effective masses and conformal mappings letgn n n denote the set of gaps of the hill operator we solve the following problems 1 find the effective massesmn 2 compare the effective massmn with the length of the gapgn and with the height of the corresponding slit on the quasimomentum plane both with fixed numbern and their sums 3 consider the problems 1 2 for more general cases the dirac operator with periodic coefficients the schrodinger operator with a limit periodic potential to obtain 1 3 we use a conformal mapping corresponding to the quasimomentum of the hill operator or the dirac operator

mkdv and its birkhoff coordinates abstract we show that the modified korteweg de vries equation with periodic boundary conditions admits global birkhoff coordinates on any soblev space h n with n 1 the construction of these coordinates involves a new lax pair for mkdv

estimates for periodic and dirichlet eigenvalues of the schrodinger operator with singular potentials abstract in this paper the periodic and the dirichlet problems for the schrodinger operator d2 dx2 v are studied for singular complex valued potentials v in the sobolev space h per 0 1 0 0 2 n 1 2 n n 2 2 v 0 v 2 n v 2 n o n 3 2 1 2 e where v k denote the fourier coefficients of v 2 the dirichlet spectrum consists of a sequence n n 1 of complex eigenvalues satisfying the asymptotics for any e 0 n n 2 2 v 0 v 2 n v 2 n 2 o n 2 1 e

spectral asymptotics for schrodinger operators with periodic point interactions abstract spectrum of the second order differential operator with periodic point interactions in l 2 r is investigated classes of unitary equivalent operators of this type are described spectral asymptotics for the whole family of periodic operators are calculated it is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely it is proven that the spectrum of the operators with anomalous spectral asymptotics when the ratio between the lengths of the bands and gaps tends to zero at infinity can be approximated by standard periodic weighted operators with step wise density functions it is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal generalized resolvent of the periodic schrodinger operator with certain energy dependent boundary conditions the operator acting in an extended hilbert space such that its resolvent restricted to l 2 r coincides with the formal resolvent is constructed explicitly

the estimates of periodic potentials in terms of effective masses letgn an an n 1 denote the gaps mn be the effective masses and n an 1 an a0 0 be the spectral bands of the hill operatort d2 dx2 v x inl2 r wherev is a 1 periodic real potential froml2 0 1 let the length gapln gn hn be the height of the corresponding slit on the quasimomentum domain and n 2 2n 1 n 0 be the band reduction let l n sqrt a n sqrt a n n geqq 1 n 1 denote the gap length for the operator sqrt t geqq 0 introduce the sequencesl ln h hn l ln n m mn and the norms left f right m 2 sum nolimits n 0 left 2 pi n right 2m f n 2 m geqq 0 m 0 the following results are obtained i the estimates of v l h 1 l 1 in terms of m 2 ii identities for the dirichlet integral of quasimomentum and integral of potentials and so on iii the generation of i ii for more general potentials

cohen macaulayness of monomial ideals and symbolic powers of stanley reisner ideals abstract we present criteria for the cohen macaulayness of a monomial ideal in terms of its primary decomposition these criteria allow us to use tools of graph theory and of linear programming to study the cohen macaulayness of monomial ideals which are intersections of prime ideal powers we can characterize the cohen macaulayness of the second symbolic power or of all symbolic powers of a stanley reisner ideal in terms of the simplicial complex these characterizations show that the simplicial complex must be very compact if some symbolic power is cohen macaulay in particular all symbolic powers are cohen macaulay if and only if the simplicial complex is a matroid complex we also prove that the cohen macaulayness can pass from a symbolic power to another symbolic powers in different ways

estimates of periodic potentials in terms of gap lengths define the hill operator t d 2 dx 2 q x in l 2 and suppose q l 2 0 1 is a 1 periodic real potential 0 1 q x dx 0 we prove the estimate 2 1 1 3 where 2 n 1 n 2 and n 0 n 1 is the gap length of t

