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

spectral asymptotics for the third order operator with periodic coefficients abstract we consider the self adjoint third order operator with 1 periodic coefficients on the real line the spectrum of the operator is absolutely continuous and covers the real line we determine the high energy asymptotics of the periodic antiperiodic eigenvalues and of the branch points of the lyapunov function furthermore in the case of small coefficients we show that either whole spectrum has multiplicity one or the spectrum has multiplicity one except for a small spectral nonempty interval with multiplicity three in the last case the asymptotics of this small interval is determined

marchenko ostrovki mapping for periodic zakharov shabat systems abstract consider the zakharov shabat or dirac operator t in the hilbert space l2 r l2 r with real periodic vector potential q q1 q2 h l2 t l2 t the spectrum of t is absolutely continuous and consists of intervals separated by gaps gn z n z n n z from the dirichlet eigenvalue mn n z of the zakharov shabat equation with dirichlet boundary conditions at 0 1 the square of the height of vertical slits on the quasimomentum domain and the points on these slits we construct the marchenko ostrovki vertical slits mapping for the periodic zakharov shabat systems h h l2 l2 using nonlinear functional analysis in hilbert spaces we show that this mapping is a real analytic isomorphism in the second part of our paper we prove a new identity for the effective masses

mourre theory for time periodic magnetic fields abstract we study the mourre theory for the floquet hamiltonian h i t h t generated by the time periodic hamiltonian h t with h t t h t which describes the schrodinger equations with general time periodic magnetic fields

periodic schrodinger operators with local defects and spectral pollution this article deals with the numerical calculation of eigenvalues of perturbed periodic schrodinger operators located in spectral gaps such operators are encountered in the modeling of the electronic structure of crystals with local defects and of photonic crystals the usual finite element galerkin approximation is known to give rise to spectral pollution in this article we give a precise description of the corresponding spurious states we then prove that the supercell model does not produce spectral pollution finally we extend results by lewin and sere on some no pollution criteria in particular we prove that using approximate spectral projectors enables one to eliminate spectral pollution in a given spectral gap of the reference periodic schrodinger operator

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

bound states for nano tubes with a dislocation abstract as a model for an interface in solid state physics we consider two real valued potentials v 1 and v 2 on the cylinder or tube s r r z where we assume that there exists an interval a 0 b 0 which is free of spectrum of v k for k 1 2 we are then interested in the spectrum of h t v t for t r where v t x y v 1 x y for x 0 and v t x y v 2 x t y for x 0 while the essential spectrum of h t is independent of t we show that discrete spectrum related to the interface at x 0 is created in the interval a 0 b 0 at suitable values of the parameter t provided v 2 has some essential spectrum in a 0 we do not require v 1 or v 2 to be periodic we furthermore show that the discrete eigenvalues of h t are lipschitz continuous functions of t if the potential v 2 is locally of bounded variation

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

structure of indicator function classes with finite vapnik chervonenkis dimensions the vapnik chervonenkis vc dimension is used to measure the complexity of a function class and plays an important role in a variety of fields including artificial neural networks and machine learning one major concern is the relationship between the vc dimension and inherent characteristics of the corresponding function class according to sauer s lemma if the vc dimension of an indicator function class f is equal to d the cardinality of the set fs1n will not be larger than d 0dcnd therefore there naturally arises a question about the vc dimension of an indicator function class what kinds of elements will be contained in the function class f if f has a finite vc dimension in this brief we answer the above question first we investigate the structure of the function class f when the cardinality of the set fs1n reaches the maximum value d 0dcnd based on the derived result we then figure out what kinds of elements will be contained in f if f has a finite vc dimension

four types of learning curves if machines are learning to make decisions given a number of examples the generalization error e t is defined as the average probability that an incorrect decision is made for a new example by a machine when trained with t examples the generalization error decreases as t increases and the curve e t is called a learning curve the present paper uses the bayesian approach to show that given the annealed approximation learning curves can be classified into four asymptotic types if the machine is deterministic with noiseless teacher signals then 1 e at 1 when the correct machine parameter is unique and 2 e at 2 when the set of the correct parameters has a finite measure if the teacher signals are noisy then 3 e at 1 2 for a deterministic machine and 4 e c at 1 for a stochastic machine

non consistent approximations of self adjoint eigenproblems application to the supercell method in this article we introduce a general theoretical framework to analyze non consistent approximations of the discrete eigenmodes of a self adjoint operator we focus in particular on the discrete eigenvalues laying in spectral gaps we first provide a priori error estimates on the eigenvalues and eigenvectors in the absence of spectral pollution we then show that the supercell method for perturbed periodic schrodinger operators falls into the scope of our study we prove that this method is spectral pollution free and we derive optimal convergence rates for the planewave discretization method taking numerical integration errors into account some numerical illustrations are provided

