eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

on new relations between spectral properties of jacobi matrices and their coefficients we study the spectral properties of jacobi matrices by using higher order trace formulae we obtain a result relating the properties of the elements of jacobi matrices and the corresponding spectral measures complicated expressions for traces of some operators can be magically simplified allowing us to apply induction arguments our theorems are generalizations of a recent result of r killip and b simon 17

schrodinger operators with sparse potentials asymptotics of the fourier transform of the spectral measure we study the pointwise behavior of the fourier transform of the spectral measure for discrete one dimensional schrodinger operators with sparse potentials we find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model we also present an improved version of known results on the spectrum of such operators

killip simon problem and jacobi flow on gmp matrices abstract one of the first and therefore most important theorems in perturbation theory claims that for an arbitrary self adjoint operator a there exists a perturbation b of hilbert schmidt class with arbitrary small operator norm which destroys completely the absolutely continuous a c spectrum of the initial operator a von neumann however if a is the discrete free 1 d schrodinger operator and b is an arbitrary jacobi matrix of hilbert schmidt class the a c spectrum remains perfectly the same that is the interval 2 2 moreover killip and simon described explicitly the spectral properties for such a b jointly with damanik they generalized this result to the case of perturbations of periodic jacobi matrices in the non degenerated case recall that the spectrum of a periodic jacobi matrix is a system of intervals of a very specific nature christiansen simon and zinchenko posed in a review dedicated to f gesztesy 2013 the following question is there an extension of the damanik killip simon theorem to the general finite system of intervals case in this paper we solve this problem completely our method deals with the jacobi flow on gmp matrices gmp matrices are probably a new object in the spectral theory they form a certain generalization of matrices related to the strong moment problem the latter ones are a very close relative of jacobi and cmv matrices the jacobi flow on them is also a probably new member of the rich family of integrable systems finally related to jacobi matrices of killip simon class analytic vector bundles and their curvature play a certain role in our construction and at least on the level of ideology this role is quite essential

an improved bl algorithm for genetic algorithm of the orthogonal packing of rectangles this paper proposes an improved bl algorithm for genetic algorithm of the orthogonal packing of rectangles some improvements on the fitness function of genetic algorithm for the orthogonal packing of rectangles are also suggested solutions of two numerical examples show the effectiveness of these improvements

an exact two dimensional non guillotine cutting tree search procedure we consider the two dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut we develop a lagrangean relaxation of a zero one integer programming formulation of the problem and use it as a bound in a tree search procedure subgradient optimization is used to optimize the bound derived from the lagrangean relaxation problem reduction tests derived from both the original problem and the lagrangean relaxation are given incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved

jost functions and jost solutions for jacobi matrices i a necessary and sufficient condition for szegő asymptotics we provide necessary and sufficient conditions for a jacobi matrix to produce orthogonal polynomials with szeg asymptotics off the real axis a key idea is to prove the equivalence of szeg asymptotics and of jost asymptotics for the weyl solution we also prove l2 convergence of szeg asymptotics on the spectrum

the absolutely continuous spectrum of one dimensional schrodinger operators with decaying potentials we investigate one dimensional schrodinger operators with asymptotically small potentials it will follow from our results that if with then is an essential support of the absolutely continuous part of the spectral measure we also prove that if then the spectrum is purely absolutely continuous on these results are optimal

sum rules for jacobi matrices and divergent lieb thirring sums abstract let e j be the eigenvalues outside 2 2 of a jacobi matrix with a n 1 l 2 and b n 0 and the density of the a c part of the spectral measure for the vector 1 we show that if b n l 4 b n 1 b n l 2 then j e j 2 5 2 and if b n l 4 b n 1 b n l 2 then 2 2 ln x 4 x 2 3 2 dx we also show that if a n 1 b n l 3 then the above integral is finite if and only if a n 1 a n b n 1 b n l 2 we prove these and other results by deriving sum rules in which the a c part of the spectral measure and the eigenvalues appear on opposite sides of the equation

schrodinger operators with many bound states consider the schrodinger operators h d2 dx2 v x we present a method for estimating the potential in terms of the negative eigenvalues of these operators among the applications are inverse lieb thirring inequalities and several sharp results concerning the spectral properties of h

on the spectral l2 conjecture 3 2 lieb thirring inequality and distributional potentials let h x2 v x be a properly defined schrodinger operator on l2 r with real potentials of the form v x q x p x the derivative is understood in the distributional sense with some p q l2 r we prove that the absolutely continuous spectrum of h fills 0 which was previously proven by deift killip for v l2 r we also refine the 3 2 lieb thirring inequality