an incremental linear time algorithm for recognizing interval graphs the fastest known algorithm for recognizing interval graphs s booth and s lucker j comput system sci 13 1976 pp 335 379 iteratively manipulates the system of all maximal cliques of the given graph in a rather complicated way in order to construct a consecutive arrangement more precisely a tree representation of all possible consecutive arrangements this paper presents a much simpler algorithm using a related but much more informative tree representation of interval graphs this tree is constructed in an incremental fashion by adding vertices to the graph in a predefined order such that adding a vertex u takes o operatorname adj u 1 amortized time

periodic weighted operators abstract consider the periodic weighted operator ty 2 2 y in l 2 r x 2 dx where the real function is 1 periodic positive and let q l 2 0 1 the spectrum of t consists of intervals separated by gaps n n 1 with the lengths n let h n be a height of the corresponding slit in the quasimomentum domain and let g n n 1 be the gap with the length g n of the operator t 0 the following results are obtained i the quasimomentum k for the weighted operator t is constructed and the basic properties of k are studied ii two sided estimates of l 2 norms of the sequences n n 1 h n 1 g n 1 in terms of 0 1 q x 2 dx iii the asymptotics of the gap length n as n are determined the proofs are based on the analysis of the quasimomentum as a conformal mapping embedding theorems and the identities between the quasimomentum and the potential in order to prove these results the asymptotics of the fundamental solutions and the lyapunov function are obtained at high energy

cohen macaulayness of large powers of stanley reisner ideals abstract we prove that for m 3 the symbolic power i m of a stanley reisner ideal is cohen macaulay if and only if the simplicial complex is a matroid similarly the ordinary power i m is cohen macaulay for some m 3 if and only if i is a complete intersection these results solve several open questions on the cohen macaulayness of ordinary and symbolic powers of stanley reisner ideals moreover they have interesting consequences on the cohen macaulayness of symbolic powers of facet ideals and cover ideals

solution of the inverse spectral problem for an impedance with integrable derivative part ii the inverse eigenvalue problem for a sturm liouville equation in impedance form with dirichlet boundary conditions on a unit interval is solved the solution of this nonlinear problem requires the investigation of a combined map from the logarithmic derivative of the impedance assumed to be square integrable to two sequences of spectral data the first is the sequence of dirichlet eigenvalues shown to be locally bounded with square roots differing from the sequence of integral multiples of by a square summable sequence the second sequence has as first term the mean of the logarithmic derivative of the impedance each term in the remainder of the sequence is the logarithm of the product of the impedance and the derivative of an eigenfunction evaluated at an endpoint it is shown that this second sequence is a locally bounded square summable sequence the combined map is a real analytic isomorphism a uniqueness result follows from the asymptotics and analyticity shown in the first paper and properties of riesz bases the existence is shown by giving a closed form of the solution it is also shown that the same map solves the inverse sturm liouville problem in impedance form with neumann boundary conditions 1993 john wiley sons inc

breather solutions in periodic media for nonlinear wave equations existence proofs for breathers are very rare in the spatially homogeneous case up to rescaling the sine gordon equation partial 2 t u partial 2 x u sin u is the only nonlinear wave equation which is known to possess breather solutions for nonlinear wave equations in periodic media no examples of breather solutions have been known so far using spatial dynamics center manifold theory and bifurcation theory for periodic systems we construct for the first time such time periodic solutions of finite energy for a nonlinear wave equation s x partial 2 t u x t partial 2 x u x t q x u x t r x u x t 3 with spatially periodic coefficients s q and r on the real axis such breather solutions play an important role in theoretical scenarios where photonic crystals are used as optical storage

