diameters and eigenvalues we derive a new upper bound for the diameter of a k regular graph g as a function of the eigenvalues of the adjacency matrix namely suppose the adjacency matrix of g has eigenvalues ai a2 a with ia i i vi21 ia i where ai k a ia21 then the diameter d g must satisfy d g we will consider families of graphs whose eigenvalues can be explicitly de termined these graphs are determined by sums or differences of vertex labels namely the pair i j being an edge depends only on the value i j or i j for directed graphs we will show that these graphs are expander graphs with small diameters by using an inequality on character sums which was recently proved by n m katz bell communications research 445 south street morristown new jersey 07960 this content downloaded from 207 46 13 111 on sat 21 may 2016 07 00 04 utc all use subject to http about jstor org terms

on the laplacian eigenvalues of a graph abstract in the note we present an upper bound for the spectral radius of laplacian matrix of a graph in terms of a 2 degree of a vertex

convergence to global equilibrium for fokker planck equations on a graph and talagrand type inequalities abstract in 2012 chow huang li and zhou 7 proposed the fokker planck equations for the free energy on a finite graph in which they showed that the corresponding fokker planck equation is a nonlinear ode defined on a riemannian manifold of probability distributions different choices for inner products result in different fokker planck equations the unique global equilibrium of each equation is a gibbs distribution in this paper we proved that the exponential rate of convergence towards the global equilibrium of these fokker planck equations the rate is measured by both the decay of the l 2 norm and that of the relative entropy with the convergence result we also prove two talagrand type inequalities relating relative entropy and wasserstein metric based on two different metrics introduced in 7 the first one is a local inequality while the second is a global inequality with respect to the lower bound metric from 7

fix points of meromorphic functions and iterated entire functions let f be a rational function of degree k and let g be a transcendental meromorphic function gross and osgood 13 proved that the composite function fog has infinitely many fix points if k 3 if k 2 then fog may have only a finite number of fix points and all such f and g have been characterized by gross and osgood there is a corresponding result for the case that f is a transcendental meromorphic function and that g is a polynomial in fact the two cases are closely connected since as pointed out by gross and yang 14 p 214 proof of theorem 23 fog has infinitely many fix points if and only if gof does this paper deals with the fix points offog wherefand g are both transcendental it has been proved in 6 that f 0 g has infinitely many fix points if f and g are entire this confirmed a conjecture of gross 111 there are a number of papers s 12 14 22 27 283 besides 6 dealing with fix points of fog for entire f and g but their methods do not seem to extend to the case of meromorphic f and in fact very little seems to be known about this case gross 1 l has asked whether fog has infinitely many fix points if f is transcendental and meromorphic and if g is transcendental and entire using some ideas from 6 we shall prove that this is indeed the case iffsatisfies a certain condition to be discussed afterward

enhancing synchronizability of diffusively coupled dynamical networks a survey in this paper we review the literature on enhancing synchronizability of diffusively coupled dynamical networks with identical nodes the last decade has witnessed intensive investigations on the collective behavior over complex networks and synchronization of dynamical systems is the most common form of collective behavior for many applications it is desired that the synchronizability the ability of networks in synchronizing activity of their individual dynamical units is enhanced there are a number of methods for improving the synchronization properties of dynamical networks through structural perturbation in this paper we survey such methods including adding removing nodes and or edges rewiring the links and graph weighting these methods often try to enhance the synchronizability through minimizing the eigenratio of the laplacian matrix of the connection graph a synchronizability measure based on the master stability function formalism we also assess the performance of the methods by numerical simulations on a number of real world networks as well as those generated through models such as preferential attachment watts strogatz and erdos renyi

on the generalized hyers ulam rassias stability in banach modules over a c algebra abstract in this paper we will investigate the generalized hyers ulam rassias stability of an n dimensional quadratic functional equation in banach modules over a c algebra and unitary elements

learning bounds via sample width for classifiers on finite metric spaces in a recent paper m anthony j ratsaby maximal width learning of binary functions theoretical computer science 411 2010 138 147 the notion of sample width for binary classifiers mapping from the real line was introduced and it was shown that the performance of such classifiers could be quantified in terms of this quantity this paper considers how to generalize the notion of sample width so that we can apply it where the classifiers map from some finite metric space by relating the learning problem to one involving the domination numbers of certain graphs we obtain generalization error bounds that depend on the sample width and on certain measures of density of the underlying metric space we also discuss how to employ a greedy set covering heuristic to bound generalization error

lower bounds of the laplacian spectrum of graphs based on diameter let g be a connected graph of order n the diameter of g is the maximum distance between any two vertices of g in the paper we will give some lower bounds for the algebraic connectivity and the laplacian spectral radius of g in terms of the diameter of g

on the stability of functional equations in banach spaces the paper is devoted to some results on the problem of s m ulam for the stability of functional equations in banach spaces the problem was posed by ulam 60 years ago

on the hyers ulam stability of linear mappings abstract let e1 be a real normed vector space and e2 a real banach space s m ulam posed the problem when does a linear mapping near an approximately linear mapping e1 e2 exist we give a new generalized solution to this problem an example illustrates when the answer to this question is negative the behaviour of bounded approximately additive mappings which do not satisfy hyers uiam stability is also investigated