an effective quasi human based heuristic for solving the rectangle packing problem abstract in this paper we introduce an effective deterministic heuristic less flexibility first for solving the classical np complete rectangle packing problem many effective heuristics implemented for this problem are cpu intensive and non deterministic in nature others including the polynomial approximation methodology j assoc comput mach 32 1 1985 130 are too laborious for practical problem sizes the technique we propose is inspired and developed by enhancing some rule of thumb guidelines resulting from the generation long work experience of human professionals in ancient days although the less flexibility first heuristic is a deterministic algorithm the results are very encouraging this algorithm can consistently produce packing densities of around 99 on most randomly generated large examples as compared with the recent results of a well known simulated annealing based rectangle packing rp algorithm ieee trans computer aided design integrated circuits systems 17 1 1998 60 the results are much better both in less dead space 2 4 vs 6 7 and much less cpu time 9 57 vs 331 78 seconds experimenting our heuristics on a public rectangle packing data set covering instances of 16 97 rectangles the average unpack ratio is quite satisfactory 0 92 for bounding boxes limited to be optimum and 2 68 for the completed packing while most cases spend only a few minutes in cpu time

spectral properties of jacobi matrices and sum rules of special form abstract in this article we relate the properties of elements of a jacobi matrix from certain class to the properties of its spectral measure the main tools we use are the so called sum rules introduced by case in orthogonal polynomials from the viewpoint of scattering theory j math phys 15 1974 2166 2174 orthogonal polynomials ii j math phys 16 1975 1435 1440 later the sum rules were efficiently applied by killip simon sum rules for jacobi matrices and their applications to spectral theory ann math 158 2003 253 321 to the spectral analysis of jacobi matrices we use a modification of the method that permits us to work with sum rules of higher orders as a corollary of the main theorem we obtain a counterpart of a result of molchanov novitskii vainberg first kdv integrals and absolutely continuous spectrum for 1 d schrodinger operator comm math phys 216 2001 195 213 for a continuous schrodinger operator on a half line

sum rules and the szegő condition for orthogonal polynomials on the real line we study the case sum rules especially c 0 for general jacobi matrices we establish situations where the sum rule is valid applications include an extension of shohat s theorem to cases with an infinite point spectrum and a proof that if lim n a n 1 and lim nb n exist and 2 then the szeg condition fails

the spectral bounds of the discrete schrodinger operator abstract let h b be a discrete schrodinger operator on l 2 z d with a potential b and a non negative coupling constant when b 0 it is well known that 0 4 d when b 0 let s b inf b and m b sup b be the bounds of the spectrum of the schrodinger operator one of the aims of this paper is to study the influence of the potential b on the bounds 0 and 4d of the spectrum of more precisely we give a necessary and sufficient condition on the potential b such that s b is strictly positive for small enough we obtain a similar necessary and sufficient condition on the potential b such that m b is lower than 4d for small enough in dimensions d 1 and d 2 the situation is more precise the following result was proved by killip and simon 2003 for d 1 in 5 then by damanik et al 2003 for d 1 and d 2 in 3 if b 0 4 d then b 0 our study on the bounds of the spectrum of b allows us to give a different and easy proof to this result

new spectral estimations for a class of integral difference operators and generalisation to higher dimensions quadratic form approach allows for new results in the analysis of a class of integral difference operators in finite domains non negativity spectral estimations a new property of legendre polynomials and establishing links with weighted mean square deviation functionals and with infinite jacobi matrices with not bounded coefficients generalisation of integral difference operators to higher dimensions is provided and application to matter relaxation in a field is considered a new class of special functions naturally appears

best first search methods for constrained two dimensional cutting stock problems best first search is a widely used problem solving technique in the field of artificial intelligence the method has useful applications in operations research as well here we describe an application to constrained two dimensional cutting stock problems of the following type a stock rectangle s of dimensions l w is supplied there are n types of demanded rectangles r1 r2 rn with the ith type having length li width wi value vi and demand constraint bi it is required to produce from the stock rectangle s ai copies of ri 1 i n to maximize a1v1 a2v2 anvn subject to the constraints ai bi only orthogonal guillotine cuts are permitted all parameters are integers a best first tree search algorithm based on wang s bottom up approach is described that guarantees optimal solutions and is more efficient than existing methods

