renormalization group analysis of differential equations subject to slowly modulated perturbations abstract the application of renormalization group rg theory to the asymptotic analysis of differential equations is considered it is found that there is a class of small structural perturbations whose effects cannot be systematically treated using the gell mann low rg approach applied in this context guided by a reinterpretation of the calculational procedure employed in this approach whose motivation is provided by the unnecessarily complicated nature of its standard interpretation we formulate a generalized perturbative analysis and an rg approach which naturally and systematically treat equations subject to perturbations of this class this formulation of rg theory is demonstrated with a number of examples for which the gell mann low formulation fails to provide a systematic theoretical framework for one representative example it is found that the wilson rg formulation also fails the implications of this failure are discussed

temporal coarse graining method to simulate the movement of atoms we propose a novel method to simulate the movement of atoms at finite temperature the main idea of our method is to derive renormalized or coarse grained in time dynamics from the euler maruyama scheme which is the standard method for solving the stochastic differential equations numerically based on this renormalization we propose a new algorithm for solving overdamped langevin equations we test our renormalization scheme on two models and demonstrate that the results obtained by this method are consistent with those obtained by the standard method our algorithm performs better than the standard scheme especially at low temperatures and with multiple processors

energy of line graphs the energy of a graph is equal to the sum of the absolute values of its eigenvalues the energy of a matrix is equal to the sum of its singular values we establish relations between the energy of the line graph of a graph g and the energies associated with the laplacian and signless laplacian matrices of g

the number of the skew eigenvalues of digraphs and their relationship with optimum skew energy abstract let g be an oriented graph of simple graph g with the orientation and vertex set v g v 1 v 2 v n the skew adjacency matrix of g is the 0 1 1 matrix s s g s i j such that s i j 1 if v i v j is an arc in g s i j 1 if v j v i is an arc in g and s i j 0 otherwise in this paper all oriented graphs with two and three skew eigenvalues are characterized also we determine the relationship between the number of skew eigenvalues and optimum skew energy of oriented graphs

numerical simulations of solutions of a two level averaged navier stokes system abstract an averaging procedure for the navier stokes equations has been proposed in an earlier article i moise r m temam renormalization group method application to navier stokes equation discrete contin dyn syst 6 1 2000 191 210 this averaging procedure is based on a two level decomposition of the solution into low and high frequencies the aim of the present article is to investigate with the help of numerical simulations the behavior of the small scales of the corresponding system space periodic solutions with a non resonant period are considered the time evolution of the averaged and standard non averaged small scales are compared at different reynolds numbers and for different values of the cut off level used to separate large and small scales of the flow variables the numerical results illustrate the efficiency of the proposed averaging procedure for the navier stokes equations the averaged small scales provide an accurate prediction of the time averaged small scales of the navier stokes solutions as the computational cost is reduced for the averaged equations long time integrations on more than 50 eddy turnover times have been performed for cut off levels ensuring a proper resolution of the large scales in these cases development of instabilities in the averaged small scale equation is observed

the skew energy of random oriented graphs abstract given a graph g let g be an oriented graph of g with the orientation and skew adjacency matrix s g the skew energy of the oriented graph g denoted by e s g is defined as the sum of the absolute values of all the eigenvalues of s g in this paper we study the skew energy of random oriented graphs and formulate an exact estimate of the skew energy for almost all oriented graphs by generalizing wigner s semicircle law moreover we consider the skew energy of random regular oriented graphs g n d and get an exact estimate of the skew energy for almost all regular oriented graphs

the energy of c4 free graphs of bounded degree abstract answering some questions of gutman we show that except for four specific trees every connected graph g of order n with no cycle of order 4 and with maximum degree at most 3 satisfies 1 n n where 1 n are the eigenvalues of g we give some general results and state two conjectures

first order approximation for quadratic dispersive equations by the renormalization group approach in this paper we study the application of the renormalization group method to a system of partial differential equations together with constructing a long time first order approximation of the solution of the system we compare the difference between the true solution and the approximate formation with the help of dispersive estimates in lp space

the skew energy of a digraph we are interested in the energy of the skew adjacency matrix of a directed graph d which is simply called the skew energy of d in this paper properties of the skew energy of d are studied in particular a sharp upper bound for the skew energy of d is derived in terms of the order of d and the maximum degree of its underlying undirected graph an infinite family of digraphs attaining the maximum skew energy is constructed moreover the skew energy of a directed tree is independent of its orientation and interestingly it is equal to the energy of the underlying undirected tree skew energies of directed cycles under different orientations are also computed some open problems are presented

