a discrete nodal domain theorem for trees let g be a connected graph with n vertices and let x x1 xn be a real vector a positive negative sign graph of the vector x is a maximal connected subgraph of g on vertices xi 0 xi 0 for an eigenvalue of a generalized laplacian of a tree we characterize the maximal number of sign graphs of an eigenvector we give an o n2 time algorithm to find an eigenvector with maximum number of sign graphs and we show that finding an eigenvector with minimum number of sign graphs is an np complete problem author s abstract

additive mappings decreasing rank one abstract let b x be the algebra of bounded operators on a real or complex banach space x and f x a subalgebra of finite rank operators a complete description of additive mappings f x f x which map rank one operators to operators of rank at most one is given

minimum vertex covers and the spectrum of the normalized laplacian on trees abstract we show that in the graph spectrum of the normalized graph laplacian on trees the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers in particular for the eigenvalue 1 its multiplicity is related to the size of a minimum vertex cover and zero entries of its eigenvectors correspond to vertices in minimum vertex covers while for eigenvalues near 1 their distance to 1 can be estimated from minimum vertex covers and for the largest eigenvalue smaller than 1 the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives

a lower bound for nodal count on discrete and metric graphs we study the number of nodal domains maximal connected regions on which a function has constant sign of the eigenfunctions of schrodinger operators on graphs under a certain genericity condition we show that the number of nodal domains of the nth eigenfunction is bounded below by n l where l is the number of links that distinguish the graph from a tree

isospectral graphs with identical nodal counts according to a recent conjecture isospectral objects have different nodal count sequences gnutzmann et al 2005 j phys a math gen 38 8921 33 we study generalized laplacians on discrete graphs and use them to construct the first non trivial counterexamples to this conjecture in addition these examples demonstrate a surprising connection between isospectral discrete and quantum graphs

local automorphisms of finitary incidence algebras abstract let r be a commutative indecomposable ring with identity and p a partially ordered set let f i p denote the finitary incidence algebra of p over r we will show that in most cases local automorphisms of f i p are actually r algebra automorphisms in fact the existence of local automorphisms which fail to be r algebra automorphisms will depend on the chosen model of set theory and will require the existence of measurable cardinals we will discuss local automorphisms of cartesian products as a special case in preparation of the general result

graph laplacians nodal domains and hyperplane arrangements eigenvectors of the laplacian of a graph g have received increasing attention in the recent past here we investigate their so called nodal domains i e the connected components of the maximal induced subgraphs of g on which an eigenvector psi does not change sign an analogue of courant s nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of psi in the spectrum this bound however is not sharp in general in this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph the class of boolean hypercubes is discussed in detail we find that despite the simplicity of this graph class for which complete spectral information is available the computations are still non trivial nevertheless we obtained some new results and a number of conjectures author s abstract

leaders in multi agent controllability under consensus algorithm and tree topology abstract by controllability of multi agent systems we mean the property that all agents can be driven to arbitrary configurations only through regulating a few leaders externally while the rest of the agents are subject to the nearest neighbor law it is known that the controllability of multi agent systems depends not only on the communication topologies among agents but also on the selection of leaders the main objective of this paper aims to characterize the virtue that leaders should have from the perspective of algebraic and graphical conditions we begin with a heuristic observation for tree graph to gain more insights necessary and sufficient conditions are then proposed in terms of downer branch and subgraphs respectively in particular it is shown for path topologies that controllability completely depends on the leaders location the obtained results may help us to pave the way towards a new method of formation and reconfiguration control in swarming robots

nodal domain counts and the chromatic number of graphs a function f defined on the vertices of a graph g induces nodal domains on the graph nodal domains of discrete and metric graphs are of growing interest among physicists and mathematicians in this paper several results regarding the nodal domain counts of discrete graphs are derived one such result is a global upper bound for the number of nodal domains of g in terms of its chromatic number another result is a criterion of resolution of laplacian isospectral graphs via their nodal counts several additional results are also shown

local derivations on operator algebras abstract when attempting to find sufficient conditions for a linear mapping to be a derivation an obvious candidate is the concept of a local derivation local derivations on operator algebras have been investigated in recent papers of kadison j algebra130 1990 494 509 and larson and sourour proc symp pure math 51 1990 187 194 a local derivation is a norm continuous linear map from an operator algebra a into an a bimodule m which agrees with some derivation at each point in the algebra we show that if a is the direct limit of finite dimensional csl algebras via extendable embeddings e g a triangular af algebra then a local derivation on a must be a derivation further we show that for many finite dimensional operator algebras any inner local derivation must be an inner derivation

social paradoxes of majority rule voting and renormalization group real space renormalization group ideas are used to study a voting problem in political science a model to construct self directed pyramidal structures from bottom up to the top is presented using majority rules it is shown that a minority and even a majority can be systematically self eliminated from top leadership provided the hierarchy has a minimal number of levels in some cases 70 of the population is found to have zero representation after six hierarchical levels results are discussed with respect to the internal operation of political organizations

a stabilization theorem for dynamics of continuous opinions abstract a stabilization theorem for processes of opinion dynamics is presented the theorem is applicable to a wide class of models of continuous opinion dynamics based on averaging like the models of hegselmann krause and weisbuch deffuant the analysis detects self confidence as a driving force of stabilization

an eigenvector interlacing property of graphs that arise from trees by schur complementation of the laplacian abstract the literature is replete with rich connections between the structure of a graph g v e and the spectral properties of its laplacian matrix l this paper establishes similar connections between the structure of g and the laplacian l of a second graph g our interest lies in l that can be obtained from l by schur complementation in which case we say that g is partially supplied with respect to g in particular we specialize to where g is a tree with points of articulation r r and consider the partially supplied graph g derived from g by taking the schur complement with respect to r in l our results characterize how the eigenvectors of the laplacian of g relate to each other and to the structure of the tree

modelling rumors the no plane pentagon french hoax case abstract the recent astonishing wide adhesion of french people to the rumor claiming no plane did crash on the pentagon on september 11 is given a generic explanation in terms of a model of minority opinion spreading using a majority rule reaction diffusion dynamics a rumor is shown to invade for sure a social group provided it fulfills simultaneously two criteria first it must initiate with a support beyond some critical threshold which however turns out to be always very low then it has to be consistent with some larger collective social paradigm of the group otherwise it just dies out both conditions were satisfied in the french case with the associated book sold at more than 200 000 copies in just a few days the rumor was stopped by the firm stand of most newspaper editors stating it is nonsense such an incredible social dynamics is shown to result naturally from an open and free public debate among friends and colleagues each one searching for the truth sincerely on a free will basis and without individual biases the polarization process appears also to be very quick in agreement with reality it is a very strong anti democratic reversal of opinion although made quite democratically the model may apply to a large range of rumors

2 local automorphisms of finite dimensional simple lie algebras abstract let f be an algebraically closed field of characteristic 0 l be a finite dimensional simple lie algebra of type a l l 1 d l l 4 e k k 6 7 8 over f a not necessarily linear map l l is called a 2 local automorphism if for every x y l there is an automorphism x y l l depending on x and y such that x x y x y x y y in this paper we prove that any 2 local automorphism of l is an automorphism

2 local derivations and automorphisms on b h abstract this paper is devoted to 2 local derivations and 2 local automorphisms on the algebra b h of all bounded linear operators on a hilbert space h we prove that every 2 local derivation on b h is a derivation a similar result is obtained for automorphisms

a necessary and sufficient condition for reaching a consensus using degroot s method abstract degroot 1974 proposed a model in which a group of k individuals might reach a consensus on a common subjective probability distribution for an unknown parameter this paper presents a necessary and sufficient condition under which a consensus will be reached by using degroot s method this work corrects an incorrect statement in the original paper about the conditions needed for a consensus to be reached the condition for a consensus to be reached is straightforward to check and yields the value of the consensus if one is reached

reaching consensus on rumors abstract an important contribution in sociophysics is the galam s model of rumors spreading this model provides an explanation of rumors spreading in a population and explains some interesting social phenomena such as the diffusion of hoaxes in this paper the model has been reformulated as a markov process highlighting the stochastic nature of the phenomena this formalization allows us to derive conditions for consensus to be reached and for the existence of some interesting phenomena such as the emergence of impasses the proposed formulation allows a deeper and more comprehensive analysis of the diffusion of rumors

opinion fluctuations and disagreement in social networks we study a tractable opinion dynamics model that generates long run disagreements and persistent opinion fluctuations our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents 1 regular agents who update their beliefs according to information that they receive from their social neighbors and 2 stubborn agents who never update their opinions and might represent leaders political parties or media sources attempting to influence the beliefs in the rest of the society when the society contains stubborn agents with different opinions the belief dynamics never lead to a consensus among the regular agents instead beliefs in the society fail to converge almost surely the belief profile keeps on fluctuating in an ergodic fashion and it converges in law to a nondegenerate random vector the structure of the graph describing the social network and the location of the stubborn agents within it shape the opinion dynamics the expected belief vector is proved to evolve according to an ordinary differential equation coinciding with the kolmogorov backward equation of a continuous time markov chain on the graph with absorbing states corresponding to the stubborn agents and hence to converge to a harmonic vector with every regular agent s value being the weighted average of its neighbors values and boundary conditions corresponding to the stubborn agents beliefs expected cross products of the agents beliefs allow for a similar characterization in terms of coupled markov chains on the graph describing the social network we prove that in large scale societies which are highly fluid meaning that the product of the mixing time of the markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large a condition of homogeneous influence emerges whereby the stationary beliefs marginal distributions of most of the regular agents have approximately equal first and second moments

local automorphisms of standard operator algebras let x be an infinite dimensional separable real or complex banach space and a a closed standard operator algebra on x then every local automorphism of a is an automorphism the assumptions of infinite dimensionality separability and closeness are all indispensable

local derivations and local automorphisms in this paper we show that if l is a completely distributive commutative subspace lattice or a j subspace lattice then the space of all bounded derivations of algl is reflexive we also study when local automorphisms on some algebras are automorphisms