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

first kdv integrals and absolutely continuous spectrum for 1 d schrodinger operator we consider 1 d schrodinger operators on l 2 r with slowly decaying potentials under some conditions on the potential related to the first integrals of the kdv equation we prove that the a c spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable examples show that for special classes of sparse potentials these results can not be improved

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

weyl s theorem in several variables in this note we consider weyl s theorem and browder s theorem in several variables the main result is as follows let t be a doubly commuting n tuple of hyponormal operators acting on a complex hilbert space if t has the quasitriangular property i e the dimension of the left cohomology for the koszul complex t is greater than or equal to the dimension of the right cohomology for t for all cn then weyl s theorem holds for t i e the complement in the taylor spectrum of the taylor weyl spectrum coincides with the isolated joint eigenvalues of finite multiplicity

on the coexistence of absolutely continuous and singular continuous components of the spectral measure for some sturm liouville operators with square summable potential abstract we prove that there exist some sturm liouville operators with square summable potentials such that the singular continuous component of the spectral measure lies on the positive half line the hausdorff dimension of the support of this singular measure can be arbitrary number from 0 to 1

l 2 bounded variation and absolutely continuous spectrum of jacobi matrices we disprove a conjecture of breuer last simon breuer et al in constr approx 32 2 221 254 2010 concerning the absolutely continuous spectrum of jacobi matrices with coefficients that obey an l2 bounded variation condition with step q we prove existence of a c spectrum on a smaller set than that specified by the conjecture and prove that our result is optimal

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

an exact algorithm for orthogonal 2 d cutting problems using guillotine cuts abstract we consider the two dimensional cutting problem which requires cutting a number of small rectangular pieces each of a given size and value from a large reatangular stock plate the objective is to maximise the value of pieces cut or for a slightly simpler problem to minimise the wastage we consider the case where there is a maximum number of times that a piece may be used in a cutting pattern we present a tree search algorithm for this problem in which the size of the tree search is limited by using a bound derived from a state space relaxation of a dynamic programming formulation of the problem a state space ascent method is used to optimise this bound the computational performance of the algorithm is investigated by tests performed on randomly generated problems with constraints of varying tightness the results indicate that the algorithm is an effective procedure capable of solving optimally practical two dimensional cutting problems of medium size

on the existence of the absolutely continuous component for the measure associated with some orthogonal systems in this article we consider two orthogonal systems sturm liouville operators and krein systems for krein systems we study the behavior of generalized polynomials at the infinity for spectral parameters in the upper half plain that makes it possible to establish the presence of absolutely continuous component of the associated measure for sturm liouville operator on the half line with bounded potential q we prove that essential support of absolutely continuous component of the spectral measure is m if and q l 2 r that holds for all boundary conditions at zero this result partially solves one open problem stated recently by s molchanov m novitskii and b vainberg we consider also some other classes of potentials

on a new class of special functions generated by integral difference operators abstract we introduce and investigate a new class of special functions n k x x 0 1 originally these functions naturally appeared upon spectral analysis of integral difference operators we discuss issues of generating functions decompositions in series links to legendre polynomials complete elliptic integrals and gauss hypergeometric functions functional relations for functions n k mostly in case n 1

large deviations and the lukic conjecture we use the large deviation approach to sum rules pioneered by gamboa nagel and rouault to prove higher order sum rules for orthogonal polynomials on the unit circle in particular we prove one half of a conjectured sum rule of lukic in the case of two singular points one simple and one double this is important because it is known that the conjecture of simon fails in exactly this case so this article provides support for the idea that lukic s replacement for simon s conjecture might be true

on the preservation of absolutely continuous spectrum for schrodinger operators abstract we present general principles for the preservation of a c spectrum under weak perturbations the schrodinger operators on the strip and on the caley tree bethe lattice are considered

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

absolutely continuous spectrum of one dimensional schrodinger operators and jacobi matrices with slowly decreasing potentials we prove that for any one dimensional schrodinger operator with potentialv x satisfying decay condition v x cx 3 4 e the absolutely continuous spectrum fills the whole positive semi axis the description of the set in on which the singular part of the spectral measure might be supported is also given analogous results hold for jacobi matrices

some jacobi matrices with decaying potential and dense point spectrum we discuss doubly infinite matrices of the formm ij i j 1 i j 1 v i ij as operators on l2 we present for each 0 examples of potentialsv n with v n o n 1 2 and wherem has only point spectrum our discussion should be viewed as a remark on the recent work of delyon kunz and souillard

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient