eigenvalue problems for nonlinear differential equations on a measure chain abstract values of are determined for which there exist positive solutions of the second order differential equation on a measure chain u t a t f u t 0 t 0 1 satisfying either the conjugate boundary conditions u 0 u 1 0 or the right focal boundary conditions u 0 u 1 0 where a and f are positive valued and both lim x 0 f x x and lim x f x x exist

on undecidability bounds for matrix decision problems in this paper we consider several reachability problems such as vector reachability membership in matrix semigroups and reachability problems in piecewise linear maps since all of these questions are undecidable in general we work on lowering the bounds for undecidability in particular we show an elementary proof of undecidability of the reachability problem for a set of 5 two dimensional affine transformations then using a modified version of a standard technique we also prove that the vector reachability problem is undecidable for two rational matrices in dimension 11 the above result can be used to show that the system of piecewise linear functions of dimension 12 with only two intervals has an undecidable set to point reachability problem we also show that the zero in the upper right corner problem is undecidable for two integral matrices of dimension 18 lowering the bound from 23

example of undecidable problems for 2 generator matrix semigroups abstract we show that the following two problems are undecidable given two square matrices decide whether the semigroup that they generate contains the zero matrix and whether it contains a matrix having a zero in the right upper corner

dynamics of continuous time quantum walks in restricted geometries we study quantum transport on finite discrete structures and we model the process by means of continuous time quantum walks a direct and effective comparison between quantum and classical walks can be attained based on the average displacement of the walker as a function of time indeed a fast growth of the average displacement can be advantageously exploited to build up efficient search algorithms by means of analytical and numerical investigations we show that the finiteness and the inhomogeneity of the substrate jointly weaken the quantum walk performance we further highlight the interplay between the quantum walk dynamics and the underlying topology by studying the temporal evolution of the transfer probability distribution and the lower bound of long time averages

existence of triple positive solutions to a class of p laplacian boundary value problems abstract by using leggett williams fixed point theorem a class of p laplacian boundary value problem is studied sufficient conditions for the existence of triple positive solutions are established

dissimilar behaviors of coherent exciton transport on scale free networks with identical degree sequence conventional wisdom is that the power law degree sequence has a profound impact on diverse dynamical processes taking place on scale free networks here we show that power law degree distribution alone is not sufficient to determine the behaviors of coherent exciton transport on scale free networks for this purpose we study coherent exciton transport on a class of scale free networks controlled by a tunable parameter q which have the same degree sequence independent of q but are very different in other structural properties determined by q such as average distance and fractality first through numerical determination of the time evolutions of return probabilities we compare in detail the dissimilarities in oscillation periods and oscillation amplitudes between two limiting cases of q 0 and q 1 second we demonstrate that in the network of q 0 up to the third generation the quantum transport problem between node groups partitioned by node degrees can be mapped onto a line furthermore we show that in the case of long time limit the transition probabilities in the networks of q 0 and q 1 exhibit different characteristic patterns with identical values finally we find that the degree averaged return probability decreases when the parameter q increases from 0 to 1 indicating that the stronger localization can be found in those networks with lower values of q our results suggest that it is far from enough to characterize the behaviors of coherent exciton transport on scale free networks merely by power law degree distribution

continuous time quantum walks on one dimensional regular networks in this paper we consider continuous time quantum walks ctqws on a one dimensional ring lattice of n nodes in which every node is connected to its 2m nearest neighbors m on either side in the framework of the bloch function ansatz we calculate the space time transition probabilities between two nodes of the lattice we find that the transport of ctqws between two different nodes is faster than that of the classical continuous time random walks ctrws the transport speed which is defined by the ratio of the shortest path length and propagating time increases with the connectivity parameter m for both ctqws and ctrws for fixed parameter m the transport of ctrws gets slower with the increase of the shortest distance while the transport speed of ctqws turns out to be a constant value in the long time limit depending on the network size n and connectivity parameter m the limiting probability distributions of ctqws show various patterns when the network size n is an even number the probability of being at the original node differs from that of being at the opposite node which also depends on the precise value of parameter m

twin solutions of boundary value problems for ordinary differential equations and finite difference equations a new twin fixed point theorem is applied first to obtain the existence of at least two positive solutions for the right focal boundary value problem y f ity 0 0 t 1 y 0 y 1 0 it is applied later to obtain the existence of at least two positive solutions for the analogous discrete right focal boundary value problem 2u k g u k 0 k 0 n u 0 u n 1 0

slow transport by continuous time quantum walks continuous time quantum walks ctqws do not necessarily perform better than their classical counterparts the continuous time random walks ctrws for one special graph where a recent analysis showed that in a particular direction of propagation the penetration of the graph is faster by ctqws than by ctrws we demonstrate that in another direction of propagation the opposite is true in this case a ctqw initially localized at one site displays a slow transport we furthermore show that when the ctqw s initial condition is a totally symmetric superposition of states of equivalent sites the transport gets to be much more rapid

triple positive solutions of three point boundary value problems for p laplacian dynamic equations on time scales in this paper we present sufficient conditions for the existence of at least three positive solutions of three point boundary value problems for p laplacian dynamic equations on a time scale to show our main results we apply a new fixed point theorem due to avery and peterson three positive fixed points of nonlinear operators on ordered banach spaces comput math appl 42 2001 313 322

properties and applications of laplacian spectra for koch networks for a network knowledge of its laplacian eigenvalues is central to understanding its structure and dynamics in this paper we study the laplacian spectra of a family of koch networks with scale free and small world properties we derive some recursive relations between the laplacian characteristic polynomials of koch networks and their subgraphs at different iterations based on the obtained recurrence relations we determine explicitly the product of all nonzero laplacian eigenvalues as well as the sum of the reciprocals of these eigenvalues then using these results we further evaluate the number of spanning trees kirchhoff index global mean first passage time and average path length of the family of koch networks finally we determine the number of spanning forests under certain conditions we expect that our method can be adapted to other types of self similar networks

coherent exciton transport and trapping on long range interacting cycles we consider coherent exciton transport modeled by continuous time quantum walks on long range interacting cycles lrics which are constructed by connecting all the two nodes of distance m in the cycle graph lric has a symmetric structure and can be regarded as the extensions of the cycle graph nearest neighboring lattice for small values of m the classical and quantum return probabilities show power law behavior p t approximately t 0 5 and pi t approximately t 1 respectively however for large values of m the classical and quantum efficiency scales as p t approximately t 1 and pi t approximately t 2 we give a theoretical explanation of this transition using the method of stationary phase approximation in the long time limit depending on the network size n and parameter m the limiting probability distributions of quantum transport show various patterns when the network size n is an even number we find an asymmetric transition probability of quantum transport between the initial node and its opposite node this asymmetry depends on the precise values of n and m finally we study the transport processes in the presence of traps and find that the survival probability decays faster on networks of large m

quantum transport on two dimensional regular graphs we study the quantum mechanical transport on two dimensional graphs by means of continuous time quantum walks and analyse the effect of different boundary conditions bcs for periodic bcs in both directions i e for tori the problem can be treated in a large measure analytically some of these results carry over to graphs which obey open boundary conditions obcs such as cylinders or rectangles under obcs the long time transition probabilities lps also display asymmetries for certain graphs as a function of their particular sizes interestingly these effects do not show up in the marginal distributions obtained by summing the lps along one direction

three positive solutions of three point boundary value problems for p laplacian dynamic equations on time scales abstract in this paper existence criteria of three positive solutions to the following boundary value problems for p laplacian dynamic equations on time scales p u t a t f u t 0 t 0 t t u 0 0 u t b 1 u 0 or u 0 b 0 u 0 u t 0 are established by using the well known leggett williams fixed point theorem an example is also given to illustrate the main results

different thresholds of bond percolation in scale free networks with identical degree sequence generally the threshold of percolation in complex networks depends on the underlying structural characterization however what topological property plays a predominant role is still unknown despite the speculation of some authors that degree distribution is a key ingredient the purpose of this paper is to show that power law degree distribution itself is not sufficient to characterize the threshold of bond percolation in scale free networks to achieve this goal we first propose a family of scale free networks with the same degree sequence and obtain by analytical or numerical means several topological features of the networks then by making use of the renormalization group technique we determine the threshold of bond percolation in our networks we find an existence of nonzero thresholds and demonstrate that these thresholds can be quite different which implies that power law degree distribution does not suffice to characterize the percolation threshold in scale free networks

existence of solutions for a first order p laplacian bvp on time scales we introduce a first order delta dynamic equation on time scales involving the one dimensional p laplacian and prove the existence of at least one positive solution an example applying our result is also given

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

coherent dynamics on hierarchical systems abstract we study the coherent transport modeled by continuous time quantum walks focussing on hierarchical structures for these we use husimi cacti lattices dual to the dendrimers we find that the transport depends strongly on the initial site of the excitation for systems of sizes n 21 we find that processes which start at central sites are nearly recurrent furthermore we compare the classical limiting probability distribution to the long time average of the quantum mechanical transition probability which shows characteristic patterns we succeed in finding a good lower bound for the space average of the quantum mechanical probability to be still or again at the initial site

multiple positive solutions for functional dynamic equations on time scales in this paper we study the following functional dynamic equation on time scales f u d t a t f u t u m t 0 email protected 0 t t u t email protected t email protected r 0 t u 0 b 0 u d 0 0 u d t 0 where f r r is an increasing homeomorphism and a positive homomorphism and f 0 0 by using the well known leggett williams fixed point theorem existence criteria for multiple positive solutions are established an example is also given to illustrate the main results

triple positive solutions of m point bvps for p laplacian dynamic equations on time scales abstract this paper is concerned with the existence of positive solutions of p laplacian dynamic equation p u t a 1 t f u t 0 subject to boundary conditions u 0 b 0 i 1 m 2 a i u i 0 u t 0 or u 0 0 u t b 1 i 1 m 2 b i u i 0 where p v v p 2 v with p 1 by using the five functionals fixed point theorem we prove that the boundary value problem has at least three positive solutions as an application an example is given to illustrate the result

distinct scalings for mean first passage time of random walks on scale free networks with the same degree sequence in general the power law degree distribution has profound influence on various dynamical processes defined on scale free networks in this paper we will show that power law degree distribution alone does not suffice to characterize the behavior of trapping problems on scale free networks which is an integral major theme of interest for random walks in the presence of an immobile perfect absorber in order to achieve this goal we study random walks on a family of one parameter denoted by q scale free networks with identical degree sequence for the full range of parameter q in which a trap is located at a fixed site we obtain analytically or numerically the mean first passage time mfpt for the trapping issue in the limit of large network order number of nodes for the whole class of networks the mfpt increases asymptotically as a power law function of network order with the exponent obviously different for different parameter q which suggests that power law degree distribution itself is not sufficient to characterize the scaling behavior of mfpt for random walks at least trapping problem performed on scale free networks

decision problems for semi thue systems with a few rules we show that the accessibility problem the common descendant problem the termination problem and the uniform termination problem are undecidable for 3 rules semi thue systems as a corollary we obtain the undecidability of the post correspondence problem for 7 rules

sturm liouville eigenvalue problems on time scales for sturm liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish rayleigh s principle our results not only unifly the corresponding theories for differential and difference equations but are also new in the discrete case

percolation in hierarchical scale free nets we study the percolation phase transition in hierarchical scale free nets depending on the method of construction the nets can be fractal or small world the diameter grows either algebraically or logarithmically with the net size assortative or disassortative a measure of the tendency of like degree nodes to be connected to one another or possess various degrees of clustering the percolation phase transition can be analyzed exactly in all these cases due to the self similar structure of the hierarchical nets we find different types of criticality illustrating the crucial effect of other structural properties aside from the scale free degree distribution of the nets

on the problem of freeness of multiplicative matrix semigroups the following problem looking as a high school exercise hides an unexpected difficulty do the matrices a 2003 andb 3505 satisfy any nontrivial equation with the multiplication symbol only this problem was mentioned as open in cassaigne et al j cassaigne t harju j karhumaki on the undecidability of freeness of matrix semigroups internat j algebra comput 9 3 4 1999 295 305 and in a book by blondel et al v blondel j cassaigne j karhumaki problem 10 3 freeness of multiplicative matrix semigroups in v blondel a megretski eds unsolved problems in mathematical systems and control theory princeton university press 2004 pp 309 314 as an intriguing instance of a natural computational problem of deciding whether a given finitely generated semigroup of 2x2 matrices is free in this paper we present a new partial algorithm for the latter which in particular easily finds that the following equation ab 1 0a 2ba 2ba 1 0 b 2a 6b 2a 2bababa 2b 2a 2bab 2 holds for the matrices above our algorithm turns out quite practical and allows us to settle also other related open questions posed in the mentioned article

inverted berezinskii kosterlitz thouless singularity and high temperature algebraic order in an ising model on a scale free hierarchical lattice small world network we have obtained exact results for the ising model on a hierarchical lattice incorporating three key features characterizing many real world networks a scale free degree distribution a high clustering coefficient and the small world effect by varying the probability p of long range bonds the entire spectrum from an unclustered non small world network to a highly clustered small world system is studied using the self similar structure of the network we obtain analytic expressions for the degree distribution p k and clustering coefficient c for all p as well as the average path length ensuremath ell for p 0 and 1 the ferromagnetic ising model on this network is studied through an exact renormalization group transformation of the quenched bond probability distribution using up to 562 phantom rule 0 2em 0ex 500 renormalized probability bins to represent the distribution for pl0 494 we find power law critical behavior of the magnetization and susceptibility with critical exponents continuously varying with p and exponential decay of correlations away from t c for p ensuremath ge 0 494 in fact where the network exhibits small world character the critical behavior radically changes we find a highly unusual phase transition namely an inverted berezinskii kosterlitz thouless singularity between a low temperature phase with nonzero magnetization and finite correlation length and a high temperature phase with zero magnetization and infinite correlation length with power law decay of correlations throughout the phase approaching t c from below the magnetization and the susceptibility respectively exhibit the singularities of mathrm exp ensuremath c sqrt t c ensuremath t and mathrm exp d sqrt t c ensuremath t with c and d positive constants with long range bond strengths decaying with distance we see a phase transition with power law critical singularities for all p and evaluate an unusually narrow critical region and important corrections to power law behavior that depend on the exponent characterizing the decay of long range interactions

double positive solutions of three point boundary value problems for p laplacian dynamic equations on time scales a new double fixed point theorem is applied to investigate the existence of at least double positive solutions of three point boundary value problems for p laplacian dynamic equations on a time scale