a generalization of the hyers ulam rassias stability of approximately additive mappings abstract in this paper we prove a generalization of the stability of approximately additive mappings in the spirit of hyers ulam and rassias

ordering trees with algebraic connectivity and diameter abstract this paper is motivated by abreu s paper which proposed finding a total ordering of trees by their algebraic connectivity is still an open problem we first discuss how to find a total ordering of trees with diameter 4 then these results are used to determine first several orderings of trees by algebraic connectivity in nonincreasing order in last we present a class of trees with large diameter whose algebraic connectivity is larger than that of a tree with diameter 3

variational problem of some quadratic functionals in complex analysis suppose we are given a pair of marked closed riemann surfacess twith the same genus we minimize the modified dirichlet integral for mappings ofsontot which is endowed with some weight in terms of conformal metric on the target surfacet in this paper it is demonstrated that the minimal harmonic map exists and it proves to be teichmuller

on the fix points of composite transcendental entire functions a transcendental entire function f z is called composite iff f z f g z for some nonlinear entire functionsf and g a point z0 in the complex plane is called a fix point of f iff f z zo gross 3 conjectured that every composite transcendental entire function must have infinitely many fix points it has been shown thus far that the conjecture is valid for finite order entire functions by several authors e g z 4 71 via different approaches in a previous paper 111 we investigated the fixpoints of a certain type of infinite order composite entire function there it is required that g be an entire function of positive finite order with a finite nevanlinna exceptional value and f be an arbitrary entire function such that the hyperorder off g is less than the order of g in this note we shall treat some other class of composite entire functions the tools are as usual based on nevanlinna s theory of meromorphic functions and an elegant result due to steinmetz s corollary i our result is the following theorem

on fuzzy approximately cubic type mapping in fuzzy banach spaces abstract the first study of the fuzzy approximately cubic mappings in nonlinear analysis is due to mirmostafaee and moslehian 2008 and a fuzzy version of a generalized hyers ulam rassias type stability for the cubic functional equation in the fuzzy normed linear space setting is presented in this note we give an alternative proof of this result using the fixed point method in probabilistic metric spaces moreover we investigate fuzzy generalized hyers ulam rassias stability of an n dimensional cubic type functional equation in fuzzy banach spaces

stability for the functional equation of cubic type abstract in this article we establish the stability of the orthogonally cubic type functional equation 1 2 for all x 1 x 2 x 3 with x i x j i j 1 2 3 where is the orthogonality in the sense of ratz and investigate the stability of the n dimensional cubic type functional equation 1 3 where n 3 is an integer

the stability of a cubic type functional equation with the fixed point alternative abstract in this note we investigate the generalized hyers ulam rassias stability for the new cubic type functional equation f x y 2 z f x y 2 z f 2 x f 2 y 2 f x y 2 f x z 2 f x z 2 f y z 2 f y z by using the fixed point alternative the first systematic study of fixed point theorems in nonlinear analysis is due to g isac and th m rassias internat j math math sci 19 1996 219 228

a lower bound for algebraic connectivity based on the connection graph stability method this paper introduces the connection graph stability method and uses it to establish a new lower bound on the algebraic connectivity of graphs the second smallest eigenvalue of the laplacian matrix of the graph that is sharper than the previously published bounds the connection graph stability score for each edge is defined as the sum of the lengths of the shortest paths making use of that edge we prove that the algebraic connectivity of the graph is bounded below by the size of the graph divided by the maximum connection graph stability score assigned to the edges

partial stabilities and partial derivations of n variable functions abstract we introduce the concepts of the partial derivations of a function from a normed algebra a 1 a n into a banach algebra b and prove the hyers ulam rassias stability of the function moreover we obtain some results about the applications of derivations such as the extended derivations and the approximately higher derivations

connection graph stability method for synchronized coupled chaotic systems this paper elucidates the relation between network dynamics and graph theory a new general method to determine global stability of total synchronization in networks with different topologies is proposed this method combines the lyapunov function approach with graph theoretical reasoning in this context the main step is to establish a bound on the total length of all paths passing through an edge on the network connection graph in particular the method is applied to the study of synchronization in rings of 2k nearest neighbor coupled oscillators a rigorous bound is given for the minimum coupling strength sufficient for global synchronization of all oscillators this bound is explicitly linked with the average path length of the coupling graph contrary to the master stability function approach developed by pecora and carroll the connection graph stability method leads to global stability of synchronization and it permits not only constant but also time dependent interaction coefficients in a companion paper blinking model and synchronization in small world networks with a time varying coupling see this issue this method is extended to the blinking model of small world networks where in addition to the fixed 2 k nearest neighbor interactions all the remaining links are rapidly switched on and off independently of each other

the generalized hyers ulam rassias stability of a cubic functional equation abstract in this paper we obtain the general solution and the generalized hyers ulam stability for a cubic functional equation f 2x y f 2x y 2f x y 2f x y 12f x

on the stability of an n dimensional cubic functional equation abstract let n 2 be an integer number in this paper we investigate the generalized hyers ulam rassias stability in banach spaces and also banach modules over a banach algebra and a c algebra and the stability using the alternative fixed point of an n dimensional cubic functional equation in banach spaces f j 1 n 1 x j 2 x n f j 1 n 1 x j 2 x n j 1 n 1 f 2 x j 2 f j 1 n 1 x j 4 j 1 n 1 f x j x n f x j x n