asymptotics of instability zones of the hill operator with a two term potential let n denote the length of the nth zone of instability of the hill operator ly y 4t cos2x 2 2cos4x y where 0 and either both t are real or both are pure imaginary numbers for even n we prove if t n are fixed then for 0 n 8 n2n n 1 2 k 1n 2 t2 2k 1 2 1 o and if t are fixed then for n n 8 2 n 2 4 n 2 2 cos 2t 1 o lognn the asymptotics for 0 for n 2m imply the following identities for squares of integers s 1k m2 is2 1 j1 jk m s 1k 2js 1 2 where 1 k m and the left sum is over all indices i1 ik such that m i1 ik m is ir 2if s r similar formulae see theorems 7 9 hold for odd n

on zeros of some entire functions we study the distribution of zeros z k for entire functions of the form sin z f 1 0 f t e iz 1 2t dt with f belonging to a space x l 1 0 1 for a large class x of spaces x including e g the spaces lp 0 1 for all p 1 oo we show that z k k k where k k z is the sequence of fourier coefficients for some function g in x and study properties of the induced mapping g f

the inverse problem for the hill operator a direct approach we consider the hill operator t yd2 dx2 v x on l2 r where v 2 l2 0 1 is an even 1 periodic real potential and r 1 0 v x dx 0 it is wellknown that the spectrum of t is absolutely continuous and consists of intervals s1 s2 such that sn any1 an any1 an an n 1 these intervals are separated by gaps g1 g2 where gn an an if a gap degenerates that is gn then the corresponding segments sn sn 1 merge let u x e v x e v be the solutions of the equation yf 00 vf ef e 2 c 1 1

inverse spectral problems for sturm liouville operators in impedance form abstract the spectral properties of sturm liouville operators with an impedance in w p 1 0 1 for some p 1 are studied in particular a complete solution of the inverse spectral problem is provided

eigenvalue asymptotics for sturm liouville operators with singular potentials abstract we derive eigenvalue asymptotics for sturm liouville operators with singular complex valued potentials from the space w 2 1 0 1 0 1 and dirichlet or neumann dirichlet boundary conditions we also give application of the obtained results to the inverse spectral problem of recovering the potential from these two spectra

level property of ordinary and symbolic powers of stanley reisner ideals abstract in this paper we prove that the t th ordinary and or symbolic power of a stanley reisner ideal is level for some positive integer t 3 if and only if i is a complete intersection and equi generated for t 2 we give a characterization of level property of the second symbolic power i 2 when is a matroid complex of dimension one

level rings and algebras with straightening laws abstract stanley 13 defined the concept of level rings intermediate between cohen macaulay and gorenstein which is of interest from viewpoints of both commutative algebra and combinatorics via the theory of canonical modules the study of level rings turns out to be one of the most important foundations of the theory of algebras with straightening laws the purpose of this paper is to state some basic results on level rings which are either stanley reisner rings or algebras with straightening laws

estimates for periodic zakharov shabat operators abstract we consider the periodic zakharov shabat operators on the real line the spectrum of this operator consists of intervals separated by gaps with the lengths g n 0 n z let n be the corresponding effective masses and let h n be heights of the corresponding slits in the quasi momentum domain we obtain a priori estimates of sequences g g n n z n n z h h n n z in terms of weighted l p norms at p 1 the proof is based on the analysis of the quasi momentum as the conformal mapping

riccati map on l02 t and its applications abstract this paper is concerned with the spectral properties of the schrodinger operator l q def d 2 d x 2 q with periodic potential q from the sobolev space h 1 t we obtain asymptotic formulas and a priori estimates for the periodic and dirichlet eigenvalues which generalize known results for the case of potentials q l 2 t the key idea is to reduce the problem to a known one the spectrum of the impedance operator via a nonlinear analytic isomorphism between l 0 2 t and the sobolev space h 0 1 t

inverse problem for periodic weighted operators define the periodic weighted operator ty 2 2y in l2 r x 2 dx suppose a function w21 r z is 1 periodic real positive 0 1 and let q l2 0 1 the spectrum of t consists of intervals n n 1 n separated by gaps n n n n 1 with the lengths n and 0 0 let m2n n 1 be the dirichlet eigenvalue of the equation y 2qy z2y y 0 y 1 0 where mn 0 introduce the lyapunov function z q for t and note that z zn q 0 for some zn n n let x z q be the solution of the equation 2q z2 z c satisfying 0 z q 0 0 z q 1 introduce the vector hn hcn hsn r2 with components hcn log 1 n 1 mn q hsn hn 2 hcn 2 1 2 sign zn mn where hn is defined by the equation cosh hn 1 n zn q 1 and coincides with the euclidien norm of the vector hn using nonlinear functional analysis in hilbert space we prove that the mapping h q h q hn 1 is a real analytic isomorphism

inverse spectral theory for uniformly open gaps in a weighted sturm liouville problem abstract motivated by a pde existence problem we study the inverse problem for a weighted sturm liouville operator l s associated with the eigenvalue problem y s x y 0 where s is a real valued periodic even function that is bounded from below by a positive constant and belongs to the l 2 based sobolev space h r 0 1 r 1 choosing gap lengths and gap midpoints as coordinates we define a spectral map g that assigns to a coefficient s the structure of the spectrum of l s we find that g is a real analytic isomorphism locally around s 1 which in particular implies the existence of coefficients s h r 0 1 r 3 2 whose spectrum features band structure with all gaps uniformly open around the gap midpoints this result paves the way for the construction of so called breathers in nonlinear wave equations with such coefficients s apart from the novelty of treating the inverse spectral problem for the full banach scale h r 0 1 r 1 the local nature of our result allows more concise and transparent proofs in particular instead of using any preliminary transformations we treat the weighted problem directly by adapting techniques used for schrodinger operators with distribution potentials

enumeration of nonisomorphic interval graphs and nonisomorphic permutation graphs abstract in this paper a general framework for enumerating every element in a graph class is given the main feature of this framework is that it is designed to enumerate only nonisomorphic graphs in a graph class applying this framework to the classes of interval graphs and permutation graphs we give efficient enumeration algorithms for these graph classes such that each element in the class is output in a polynomial time delay the experimental results are also given the catalogs of graphs in these graph classes are also provided

cohen macaulayness for symbolic power ideals of edge ideals abstract let s k x 1 x n be a polynomial ring over a field k let i g s denote the edge ideal of a graph g we show that the lth symbolic power i g l is a cohen macaulay ideal i e s i g l is cohen macaulay for some integer l 3 if and only if g is a disjoint union of finitely many complete graphs when this is the case all the symbolic powers i g l are cohen macaulay ideals similarly we characterize graphs g for which s i g l has flc as an application we show that an edge ideal i g is complete intersection provided that s i g l is cohen macaulay for some integer l 3 this strengthens the main theorem in crupi et al 2010 3

the marchenko ostrovski mapping and the trace formula for the camassa holm equation we consider the periodic weighted operator ty 2 2y 14 4 in l2 r 2dx where is a 1 periodic positive function satisfying q l2 0 1 the spectrum of t consists of intervals separated by gaps in the first part of the paper we construct the marchenko ostrovski mapping q h q and solve the corresponding inverse problem for our approach it is essential that the mapping h has the factorization h q h0 v q where q v q is a certain nonlinear mapping and v h0 v is the marchenko ostrovski mapping for the hill operator moreover we solve the inverse problem for the gap length mapping in the second part of this paper we derive the trace formula for t

linear time automorphism algorithms for trees interval graphs and planar graphs an algorithm based upon edmonds s procedure for testing isomorphism of trees is extended to answer various questions concerning automorphisms of a labeled forest this and linear pattern matching techniques are used to build efficient algorithms which find the automorphism partition and a set of generators for the automorphism group determine the order of the automorphism group and compute a coding for forests interval graphs outerplanar graphs and planar graphs

cohen macaulayness of powers of two dimensional squarefree monomial ideals abstract two dimensional squarefree monomial ideals can be seen as the stanley reisner ideals of graphs the main results of this paper are combinatorial characterizations for the cohen macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph