we prove that it is an euler equation on the diffeomorphism group of the circle corresponding to a natural right invariant sobolev metric we show that the equation is bihamiltonian and admits both cusped and smooth traveling wave solutions which are natural candidates for solitons we also prove that it is locally well posed and establish results on the lifespan of its solutions throughout the paper we argue that despite similarities to the kdv ch and hs equations the new equation manifests several distinctive features that set it apart from the other three

initial ideals veronese subrings and rates of algebras abstract let s be a polynomial ring over an infinite field and let i be a homogeneous ideal of s let t d be a polynomial ring whose variables correspond to the monomials of degree d in s we study the initial ideals of the ideals v d i t d that define the veronese subrings of s i in suitable orders they are easily deduced from the initial ideal of i we show that in v d i is generated in degree max reg i d 2 where reg i is the regularity of the ideal i in other words the d th veronese subrings of any commutative graded ring s i has a grobner basis of degree max i d 2 we also give bounds on the regularity of i in terms of the degrees of the generators of in i and some combinatorial data this implies a version of backelin s theorem that high veronese subrings of any ring are homogeneous koszul algebras in the sense of priddy trans amer math soc 152 1970 39 60 we also give a general obstruction for a homogeneous ideal i s to have an initial ideal in i that is generated by quadrics beyond the obvious requirement that i itself should be generated by quadrics and the stronger statement that s i is koszul we use the obstruction to show that in certain dimensions a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics for the application to backelin s theorem we require a result of backelin whose proof has never appeared we give a simple proof of a sharpened version bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials following a method of bruns and herzog

koszul homology and syzygies of veronese subalgebras a graded k algebra r has property np if it is generated in degree 1 has relations in degree 2 and the syzygies of order p on the relations are linear the green lazarsfeld index of r is the largest p such that it satisfies the property np our main results assert that under a mild assumption on the base field the cth veronese subring of a polynomial ring has green lazarsfeld index c 1 the same conclusion also holds for an arbitrary standard graded algebra provided c gg 0

stability of periodic peakons for the modified μ camassa holm equation abstract considered herein is the dynamical stability of periodic peaked solitons for the modified camassa holm equation with the cubic nonlinearity the equation is a version of the modified camassa holm equation and is integrable with the lax pair and bi hamiltonian structure the equation admits the periodic peakons it is shown that the periodic peakons are dynamically stable under small perturbations in the energy space by finding inequalities related to the first three conservation laws with global maximum and minimum of the solution

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

local well posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude abstract we study the local well posedness of a periodic nonlinear equation for surface waves of moderate amplitude in shallow water we use an approach proposed by kato which is based on semigroup theory for quasi linear equations we also show that singularities for the model equation can occur only in the form of wave breaking in particular surging breakers

wave breaking phenomena for the nonlocal whitham type equations abstract in this paper the formation of singularities for the nonlocal whitham type equations is studied it is shown that if the lowest slope of flows can be controlled by its highest value with the bounded whitham type integral kernel initially then the finite time blow up will occur in the form of wave breaking this refined wave breaking result is established by analyzing the monotonicity and continuity properties of a new system of the riccati type differential inequalities involving the extremal slopes of flows our theory is illustrated via the whitham equation camassa holm equation degasperis procesi equation and their versions as well as hyperelastic rod equation

computing graded betti tables of toric surfaces we present various facts on the graded betti table of a projectively embedded toric surface expressed in terms of the combinatorics of its defining lattice polygon these facts include explicit formulas for a number of entries as well as a lower bound on the length of the quadratic strand that we conjecture to be sharp and prove to be so in several special cases we also present an algorithm for determining the graded betti table of a given toric surface by explicitly computing its koszul cohomology and report on an implementation in sagemath it works well for ambient projective spaces of dimension up to roughly 25 depending on the concrete combinatorics although the current implementation runs in finite characteristic only as a main application we obtain the graded betti table of the veronese surface nu 6 p 2 subset of p 27 in characteristic 40 009 this allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary veronese surface nu d p 2

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

well posedness wave breaking and peakons for a modified μ camassa holm equation considered herein is a modified periodic camassa holm equation with cubic nonlinearity which is called the modified camassa holm equation the proposed equation is shown to be formally integrable with the lax pair and bi hamiltonian structure local well posedness of the initial value problem to the modified camassa holm equation in the besov space is established existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are then investigated it is shown that the equation admits a single peaked soliton and multi peakon solutions with a similar character of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and several wave breaking mechanisms for solutions with certain initial profiles are described in detail

on the blow up structure for the generalized periodic camassa holm and degasperis procesi equations considered herein are the generalized camassa holm and degasperis procesi equations in the spatially periodic setting the precise blow up scenarios of strong solutions are derived for both of equations several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established the exact blow up rates are also determined finally geometric descriptions of these two integrable equations from non stretching invariant curve flows in centro equiaffine geometries pseudo spherical surfaces and affine surfaces are given

isotropic fading vector broadcast channels the scalar upper bound and loss in degrees of freedom we propose a scalar upper bound on the capacity region of the isotropic fading vector broadcast channel in terms of the capacity region of a scalar fading broadcast channel the scalar upper bound is applicable to the broad class of isotropic fading broadcast channels regardless of the distribution of the users channel magnitudes the distribution of the additive noise experienced by each user or the amount of channel knowledge available at the receiver using this upper bound we prove the optimality of the alamouti scheme in a broadcast setting extend the recent results on the capacity of nondegraded fading scalar broadcast channels to nondegraded fading vector broadcast channels and determine the capacity region of a fading vector gaussian broadcast channel with channel magnitude feedback we also provide an example of a rayleigh fading broadcast channel with no channel state information available to the receiver csir where the bound on the capacity region obtained by a naive application of the scalar upper bound is provably loose because it fails to account for the additional loss in degrees of freedom due to lack of channel knowledge at the receiver a tighter upper bound is obtained by separately accounting for the loss in degrees of freedom due to lack of csir before applying the scalar upper bound

all abelian quotient c i singularities admit projective crepant resolutions in all dimensions abstract for gorenstein quotient spaces c d g a direct generalization of the classical mckay correspondence in dimensionsd 4 would primarily demand the existence of projective crepant desingularizations since this turned out to be not always possible reid asked about special classes of such quotient spaces that would satisfy the above property we prove that the underlying spaces of all gorenstein abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space have torus equivariant projective crepant resolutions in all dimensions we use techniques from toric and discrete geometry

continuous properties of the data to solution map for a generalized μ camassa holm integrable equation this work studies a generalized type integrable equation with both quadratic and cubic nonlinearities the camassa holm and modified camassa holm equations are members of this family of equations it has been shown that the cauchy problem for this generalized camassa holm integrable equation is locally well posed for initial data u0 hs s 5 2 in this work we further investigate the continuity properties to this equation it is proved in this work that the data to solution map of the proposed equation is not uniformly continuous it is also found that the solution map is holder continuous in the hr topology when 0 r 5 2 in this work we further investigate the continuity properties to this equation it is proved in this work that the data to solution map of the proposed equation is not uniformly continuous it is also found that the solution map is holder continuous in the hr topology when 0 r s with holder exponent depending on both s and r

zero forcing methods for downlink spatial multiplexing in multiuser mimo channels the use of space division multiple access sdma in the downlink of a multiuser multiple input multiple output mimo wireless communications network can provide a substantial gain in system throughput the challenge in such multiuser systems is designing transmit vectors while considering the co channel interference of other users typical optimization problems of interest include the capacity problem maximizing the sum information rate subject to a power constraint or the power control problem minimizing transmitted power such that a certain quality of service metric for each user is met neither of these problems possess closed form solutions for the general multiuser mimo channel but the imposition of certain constraints can lead to closed form solutions this paper presents two such constrained solutions the first referred to as block diagonalization is a generalization of channel inversion when there are multiple antennas at each receiver it is easily adapted to optimize for either maximum transmission rate or minimum power and approaches the optimal solution at high snr the second known as successive optimization is an alternative method for solving the power minimization problem one user at a time and it yields superior results in some e g low snr situations both of these algorithms are limited to cases where the transmitter has more antennas than all receive antennas combined in order to accommodate more general scenarios we also propose a framework for coordinated transmitter receiver processing that generalizes the two algorithms to cases involving more receive than transmit antennas while the proposed algorithms are suboptimal they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity

non uniform dependence on initial data for the modified μ camassa holm equation considered herein is the initial value problem for the modified camassa holm equation it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs s for s 5 2 on the circle based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

on a completely integrable nonlinear hyperbolic variational equation abstract we show that the nonlinear partial differential equation u t uu x xx 1 2 u x 2 x is a completely integrable bi variational bi hamiltonian system the corresponding equation for w u xx belongs to the harry dym hierarchy this equation arises in two different physical contexts in two nonequivalent variational forms it describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field and it is the high frequency limit of the camassa holm equation which is an integrable model equation for shallow water waves using the bi hamiltonian structure we derive a recursion operator a lax pair and an infinite family commuting hamiltonian flows together with the associated conservation laws we also give the transformation to action angle coordinates smooth solutions of the partial differential equation break down because their derivative blows up in finite time nevertheless the hamiltonian structure and complete integrability appear to remain valid globally in time even after smooth solutions break down we show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions we compute the bi hamiltonian structure on this invariant manifold which is obtained by restricting the bi hamiltonian structure of the partial differential equation

holder continuity on μ b equation abstract this paper is concerned with the holder continuous on initial data for the b equation on the circle it is shown that the data to solution for the b equation is holder continuous from bounded set of sobolev spaces with exponent s 3 2 measured in a weaker sobolev norm with index r s in periodic case

the degasperis procesi equation as a non metric euler equation in this paper we present a geometric interpretation of the degasperis procesi dp equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle we also show that for any evolution in the family of b equations there is neither gain nor loss of the spatial regularity of solutions this in turn allows us to view the dp and the camassa holm equation as an ode on the frechet space of all smooth functions on the circle

blow up solutions and peakons to a generalized μ camassa holm integrable equation considered here is a generalized type integrable equation which can be regarded as a generalization to both the camassa holm and modified camassa holm equations it is shown that the proposed equation is formally integrable with the lax pair and the bi hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary gravity waves local well posedness of the cauchy problem in the suitable sobolev space is established by the viscosity method existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are investigated it is found that the equation admits single and multi peaked traveling wave solutions the effects of varying camassa holm and modified camassa holm nonlocal nonlinearities on blow up criteria and wave breaking are illustrated in detail our analysis relies on the method of characteristics and conserved quantities and is proceeded with a priori differential estimates

euler equations on homogeneous spaces and virasoro orbits abstract we show that the following three systems related to various hydrodynamical approximations the korteweg de vries equation the camassa holm equation and the hunter saxton equation have the same symmetry group and similar bihamiltonian structures it turns out that their configuration space is the virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right invariant metrics on this group or on appropriate homogeneous spaces in particular we describe how arnold s approach to the euler equations as geodesic flows of one sided invariant metrics extends from lie groups to homogeneous spaces we also show that the above three cases describe all generic bihamiltonian systems which are related to the virasoro group and can be integrated by the translation argument principle they correspond precisely to the three different types of generic virasoro orbits finally we discuss interrelation between the above metrics and kahler structures on virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

k symplectic structures the basic properties of k symplectic structures are introduced and developed in analogy with the well known symplectic differential geometry examples of such structures related to k symplectic lie algebras are given

blow up phenomena and persistence properties for an integrable two component peakon system abstract in this paper we are concerned with an integrable two component peakon system which was proposed by xia qiao and zhou we present a precise blow up scenario and a new blow up result for strong solutions to the system moreover we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically respectively

global existence and blow up for a weakly dissipative μdp equation abstract in this paper we study a weakly dissipative variant of the periodic degasperis procesi equation we show the local well posedness of the associated cauchy problem in h s s s 3 2 and discuss the precise blow up scenario for s 3 we also present explicit examples for globally existing solutions and blow up

convex normality of rational polytopes with long edges abstract we introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths guaranteeing the property as an application we show that if every edge of a lattice d polytope p has lattice length 4 d d 1 then p is normal this answers in the positive a question raised in 2007 if p is a lattice simplex whose edges have lattice lengths d d 1 then p is even covered by lattice parallelepipeds for the approach developed here it is necessary to involve rational polytopes even for the application to lattice polytopes

polytopal linear groups the main objects of this paper are the graded automorphisms of w x polytopal semigroup rings i e semigroup rings k s where k is a field p and s is the semigroup associated with a lattice polytope p bruns p w x w x gubeladze and trung bgt the generators of k s correspond bijecp tively to the lattice points in p and their relations are the binomials representing the affine dependencies of the lattice points the simplest examples of such rings are the polynomial rings w x k x x with the standard grading they are associated to the unit 1 n w x simplices d the graded automorphism group gl k of k x x ny1 n 1 n is generated by diagonal and elementary automorphisms our main result is a generalization of this classical fact to the graded automorphism group g p of an arbitrary polytopal semigroup ring it says that each automork phism g g g p has a non unique normal form as a composition of toric k and elementary automorphisms and symmetries of the underlying poly tope in view of this analogy we call the groups g p polytopal linear k groups as in the case of gl k the elementary and toric automorphisms n generate g p if and only if this group is connected the elementary k automorphisms are defined in terms of so called column structures on p

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

high rank linear syzygies on low rank quadrics we study the linear syzygies of a homogeneous ideal i subseteq s rm sym bf k v focussing on the graded betti numbers b i i 1 rm dim bf k rm tor i s i bf k i 1 for a variety x and divisor d with v h 0 d what conditions on d ensure that b i i 1 ne 0 eisenbud has shown that a decomposition d sim a b such that a and b have at least two sections gives rise to determinantal equations and corresponding syzygies in i x and conjectured that if i 2 is generated by quadrics of rank le 4 then the last nonvanishing b i i 1 is a consequence of such equations we describe obstructions to this conjecture and prove a variant the obstructions arise from toric specializations of the rees algebra of koszul cycles and we give an explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank this gives one answer to a question posed by eisenbud and koh

on the log concavity of hilbert series of veronese subrings and ehrhart series for every positive integer n consider the linear operator un on polynomials of degree at most d with integer coefficients defined as follows if we write frac h t 1 t d 1 sum m geq 0 g m t m for some polynomial g m with rational coefficients then frac rm u n h t 1 t d 1 sum m geq 0 g nm t m we show that there exists a positive integer nd depending only on d such that if h t is a polynomial of degree at most d with nonnegative integer coefficients and h 0 1 then for n nd unh t has simple real negative roots and positive strictly log concave and strictly unimodal coefficients applications are given to ehrhart polynomials and unimodular triangulations of dilations of lattice polytopes as well as hilbert series of veronese subrings of cohen macauley graded rings

orbital stability of periodic peakons to a generalized μ camassa holm equation in this paper we study the orbital stability of the periodic peaked solitons of the generalized camassa holm equation with nonlocal cubic and quadratic nonlinearities the equation is a version of a linear combination of the camassa holm equation and the modified camassa holm equation it is also integrable with the lax pair and bi hamiltonian structure and admits the single peakons and multi peakons by constructing an inequality related to the maximum and minimum of solutions with the conservation laws we prove that even in the case that the camassa holm energy counteracts in part the modified camassa holm energy the shapes of periodic peakons are still orbitally stable under small perturbations in the energy space

toric ideals generated by quadratic binomials abstract a combinatorial criterion for the toric ideal arising from a finite graph to be generated by quadratic binomials is studied such a criterion guarantees that every koszul algebra generated by squarefree quadratic monomials is normal we present an example of a normal non koszul squarefree semigroup ring whose toric ideal is generated by quadratic binomials as well as an example of a non normal koszul squarefree semigroup ring whose toric ideal possesses no quadratic grobner basis in addition all the affine semigroup rings which are generated by squarefree quadratic monomials and which have 2 linear resolutions will be classified moreover it is shown that the toric ideal of a normal affine semigroup ring generated by quadratic monomials is generated by quadratic binomials if its underlying polytope is simple

on the equations defining toric l c i singularities based on nakajima s classification theorem we describe the precise form of the binomial equations which determine toric locally complete intersection l c i singularities

normality and covering properties of affine semigroups s x gp s mx s for some m 0 one calls s normal if s s for simplicity we will often assume that gp s z this is harmless because we can replace z by gp s if necessary the rank of s is the rank of gp s we will only be interested in the case in which s s 0 such affine semigroups will be called positive the positivity of s is equivalent to the pointedness of the cone c s r s generated by s in r one has ranks dimc s all the cones appearing in this paper have their apex at the origin the normality of s can now be characterized geometrically one obviously has s c s z and so s is normal if and only if s c s z conversely every convex pointed finitely generated rational cone c r yields a normal affine semigroup s c c z this semigroup is finitely generated by gordan s lemma if we simply speak of a cone c in the following then it is always assumed that c is convex pointed finitely generated and rational since dimc s ranks we often call the dimension of a cone its rank each positive affine semigroup s can be embedded into z m ranks one uses m linearly independent integral linear forms representing support hyperplanes of c s therefore every element can be written as a sum of irreducible elements and the set of irreducible elements of s is finite since s is finitely generated for a cone c the set of irreducible elements of s c is often called the hilbert basis hilb c of c in the combinatorial literature a convention we will follow more generally we also call the set of irreducible elements of a positive affine semigroup s its hilbert basis and denote it by hilb s this paper is devoted to a discussion of sufficient and potentially necessary conditions for the normality of s in terms of combinatorial properties of hilb s as far as the necessity is concerned these conditions can of course be formulated in terms of hilbert bases of rational cones the first property under consideration is the existence of a unimodular hilbert cover

on the solutions of a model equation for shallow water waves of moderate amplitude abstract this paper is concerned with the cauchy problem of a model equation for shallow water waves of moderate amplitude which was proposed by a constantin and d lannes the hydrodynamical relevance of the camassa holmand degasperis procesi equations arch ration mech anal 192 2009 165 186 first the local well posedness of the model equation is obtained in besov spaces b p r s p r 1 s max 3 2 1 1 p which generalize the sobolev spaces h s by using littlewood paley decomposition and transport equation theory second the local well posedness in critical case with s 3 2 p 2 r 1 is considered moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time finally persistence properties on strong solutions are also investigated

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

on blow up of solutions to the two component π camassa holm system abstract we investigate cauchy problem of the two component camassa holm system which arises from geodesic flow on a semidirect product lie group of the circle the precise blow up scenarios of strong solutions are derived for the system then several criteria to guarantee blow up of strong solutions are presented finally the exact blow up rate is determined

integrable equations arising from motions of plane curves the motion of plane curves in klein geometry is studied it is shown that the kdv harry dym sawada kotera burgers the defocusing mkdv hierarchies the camassa holm and the kaup kupershmidt equation naturally arise from the motions of plane curves in sl 2 sim 2 sa 2 and a 2 geometries these local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area motions of curves in euclidean special linear and similarity geometries corresponding to the traveling wave solutions of the mkdv kdv and burgers equations are discussed