submatrices of non tree realizable distance matrices abstract we deal with distance matrices of real this means not necessarily integer numbers it is known that a distance matrix d of order n is tree realizable if and only if all its principal submatrices of order 4 are tree realizable we discuss bounds for the number denoted qi d of non tree realizable principal submatrices of order i 4 of a non tree realizable distance matrix d of order n i and we construct some distance matrices which meet extremal conditions on qi d our starting point is a proof that a non tree realizable distance matrix of order 5 has at least two non tree realizable principal submatrices of order 4 optimal realizations by graphs with circuits of distance matrices which are not tree realizable are not yet as well known as optimal realizations which are trees using as starting point the optimal realization of the arbitrary distance matrix of order 4 we investigate optimal realizations of non tree realizable distance matrices with the minimum number of non tree realizable principal submatrices of order 4

characterization of the spectrum of the landau hamiltonian with delta impurities we consider a random schrodinger operator in an external magnetic field the random potential consists of delta functions of random strengths situated on the sites of a regular two dimensional lattice we characterize the spectrum in the lowest n landau bands of this random hamiltonian when the magnetic field is sufficiently strong depending on n we show that the spectrum in these bands is entirely pure point that the energies coinciding with the landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length by relating the hamiltonian to a lattice operator we are able to use the aizenman molchanov method to prove localization

the spectrum of a magnetic schrodinger operator with randomly located delta impurities we consider a single band approximation to the random schrodinger operator in an external magnetic field the spectrum of such an operator has been characterized in the case where delta impurities are located on the sites of a lattice in this paper we generalize these results by letting the delta impurities have random positions as well as strengths they are located in squares of a lattice with a general bounded distribution we characterize the entire spectrum of this operator when the magnetic field is sufficiently high we show that the whole spectrum is pure point the energy coinciding with the first landau level is infinitely degenerate and that the eigenfunctions corresponding to other landau band energies are exponentially localized

a chain of evolution algebras abstract we introduce a notion of chain of evolution algebras the sequence of matrices of the structural constants for this chain of evolution algebras satisfies an analogue of chapman kolmogorov equation we give several examples time homogenous time non homogenous periodic etc of such chains for a periodic chain of evolution algebras we construct a continuum set of non isomorphic evolution algebras and show that the corresponding discrete time chain of evolution algebras is dense in the set we obtain a criteria for an evolution algebra to be baric and give a concept of a property transition for several chains of evolution algebras we describe the behavior of the baric property depending on the time for a chain of evolution algebras given by the matrix of a two state evolution we define a baric property controller function and under some conditions on this controller we prove that the chain is not baric almost surely with respect to lebesgue measure we also construct examples of the almost surely baric chains of evolution algebras we show that there are chains of evolution algebras such that if it has a unique resp infinitely many absolute nilpotent element at a fixed time then it has unique resp infinitely many absolute nilpotent element any time also there are chains of evolution algebras which have not such property for an example of two dimensional chain of evolution algebras we give the full set of idempotent elements and show that for some values of parameters the number of idempotent elements does not depend on time but for other values of parameters there is a critical time t c such that the chain has only two idempotent elements if time t t c and it has four idempotent elements if time t t c

localization for the anderson model on a strip with singular potentials abstract we prove exponential localization in the anderson model on a one dimensional strip for any potential whose probability distribution is not concentrated on a single point and possesses some finite moment in particular we obtain localization for bernoulli potentials on a strip

stochastic matrices arising from genetic inheritance we establish a connection between two different approaches that can be used to describe the same genetic situation we focus on the notion of coalgebra with genetic realization introduced by tian and li and show how their structure constants can be used to characterize different stochastic processes related to the genetic inheritance

trees tight extensions of metric spaces and the cohomological dimension of certain groups a note on combinatorial properties of metric spaces abstract the concept of tight extensions of a metric space is introduced the existence of an essentially unique maximal tight extension t x the tight span being an abstract analogon of the convex hull is established for any given metric space x and its properties are studied applications with respect to 1 the existence of embeddings of a metric space into trees 2 optimal graphs realizing a metric space and 3 the cohomological dimension of groups with specific length functions are discussed

non perturbative anderson localization in heavy tailed potentials via large deviations moment analysis we study a class of anderson hamiltonians with heavy tailed independent and identically distributed random potential on graphs with sub exponential growth of the balls and of the number of self avoiding paths connecting pairs of points we show that for a class of marginal distributions anderson localization occurs non perturbatively i e for any nonzero amplitude of the potential like in one dimensional systems the proof is based on the moment analysis of the green functions via large deviations estimates

localization criteria for anderson models on locally finite graphs we prove spectral and dynamical localization for anderson models on locally finite graphs using the fractional moment method our theorems extend earlier results on localization for the anderson model on d we establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder

hyperconvexity and tight span theory for diversities abstract the tight span or injective envelope is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded the concept has stimulated a large body of theory and has applications to metric classification and data visualisation here we introduce a generalisation of metrics called diversities and demonstrate that the rich theory associated to metric tight spans and hyperconvexity extends to a seemingly richer theory of diversity tight spans and hyperconvexity

markov processes of cubic stochastic matrices quadratic stochastic processes abstract we consider markov processes of cubic stochastic in a fixed sense matrices which are also called quadratic stochastic process qsps a qsp is a particular case of a continuous time dynamical system whose states are stochastic cubic matrices satisfying an analogue of the kolmogorov chapman equation kce since there are several kinds of multiplications between cubic matrices we have to fix first a multiplication and then consider the kce with respect to the fixed multiplication moreover the notion of stochastic cubic matrix also varies depending on the real models of application the existence of a stochastic at each time solution to the kce provides the existence of a qsp in this paper our aim is to construct qsps for two specially chosen notions of stochastic cubic matrices and two multiplications of such matrices known as maksimov s multiplications we construct a wide class of qsps and give some time dependent behavior of such processes we give an example with applications to the biology constructing a qsp which describes the time behavior dynamics of a population with the possibility of twin births

classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field abstract evolution algebras were introduced into genetics to deal with the mechanism of inheritance of asexual organisms their distribution into isotopism classes is uniquely related with the mutation of alleles in non mendelian genetics this paper deals with such a distribution by means of computational algebraic geometry we focus in particular on the two dimensional case which is related to the asexual reproduction processes of diploid organisms specifically we determine the existence of four isotopism classes whatever the base field is and we characterize the corresponding isomorphism classes

microlocalization percolation and anderson localization for the magnetic schrodinger operator with a random potential abstract we study the spectral properties of the magnetic schrodinger operator with a random potential using results from microlocal analysis and percolation we show that away from the landau levels the spectrum is almost surely pure point with at least exponentially decaying eigenfunctions moreover it is shown that the rate of decay is proportional to b

on the ergodic principle for markov and quadratic stochastic processes and its relations in the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated markov process satisfies one

construction of flows of finite dimensional algebras abstract recently we introduced the notion of flow depending on time of finite dimensional algebras a flow of algebras fa is a particular case of a continuous time dynamical system whose states are finite dimensional algebras with cubic matrices of structural constants satisfying an analogue of the kolmogorov chapman equation kce since there are several kinds of multiplications between cubic matrices one has fix a multiplication first and then consider the kce with respect to the fixed multiplication the existence of a solution for the kce provides the existence of an fa in this paper our aim is to find sufficient conditions on the multiplications under which the corresponding kce has a solution mainly our conditions are given on the algebra of cubic matrices acm considered with respect to a fixed multiplication of cubic matrices under some assumptions on the acm e g power associative unital associative commutative we describe a wide class of fas which contain algebras of arbitrary finite dimension in particular adapting the theory of continuous time markov processes we construct a class of fas given by the matrix exponent of cubic matrices moreover we remarkably extend the set of fas given with respect to the maksimov s multiplications of our paper 8 for several fas we study the time dependent behavior dynamics of the algebras we derive a system of differential equations for fas

a characterization of the anderson metal insulator transport transition we investigate the anderson metal insulator transition for random schrodinger operators we define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the hilbert schmidt norm we introduce a local transport exponent e and set the weak metallic transport region to be the part of the spectrum with nontrivial transport i e e 0 we prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent e provides a characterization of the metal insulator transport transition moreover we show that if there is such a transition then e has to be discontinuous at a transport mobility edge more precisely we show that if the transport is nontrivial then e 1 2d where d is the space dimension these results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors bootstrap multiscale analysis we also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis proving that the multiscale analysis is valid all the way up to a transport mobility edge

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

localization and universality of poisson statistics for the multidimensional anderson model at weak disorder we prove anderson localization with the mean field lyapunov exponent and poisson statistics for eigenvalue spacing for the multi dimensional anderson model at weak disorder these results are obtained by developing the supersymmetric formalism initiated in w1 see also sjw rid

anderson localization for generic deterministic operators abstract we consider a class of ensembles of lattice schrodinger operators with deterministic potentials including quasi periodic potentials with diophantine frequencies depending upon an infinite number of parameters in an auxiliary measurable space using a variant of the multi scale analysis we prove anderson localization for generic ensembles in the strong disorder regime and establish an analog of minami type bounds for spectral spacings

landau hamiltonians with random potentials localization and the density of states we prove the existence of localized states at the edges of the bands for the two dimensional landau hamiltonian with a random potential of arbitrary disorder provided that the magnetic field is sufficiently large the corresponding eigenfunctions decay exponentially with the magnetic field and distance we also prove that the integrated density of states is lipschitz continuous away from the landau energies the proof relies on a wegner estimate for the finite area magnetic hamiltonians with random potentials and exponential decay estimates for the finitearea green s functions the proof of the decay estimates for the green s functions uses fundamental results from two dimensional bond percolation theory

localization for random and quasiperiodic potentials a survey is made of some recent mathematical results and techniques for schrodinger operators with random and quasiperiodic potentials a new proof of localization for random potentials established in collaboration with h von dreifus is sketched

dynamical localization for discrete and continuous random schrodinger operators we show for a large class of random schrodinger operators h on and on that dynamical localization holds i e that with probability one for a suitable energy interval i and for q a positive real here is a function of sufficiently rapid decrease and p i h is the spectral projector of h corresponding to the interval i the result is obtained through the control of the decay of the eigenfunctions of h and covers in the discrete case the anderson tight binding model with bernoulli potential dimension 1 or singular potential 1 and in the continuous case anderson as well as random landau hamiltonians

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

2 dimensional localization of acoustic waves in random perturbation of periodic media abstract in this work we study internal band edges localization of acoustic waves in 2 dimensional space obtained by random perturbation of some periodic media our results rely on the study of lifshitz tails for the integrated density of states for random acoustic operators of the form a 1 localization is then deduced by the standard multiscale argument

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

upper bounds in quantum dynamics of course the case of main interest is where h is given by l2 rd or 2 zd h is a schr dinger operator of the form a v and ip 0 is a localized wavepacket under the time evolution 1 the wavepacket i t will in general spread out with time finding quantitative bounds for this spreading in concrete cases and developing methods for proving such bounds has been the objective of a great number of papers in recent years one is often interested in the growth behavior of the moments of the position operator that is the function 11 x p 2 i 2 these questions are particularly complicated and interesting in situations where singular continuous spectra occur the discovery of the occurrence of such spectra for many models of physical relevance has triggered some activity on the quantum dynamical side for models with absolutely continuous spectra on the other hand quantum dynamical questions have been investigated much earlier in the context of scattering theory and have been answered to a great degree of satisfaction models with point spectrum or more restrictively models exhibiting anderson localization have also been investigated from a quantum dynamical point of view while it is not true in general that spectral localization implies dynamical localization 26 73 it has been shown that the three main approaches to spectral localization the fractional moment method 3 4 5 multi scale analysis 28 27 75 and the bourgain goldstein schlag method 12 37 yield dynamical localization as a by product 2 13 22 31 34 quantum dynamics for schr dinger operators with singular continuous spectra is much less understood the two primary examples motivated by physics are the harper model also known as the almost mathieu operator at critical cou pling which describes a bloch electron in a constant magnetic field with irrational flux through a unit cell and the fibonacci model which describes a standard one dimensional quasicrystal both models have been studied heavily in both the physics and mathematics communities

a canonical decomposition theory for metrics on a finite set abstract we consider specific additive decompositions d d 1 d n of metrics defined on a finite set x where a metric may give distance zero to pairs of distinct points the simplest building stones are the slit metrics associated to splits i e bipartitions of the given set x while an additive decomposition of a hamming metric into split metrics is in no way unique we achieve uniqueness by restricting ourselves to coherent decompositions that is decompositions d d 1 d n such that for every map f x r with f x f y d x y for all x y x there exist maps f 1 f n x r with f f 1 f n and f i x f i y d i x y for all i 1 n and all x y x these coherent decompositions are closely related to a geometric decomposition of the injective hull of the given metric a metric with a coherent decomposition into a weighted sum of split metrics will be called totally split decomposable tree metrics and more generally the sum of two tree metrics are particular instances of totally split decomposable metrics our main result confirms that every metric admits a coherent decomposition into a totally split decomposable metric and a split prime residue where all the split summands and hence the decomposition can be determined in polynomial time and that a family of splits can occur this way if and only if it does not induce on any four point subset all three splits with block size two

the quantum hall effect for electrons in a random potential we assume the existence of sufficiently localised states near the edges of each landau band we then prove that the hall conductivity is quantised and the parallel conductivity vanishes when the filling factor stays close to an integer the hall integer is a topological invariant given by the landau band index

upper bounds on the density of states of single landau levels broadened by gaussian random potentials we study a nonrelativistic charged particle on the euclidean plane r2 subject to a perpendicular constant magnetic field and an r2 homogeneous random potential in the approximation that the corresponding random landau hamiltonian on the hilbert space l2 r2 is restricted to the eigenspace of a single but arbitrary landau level for a wide class of r2 homogeneous gaussian random potentials we rigorously prove that the associated restricted integrated density of states is absolutely continuous with respect to the lebesgue measure we construct explicit upper bounds on the resulting derivative the restricted density of states as a consequence any given energy is seen to be almost surely not an eigenvalue of the restricted random landau hamiltonian

anderson localization for the almost mathieu equation ii point spectrum for lambda 2 we prove that for any 2 and a e the pure point spectrum of the almost mathieu operator h n n 1 n 1 cos 2 n n contains the essential closure hat sigma of the spectrum corresponding eigenfunctions decay exponentially the singular continuous component if it exists is concentrated on a set of zero measure which is nowhere dense in hat sigma

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

graded cluster expansion for lattice systems in this paper we develop a general theory which provides a unified treatment of two apparently different problems the weak gibbs property of measures arising from the application of renormalization group maps and the mixing properties of disordered lattice systems in the griffiths phase we suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely large regions are widely separated we then show how it is possible to construct a convergent multi scale cluster expansion

quantum particle in a hierarchical potential with tunnelling over arbitrarily large scales the authors introduce a notion of hierarchical potential by iterating a basic elementary geometric construction over an increasing sequence of length scales they then show that a quantum particle in such a potential exhibits a subdiffusive behaviour characterised by r2 or c ln t beta if the initial state is a normalised wavepacket superposition of states of sufficiently low energy

an explicit computation of the injective hull of certain finite metric spaces in terms of their associated buneman complex an explicit computation of the injective hull of certain finite metric spaces in terms of their associated buneman complex

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

localization for random schrodinger operators with correlated potentials we prove localization at high disorder or low energy for lattice schrodinger operators with random potentials whose values at different lattice sites are correlated over large distances the class of admissible random potentials for our multiscale analysis includes potentials with a stationary gaussian distribution whose covariance functionc x y decays as x y where 0 can be arbitrarily small and potentials whose probability distribution is a completely analytical gibbs measure the result for gaussian potentials depends on a multivariable form of nelson s best possible hypercontractive estimate

flow of finite dimensional algebras abstract each finite dimensional algebra can be identified to the cubic matrix given by structural constants defining the multiplication between the basis elements of the algebra in this paper we introduce the notion of flow depending on time of finite dimensional algebras this flow can be considered as a particular case of continuous time dynamical system whose states are finite dimensional algebras with matrices of structural constants satisfying an analogue of kolmogorov chapman equation these flows of algebras fas can also be considered as deformations of algebras with the rule the evolution equation given by kolmogorov chapman equation we mainly use the multiplications of cubic matrices which were introduced by maksimov and consider kolmogorov chapman equation with respect to these multiplications if all cubic matrices of structural constants are stochastic there are several kinds of stochasticity then the corresponding fa is called stochastic fa sfa we define sfa generated by known quadratic stochastic processes for some multiplications of cubic matrices we reduce kolmogorov chapman equation given for cubic matrices to the equation given for square matrices using this result many fas are given time homogeneous time non homogeneous periodic etc for a periodic fa we construct a continuum set of finite dimensional algebras and show that the corresponding discrete time fa is dense in the set moreover we give a construction of an fa which contains algebras of arbitrary finite dimension for several fas we describe the time depending behavior dynamics of the properties to be baric limit algebras commutative evolution algebras or associative algebras

electron states in a magnetic field and random impurity potential use of the theory of entire functions the authors determine the wavefunction of an electron in the presence of a transverse magnetic field and randomly located delta function impurities in the x y plane within the subspace of the lowest landau level the wavefunction contains as a factor an entire function of z x iy which vanishes at all impurity sites the question of whether such states are extended is related to the rate of growth of an entire function in terms of the distribution of its zeros for a homogeneous distribution of impurities the corresponding weierstrass product is of order 2 and of finite type this rate of growth can be exactly compensated by the gaussian term due to the presence of the magnetic field such that there is a critical field beyond which extended states may exist if the impurities are located on the sites of a square lattice the extended states are given in closed form in terms of the weierstrass sigma function

evolution algebras of arbitrary dimension and their decompositions abstract we study evolution algebras of arbitrary dimension we analyze in deep the notions of evolution subalgebras ideals and non degeneracy and describe the ideals generated by one element and characterize the simple evolution algebras we also prove the existence and unicity of a direct sum decomposition into irreducible components for every non degenerate evolution algebra when the algebra is degenerate the uniqueness cannot be assured the graph associated to an evolution algebra relative to a natural basis will play a fundamental role to describe the structure of the algebra concretely a non degenerate evolution algebra is irreducible if and only if the graph is connected moreover when the evolution algebra is finite dimensional we give a process called the fragmentation process to decompose the algebra into irreducible components

infinite degeneracy for a landau hamiltonian with poisson impurities we consider a single band approximation to the random schrodinger operator in an external magnetic field the random potential consists of delta functions of random strengths whose positions have a poisson distribution we prove that if the magnetic field is sufficiently high compared to the density of scatterers then with probability one there exists an infinitely degenerate eigenenergy coinciding with the first landau level in the absence of a random potential

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

new special cases of the quadratic assignment problem with diagonally structured coefficient matrices abstract we consider new polynomially solvable cases of the well known quadratic assignment problem involving coefficient matrices with a special diagonal structure by combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the qap where one of the two coefficient matrices involved is a robinson matrix with an additional structural property this matrix can be represented as a conic combination of cut matrices in a certain normal form the other matrix is a conic combination of a monotone anti monge matrix and a down benevolent toeplitz matrix we consider the recognition problem for the special class of robinson matrices mentioned above and show that it can be solved in polynomial time

the density of states in the anderson model at weak disorder a renormalization group analysis of the hierarchical model we study the density of states in a hierarchical approximation of the anderson tight binding model at weak disorder using a renormalization group approach since the laplacian term in our model is hierarchical the renormalization group transformations act essentially on the local potential distribution and the energy technically we use the supersymmetric replica trick and study the averaged green s function starting with a gaussian distribution with small variance we find that the density of states is analytic as soon as the variance of the potential is turned on except possibly near the band edge where we can show this only for 2 which corresponds tod 4 moreover it is perturbatively close to the free one except near the eigenvalues of the hierarchical laplacian where it is given up to perturbative corrections by the rescaled potential distribution

on a class of metrics related to graph layout problems we examine the metrics that arise when a finite set of points is embedded in the real line in such a way that the distance between each pair of points is at least 1 these metrics are closely related to some other known metrics in the literature and also to a class of combinatorial optimization problems known as graph layout problems we prove several results about the structure of these metrics in particular it is shown that their convex hull is not closed in general we then show that certain linear inequalities define facets of the closure of the convex hull finally we characterize the unbounded edges of the convex hull and of its closure

inferring evolutionary trees with strong combinatorial evidence we consider the problem of inferring the evolutionary tree of a set of n species we propose a quartet reconstruction method which specifically produces trees whose edges have strong combinatorial evidence let q be a set of resolved quartets defined on the studied species the method computes the unique maximum subset q of q which is equivalent to a tree and outputs the corresponding tree as an estimate of the species phylogeny we use a characterization of the subset q due to bandelt and dress adv appl math 7 1986 309 343 to provide an o n4 incremental algorithm for this variant of the np hard quartet consistency problem moreover when chosing the resolution of the quartets by the four point method fpm and considering the cavender farris model of evolution we show that the convergence rate of the q method is at worst polynomial when the maximum evolutive distance between two species is bounded we complete these theoretical results by an experimental study on real and simulated data sets the results show that i as expected the strong combinatorial constraints it imposes on each edge leads the q method to propose very few incorrect edges ii more surprisingly the method infers trees with a relatively high degree of resolution

localization of classical waves i acoustic waves we consider classical acoustic waves in a medium described by a position dependent mass density x we assume that x is a reandom perturbation of a periodic function 0 x and that the periodic acoustic operator a 0 nabla cdot tfrac 1 varrho 0 x nabla has a gap in the spectrum we prove the existence of localized waves i e finite energy solutions of the acoustic equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times with probability one localization of acoustic waves is a consequence of anderson localization for the self adjoint operators a nabla cdot tfrac 1 varrho 0 x nabla onl2 d we prove that in the random medium described by x the random operatora exhibits anderson localization inside the gap in the spectrum ofa0 this is shown even in situations when the gap is totally filled by the spectrum of the random opertor we can prescribe random environments that ensure localization in almost the whole gap

algebraic computation of genetic patterns related to three dimensional evolution algebras the mitosis process of an eukaryotic cell can be represented by the structure constants of an evolution algebra any isotopism of the latter corresponds to a mutation of genotypes of the former this paper uses computational algebraic geometry to determine the distribution of three dimensional evolution algebras over any field into isotopism classes and hence to describe the spectrum of genetic patterns of three distinct genotypes during a mitosis process their distribution into isomorphism classes is also determined in case of dealing with algebras having a one dimensional annihilator

upper bound on the energy gap in terms of the localization in tight binding systems we present an upper bound on the energy gap above the ground energy in terms of the variance of the particle position in one particle tight binding systems our inequality gives a complementary relationship between the energy gap and the localization of the particle that is the energy gap narrows as the ground state spreads spatially while the energy gap can broaden as the ground state becomes localized we first consider the ground state in one dimensional tight binding systems and relate the energy gap to the variance of the particle position in the ground state we then extend the result to higher dimensional systems we compare our upper bound with the exact energy gap as calculated numerically in one and two dimensions this comparison confirms that our upper bound places a tight restriction on the energy gap