upper bound of skew energy of an oriented graph in terms of its skew rank abstract let g be an oriented graph with skew adjacency matrix s g the skew energy e s g of g is the sum of the norms of all eigenvalues of s g and the skew rank r s g of g is the rank of s g recently tian and wong gave a lower bound of e s g in terms of its skew rank they proved that r s g e s g and they characterized the oriented graphs which satisfy the equality in this paper we aim to establish an upper bound for skew energy of an oriented graph in terms of its skew rank and maximum vertex degree it is proved that e s g r s g for an arbitrary oriented graph g with maximum degree and the upper bound is attained if and only if g is the disjoint union of r s g 2 copies of k together with some isolated vertices where the orientation is switching equivalent to the elementary orientation of k which assigns all edges the same direction from a chromatic set to another one

optimum skew energy of a tournament abstract let g be a simple undirected graph and g the corresponding oriented graph of g with the orientation the skew energy of g denoted by e g is defined as the sum of the singular values of its skew adjacency matrix s g in 2010 adiga et al proved e g n where is the maximum degree of g of order n in this paper we characterize the skew energy of a tournament and present some properties about an optimum skew energy tournament

information loss under coarse graining a geometric approach we use information geometry in which the local distance between models measures their distinguishability from data to quantify the flow of information under the renormalization group we show that information about relevant parameters is preserved with distances along relevant directions maintained under flow by contrast irrelevant parameters become less distinguishable under the flow with distances along irrelevant directions contracting according to renormalization group exponents we develop a covariant formalism to understand the contraction of the model manifold we then apply our tools to understand the emergence of the diffusion equation and more general statistical systems described by a free energy our results give an information theoretic justification of universality in terms of the flow of the model manifold under coarse graining

on a certain renormalization group method in this paper a mathematical study of the renormalization group method recently introduced by chen goldenfeld and oono phys rev e 54 376 1997 phys rev lett 73 1311 1994 nato adv study inst ser ser b 284 375 1991 is given for the case of autonomous nonlinear systems of differential equations we also observe that the approximation results obtained by this method by chen goldenfeld and ono nato adv study inst ser ser b 284 375 1991 are valid over long time intervals moreover a connection between this method and the classical poincare dulac normal forms and the averaging method is briefly discussed

renormalization group theory for perturbed evolution equations we show that proto rg renormaliation group theory can be used to give a systematic description of the evolution of soltion of perturbed equations the equations describing the deformation of its shape as the effect of perturbation are proto rg equations the rg approach may be simpler than inverse scattering theory ist and another approaches because it dose not rely on any knowledge of ist it is very concise and easy to understand

on the laplacian energy like invariant abstract let g be a connected graph of order n with laplacian eigenvalues 1 2 n 1 n 0 the laplacian energy like invariant of the graph g is defined as lel lel g i 1 n 1 i lower and upper bounds for lel are obtained in terms of n number of edges maximum vertex degree and number of spanning trees

5 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be the corresponding oriented graph of g with the orientation the skew energy of g denoted by e s g is defined as the sum of the singular values of the skew adjacency matrix s g in 2010 adiga et al certified that s g n where is the maximum degree of g of order n it has been shown that every 5 regular oriented graph with optimum skew energy has even neighborhood property that is each pair of neighborhoods of a graph have even number of common vertices in this paper we characterize all connected 5 regular graphs of order n with this property moreover we determine all connected 5 regular oriented graphs of order n with maximum skew energy

a path integral method for coarse graining noise in stochastic differential equations with multiple time scales abstract we present a new path integral method to analyze stochastically perturbed ordinary differential equations with multiple time scales the objective of this method is to derive from the original system a new stochastic differential equation describing the system s evolution on slow time scales for this purpose we start from the corresponding path integral representation of the stochastic system and apply a multi scale expansion to the associated path integral kernel of the corresponding lagrangian as a concrete example we apply this expansion to a system that arises in the study of random dispersion fluctuations in dispersion managed fiber optic communications moreover we show that for this particular example the new path integration method yields the same result at leading order as an asymptotic expansion of the associated fokker planck equation

lie equations for asymptotic solutions of perturbation problems of ordinary differential equations lie theory is applied to perturbation problems of ordinary differential equations to construct approximate solutions and invariant manifolds according to the renormalization group approach of iwasa and nozaki a method to construct asymptotic solutions invariant under the renormalization group prog theor phys 116 605 2006 it is proved that asymptotic behavior of solutions is obtained from the lie equations even if original equations have no symmetries normal forms of the lie equations are introduced to investigate the existence of invariant manifolds

skew rank of an oriented graph in terms of matching number abstract an oriented graph g is a digraph without loops and multiple arcs where g is called the underlying graph of g let s g denote the skew adjacency matrix of g the rank of s g is called the skew rank of g denoted by sr g which is even since s g is skew symmetric li and yu 2015 12 proved that the skew rank of an oriented unicyclic graph g is either 2 m g 2 or 2 m g where m g denotes the matching number of g in this paper we extend this result to general cases it is proved that the skew rank of an oriented connected graph g is an even integer satisfying 2 m g 2 g sr g 2 m g where g e g v g 1 is the number of fundamental cycles also called the first betti number besides the oriented graphs satisfying sr g 2 m g 2 g are characterized definitely

krein space based robust h fault estimation for two dimensional uncertain linear discrete time varying systems abstract in this study the robust h fault estimation problem for two dimensional linear time varying systems with norm bounded unknown input measurement noise and time varying process uncertainty is investigated by introducing an equivalent auxiliary system and a new certain indefinite quadratic form performance function the system uncertainty can be appropriately considered into the new performance function and the fault estimator design is converted to the minimization problem of a quadratic form based on the partially equivalence property between the deterministic quadratic form problem and the krein space estimation theory the two dimensional h fault estimation problem can be solved via signal deconvolution in krein space through employing projection operation and riccati like difference equation in two dimensions both the recursive form fault estimator and the explicit condition for existence of the estimator are derived one darboux equation example is provided to illustrate the effectiveness of the proposed fault estimator

on the mixed adjacency matrix of a mixed graph a mixed graph is a graph with edges and arcs which can be considered as a hybrid of an undirected graph and a directed graph in this paper we define the mixed adjacency matrix and the mixed energy of a mixed graph the mixed adjacency matrix generalizes both the adjacency matrix of an undirected graph and the skew adjacency matrix of a digraph then we compute the characteristic polynomial of the mixed adjacency matrix of a mixed graph and deduce some basic results from it furthermore we give bounds to the mixed energy of a general mixed graph and we compute the mixed energy of some special mixed graphs at the end of the paper we introduce mixed unitary cayley graphs and compute their spectra

stability of an n 2 dimensional invariant torus in the kuramoto model at small coupling abstract when the natural frequencies are allocated symmetrically in the kuramoto model there exists an invariant torus of dimension n 2 1 n is the population size a global phase shift invariance allows us to reduce the model to n 1 dimensions using the phase differences and doing so the invariant torus becomes n 2 dimensional by means of perturbative calculations based on the renormalization group technique we show that this torus is asymptotically stable at small coupling if n is odd if n is even the torus can be stable or unstable depending on the natural frequencies and both possibilities persist in the small coupling limit

strongly regular graphs with maximal energy the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix koolen and moulton have proved that the energy of a graph on n vertices is at most n 1 n 2 and that equality holds if and only if the graph is strongly regular with parameters n n n 2 n 2 n 4 n 2 n 4 such graphs are equivalent to a certain type of hadamard matrices here we survey constructions of these hadamard matrices and the related strongly regular graphs

approximation of center manifolds on the renormalization group method the renormalization group rg method for differential equations is one of the perturbation methods for obtaining approximate solutions this article shows that the rg method is effectual for obtaining an approximate center manifold and an approximate flow on it when applied to equations having a center manifold

stochastic navier stokes equation and renormalization group theory abstract we present an application of a certain renormalization group rng theory developed by chen goldenfeld and oono to the stochastic two dimensional navier stokes equation with additive gaussian noise and periodic boundary conditions this article is a generalization of results of moise and temam which were for the deterministic equation because the classical rng theory e g in a famous work by yakhot and orszag utilizes significantly on adding noise terms our work could be another step towards a rigorous understanding of their method

oriented unicyclic graphs with the first n 92 largest skew energies abstract let g be an oriented graph obtained by assigning an orientation to the edge set of a simple undirected graph g such that g becomes a directed graph let s g be the skew adjacency matrix of g the skew energy of g is defined as the sum of the absolute values of all eigenvalues of s g in this paper we provide a new method to compare the skew energies of two oriented graphs whose skew characteristic polynomials satisfy a given recurrence relation and determine the oriented unicyclic graphs of order n with the first n 9 2 largest skew energies for n 31

3 regular mixed graphs with optimum hermitian energy abstract let g be a simple undirected graph and g be a mixed graph of g with the generalized orientation and hermitian adjacency matrix h g then g is called the underlying graph of g the hermitian energy of the mixed graph g denoted by e h g is defined as the sum of all the singular values of h g a k regular mixed graph on n vertices having hermitian energy n k is called a k regular optimum hermitian energy mixed graph liu and li 2015 11 proposed the problem of determining all the k regular connected optimum hermitian energy mixed graphs this paper is to give a solution to the problem for the case k 3

simplified renormalization group method for ordinary differential equations abstract the renormalization group rg method for differential equations is one of the perturbation methods which allows one to obtain invariant manifolds of a given ordinary differential equation together with approximate solutions to it this article investigates higher order rg equations which serve to refine an error estimate of approximate solutions obtained by the first order rg equations it is shown that the higher order rg equation maintains the similar theorems to those provided by the first order rg equation which are theorems on well definedness of approximate vector fields and on inheritance of invariant manifolds from those for the rg equation to those for the original equation for example since the higher order rg equation is defined by using indefinite integrals and is not unique for the reason of the undetermined integral constants the simplest form of rg equation is available by choosing suitable integral constants it is shown that this simplified rg equation is sufficient to determine whether the trivial solution to time dependent linear equations is hyperbolically stable or not and thereby a synchronous solution of a coupled oscillators is shown to be stable

3 regular digraphs with optimum skew energy abstract the skew energy of a digraph d is defined as the sum of the singular values of its skew adjacency matrix s d in this paper we first interpret the entries of the power of the skew adjacency matrix of a digraph in terms of the number of its walks and then focus on the question posed by adiga et al c adiga r balakrishnan wasin so the skew energy of a graph linear algebra appl 432 2010 1825 1835 of determining all 3 regular connected digraphs with optimum skew energy

bicyclic oriented graphs with skew rank 2 or 4 the skew rank of oriented graph g denoted by sr g is the rank of the skew adjacency matrix of g the skew rank is even since the skew adjacency matrix is skew symmetric in this paper we characterize the bicyclic oriented graphs with skew rank 2 or 4

the renormalization group a perturbation method for the graduate curriculum in this paper the renormalization group rg method of chen goldenfeld and oono phys rev lett 73 1994 pp 1311 1315 phys rev e 54 1996 pp 376 394 is presented in a pedagogical way to increase its visibility in applied mathematics and to argue favorably for its incorporation into the corresponding graduate curriculum the method is illustrated by some linear and nonlinear singular perturbation problems

singular perturbation analysis of a reduced model for collective motion a renormalization group approach in a system of noisy self propelled particles with interactions that favor directional alignment collective motion will appear if the density of particles is beyond a critical density starting with a reduced model for collective motion we determine how the critical density depends on the form of the initial perturbation specifically we employ a renormalization group improved perturbative method to analyze the model equations and show analytically up to first order in the perturbation parameter how the critical density is modified by the strength of the initial angular perturbation in the system

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

renormalization group derivation of phase equations abstract phase equations describing the evolution of large scale modulation of spatially periodic patterns in two dimensional systems are derived by employing the renormalization group method a general formula for phase diffusion coefficients is given under certain conditions

4 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be an oriented graph of g with the orientation and skew adjacency matrix s g the skew energy of the oriented graph g denoted by e s g is defined as the sum of the absolute values of all the eigenvalues of s g in this paper we characterize the underlying graphs of all 4 regular oriented graphs with optimum skew energy and give orientations of these underlying graphs such that the skew energy of the resultant oriented graphs indeed attain optimum it should be pointed out that there are infinitely many 4 regular connected optimum skew energy oriented graphs while the 3 regular case only has two graphs k 4 the complete graph on 4 vertices and q 3 the hypercube

renormalization group theory for global asymptotic analysis we show with several examples that renormalization group rg theory can be used to understand singular and reductive perturbation methods in a unified fashion amplitude equations describing slow motion dynamics in nonequilibrium phenomena are rg equations the renormalized perturbation approach may be simpler to use than other approaches because it does not require the use of asymptotic matching and yields practically superior approximations

hermitian adjacency matrices and hermitian energies of mixed graphs abstract a complex adjacency matrix of a mixed graph is introduced in the present paper which is a hermitian matrix and called the hermitian adjacency matrix it incorporates both adjacency matrix of an undirected graph and skew adjacency matrix of an oriented graph some of its properties are studied furthermore properties of its characteristic polynomial are studied cospectral problems among mixed graphs including mixed graphs and their underlying graphs oriented graphs and their underlying graphs are studied we give equivalent conditions for a mixed graph especially oriented graph that share the same spectrum with its underlying graph as a consequence we reconfirm a conjecture which was proposed by cui and hou in ref 8 we also show that the spectrum of the hermitian matrix of a mixed graph is invariant when changing the value of any its cut edge if any correspondingly an energy of a mixed graph is introduced and called the hermitian energy it incorporates both the energy of an undirected graph and the skew energy of an oriented graph some of its bounds are given especially the mixed graphs with optimal upper bound of hermitian energy are characterized an infinite family of mixed graphs attaining the maximum hermitian energy is constructed moreover the hermitian energy of a mixed tree is showed to be equal to the energy of its underlying tree finally the integral formula for hermitian energy of a mixed graph is given

