stability for quadratic functional equation in the spaces of generalized functions in this paper we consider the general solution of quadratic functional equation f ax y f ax y f x y f x y 2 a2 1 f x for any integer a with a 1 0 1 moreover we reformulate and prove the hyers ulam rassias stability theorem of the above equation in the spaces of tempered distributions and fourier hyperfunctions the generalized hyers ulam stability originated from the th m rassias s stability theorem that appeared in his paper th m rassias on the stability of the linear mapping in banach spaces proc amer math soc 72 1978 297 300

star partial order on b h abstract let h be an infinite dimensional complex hilbert space and let b h be the algebra of all bounded linear operators on h in the paper the equivalent definition of the star partial order on b h using selfadjoint idempotent operators is introduced also some properties of the generalized concept of order relations on b h defined with the help of idempotent operators are investigated

a directed network of greek and roman mythology abstract we construct a directed network using a dictionary of greek and roman mythology in which the nodes represent the entries listed in the dictionary and we make directional links from an entry to other entries that appear in its explanatory part we find that this network is clearly not a random network but a directed scale free network in which the distributions of out degree and in degree follow a power law with exponents out 3 0 and in 2 5 respectively also we measure several quantities which describe the topological properties of the network and compare it to that of other real networks

local world evolving networks with tunable clustering abstract we propose an extended local world evolving network model including a triad formation tf step in the process of network evolution random fluctuation in the number of new edges is involved we derive analytical expressions for degree distribution clustering coefficient and average path length our model can unify the generic properties of real life networks scale free degree distribution high clustering and small inter node separation moreover in our model the clustering coefficient is tunable simply by changing the expected number of tf steps after a single local preferential attachment step

hyers ulam stability of non autonomous systems in terms of boundedness of cauchy problems in this paper the concepts of hyers ulam stability are generalized for non autonomous linear differential systems we prove that the k periodic linear differential matrix system z t a t z t t r is hyers ulam stable if and only if the matrix family l e k 0 has no eigenvalues on the unit circle i e we study the hyers ulam stability in terms of dichotomy of the differential matrix system z t a t z t t r furthermore we relate hyers ulam stability of the system z t a t z t t r to the boundedness of solution of the following cauchy problem y t a t y t t t 0 y 0 x x 0 where a t is a square matrix for any t r t is a bounded function and x x 0 c m

effect of node deleting on network structure abstract the ever increasing knowledge of the structure of various real world networks has uncovered their complex multi mechanism governed evolution processes therefore a better understanding of the structure and evolution of these networked complex systems requires us to describe such processes in a more detailed and realistic manner in this paper we introduce a new type of network growth rule which comprises addition and deletion of nodes and propose an evolving network model to investigate the effect of node deleting on network structure it is found that with the introduction of node deleting network structure is significantly transformed in particular degree distribution of the network undergoes a transition from scale free to exponential forms as the intensity of node deleting increases at the same time nontrivial disassortative degree correlation develops spontaneously as a natural result of network evolution in the model we also demonstrate that node deleting introduced in the model does not destroy the connectedness of a growing network so long as the increasing rate of edges is not excessively small in addition it is found that node deleting will weaken but not eliminate the small world effect of a growing network and generally it will decrease the clustering coefficient in a network

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

exploring complex networks by walking on them we carry out a comparative study of the problem of a walker searching several typical complex networks the search efficiency is evaluated for various strategies having no knowledge of the global properties of the underlying networks and the optimal path between any two given nodes it is found that the best search strategy is the self avoiding random walk the preferentially self avoiding random walk does not help in improving the search efficiency further in return topological information of the underlying networks may be drawn by comparing the results of the different search strategies

complex earthquake networks hierarchical organization and assortative mixing to characterize the dynamical features of seismicity as a complex phenomenon the seismic data are mapped to a growing random graph which is a small world scale free network here hierarchical and mixing properties of such a network are studied the clustering coefficient is found to exhibit asymptotic power law decay with respect to connectivity showing hierarchical organization this structure is supported by not only main shocks but also small shocks and may have its origin in the combined effect of vertex fitness and deactivation by stress release at faults the nearest neighbor average connectivity and the pearson correlation coefficient are also calculated it is found that the earthquake network has assortative mixing this is a main difference of the earthquake network from the internet with disassortative mixing physical implications of these results are discussed

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

lattice properties of the order for complex matrices abstract it is shown that the set c m n of complex m n matrices forms a lower semilattice under the partial ordering a b defined by a a a b aa ba where a denotes the conjugate transpose of a as a special case of a result for division rings it is further shown that over any field f for m n 2 and any proper involution of f2 2 the corresponding intersections a b all exist

generation of scale free networks using a simple preferential rewiring dynamics abstract we propose a simple dynamical process for non growing networks where steady states in the long time limit exhibit power law degree distributions with the exponent 2 at each time step two nodes i and j are randomly selected and one incoming link to i is redirected to j with the rewiring probability r determined only by degrees of two nodes k i and k j while higher degree nodes are preferred to get another link this is an application of the general model introduced earlier s ree phys rev e 73 2006 026115 to take the structure of networks into account we also consider three possible distinctions for the model i how we choose a rewiring link out of all incoming links to i three cases ii whether links are directed or not two cases iii types of networks considering the existence of self loops and multiple links two cases as a result we specify the total of 12 different cases of the model we then observe numerically that networks will evolve to steady states with power law degree distributions when parameters of the model satisfy certain conditions this work is from an effort to find a simple model of the network dynamics generating scale free networks and has a potential to become an underlying mechanism for wide range of scale free non growing networks

stability of functional equations in the spaces of distributions and hyperfunctions making use of the fundamental solution of the heat equation we prove the stability theorems of quadratic functional equation and d alembert equation in the spaces of schwartz distributions and sato hyperfunctions

the supremum of linear operators for the order abstract let b h be the algebra of all bounded linear operators on a complex hilbert space h in 1978 drazin introduced a partial order on b h for a b b h a b if and only if a a a b and aa ba given a b b h let a b be the least upper bound supremum for a and b with respect to the order in this paper we present some necessary and sufficient conditions for which a b exists and give an explicit representation of a b if a b exists

voter model dynamics in complex networks role of dimensionality disorder and degree distribution we analyze the ordering dynamics of the voter model in different classes of complex networks we observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks we also find that when there is no ordering in the system the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity the existence of hubs i e highly connected nodes in the network modifies the linear system size scaling law of the survival time the size of an ordered domain is sensitive to the network disorder and the average degree decreasing with both however it seems not to depend on network size and on the heterogeneity of the degree distribution

stability of multi additive mappings in non archimedean normed spaces abstract a function f v n w where v is a commutative semigroup w is a linear space and n 1 is an integer is called multi additive if it is additive in each variable in this paper we prove the generalized hyers ulam stability of multi additive mappings in non archimedean normed spaces using the so called direct method

feature of topological properties in an earthquake network abstract earthquake network is investigated by considering the volume resolution and the temporal causality we perform the numerical computations for network metrics from seismic time series data taken in japan for our case we mainly treat the feature of the topological properties via various statistical quantities such as the characteristic path length the average clustering coefficient the small worldness the cost efficiency the global efficiency the modularity and the assortativity the universal and irregular properties of statistical quantities in earthquake network do not find unambiguously but it may be inferred that these topological properties improve by implementing the method and its technique from registered data of earthquake networks

scale free networks with a large to hypersmall world transition abstract recently there has been a tremendous interest in models of networks with a power law distribution of degree so called scale free networks it has been observed that such networks normally have extremely short path lengths scaling logarithmically or slower with system size as an exotic and counterintuitive example we propose a simple stochastic model capable of generating scale free networks with linearly scaling distances furthermore by tuning a parameter the model undergoes a phase transition to a regime with extremely short average distances apparently slower than log log n which we call a hypersmall world regime we characterize the degree degree correlation and clustering properties of this class of networks

hyers ulam rassias stability of cauchy equation in the space of schwartz distributions abstract we reformulate and prove the hyers ulam rassias stability of cauchy equation in the space of schwartz tempered distributions and fourier hyperfunctions

topology and correlations in structured scale free networks we study a recently introduced class of scale free networks showing a high clustering coefficient and nontrivial connectivity correlations we find that the connectivity probability distribution strongly depends on the fine details of the model we solve exactly the case of low average connectivity providing also exact expressions for the clustering and degree correlation functions the model also exhibits a lack of small world properties in the whole parameter range we discuss the physical properties of these networks in the light of the present detailed analysis

some properties on the star order of bounded operators abstract in this paper we give various characterizations of star order and present some new relationships between the logic order and the star order we present some necessary and sufficient conditions for which the infimum or upper bound exists with respect to given orders

star order automorphisms on the poset of type 1 operators abstract let h be a complex infinite dimensional hilbert space and b h the algebra of all bounded linear operators on h the star partial order is defined by a b if and only if a a a b and a a a b for any a and b in b h we give a type decomposition of operators with respect to star order for any a b h there are unique type 1 operator a 1 which is 0 or the supremum of those rank 1 operators less than a 1 and type 2 operator a 2 which is not greater than any rank 1 operator under star order such that a i a i 1 2 and a a 1 a 2 moreover we determine all automorphisms on the poset of type 1 operators as a consequence we characterize continuous automorphisms on b h

small world structure of earthquake network discoveries of the scale free and small world features are reported on the network constructed from the seismic data it is shown that the connectivity distribution decays as a power law and the value of the degrees of separation i e the characteristic path length between two earthquakes as the vertices chosen at random takes a small value between 2 and 3 the clustering coefficient is also calculated and is found to be about 10 times larger than that in the case of the completely random network these features highlight a novel aspect of seismicity as a complex phenomenon

finite data size scaling of clustering in earthquake networks abstract an earthquake network is known to be of the small world type the values of the network s characteristics however depend not only on the cell size i e the scale of coarse graining needed for constructing the network but also on the size of a seismic data set here discovery of a scaling law for the clustering coefficient in terms of the data size which is referred to here as finite data size scaling is reported its universality is shown to be supported by the detailed analysis of the data taken from california japan and iran effects of setting a threshold of magnitude are also discussed

stability of approximately quadratic schwartz distributions abstract we consider approximately quadratic distributions in the spirit of the well known ulam problem and prove that every approximately quadratic distribution can be approximated by a quadratic function

a new deterministic complex network model with hierarchical structure abstract we introduce a new simple pseudo tree like network model deterministic complex network dcn the proposed dcn model may simulate the hierarchical structure nature of real networks appropriately and have the unique property of skipping the levels which is ubiquitous in social networks our results indicate that the dcn model has a rather small average path length and large clustering coefficient leading to the small world effect strikingly our dcn model obeys a discrete power law degree distribution p k k with exponent approaching 1 0 we also discover that the relationship between the clustering coefficient and degree follows the scaling law c k k 1 which quantitatively determines the dcn s hierarchical structure

approximately additive schwartz distributions generalizing the hyers ulam rassias stability theorem th m rassias on the stability of linear mapping in banach spaces proc amer math soc 72 1978 297 300 to the space of schwartz distributions we introduce a concept of approximately additive schwartz distributions and prove that every approximately additive distribution can be approximated by linear functions

nonextensivity measure for earthquake networks studying earthquakes and the associated geodynamic processes based on the complex network theory enables us to learn about the universal features of the earthquake phenomenon in addition we can determine new indices for identification of regions geophysically it was found that earthquake networks are scale free and its degree distribution obeys the power law here we claim that the q exponential function is better than power law model for fitting the degree distribution we also study the behavior of q parameter nonextensivity measure with respect to resolution it was previously asserted in eur phys j b 2012 85 23 that the topological characteristics of earthquake networks are dependent on each other for large values of the resolution a peak in the plot of q against resolution determines the beginning of the assertion range

scaling relation for earthquake networks abstract the scaling relation 2 1 for the exponents of the power law connectivity distribution and the power law eigenvalue distribution of the adjacency matrix is theoretically predicted to be fulfilled by a locally treelike scale free network in the effective medium approximation i e an analog of the mean field approximation here it is shown that such a relation holds well for the reduced simple earthquake networks i e the network without tadpole loops and multiple edges constructed from the seismic data taken from california and japan this validates the goodness of the effective medium approximation in the earthquake networks and is consistent with the hierarchical organization of the networks the present result may be useful for modeling seismicity on complex networks