on eigenfunctions of markov processes on trees we begin by studying the eigenvectors associated to irreducible finite birth and death processes showing that the i nontrivial eigenvector i admits a succession of i decreasing or increasing stages each of them crossing zero imbedding naturally the finite state space into a continuous segment one can unequivocally define the zeros of i which are interlaced with those of i 1 these kind of results are deduced from a general investigation of minimax multi sets dirichlet eigenproblems which leads to a direct construction of the eigenvectors associated to birth and death processes this approach can be generically extended to eigenvectors of markov processes living on trees this enables to reinterpret the eigenvalues and the eigenvectors in terms of the previous dirichlet eigenproblems and a more general conjecture is presented about related higher order cheeger inequalities finally we carefully study the geometric structure of the eigenspace associated to the spectral gap on trees

interlacing property of zeros of eigenvectors of schrodinger operators on trees we prove an analogue for trees of sturm s oscillation theorem on the interlacing property of zeros of eigenfunctions of a schrodinger operator let be a finite tree and a combinatorial schrodinger operator on if the eigenvectors of are ordered according to increasing eigenvalues and the vertices corresponding to zero coordinates are of degree at most two then the zeros of the linear extensions of eigenvectors have the interlacing property

2 local derivations on matrix algebras over commutative regular algebras abstract the paper is devoted to 2 local derivations on matrix algebras over commutative regular algebras we give necessary and sufficient conditions on a commutative regular algebra to admit 2 local derivations which are not derivations we prove that every 2 local derivation on a matrix algebra over a commutative regular algebra is a derivation we apply these results to 2 local derivations on algebras of measurable and locally measurable operators affiliated with type i von neumann algebras

2 local triple homomorphisms on von neumann algebras and jbw triples abstract we prove that every not necessarily linear nor continuous 2 local triple homomorphism from a jbw triple into a jb triple is linear and a triple homomorphism consequently every 2 local triple homomorphism from a von neumann algebra respectively from a jbw algebra into a c algebra respectively into a jb algebra is linear and a triple homomorphism

permanental roots and the star degree of a graph abstract for undirected graphs without loops or multiple edges we define the star degree of a graph and prove that it is equal to the multiplicity of the root 1 of per xi b where b d a considering bipartite graphs we prove that per xi b per xi l where l d a and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per xi l

two stage predictions of evolutionary dynamics during the rumor dissemination abstract the spreading of rumor is the common type of collective actions as a macro social phenomenon the rumor spreading should be supported by the micro level actions and interactions of people agents therefore the agent based modeling is applied to modeling the micro level mechanism of rumor spreading which has more significance in the big data age the outcomes of simulations indicate that there exist a stage transfer and the turning point d which divides the whole spreading process into the linear od and nonlinear de stages in the linear stage od the spreading percentage increases rapidly into a relatively high level of spreading percentage however after the turning point d it increases slowly until the 100 is reached in the nonlinear stage de for the linear stage od we estimated the linear dynamics with the linear ols regression and it suggests that the adjusted r2 is as high as over 98 99 which is infinitely close to 100 for the nonlinear stage de we estimated the nonlinear stage via adding the quadratic term of time into the regression model it suggests that the adjusted r2 is as high as over 98 as well which is also close to 100 based on the predictions of linear and nonlinear stages with the highest accuracies we combined the linear and nonlinear stages oe od de to predict the whole process of rumor spreading eventually the smooth predicted curves spcs can be obtained to better fit the evolutionary dynamic of spreading curves in reality

anticonformity or independence insights from statistical physics the aim of this paper is to examine how different types of social influence introduced on the microscopic individual level manifest on the macroscopic level i e in the society the inspiration for this task came mainly from two sources social psychology that recognize two different types of nonconformity anticonformity and independence and the observation related to the agent based modeling that was verbalized in 2002 by macy and willer that there was a little effort to provide analysis of how results differ depending on the model designs to achieve the goal we propose a generalized model of opinion dynamics that as a special cases reduces to the linear voter model sznajd model q voter model and the majority rule we use the model to examine the differences that appear at the macroscopic level under the influence of two types of nonconformity introduced on the microscopic level we answer the question if the observed differences are universal or model dependent

on the eigenvectors belonging to the minimum eigenvalue of an essentially nonnegative symmetric matrix with bipartite graph we prove that if b is an essentially nonnegative symmetric matrix with minimum eigenvalue m b and if the graph of b is connected and bipartite with bipartition v1 v2 then there exists an eigenvector x belonging to m b such that xi 0 for i v1 and xi 0 for i v2 we then establish two facts concerning essentially nonnegative symmetric matrices with bipartite graph and point out how the above result can be used in conjunction with these facts to give a proof not requiring the perron frobenius theory of the following theorem of constantine 2 among all essentially nonnegative symmetric n by n matrices with minimum diagonal entry at least and maximum off diagonal entry at most m the matrix with smallest minimum eigenvalue is man in where an is the adjacency matrix of the complete bipartite graph k n2 n2