amplitude equation for lattice maps a renormalization group approach we consider the development of instabilities of homogeneous stationary solutions of discrete time lattice maps under some generic hypotheses we derive an amplitude equation which is the space time continuous ginzburg landau equation using dynamical renormalization group methods we control the accuracy of this approximation in a large ball of its basin of attraction

amplitude modulation for the swift hohenberg and kuramoto sivashinski equations employing a harmonic balance technique inspired from the methods of renormalization group and multiple scales r e o malley jr and e kirkinis a combined renormalization group multiple scale method for singularly perturbed problems stud appl math 124 4 383 410 2010 we derive the amplitude equations for the swift hohenberg and kuramoto sivashinski equations to arbitrary order in the context of roll patterns this new and straightforward derivation improves previous attempts and can be carried out with symbolic computation that minimizes effort and avoids error

graph energy change due to edge deletion the energy of a graph is the sum of the singular values of its adjacency matrix we are interested in how the energy of a graph changes when edges are deleted examples show that all cases are possible increased decreased unchanged our goal is to find possible graph theoretical descriptions and to provide an infinite family of graphs for each case the main tool is a singular value inequality for complementary submatrices and its equality case

edge addition singular values and energy of graphs and matrices the energy of a graph matrix is the sum of the absolute values of its eigenvalues we investigate the result of duplicating removing an edge to the energy of a graph we also deal with the problem that which graphs g have the property that if the edges of g are covered by some subgraphs then the energy of g does not exceed the sum of the subgraphs energies the problems are addressed in the general setting of energy of matrices which leads us to consider the singular values too among the other results it is shown that the energy of a complete multipartite graph increases if a new edge added or an old edge is deleted

the energy of graphs and matrices abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman let 2 m n a be an m n nonnegative matrix with maximum entry and a 1 n extending previous results of koolen and moulton for graphs we prove that e a a 1 m n m 1 a 2 2 a 1 2 m n n m m 2 furthermore if a is any nonconstant matrix then e a 1 a a 2 2 1 2 a 2 a finally we note that wigner s semicircle law implies that e g 4 3 o 1 n 3 2 for almost all graphs g

application of the renormalization group method to the reduction of transport equations we first give a comprehensive review of the renormalization group method for global and asymptotic analysis putting an emphasis on the relevance to the classical theory of envelopes and on the importance of the existence of invariant manifolds of the dynamics under consideration we clarify that an essential point of the method is to convert the problem from solving differential equations to obtaining suitable initial or boundary conditions the rg equation determines the slow motion of the would be integral constants in the unperturbative solution on the invariant manifold the rg method is applied to derive the navier stokes equation from the boltzmann equation as an example of the reduction of dynamics we work out how to obtain the transport coefficients in terms of the one body distribution function

on oriented graphs whose skew spectral radii do not exceed 2 abstract let s g be the skew adjacency matrix of the oriented graph g of order n and 1 2 n be all eigenvalues of s g the skew spectral radius s g of g is defined as max 1 2 n in this paper we investigate oriented graphs whose skew spectral radii do not exceed 2

bicyclic oriented graphs with the second largest skew energy abstract let g be an oriented graph and p s g x det x i s g be the skew characteristic polynomial of its skew adjacency matrix s g the skew energy of g is defined to be the sum of the absolute values of eigenvalues of s g in this paper we firstly find a novel relation between the coefficients of skew characteristic polynomial of unicyclic graph with the third largest skew energy and bicyclic oriented graphs and then use it along with other techniques to characterize the bicyclic oriented graphs with the second largest skew energy

applications of a theorem by ky fan in the theory of graph energy abstract the energy of a graph g is equal to the sum of the absolute values of the eigenvalues of g which in turn is equal to the sum of the singular values of the adjacency matrix of g let x y and z be matrices such that x y z the ky fan theorem establishes an inequality between the sum of the singular values of z and the sum of the sum of the singular values of x and y this theorem is applied in the theory of graph energy resulting in several new inequalities as well as new proofs of some earlier known inequalities

on the average energy of circulant graphs we obtain upper and lower bounds on the average energy of circulant graphs with n vertices and regularity d the average is taken over all representations of such graphs by circulant adjacency matrices