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

on resonances and the formation of gaps in the spectrum of quasi periodic schrodinger equations we consider one dimensional difference schrodinger equations h x n n 1 n 1 v x n n e n n z x 0 1 with real analytic potential function v x if l e w o is greater than 0 for all e e e and some diophantine 0 then the integrated density of states is absolutely continuous for almost every close to 0 as shown by the authors in earlier work in this paper we establish the formation of a dense set of gaps in spec h x w e e our approach is based on an induction on scales argument and is therefore both constructive as well as quantitative resonances between eigenfunctions of one scale lead to pre gaps at a larger scale to pass to actual gaps in the spectrum we show that these pre gaps cannot be filled more than a finite and uniformly bounded number of times to accomplish this one relates a pre gap to pairs of complex zeros of the dirichlet determinants off the unit circle amongst other things we establish a nonperturbative version of the co variant parametrization of the eigenvalues and eigenfunctions via the phases in the spirit of sinai s perturbative description of the spectrum via his function a this allows us to relate the gaps in the spectrum with the graphs of the eigenvalues parametrized by the phase our infinite volume theorems hold for all diophantine frequencies up to a set of hausdorff dimension zero

effective masses for laplacians on periodic graphs abstract we consider laplacians on both periodic discrete and periodic metric equilateral graphs their spectrum consists of an absolutely continuous part which is a union of non degenerate spectral bands and flat bands i e eigenvalues of infinite multiplicity we estimate effective masses associated with the ends of each spectral band in terms of geometric parameters of the graphs moreover in the case of the bottom of the spectrum we determine two sided estimates on the effective mass in terms of geometric parameters of the graphs the proof is based on floquet theory factorization of fiber operators perturbation theory and the relation between effective masses for laplacians on discrete and metric graphs obtained in our paper

exponential or polynomial learning curves case based studies learning curves exhibit a diversity of behaviors such as phase transition however the understanding of learning curves is still extremely limited and existing theories can give the impression that without empirical studies e g cross validation one can probably do nothing more than qualitative interpretations in this note we propose a theory of learning curves based on the idea of reducing learning problems to hypothesistesting ones this theory provides a simple approach that is potentially useful for predicting and interpreting a diversity of learning curve behaviors qualitatively and quantitatively and it applies to finite training sample size and finite learning machine and for learning situations not necessarily within the bayesian framework we illustrate the results by examining some exponential learning curve behaviors observed in cohn and tesauro 1992 s experiment

the essential numerical range for unbounded linear operators abstract we introduce the concept of essential numerical range w e t for unbounded hilbert space operators t and study its fundamental properties including possible equivalent characterizations and perturbation results many of the properties known for the bounded case do not carry over to the unbounded case and new interesting phenomena arise which we illustrate by some striking examples a key feature of the essential numerical range w e t is that it captures spectral pollution in a unified and minimal way when approximating t by projection methods or domain truncation methods for pdes

how bad may learning curves be in this paper we motivate the need for estimating bounds on learning curves of average case learning algorithms when they perform the worst on training samples we then apply the method of reducing learning problems to hypothesis testing ones to investigate the learning curves of a so called ill disposed learning algorithm in terms of a system complexity the boolean interpolation dimension since the ill disposed algorithm behaves worse than ordinal ones and the boolean interpolation dimension is generally bounded by the number of system weights the results can apply to interpreting or to bounding the worst case learning curve in real learning situations this study leads to a new understanding of the worst case generalization in real learning situations which differs significantly from that in the uniform learnable setting via vapnik chervonenkis vc dimension analysis we illustrate the results with some numerical simulations

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

are the maximum shortening velocity and the shape parameter in a hill type model of whole muscle related to activation abstract mathematical models of the inter relationship of muscle force velocity and activation are useful in forward dynamic simulations of human movement tasks the objective of this work was to determine whether the parameters maximum shortening velocity v max and shape parameter k of a hill type muscle model interrelating muscle force velocity and activation are themselves dependent on the activation to fulfill this objective surface emg signals from four muscles as well as the kinematics and kinetics of the arm were recorded from 14 subjects who performed rapid release elbow extension tasks at 25 50 75 and 100 activation mvc the experimental elbow flexion angle was tracked by a forward dynamic simulation of the task in which v max and k of the triceps brachii were varied at each activation level to minimize the difference between the simulated and experimental elbow flexion angle because a preliminary analysis demonstrated no dependency of k on activation additional simulations were performed with constant k values of 0 15 0 20 and 0 25 the optimized values of v max normalized to the average value within a subject were then regressed onto the activation normalized v max depended significantly on the activation p k furthermore the estimated v max values were not sensitive to the selected k value the results support the use of hill type models in which v max depends on activation in forward dynamic simulations modeling muscles with mixed fiber type composition recruited in the range of 25 100 activation the use of more accurate models will lend greater confidence to the results of forward dynamic simulations

on approximation of the eigenvalues of perturbed periodic schrodinger operators this paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic schrodinger operator perturbed by a fast decreasing potential we use a recently developed technique the so called quadratic projection method in order to achieve convergence free from spectral pollution we describe the theoretical foundations of the method in detail and illustrate its effectiveness by several examples

lattice dislocations in a 1 dimensional model the spectral properties of the schrodinger operator t t d2 dx2 q x t in l2 are studied where the potential q is defined by q p x t x 0 and q p x x 0 p is a 1 periodic potential and t is the dislocation parameter for each t the absolutely continuous spectrum ac t t ac t 0 consists of intervals which are separated by the gaps n t t n t 0 n n n 1 we prove in each gap n n 1 there exist two unique states an eigenvalue and a resonance n t of the dislocation operator such that n 0 n and the point n t runs clockwise around the gap n changing the energy sheet whenever it hits n making n 2 complete revolutions in unit time on the first sheet n t is an eigenvalue and on the second sheet n t is a resonance in general these motions are not monotonic there exists a unique state 0 t in the basic gap 0 t t 0 t 0 0 the asymptotics of n t as n is determined

determination of subject specific model parameters for visco elastic elements the determination of subject specific model parameter values is necessary in order for a computer simulation model of human motion to be evaluated quantitatively this study used an optimisation procedure along with a kinematically driven simulation model of the contact phase in running jumps to determine the elastic parameters of segmental wobbling masses and the foot ground interface kinetic and kinematic data were obtained on running jumps for height and distance performed by an elite male high jumper stiffness and damping coefficients of the visco elastic elements in the model were varied until the difference between simulation and performance was minimised percentage differences of 6 and 9 between the simulated and recorded performances were obtained in the jumps for height and distance respectively when the parameters obtained from the jump for height were used in a simulation of the jump for distance and vice versa there was poor agreement with the recorded jump on the other hand a common set of visco elastic parameters were obtained using the data from both recorded jumps resulting in a mean difference of only 8 made up of 7 and 10 between simulation and performance that was almost as good as the individual matches simulations were not overly sensitive to perturbations of the common set of visco elastic parameters it is concluded that subject specific elastic parameters should be calculated from more than a single jump in order to provide a robust set of values that can be used in different simulations

eigenvalue branches of the schrodinger operatorh λw in a gap of σ h the authors study the eigenvalue branches of the schrodinger operatorh w in a gap of h in particular they consider questions of asymptotic distribution of eigenvalues and bounds on the number of branches they also address the completeness problem

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

how tight are the vapnik chervonenkis bounds we describe a series of numerical experiments that measure the average generalization capability of neural networks trained on a variety of simple functions these experiments are designed to test the relationship between average generalization performance and the worst case bounds obtained from formal learning theory using the vapnik chervonenkis vc dimension blumer et al 1989 haussler et al 1990 recent statistical learning theories tishby et al 1989 schwartz et al 1990 suggest that surpassing these bounds might be possible if the spectrum of possible generalizations has a gap near perfect performance we indeed find that in some cases the average generalization is significantly better than the vc bound the approach to perfect performance is exponential in the number of examples m rather than the 1 m result of the bound however in these cases we have not found evidence of the gap predicted by the above statistical theories in other cases we do find the 1 m behavior of the vc bound

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 problem and the trace formula for the hill operator ii consider the hill operator t d 2 dx 2 q x on l 2 mathbb r where q in l 2 0 1 is a 1 periodic real potential and int 0 1q x dx 0 the spectrum of t is absolutely continuous and consists of intervals separated by gaps gamma n a n a n n geq 1 let mu n n geq 1 be the dirichlet eigenvalue of the equation y qy ly on the interval 0 1 introduce the vector g n g cn g sn in mathbb r 2 with components g cn a n a n 2 mu n and g sn gamma n 2 4 g cn 2 s n where the sign s n or s n for all n geq 1 using nonlinear functional analysis in hilbert spaces we show that the mapping g q to g q g n 1 iy in ell 2 oplus ell 2 is a real analytic isomorphism in the second part a trace formula for q in l 2 0 1 is proved

estimates for the hill operator ii consider the hill operator ty y q t y in l2 r where the real potential q is 1 periodic and q q l2 0 1 the spectrum of t consists of spectral bands separated by gaps n n 1 with length n 0 we obtain two sided estimates of the gap lengths n2 n 2 in terms of 01q t 2dt moreover we obtain the similar two sided estimates for spectral data the height of the corresponding slit on the quasimomentum domain action variables for the kdv equation and so on in order prove this result we use the analysis of a conformal mapping corresponding to quasimomentum of the hill operator that makes it possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory then the proof is based on the analysis of the conformal mapping the embedding theorems and the identities furthermore we obtain the similar two sided estimates for potentials which have p 2 derivatives

inverse problem for the discrete 1d schrodinger operator with small periodic potentials consider the discrete 1d schrodinger operator on with an odd 2k periodic potential q for small potentials we show that the mapping q heights of vertical slits on the quasi momentum domain similar to the marchenko ostrovski maping for the hill operator is a local isomorphism and the isospectral set consists of 2 k distinct potentials finally the asymptotics of the spectrum are determined as q 0

absolute continuity in periodic waveguides we study second order elliptic operators with periodic coefficients in two dimensional simply connected periodic waveguides with the dirichlet or neumann boundary conditions it is proved that under some mild smoothness restrictions on the coefficients such operators have purely absolutely continuous spectra the proof follows a method suggested previously by a morame to tackle periodic operators with variable coefficients in dimension 2 2000 mathematical subject classification 35j10 35p05 35j25

a physiologically based criterion of muscle force prediction in locomotion abstract the inversely nonlinear relationship of muscle contraction force and the possible contraction duration i e endurance is utilized in a method to mathematically predict individual muscle forces the method uses a nonlinear optimization scheme to solve the redundant distribution problem at the joints of interest the method is demonstrated at the elbow during isometric contraction and in the lower extremity during locomotion during gait the observed muscle activity pattern in the lower extremity as determined by emg shows substantial agreement with that activity pattern predicted when endurance is used as the optimization criterion the importance of selecting muscle prediction criteria based on sound physiological bases rather than on an arbitrary or mathematically convenient basis is emphasized

a dirichlet to neumann approach for the exact computation of guided modes in photonic crystal waveguides this work deals with one dimensional infinite perturbation namely line defects in periodic media in optics such defects are created to construct an open waveguide that concentrates light the existence and computation of the eigenmodes is a crucial issue this is related to a self adjoint eigenvalue problem associated to a pde in an unbounded domain in the directions orthogonal to the line defect which makes both the analysis and the computations more complex using a dirichlet to neumann approach we show that this problem is equivalent to one set on a small neighborhood of the defect contrary to existing methods this one is exact but there is a price to be paid the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature 2013 society for industrial and applied mathematics

global optimization of statistical functions with simulated annealing abstract many statistical methods rely on numerical optimization to estimate a model s parameters unfortunately conventional algorithms sometimes fail even when they do converge there is no assurance that they have found the global rather than a local optimum we test a new optimization algorithm simulated annealing on four econometric problems and compare it to three common conventional algorithms not only can simulated annealing find the global optimum it is also less likely to fail on difficult functions because it is a very robust algorithm the promise of simulated annealing is demonstrated on the four econometric problems

on the resonances and eigenvalues for a 1d half crystal with localised impurity we consider the schrdinger operator h on the half line with a periodic potential p plus a compactly supported potential q for generic p its essential spectrum has an infinite sequence of open gaps we determine the asymptotics of the resonance counting function and show that for sufficiently high energy each non degenerate gap contains exactly one eigenvalue or antibound state giving asymptotics for their positions conversely for any potential q and for any sequences n 0 1 and xn 0 there exists a potential p such that xn is the length of the n th gap n and h has exactly n eigenvalues and 1 n antibound state in each high energy gap moreover we show that between any two eigenvalues in a gap there is an odd number of antibound states and hence deduce an asymptotic lower bound on the number of antibound states in an adiabatic limit

towards more practical average bounds on supervised learning in this paper we describe a method which enables us to study the average generalization performance of learning directly via hypothesis testing inequalities the resulting theory provides a unified viewpoint of average case learning curves of concept learning and regression in realistic learning problems not necessarily within the bayesian framework the advantages of the theory are that it alleviates the practical pessimism frequently claimed for the results of the vapnik chervonenkis vc theory and its alike and provides general insights into generalization besides the bounds on learning curves are directly related to the number of adjustable system weights although the theory is based on an approximation assumption and cannot apply to the worst case learning setting the precondition of the assumption is mild and the approximation itself is only a sufficient condition for the validity of the theory we illustrate the results with numerical simulations and apply the theory to examining the generalization ability of combination of neural networks

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

on the practical applicability of vc dimension bounds this article addresses the question of whether some recent vapnik chervonenkis vc dimension based bounds on sample complexity can be regarded as a practical design tool specifically we are interested in bounds on the sample complexity for the problem of training a pattern classifier such that we can expect it to perform valid generalization early results using the vc dimension while being extremely powerful suffered from the fact that their sample complexity predictions were rather impractical more recent results have begun to improve the situation by attempting to take specific account of the precise algorithm used to train the classifier we perform a series of experiments based on a task involving the classification of sets of vowel formant frequencies the results of these experiments indicate that the more recent theories provide sample complexity predictions that are significantly more applicable in practice than those provided by earlier theories however we also find that the recent theories still have significant shortcomings

convergence of the supercell method for defect modes calculations in photonic crystals we present a rigorous study of the convergence of the supercell method used for determing defect modes in photonic crystals with compactly supported perturbations transverse electric and transverse magnetic polarized waves are investigated in 2 d structures we prove an exponential convergence of the defect frequencies with the supercell size and give a justification of the quasi independence of the corresponding eigenfunctions on the wave vector we also give a characterization of the supercell eigenvalues corresponding to the background photonic crystal

random hamiltonians ergodic in all but one direction letv 1 andv 2 be two ergodic random potentials on d we consider the schrodinger operatorh h0 v withh0 and forx x1 x2 v omega x left begin array 20 c v omega 1 x if x 1 over smash scriptstyle vphantom x 0 end array right we prove certain ergodic properties of the spectrum for this model and express the integrated density of states in terms of the density of states of the stationary potentialsv 1 andv 2 finally we prove the existence of the density of surface states forh

spectral pollution and how to avoid it this paper devoted to the study of spectral pollution contains both abstract results and applications to some self adjoint operators with a gap in their essential spectrum occuring in quantum mechanics first we consider galerkin basis which respect the decomposition of the ambient hilbert space into a direct sum h ph oplus 1 p h given by a fixed orthogonal projector p and we localize the polluted spectrum exactly this is followed by applications to periodic schrodinger operators pollution is absent in a wannier type basis and to dirac operator several natural decompositions are considered in the second part we add the constraint that within the galerkin basis there is a certain relation between vectors in ph and vectors in 1 p h abstract results are proved and applied to several practical methods like the famous kinetic balance of relativistic quantum mechanics

regularity of the surface density of states abstract we prove that the integrated surface density of states of continuous or discrete anderson type random schrodinger operators is a measurable locally integrable function rather than a signed measure or a distribution this generalizes our recent results on the existence of the integrated surface density of states in the continuous case and those of a chahrour in the discrete case the proof uses the new l p bound on the spectral shift function recently obtained by combes hislop and nakamura also we provide a simple proof of their result on the holder continuity of the integrated density of bulk states

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

the lyapunov function for schrodinger operators with a periodic 2 2 matrix potential abstract we consider the schrodinger operator on the real line with a 2 2 matrix valued 1 periodic potential the spectrum of this operator is absolutely continuous and consists of intervals separated by gaps we define a lyapunov function which is analytic on a two sheeted riemann surface on each sheet the lyapunov function has the same properties as in the scalar case but it has branch points which we call resonances we prove the existence of real as well as non real resonances for specific potentials we determine the asymptotics of the periodic and the anti periodic spectrum and of the resonances at high energy we show that there exist two type of gaps 1 stable gaps where the endpoints are the periodic and the anti periodic eigenvalues 2 unstable resonance gaps where the endpoints are resonances i e real branch points of the lyapunov function we also show that periodic and anti periodic spectrum together determine the spectrum of the matrix hill operator

cantor spectrum for the almost mathieu operator in this paper we use results on reducibility localization and duality for the almost mathieu operator left h b phi x right n x n 1 x n 1 b cos left 2 pi n omega phi right x n on l 2 and its associated eigenvalue equation to deduce that for b 0 2 and diophantine the spectrum of the operator is a cantor subset of the real line this solves the so called ten martini problem for these values of b and moreover we prove that for b 0 small or large enough all spectral gaps predicted by the gap labelling theorem are open

anderson transitions for a family of almost periodic schrodinger equations in the adiabatic case this work is devoted to the study of a family of almost periodic one dimensional schrodinger equations using results on the asymptotic behavior of a corresponding monodromy matrix in the adiabatic limit we prove the existence of an asymptotically sharp anderson transition in the low energy region more explicitly we prove the existence of energy intervals containing only singular spectrum and of other energy intervals containing absolutely continuous spectrum the zones containing singular spectrum and those containing absolutely continuous are separated by asymptotically sharp transitions the analysis may be viewed as utilizing a complex wkb method for adiabatic perturbations of periodic schrodinger equations the transition energies are interpreted in terms of phase space tunneling

a variational approach to dislocation problems for periodic schrodinger operators abstract as a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potential v v x y on r 2 with period lattice z 2 by setting w t x y v x t y for x 0 and w t x y v x y for x 0 for t 0 1 for lipschitz continuous v it is shown that the schrodinger operators h t w t have spectrum surface states in the spectral gaps of h 0 for suitable t 0 1 we also discuss the density of these surface states as compared to the density of the bulk our approach is variational and it is first applied to the well known dislocation problem korotyaev 2000 2005 15 16 on the real line we then proceed to the dislocation problem for an infinite strip and for the plane in appendix a we discuss regularity properties of the eigenvalue branches in the one dimensional dislocation problem for suitable classes of potentials

a theoretical analysis of preferred pedaling rate selection in endurance cycling one objective of this study was to investigate whether neuromuscular quantities were associated with preferred pedaling rate selection during submaximal steady state cycling from a theoretical perspective using a musculoskeletal model with an optimal control analysis specific neuromuscular quantities of interest were the individual muscle activation force stress and endurance to achieve this objective a forward dynamic model of cycling and optimization framework were used to simulate pedaling at three different rates of 75 90 and 105 rpm at 265 w the pedaling simulations were produced by optimizing the individual muscle excitation timing and magnitude to reproduce experimentally collected data the results from these pedaling simulations indicated that all neuromuscular quantities were minimized at 90 rpm when summed across muscles in the context of endurance cycling these results suggest that minimizing neuromuscular fatigue is an important mechanism in pedaling rate selection a second objective was to determine whether any of these quantities could be used to predict the preferred pedaling rate by using the quantities with the strongest quadratic trends as the performance criterion to be minimized in an optimal control analysis these quantities were analyzed to assess whether they could be further minimized at 90 rpm and produce normal pedaling mechanics the results showed that both the integrated muscle activation and average endurance summed across all muscles could be further minimized at 90 rpm indicating that these quantities cannot be used individually to predict preferred pedaling rates

edge states in ordinary differential equations for dislocations in this article we study schrodinger operators on the real line when the external potential represents a dislocation in a periodic medium we study how the spectrum varies with the dislocation parameter we introduce several integer valued indices including the chern number for bulk indices and various spectral flows for edge indices we prove that all these indices coincide providing a proof of a bulk edge correspondence in this case the study is also made for dislocations in dirac models on the real line we prove that 0 is always an eigenvalue of such operators

on optimization problems with prescribed rearrangements on considere certains resultats concernant des fonctions avec des distributions imposees et des prroblemes d optimisation qui y sont relies

eigenvalues in gaps and decoupling by neumann boundary conditions abstract continuing earlier investigations we show the existence of eigenvalues of schrodinger operators h w r inside a spectral gap of h v under suitable conditions on v and w our method of proof is based on a decoupling of regions in r v by means of dirichlet or neumann boundary conditions where in particular the use of neumann boundary conditions in the decoupling process leads to a new asymptotic results for eigenvalue counting functions

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

on the existence of eigenvalues of the schrodinger operatorh λw in a gap of σ h the authors prove a number of results asserting the existence of eigenvalues of the schrodinger operatorh w in a gap of h whereh v andv andw are bounded the existence of these eigenvalues is an important element in the theory of the colour of crystals the basic theorems are proved in v stronger results forv 1 are also presented

bivariate optimization of pedalling rate and crank arm length in cycling the contribution of this paper is a bivariate optimization of cycling performance relying on a biomechanical model of the lower limb a cost function derived from the joint moments developed during cycling is computed at constant average power both pedalling rate i e rpm and crank arm length are systematically varied to explore the relation between these variables and the cost function a crank arm length of 170 mm and pedalling rate of 100 rpm correspond closely to the cost function minimum in cycling situations where the rpm deviates from 100 rpm however crank arms of length other than 170 mm yield minimum cost function values in addition the sensitivity of optimization results to both increased power and anthropometric parameter variations is examined at increased power the cost function minimum is more strongly related to the pedalling rate with higher pedalling rates corresponding to the minimum anthropometric parameter variations influence the results significantly in general it is found that the cost function minimum for tall people occurs at longer crank arm lengths and lower pedalling rates than the length and rate for short people

conformal spectral theory for the monodromy matrix for any n x n monodromy matrix we define the lyapunov function which is analytic on an associated n sheeted riemann surface on each sheet the lyapunov function has the standard properties of the lyapunov function for the hill operator the lyapunov function has real or complex branch points which we call resonances we determine the asymptotics of the periodic anti periodic spectrum and of the resonances at high energy we show that the endpoints of each gap are periodic anti periodic eigenvalues or resonances real branch points moreover the following results are obtained 1 we define the quasimomentum as an analytic function on the riemann surface of the lyapunov function various properties and estimates of the quasimomentum are obtained 2 we construct the conformal mapping with imaginary part given by the lyapunov exponent and we obtain various properties of this conformal mapping which are similar to the case of the hill operator 3 we determine various new trace formulae for potentials and the lyapunov exponent 4 we obtain a priori estimates of gap lengths in terms of the dirichlet integral we apply these results to the schrodinger operators and to first order periodic systems on the real line with a matrix valued complex self adjoint periodic potential

simultaneous quantization of edge and bulk hall conductivity the edge hall conductivity is shown to be an integer multiple of e 2 h which is almost surely independent of the choice of the disordered configuration its equality to the bulk hall conductivity given by the kubo chern formula follows from k theoretic arguments this leads to quantization of the hall conductance for any redistribution of the current in the sample it is argued that in experiments at most a few per cent of the total current can be carried by edge states

quantifying inductive bias ai learning algorithms and valiant s learning framework abstract we show that the notion of inductive bias in concept learning can be quantified in a way that directly relates to learning performance in the framework recently introduced by valiant our measure of bias is based on the growth function introduced by vapnik and chervonenkis and on the vapnik chervonenkis dimension we measure some common language biases including restriction to conjunctive concepts conjunctive concepts with internal disjunction k dnf and k cnf concepts we also measure certain types of bias that result from a preference for simpler hypotheses using these bias measurements we analyze the performance of the classical learning algorithm for conjunctive concepts from the perspective of valiant s learning framework we then augment this algorithm with a hypothesis simplification routine that uses a greedy heuristic and show how this improves learning performance on simpler target concepts improved learning algorithms are also developed for conjunctive concepts with internal disjunction k dnf and k cnf concepts we show that all our algorithms are within a logarithmic factor of optimal in terms of the number of examples they require to achieve a given level of learning performance in the valiant framework our results hold for arbitrary attribute based instance spaces defined by either tree structured or linear attributes

learning from examples in large neural networks a statistical mechanical theory of learning from examples in layered networks at finite temperature is studied when the training error is a smooth function of continuously varying weights the generalization error falls off asymptotically as the inverse number of examples by analytical and numerical studies of single layer perceptrons we show that when the weights are discrete the generalization error can exhibit a discontinuous transition to perfect generalization for intermediate sizes of the example set the state of perfect generalization coexists with a metastable spin glass state