on the l p solvability of higher order parabolic and elliptic systems with bmo coefficients we prove the solvability in sobolev spaces for both divergence and non divergence form higher order parabolic and elliptic systems in the whole space on a half space and on a bounded domain the leading coefficients are assumed to be merely measurable only in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders for the proof we develop a set of new techniques to produce mean oscillation estimates for systems on a half space

lipschitz regularity of solutions for mixed integro differential equations we establish new holder and lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro differential equations by the classical ishii lions s method we thus extend the holder regularity results recently obtained by barles chasseigne and imbert 2011 in addition we deal with a new class of nonlocal equations that we term mixed integro differential equations these equations are particularly interesting as they are degenerate both in the local and nonlocal term but their overall behavior is driven by the local nonlocal interaction e g the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one

the solvability of a strongly coupled nonlocal system of equations abstract we prove existence and uniqueness of strong pointwise solutions to a linear nonlocal strongly coupled hyperbolic system of equations posed on all of euclidean space the system of equations comes from a linearization of a nonlocal model of elasticity in solid mechanics it is a nonlocal analogue of the navier lame system of classical elasticity we use a well known semigroup technique that hinges on the strong solvability of the corresponding steady state elliptic system the leading operator is an integro differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field for an operator possessing an asymmetric kernel comparable to that of the fractional laplacian we prove the l 2 solvability of the elliptic system in a bessel potential space using the fourier transform and a priori estimates this l 2 solvability together with the hille yosida theorem is used to prove the well posedness of the wave type time dependent problem for the fractional laplacian kernel we extend the solvability to l p spaces using classical multiplier theorems

on lp estimates for a class of non local elliptic equations abstract we consider non local elliptic operators with kernel k y a y y d where 0 2 is a constant and a is a bounded measurable function by using a purely analytic method we prove the continuity of the non local operator l from the bessel potential space h p to l p and the unique strong solvability of the corresponding non local elliptic equations in l p spaces as a byproduct we also obtain interior l p estimates the novelty of our results is that the function a is not necessarily to be homogeneous regular or symmetric an application of our result is the uniqueness for the martingale problem associated to the operator l

levy processes and fourier multipliers abstract we study fourier multipliers which result from modulating jumps of levy processes using the theory of martingale transforms we prove that these operators are bounded in l p r d for 1 p and we obtain the same explicit bound for their norm as the one known for the second order riesz transforms

on the nullity of a graph with cut points let g be a simple graph of order n and a g be its adjacency matrix the nullity of a graph g denoted by g is the multiplicity of the eigenvalue zero in the spectrum of a g denote by ck and lk the set of all connected graphs with k induced cycles and the set of line graphs of all graphs in ck respectively in 1998 sciriha i sciriha on singular line graphs of trees congr numer 135 1998 73 91 show that the order of every tree whose line graph is singular is even then gutman and sciriha i gutman i sciriha on the nullity of line graphs of trees discrete math 232 2001 35 45 show that the nullity set of l0 is 0 1 in this paper we investigate the nullity of graphs with cut points and deduce some concise formulas then we generalize scirihas result showing that the order of every graph g is even if such a graph g satisfies that g ck and l g k 1 and the nullity set of lk is 0 1 k k 1 for any given k where l g denotes the line graph of the graph g

an lp theory for a class of non local elliptic equations related to nonsymmetric measurable kernels abstract we study the integro differential operators l with kernels k y a y j y where j y is rotationally invariant and j y d y is a levy measure on r d i e r d 1 y 2 j y d y and a y is an only measurable function with positive lower and upper bounds under few additional conditions on j y we prove the unique solvability of the equation l u u f in l p spaces and present some l p estimates of the solutions

the nullity of k 1 cyclic graphs abstract a simple connected graph is called a k cyclic graph provided that the number of edges equals its order plus k 1 in this paper we define a class of k 1 cyclic graph denoted by k the singularity of graph is studied also the k 1 cyclic graphs that contain graph as an induced subgraph are paid much attention we characterize those graphs that have extreme nullities the results generalize the corresponding theorems on unicyclic and bicyclic graphs

relationship between the rank and the matching number of a graph abstract given a simple graph g let a g be its adjacency matrix and g be its matching number the rank of g written as r g refers to the rank of a g in this paper some relations between the rank and the matching number of a graph are studied firstly it is proved that 2 d g r g 2 g n o where d g and no are respectively the dimension of cycle space and the number of odd cycles of g secondly sharp lower bounds on both r g g and r g g are determined all the corresponding extremal graphs are characterized respectively

lp estimates for time fractional parabolic equations in divergence form with measurable coefficients abstract in this paper we establish l p estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean oscillations with respect to the other variables the corresponding results for equations on a half space are also derived

on the nullity of bipartite graphs abstract the nullity of a graph is defined to be the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph in this paper we obtain the nullity set of bipartite graphs of order n and characterize the bipartite graphs with nullity n 4 and the regular bipartite graphs with nullity n 6

trees with maximum nullity abstract the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum among n vertex trees the star has greatest nullity equal to n 2 we generalize this by showing that among n vertex trees whose vertex degrees do not exceed a certain value d the greatest nullity is n 2 n 1 d methods for constructing such maximum nullity trees are described

parabolic equations with vmo coefficients in sobolev spaces with mixed norms an lq lp theory of divergence and non divergence form parabolic equations is presented the main coefficients are supposed to belong to the class vmox which in particular contains all measurable functions depending only on t the method of proving simplifies the methods previously used in the case p q

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

characterization of graphs whose signature equals the number of odd cycles abstract let g be a simple graph with vertex set v g and edge set e g the signature s g of g is the difference between the number of positive eigenvalues and the number of negative eigenvalues of the adjacency matrix a g in 20 it was proved that c 1 g s g c 1 g where c 1 g denotes the number of odd cycles in g a problem arises naturally what graphs have signature attaining the upper bound c 1 g resp the lower bound c 1 g in this paper we focus our attention on this problem characterizing graphs g whose signature equals c 1 g resp c 1 g

aleksandrov bakelman pucci type estimates for integro differential equations in this work we provide an aleksandrov bakelman pucci type estimate for a certain class of fully nonlinear elliptic integro differential equations the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal fractional order setting and on the use of riesz potentials to interpret second derivatives as fractional order operators this result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations furthermore this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the l and ln norms of the right hand side in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions these results appear to be new even for the linear case of the relevant equations

phase structure of systems with infinite numbers of absorbing states critical properties of systems exhibiting phase transitions into phases with infinite numbers of absorbing states are studied we analyze a non markovian langevin equation recently proposed to describe the critical behavior of such systems and also introduce and study a non markovian discrete model which is argued to present the same critical features on the basis of mean field analysis monte carlo simulations and theoretical arguments we conclude that the phenomenology of the non markovian models closely parallels that of systems with many absorbing states in one and two dimensions the bulk or static critical properties of these systems fall in the directed percolation dp universality class by contrast the critical properties associated with the spread of an initially localized seed exhibit a more complex behavior depending on parameter values they can both in one and two dimensions fall either in the dynamical percolation or dp universality class or else exhibit apparently nonuniversal exponents in contrast to previous results however the nonuniversal exponents in 2d are found to satisfy a scaling law which implies that a particular linear combination of them is universal and assumes dp values these results demonstrate the efficacy of the non markovian approach for understanding systems with many absorbing states which are difficult to analyze in their original microscopic formulation

harnack inequalities for non local operators of variable order we consider harmonic functions with respect to the operator formula math under suitable conditions on n x h we establish a harnack inequality for functions that are nonnegative and harmonic in a domain the operator l is allowed to be anisotropic and of variable order

global lorentz estimates with variable power to linear parabolic integro differential equations abstract we shall establish a global estimate in the framework of the lorentz spaces for variable power of the fractional derivative of solutions to the cauchy dirichlet problems of linear parabolic integro differential equations with smooth coefficients m t x y 1 2 in a bounded domain here we assume that some higher order derivatives of m t x y in y are merely measurable in the time variable t and satisfy uniformly continuous in the spatial variables x the boundaries of domains are c 1 1 and the variable exponents satisfy log holder continuity

hs p regularity theory for a class of nonlocal elliptic equations abstract in this paper we study the regularity of weak solutions to a class of nonlocal elliptic equations in bessel potential spaces h s p our main results can be seen as an extension of the well known w 1 p regularity theory for local second order elliptic equations in divergence form to the nonlocal setting

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

forward self similar solutions of the fractional navier stokes equations abstract we study forward self similar solutions to the 3 d navier stokes equations with the fractional diffusion first we construct a global time forward self similar solutions to the fractional navier stokes equations with 5 6 1 for arbitrarily large self similar initial data by making use of the so called blow up argument moreover we prove that this solution is smooth in r 3 0 in particular when 1 we prove that the solution constructed by korobkov and tsai 2016 16 satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system which implies this solution has the same properties as a solution which was constructed in jia and sverak 2014 13

on the nullity of graphs with pendant trees the nullity of a graph is defined as the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph we investigate a class of graphs with pendant trees and express the nullity of such graph in terms of that of its subgraphs as an application of our results we characterize unicyclic graphs with a given nullity

on the nullity of bicyclic graphs the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum in this paper we obtain the nullity set of n vertex unicyclic graphs and characterize the unicyclic graphs with extremal nullity

on the nullity of graphs with pendent vertices abstract the nullity of a graph g denoted by g is the multiplicity of the eigenvalue zero in its spectrum cheng and liu b cheng b liu on the nullity of graphs electron j linear algebra 16 2007 60 67 characterized the extremal graphs attaining the upper bound n 2 and the second upper bound n 3 in this paper as the continuance of it we determine the extremal graphs with pendent vertices achieving the third upper bound n 4 and fourth upper bound n 5 we then proceed recursively to construct all graphs with pendent vertices which satisfy g 0 our results provide a unified approach to determine n vertex unicyclic respectively bicyclic and tricyclic graphs which achieve the maximal and second maximal nullity and characterize n vertex extremal trees attaining the second and third maximal nullity as a consequence we respectively determine the nullity sets of trees unicyclic graphs bicyclic graphs and tricyclic graphs on n vertices

on the cauchy problem for integro differential operators in sobolev classes and the martingale problem abstract the existence and uniqueness in sobolev spaces of solutions of the cauchy problem to parabolic integro differential equation with variable coefficients of the order 0 2 is investigated the principal part of the operator has kernel m t x y y d with a bounded nondegenerate m holder in x and measurable in y the lower order part has bounded and measurable coefficients the result is applied to prove the existence and uniqueness of the corresponding martingale problem

critical exponents for a dimer dimer irreversible surface reaction abstract a dimer dimer reaction model of the type a2 b2 2ab is investigated by means of computer simulations the desorption of b2 dimer with probability p is studied here we investigate the critical behaviour the critical exponents of our model do not fall into the reggeon field theory or directed percolation universality class the universality class of the reaction changes with changes in p

the three species monomer monomer model in the reaction controlled limit we study the one dimensional three species monomer monomer reaction model in the reaction controlled limit using mean field theory and dynamic monte carlo simulations the phase diagram consists of a reactive steady state bordered by three equivalent adsorbing phases where the surface is saturated with one monomer species the transitions from the reactive phase are all continuous while the transitions between adsorbing phases are first order bicritical points occur where the reactive phase simultaneously meets two adsorbing phases the transitions from the reactive to an adsorbing phase show directed percolation critical behaviour while the universal behaviour at the bicritical points is in the even branching annihilating random walk class the results are contrasted and compared with previous results for the adsorption controlled limit of the same model

nonuniversal critical spreading in two dimensions continuous phase transitions are studied in a two dimensional nonequilibrium model with an infinite number of absorbing configurations spreading from a localized source is characterized by nonuniversal critical exponents which vary continuously with the density ensuremath varphi in the surrounding region the exponent ensuremath delta changes by more than an order of magnitude and ensuremath eta changes sign the location of the critical point also depends on ensuremath varphi which has important implications for scaling as expected on the basis of universality the static critical behavior belongs to the directed percolation class textcopyright 1996 the american physical society

on the null spaces of acyclic and unicyclic singular graphs for acyclic and unicyclic graphs we determine a necessary and sufficient condition for a graph g to be singular further it is shown that this characterization can be used to construct a basis for the null space of g