on mott s formula for the ac conductivity in the anderson model we study the ac conductivity in linear response theory in the general framework of ergodic magnetic schrodinger operators for the anderson model if the fermi energy lies in the localization regime we prove that the ac conductivity is bounded from above by c 2 log 1 d 2 at small frequencies this is to be compared to mott s formula which predicts the leading term to be c 2 log 1 d 1

constrained evolution algebras and dynamical systems of a bisexual population abstract consider a bisexual population such that the set of females can be partitioned into finitely many different types indexed by 1 2 n and similarly that the male types are indexed by 1 2 recently an evolution algebra of bisexual population was introduced by identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra in this paper we study constrained evolution algebra of bisexual population in which type 1 of females and males have preference for such algebras sets of idempotent and absolute nilpotent elements are known we consider two particular cases of this algebra giving more constraints on the structural constants of the algebra by the first our constraint we obtain an n dimensional algebra with a matrix of structural constants containing only 0 and 1 in the second case we consider n 2 but with general constraints in both cases we study dynamical systems generated by the quadratic evolution operators of corresponding constrained algebras we find all fixed points limit points and some 2 periodic points of the dynamical systems moreover we study several properties of the constrained algebras connecting them to the dynamical systems we give some biological interpretation of our results

simplicity of eigenvalues in the anderson model we give a transparent and intuitive proof that all eigenvalues of the anderson model in the region of localization are simple

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

remark on the absence of absolutely continuous spectrum for d dimensional schrodinger operators with random potential for large disorder or low energy we show that there is no absolutely continuous part in the spectrum of the anderson tight binding model for large disorder or low energy the proof is based on the exponential decay of the green s function proved by frohlich and spencer the extension of this result to the continuous case is also discussed

localization for some continuous random schrodinger operators we study the spectrum of random schrodinger operators acting onl2 r d of the following type h delta w sum x in mathbb z d t x v x the t x x in mathbb z d are i i d random variables under weak assumptions onv we prove exponential localization forh at the lower edge of its spectrum in order to do this we give a new proof of the wegner estimate that works without sign assumptions onv

chain of evolution algebras of chicken population abstract recently ladra and rozikov introduced a notion of evolution algebra of a chicken population eacp the algebra is given by a rectangular matrix of structural constants in this paper we introduce a notion of chain of evolution algebras of a chicken population ceacp the sequence of matrices of the structural constants for this ceacp satisfies an analogue of chapman kolmogorov equation with a specific multiplication defined for rectangular matrices we give several examples time homogeneous time non homogeneous periodic etc of such chains we construct some periodic 3 dimensional ceacp which contains a continuum set of non isomorphic eacp and show that the corresponding discrete time ceacp is dense in the set moreover we study time depending dynamics of 2 and 3 dimensional ceacp to be isomorphic to a fixed algebra

diffusion in the mean for an ergodic schrodinger equation perturbed by a fluctuating potential diffusive scaling of position moments and a central limit theorem are obtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a random potential consisting of a large static part and a small part that fluctuates stochastically in time if the static random potential is strong enough to induce complete localization in the absence of time dependent noise then the diffusion constant is shown to go to zero proportional to the square of the strength of the time dependent part

two parameter spectral averaging and localization for non monotonic random schrodinger operators we prove exponential localization at all energies for two types of one dimensional random schr6dinger operators the poisson model and the random displacement model as opposed to anderson type models these operators are not monotonic in the random parameters therefore the classical one parameter version of spectral averaging as used in localization proofs for anderson models breaks down we use the new method of two parameter spectral averaging and apply it to the poisson as well as the displacement case in addition we apply results from inverse spectral theory which show that two parameter spectral averaging works for sufficiently many energies all but a discrete set to conclude localization at all energies