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

reduction of a nilpotent fuzzy matrix abstract a problem of reducing fuzzy matrices is considered and some results are obtained with regard to nilpotent matrices first reduction of irreflexive and transitive matrices is examined and then the results are applied to nilpotent matrices nilpotent matrices are matrices which represent acyclic graphs and the graphs are used to represent consistent systems the results are useful for simplification of various systems with fuzzy transitivity

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

a direct branching algorithm for checking the equivalence of two deterministic pushdown transducers one of which is real time strict abstract a new direct branching algorithm is presented for checking the equivalence of deterministic pushdown transducers one of whose associated deterministic pushdown automata dpdas is real time strict it is a straightforward extension of the corresponding algorithm for dpdas tomita 1982 and then proves the general applicability of our approach

linear objective function optimization with fuzzy relation equation constraints regarding max av composition in this paper an optimization model with a linear objective function subject to a system of the fuzzy relation equations with max av composition is presented the solution set of such a fuzzy relation equations is a non convex set in this paper firstly we discuss the feasible solution set with two schemes and secondly study relationship between maximum and minimum points and also the feasible points as well furthermore an algorithm and few concrete examples are presented in order to optimize linear objective function

szpilrajn s theorem on fuzzy orderings fuzzy relation matrices are considered and the well known szpilrajn theorem on orderings is generalized to fuzzy orderings zadeh has already shown a generalized theorem but another generalization is given in matrix form some interesting operations on fuzzy matrices are introduced and their properties are mentioned briefly the theorem is represented in terms of the matrix operations the operations have many properties and are useful in the various applications of fuzzy matrices or fuzzy relations

convergence of the power sequence of a nearly monotone increasing fuzzy matrix abstract by introducing a new classification the essential role of the principal diagonal elements for the convergence of the power sequence of a fuzzy matrix a is exported at the level of a itself the established theorems cover those of monotone or nearly monotone increasing fuzzy matrices the convergence index of matrices of this type is always less than or equal to n the results can be considered to be a generalization of the results obtained by some authors on the one hand and on the other it is more important that the results have laid the foundation for studying the convergence and oscillation for power sequence of an arbitrary fuzzy matrix in general furthermore as a typical case which needs to be considered the necessary and sufficient conditions for an increasing fuzzy matrix a to have the property a n 1 a n a n 1 were established and the typical form was represented

fixed points of functions with max weighted quasi arithmetic mean operator abstract let 0 1 and f be a continuous strictly monotone real valued function the weighted quasi arithmetic mean of two numbers a b is defined by a b f 1 f a 1 f b let a a i j be an n n real matrix and x x 1 x 2 x n t r n we construct a function a 1 a 2 a n a r n r n by j a x max 1 l n x l a l j for all j 1 2 n in this paper we show that a has a unique fixed point x a moreover it can be shown that for each x r n the sequence x a k generated by the following iterative scheme x a 0 x and x a k a x a k 1 for all k 1 converges to the unique fixed point x a besides some properties of the fixed point are derived as an application our results imply that the max weighted quasi arithmetic mean powers of any matrix are always convergent the continuity of the function m n n r r n defined by a x a is proposed as well

sufficient conditions for the ergodicity of fuzzy markov chains abstract analogous to the traditional markov chains which have been studied extensively and have many successful applications fuzzy markov chains have been proposed for decision making in an environment of uncertainty and imprecision for decades it is known that results of fuzzy markov chains depend on the transition matrix as well as the algebraic composition involved in their study of max min fuzzy markov chains avrachenkov and sanchez raised an open question for finding conditions to ensure the ergodicity of max min fuzzy markov chains in this paper we provide sufficient conditions for the ergodicity of both max min and max product fuzzy markov chains it is not surprising that such sufficient conditions are very different because of the max min and max product compositions

a note on the power sequence of a fuzzy matrix abstract the properties of power sequence of an n n fuzzy matrix under some different pairs of fuzzy operations have been studied especially under min max composition the properties can be deduced from the results in this paper furthermore some results were established in the sense of max t composition when t is a commonly used product type triangular norm

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

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

an extended direct branching algorithm for checking equivalence of deterministic pushdown automata abstract this paper extends the direct branching algorithm of 25 for checking equivalence of deterministic pushdown automata it does so by providing a technique called halting for dealing with nodes with unbounded degree in the comparison tree this may occur when a skipping step may be applied infinitely many times to a certain node as a result of infinite sequences of e moves this extension allows the algorithm to check equivalence of two deterministic pushdown automata when none of them is real time but in a certain condition that properly contains a case where one of them is real time strict

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

transitivity of generalized fuzzy matrices generalized fuzzy matrices are considered as matrices over a special type of semiring which is called a path algebra transitivity of generalized fuzzy matrices is defined and some properties of transitivity are shown especially properties of the composition of transitive fuzzy matrices representing transitive fuzzy relations are examined by introducing a third operation on the semiring with two operations the transitivity is closely related to metrics distances ultrametrics and so on

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

on the power sequence of a fuzzy matrix iii a detailed study on the power sequence of matrices of commonly used types abstract focusing on the behavior of the principal diagonal elements of ak a new classification has been introduced which is called circularly k dominating in turns out that the convergence index or the oscillating index of the power sequence of an n n fuzzy matrix of the circularly k dominating type is bounded by nk n k from above and if it is oscillating then the period index pa is a factor of k the fuzzy matrices of the 2 dominating type were discussed in detail it was shown that the 2 dominating type is a more general class than those have been discussed before and the results established for matrices of 2 dominating type is as good as the results obtained for controllable matrices therefore most commonly used types of fuzzy matrices can be examined under the framework of 2 dominating matrices and the convergence index or oscillating index can be estimated based on the results

on some decision questions concerning pushdown machines abstract certain decision questions e g equivalence ambiguity single valuedness monotonicity etc concerning pushdown transducers and acceptors are investigated examples of results shown are the following 1 1 there is an algorithm to decide given two deterministic pushdown transducers one of which has a finite turn counter for a pushdown stack whether they are equivalent i e they define the same input output relation 2 2 there are subclasses of pushdown acceptors which have a decidable ambiguity problem but an undecidable inherent ambiguity problem it seems that these classes are the first known to exhibit this property 3 3 there is an algorithm to decide given a homomorphism h and a language l accepted by a finite turn nondeterministic counter automaton whether h is one to one on l in contrast there is no such algorithm if l is accepted by a 1 turn deterministic pushdown automaton or by an unrestricted deterministic counter automaton

a direct branching algorithm for checking equivalence of some classes of deterministic pushdown automata a new direct algorithm is presented for checking equivalence of some classes of deterministic pushdown automata dpda s after korenjak and hopcroft s branching algorithm it is not only powerful enough to be applicable to two dpda s accepting by empty stack one of which is real time but also simple even for dpda s in lower subclasses this is the first time the branching algorithm has been used to give such a general decision procedure without ever mixing the two languages in question in other words it deals with only the equivalence equation whose left hand side consists of a pure reachable configuration of one dpda and whose right hand side that of the other

on the power sequence of a fuzzy matrix with convex combination of max product and max min operations in the literature the powers of a fuzzy matrix with max min max product max archimedean t norm max t norm max arithmetic mean compositions have been studied it turns out that the limiting behavior of the powers of a fuzzy matrix depends on the composition involved in this paper we consider the powers of a fuzzy matrix with convex combination of max product and max min operations we show that the powers of a fuzzy matrix a with such operations are asymptotical p period if and only if the powers of an associated boolean matrix a are p periodic moreover necessary and sufficient conditions for such a fuzzy matrix to be nilpotent are proposed

on transductions of formal power series over complete semirings abstract rational and pushdown transductions of formal languages are generalized to formal power series with coefficients in a complete semiring a characterization similar to nivat s theorem is given commutativity requirements for the coefficients are especially studied

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

l a l b decidability results from complete formal systems abstract the equivalence problem for deterministic pushdown automata is shown to be decidable we exhibit a complete formal system for deducing equivalent pairs of deterministic rational boolean series on the alphabet associated with a dpda m we then extend the result to deterministic pushdown transducers from a free monoid into an abelian group a general algebraic and logical framework inspired by harrison et al theoret comput sci 9 1979 173 205 conrcelle theoret comput sci 6 1978 255 279 and meitus kybernetika 5 1989 14 25 in russian is developed

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

modifying the power method in max algebra in the max algebra system the eigenequation for an n n irreducible nonnegative matrix a aij is a x a x here a x i maxjaijxj and a is the maximum circuit geometric mean the complexity of the power method given in l elsner p van den driessche linear algebra appl to appear to compute a and x is considered under some assumptions on the critical matrix it is shown that the algorithm may have time complexity o n4 a modified power method based on karp s formula is presented for this new algorithm with no assumptions on the critical matrix a and x can be computed in o n3 time furthermore this algorithm can be used to compute all linearly independent eigenvectors corresponding to a

on the sequence of consecutive powers of a fuzzy matrix with max archimedean t norms in the literature the limiting behavior of powers of a fuzzy matrix has been studied with max product composition pang and guu proposed a simple and effective characterization for the limiting behavior with the notion of asymptotic period in this paper we shall extend pang and guu s work to the realm with the max archimedean t norm composition to which the max product belongs moreover some sufficient conditions for the powers to converge in finitely many steps will be established

convergence of max arithmetic mean powers of a fuzzy matrix fuzzy matrices provide convenient representations for fuzzy relations on finite universes in the literature the powers of a fuzzy matrix with max min max product max archimedean t norm compositions have been studied it turns out that the limiting behavior of the powers of a fuzzy matrix depends on the composition involved in this paper the max arithmetic mean composition is considered for the fuzzy relations we show that the max arithmetic mean powers of a fuzzy matrix always are convergent

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

on the oscillating power sequence of a fuzzy matrix abstract the properties of an oscillating power sequence of an n n fuzzy matrix have been studied the necessary and sufficient conditions have been established for a power sequence to be oscillatory furthermore the oscillation index was detailed to some degree especially for some typical boolean matrices it is also shown that the period index set of the power sequences of fuzzy matrices of order n is not bounded from above by a power of n for all integers n

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

reconstruction of radial dirac and schrodinger operators from two spectra abstract we solve the inverse problem of reconstructing radial dirac and schrodinger operators acting in the unit ball of r 3 from two spectra of their one dimensional parts corresponding to a fixed nonzero angular momentum we give a complete description of the spectral data prove existence and uniqueness of solutions to the inverse problem and present the reconstruction algorithm

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

on the convergence of powers of interval matrices abstract we give a necessary and sufficient condition for the sequence a k of the powers of an interval matrix a to converge to the null matrix o we construct a point matrix b which has spectral radius b less than one if a k converges

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

sufficient conditions for the stability of linear takagi sugeno free fuzzy systems in this paper we will study the stability issues of the linear takagi sugeno t s free fuzzy systems based on matrix norm we propose a new sufficient condition for the linear t s free fuzzy system to be globally asymptotically stable we then study the stability analysis in the case of systems with consequent parameter uncertainty based on mayer s convergent theorem we propose a sufficient condition which is easily implemented for the systems with consequent parameter uncertainty to be globally asymptotically stable

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

on the max contraction powers of a fuzzy matrix abstract in the literature the powers of a fuzzy matrix with max arithmetic mean convex combination of max min and max arithmetic mean convex combination of max product and max min compositions have been studied it turns out that the limiting behavior of the powers of a fuzzy matrix depends on the composition involved in this paper we consider the max contraction powers of a fuzzy matrix which is an extension of the max arithmetic mean convex combination of max min and max arithmetic mean convex combination of max product and max arithmetic mean operation we show that the powers of such a fuzzy matrix are always convergent and the limit matrix has the feature that all elements of each column are identical

convergence eigen fuzzy sets and stability analysis of relational matrices abstract if stable control applications are formulated with a max product composition then instability in the relational matrix under any conditions is undesirable this paper reviews the stability analysis of relational matrices combined with the max min composition 19 and then presents an analysis of the stability of relational matrices combined with the max product composition this analysis includes results defining the convergence properties of the relational matrix determination of the eigen fuzzy sets of the stable matrices and algorithmic solutions for several important cases a method is also presented to stabilize unstable relational matrices for some control applications

criteria for existence of riesz bases consisting of root functions of hill and 1d dirac operators abstract we study the system of root functions srf of hill operator l y y v y with a singular complex valued potential v h per 1 and the srf of 1d dirac operator l y i 1 0 0 1 d y d x v y with matrix l 2 potential v 0 p q 0 subject to periodic or anti periodic boundary conditions series of necessary and sufficient conditions in terms of fourier coefficients of the potentials and related spectral gaps and deviations for srf to contain a riesz basis are proven equiconvergence theorems are used to explain basis property of srf in l p spaces and other rearrangement invariant function spaces

a note on the sequence of consecutive powers of a nonnegative matrix in max algebra abstract in this note we shall consider the sequence of consecutive powers in max algebra of a nonnegative matrix with each element less than or equal to 1 the notion of asymptotic period is defined to study the limiting behavior of this sequence a simple and effective characterization for the asymptotic period of the sequence is given

the non linear stability of front solutions for parabolic partial differential equations for the ginzburg landau equation and similar reaction diffusion equations on the line we show convergence of complex perturbations of front solutions towards the front solutions by exhibiting a coercive functional

a weaker sufficient condition for the equivalence of a pair of dpda s to be decidable abstract this short paper further extends the result of tomita 1984 to have a weaker sufficient condition for the equivalence of a pair of non real time deterministic pushdown automata to be decidable

convergence of powers for a fuzzy matrix with convex combination of max min and max arithmetic mean operations fuzzy matrices have been proposed to represent fuzzy relations on finite universes since thomason s paper in 1977 showing that powers of a max min fuzzy matrix either converge or oscillate with a finite period conditions for limiting behavior of powers of a fuzzy matrix have been studied it turns out that the limiting behavior depends on the algebraic operations employed which usually in the literature includes max min max product max archimedean t norm max t norm max arithmetic mean operations respectively in this paper we consider the powers of a fuzzy matrix with convex combination of max min and max arithmetic mean operations we show that the powers of such a fuzzy matrix are always convergent

smoothness of schrodinger operator potential in the case of gevrey type asymptotics of the gaps abstract consider the schrodinger equation y v x y y with a periodic real valued l 2 potential v of period 1 v x m v m exp 2 imx let n n be its zones of instability i e n n are pairs of periodic and antiperiodic eigenvalues and b 0 1 determines gevrey classes theorem if n 1 n n 2 1 n 2s e 2an b s 0 a0 then v x is a gevrey function and moreover m v m 2 1 m 2s e 2a m b

note on convergence of powers of a fuzzy matrix we show by example that a theorem in thomason j math anal appl 57 1977 476 also recently quoted in imai et al fuzzy sets and systems 109 2000 405 about the convergence of fuzzy matrices is false

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

modulating pulse solutions for a class of nonlinear wave equations we consider modulating pulse solutions for a nonlinear wave equation on the infinite line such a solution consists of a permanent pulse like envelope steadily advancing in the laboratory frame and modulating an underlying wave train the problem is formulated as an infinite dimensional dynamical system with one stable one unstable and infinitely many neutral directions using a partial normal form and invariant manifold theory we establish the existence of modulating pulse solutions which decay to small amplitude disturbances at large distances

canonical form of a transitive fuzzy matrix some properties of a transitive fuzzy matrix are examined and the canonical form of the transitive matrix is given using the properties as a special case an open problem concerning idempotent matrices of kim and roush is solved in our results a nilpotent matrix and a symmetric matrix play an important role we decompose a transitive matrix into the sum of a nilpotent matrix and a symmetric matrix then we obtain a canonical form of the transitive matrix

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

multiplicities of the eigenvalues of periodic dirac operators abstract let us consider the dirac operator l ij d dx u j 1 0 0 1 u 0 a cos 2 x a cos 2 x 0 where a 0 is real on i 0 1 with boundary conditions bc per i e f 1 f 0 and bc per i e f 1 f 0 f f 1 f 2 h 1 i then l bc l bc and all per l u are of multiplicity 2 while per l u are simple theorem 15 this is an analogue of ince s statement for mathieu hill operator links between the spectra of dirac and hill operators lead to detailed information about the spectra of hill operators with potentials of the ricatti form v p p 2 section 3 it helps to get analogues of grigis results ann sci ecole norm sup 4 20 1987 641 on the zones of instability of hill operators with polynomial potentials and their asymptotics for the case of dirac operators as well section 4 2

modulating pulse solutions to quadratic quasilinear wave equations over exponentially long length scales this paper presents an existence proof for modulating pulse solutions to a wide class of quadratic quasilinear klein gordon equations of the form partial t 2 u partial x 2 u u f 1 u partial x u partial t u partial x 2 u f 2 u partial x u partial t u modulating pulse solutions consist of a pulse like envelope advancing in the laboratory frame and modulating an underlying wave train they are also referred to as moving breathers since they are time periodic in a moving frame of reference the problem is formulated as an infinite dimensional dynamical system with three stable three unstable and infinitely many neutral directions by transforming part of the equation into a normal form with an exponentially small remainder term and using a generalisation of local invariant manifold theory to the quasilinear setting we prove the existence of small amplitude modulating pulses on domains in space whose length is exponentially large compared to the magnitude of the pulse

nivat s theorem for pushdown transducers abstract the notions of an extended pushdown automaton pda and a pushdown transducer pdt are formalized in matrix notation by generalizing the concept of a pushdown transition matrix the resulting matrices form a semiring the equivalence to nonextended pdas and pdts is shown the semiring structure gives a new proof that the types reset pushdown and one counter are real time types moreover it allows the construction of the functional composition of a pushdown transduction and a finite transducer mapping as an important application nivat s theorem is generalized to rational and polynomial pdts over arbitrary semirings examples include the mirror image of the input and the dyck reduction

analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems a method for approximate solution of spectral problems for sturm liouville equations based on the construction of the delsarte transmutation operators is presented in fact the problem of numerical approximation of solutions and eigenvalues is reduced to approximation of a primitive of the potential by a finite linear combination of generalized wave polynomials introduced in khmelnytskaya et al 2013 and kravchenko and torba 2014 the method allows one to compute both lower and higher eigendata with an extreme accuracy

new infinite families of exact sums of squares formulas jacobi elliptic functions and ramanujan s tau function in this paper we give two infinite families of explicit exact formulas that generalize jacobi s 1829 4 and 8 squares identities to 4n2 or 4n n 1 squares respectively without using cusp forms our 24 squares identity leads to a different formula for ramanujan s tau function n when n is odd these results arise in the setting of jacobi elliptic functions jacobi continued fractions hankel or turanian determinants fourier series lambert series inclusion exclusion laplace expansion formula for determinants and schur functions we have also obtained many additional infinite families of identities in this same setting that are analogous to the function identities in appendix i of macdonald s work macdonald i g 1972 invent math 15 91 143 a special case of our methods yields a proof of the two conjectured kac v g and wakimoto m 1994 in progress in mathematics eds brylinski j l brylinski r guillemin v kac v birkhauser boston boston ma vol 123 pp 415 456 identities involving representing a positive integer by sums of 4n2 or 4n n 1 triangular numbers respectively our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series gustafson s cl nonterminating 6 5 summation theorem and andrews basic hypergeometric series proof of jacobi s 4 and 8 squares identities we have elsewhere applied symmetry and schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n n 1 squares respectively our sums of more than 8 squares identities are not the same as the formulas of mathews 1895 glaisher 1907 ramanujan 1916 mordell 1917 1919 hardy 1918 1920 kac and wakimoto and many others

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

divergence of spectral decompositions of hill operators with two exponential term potentials abstract we consider the hill operator l y y v x y 0 x subject to periodic or antiperiodic boundary conditions bc with potentials of the form v x a e 2 i r x b e 2 i s x a b 0 r s n r s it is shown that the system of root functions does not contain a basis in l 2 0 c if bc are periodic or if bc are antiperiodic and r s are odd or r 1 and s 3

powers of matrices over distributive lattices a review in this paper we review the results on powers of matrices over a distributive lattice we stress the significance of the graph theoretical approach and show how several previous results for matrices of special types can be obtained in a unified way

on the asymptotic stability of nonnegative matrices in max algebra abstract in the max algebra system for an n n nonnegative matrix a a ij the eigenequation for max eigenvalue and corresponding max eigenvector x is a x x where a x i max 1 j n a ij x j and a is the maximum circuit geometric mean it is shown that the following conditions are mutually equivalent i a r n ii a 1 iii a lim k a k 0 where a max x 1 x 0 a x and a limsup k a k 1 k

convergence of products of fuzzy matrices convergence of powers of a fuzzy matrix has been studied in the literature in this paper we shall extend the scope to the products of a finite number of fuzzy matrices we shall demonstrate that the outcomes of products of fuzzy matrices will be more complicated than the cases of powers of a fuzzy matrix two notions of convergence are established weak convergence and strong convergence the main results to guarantee the weak convergence of products of fuzzy matrices are due to the concepts of transitive and compact properties of fuzzy matrices which can be viewed as a generalization of the transitive and compact properties of a fuzzy matrix sufficient conditions similar to the pinching theorem in calculus are given as well for strong convergence

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 max nilpotent t norm powers of a fuzzy matrix fuzzy matrices have been proposed to represent fuzzy relations in finite universes various studies have evaluated the powers of a fuzzy matrix with max min max product max archimedean t norm max t norm max arithmetic mean compositions indicating that the limiting behavior of the powers of a fuzzy matrix depends on its composition in this paper max nilpotent composition is considered for the fuzzy relations we demonstrate that the max nilpotent powers of a fuzzy matrix either converge or oscillate in a finite period moreover the max nilpotent t norm powers of a fuzzy matrix a are p periodic if and only if the powers of an associated boolean matrix a c are p periodic finally necessary and sufficient conditions are proposed for nilpotent fuzzy matrices that exhibit max nilpotent composition

oscillation problems for hill s equation with periodic damping abstract this paper deals with the second order linear differential equation x a t x b t x 0 where a and b are periodic coefficients the main purpose is to present new criteria which guarantee that all nontrivial solutions are nonoscillatory and that those are oscillatory our nonoscillation theorem and oscillation theorem are proved by using the riccati technique in our theorem the composite function of an indefinite integral of b and a suitable multiple valued continuously differentiable function are focused and the composite function of them plays an important role the results obtained here include a result by kwong and wong 15 and a result by sugie and matsumura 26 an application to a equation of whittaker hill type is given to show the usefulness of our results finally simulations are also attached to illustrate that our oscillation criterion is sharp

convergence of powers of a fuzzy transitive matrix convergence of powers of a fuzzy transitive matrix is considered and some conditions for convergence are given under the max min operation on fuzzy matrices a fuzzy transitive matrix is a matrix which represents a fuzzy transitive relation and has many interesting properties the conditions for convergence of fuzzy matrices are examined under a special operation which is essential to reduction of fuzzy matrices or fuzzy systems given results show graph theoretic properties of fuzzy matrices and are useful when we consider convergence of systems represented by fuzzy matrices

convergence of spectral decompositions of hill operators with trigonometric polynomial potentials we consider the hill operator ly y prime prime v x y quad0 leq x leq pi subject to periodic or antiperiodic boundary conditions with potentials v which are trigonometric polynomials with nonzero coefficients of the form then the system of eigenfunctions and at most finitely many associated functions is complete but it is not a basis in l 2 0 pi mathbb c if a b in the case i if a b and neither b 2 4b nor a 2 4a is an integer square in the case iii and it is never a basis in the case ii subject to periodic boundary conditions

refined asymptotics of the spectral gap for the mathieu operator abstract the mathieu operator l y y 2 a cos 2 x y a c a 0 considered with periodic or anti periodic boundary conditions has close to n 2 for large enough n two periodic if n is even or anti periodic if n is odd eigenvalues n n for fixed a we show that n n 8 a 4 n n 1 2 1 a 2 4 n 3 o 1 n 4 n this result extends the asymptotic formula of harrell avron simon by providing more asymptotic terms

on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

whittaker hill equation and semifinite gap schrodinger operators a periodic one dimensional schrodinger operator is called semifinite gap if every second gap in its spectrum is eventually closed we construct explicit examples of semifinite gap schrodinger operators in trigonometric functions by applying darboux transformations to the whittaker hill equation we give a criterion for the regularity of the corresponding potentials and investigate the spectral properties of the new operators

eigenvalue asymptotics for dirac bessel operators in this paper we establish the eigenvalue asymptotics for non self adjoint dirac bessel operators on 0 1 with arbitrary real angular momenta and square integrable potentials which gives the first step for solution of the related inverse problem the approach is based on a careful examination of the corresponding characteristic functions and their zero distribution