a free boundary problem for the laplacian with a constant bernoulli type boundary condition we study a free boundary problem for the laplace operator where we impose a bernoulli type boundary condition we show that there exists a solution to this problem we use a beurling s technique by defining two classes of sub and supersolutions and a perron argument we try to generalize here a previous work of a henrot and h shahgholian we extend these results in different directions

characterization of graphs having extremal randic indices the higher randic index rt g of a simple graph g is defined as rt g i1i2 it 11 i1 i2 it 1 where i denotes the degree of the vertex i and i1i2 it 1 runs over all paths of length t in g in j a rodriguez a spectral approach to the randic index linear algebra appl 400 2005 339 344 the lower and upper bound on r1 g was determined in terms of a kind of laplacian spectra and the lower and upper bound on r2 g were done in terms of kinds of adjacency and laplacian spectra in this paper we characterize the graphs which achieve the upper or lower bounds of r1 g and r2 g respectively

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

partially overdetermined elliptic boundary value problems we consider semilinear elliptic dirichlet problems in bounded domains overdetermined with a neumann condition on a proper part of the boundary under different kinds of assumptions we show that these problems admit a solution only if the domain is a ball when these assumptions are not fulfilled we discuss possible counterexamples to symmetry we also consider neumann problems overdetermined with a dirichlet condition on a proper part of the boundary and the case of partially overdetermined problems on exterior domains

characteristics polynomial of normalized laplacian for trees here we find the characteristics polynomial of normalized laplacian of a tree the coefficients of this polynomial are expressed by the higher order general randic indices for matching whose values depend on the structure of the tree we also find the expression of these indices for starlike tree and a double starlike tree hm p q moreover we show that two cospectral hm p q of the same diameter are isomorphic

concavity properties for elliptic free boundary problems abstract we prove some concavity properties connected to nonlinear bernoulli type free boundary problems in particular we prove a brunn minkowski inequality and an urysohn s type inequality for the bernoulli constant and we study the behavior of the free boundary with respect to the given boundary data moreover we prove a uniqueness result regarding the interior problem

serrin type overdetermined problems an alternative proof we prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations namely the hessian equations in the case of the poisson equation our proof is alternative to the proofs proposed by serrin moving planes and by weinberger moreover our proof makes no direct use of the maximum principle while it sheds light on a relation between the serrin problem and the isoperimetric inequality

on quasilinear elliptic equations in domains with coninal boundary points on etudie les solutions de l equation pseudo laplace au div u p 2 u f 1 p pres de points limites coniques

existence and geometric properties of solutions of a free boundary problem in potential theory let 0 g h l rn n 2 be two given density functions at least one of them bounded away from zero outside a compact set and g holder continuous we prove that for any compactly supported positive measure which is sufficiently concentrated e g has sufficiently high n 1 dimensional density there exists a bounded open set rn such that the newtonian potential of the measure hln ghn 1 agrees with that of outside some regularity of is obtained as well as several results on the geometry of example if h and g are constant then for any x the inward normal ray of at x if it exists intersects the closed convex hull of typeset by ams tex 1 2 bjorn gustafsson and henrik shahgholian

second order randic index of fluoranthene type benzenoid systems fluoranthene type benzenoid system are polycyclic conjugated molecules consisting of two benzenoid fragments connected by two carbon carbon bonds so as to form a five membered ring fluoranthene type benzenoid system and its congeners are compounds which in view of their chemical and physical properties belong among benzenoid hydrocarbons let f denotes a fluoranthene type benzenoid system whose second order randic index is denoted by r2 f in this paper we give the expression of r2 f in terms of their inlet features and we find the minimal and maximal value of the second order randic index over the set of cata catacondensed fluoranthene type benzenoid systems and characterize their corresponding graphs

extremality of degree based graph entropies abstract many graph invariants have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy we study graph entropies which are based on vertex degrees by using so called information functionals when considering shannon entropy based graph measures there has been very little work to find their extremal values the main contribution of this paper is to prove some extremal values for the underlying graph entropy of certain families of graphs and to find the connection between the graph entropy and the sum of degree powers further conjectures to determine extremal values of graph entropies are given

the universal regular quotient of the chow group of points on projective varieties let x be a projective variety of dimension n de ned over an algebraically closed eld k for x irreducible and non singular matsusaka ma constructed an abelian variety alb x and a morphism a x alb x called the albanese variety and mapping respectively depending on the choice of a base point on x which is universal among the morphisms to abelian varieties see lang la serre se for other constructions over the eld of complex numbers the existence of alb x and a was known before and has a purely hodge theoretic description see igusa i for the hodge theoretic construction incidentally the terminology albanese variety was introduced by a weil for reasons explained in his commentary on the article 1950a of volume i of his collected works see w one of which is that the paper alb of albanese de nes it for a surface as a quotient of the group of 0 cycles of degree 0 modulo an equivalence relation let chn x deg 0 denote the chow group of 0 cycles of degree 0 on x modulo rational equivalence when x is irreducible and non singular a remarkable feature of the albanese morphism a is that it factors through a regular homomorphism u chn x deg 0 alb x that is a homomorphism which when composed with the cycle map c x chn x deg 0 gives an algebraic morphism this follows immediately from the fact that an abelian variety does not contain any rational curve thus one can reformulate matsusaka s theorem as the invent math 135 595 664 1999 doi 10 1007 s002229900890

the general connectivity indices of fluoranthene type benzenoid systems the general connectivity index r g of a graph g v e is defined as r g u v e g d u d v where u v is an edge of g du and dv denote the degrees of the vertices u and v respectively and 0 is an arbitrary real number in this paper we give the expressions for computing the general connectivity indices of fluoranthene type benzenoid systems and we determine the extremal values of r g in f benzenoid systems with h hexagons for some real number especially the extremal values of r g in cata catacondensed fluoranthene type benzenoid systems with h hexagons were studied

new lower bounds for the randic spread abstract let g v g e g be an n m graph the randic spread of g s r g is defined as the maximum distance of its randic eigenvalues disregarding the randic spectral radius of g in this work we use numerical inequalities and bounds for the matricial spread to obtain relations between this spectral parameter and some structural and algebraic parameters of the underlying graph such as the sequence of vertex degrees the nullity randic index generalized randic indices and its independence number in the last section a comparison is presented for regular graphs

a free boundary problem for p laplacian in the plane abstract we consider the following free boundary problem in an unbounded domain in two dimensions p u 0 in u 0 u n g 0 on j 0 u 1 u n g 1 on j 1 where j 0 j 1 we prove that if 0 u 1 in j i is the graph of a function in c loc 1 r and g i is a constant for each i 0 1 then the free boundary must be two parallel straight lines and the solution u must be a linear function the proof is based on maximum principle

sharp lower bounds on the laplacian eigenvalues of trees abstract let 1 t and 2 t be the largest and the second largest laplacian eigenvalues of a tree t we obtain the following sharp lower bound for 1 t 1 t max d i m i 1 d i m i 1 2 4 d i m i 1 2 v i v wheredi and mi denote the degree of vertex vi and the average of the degrees of the vertices adjacent to vertex vi respectively equality holds if and only if t is a tree t di dj where t di dj is formed by joining the centres of di copies of k1 dj 1 to a new vertex vi that is t di dj vi dik1 dj 1 let v1 be the highest degree vertex of degree d1 and v2 be the second highest degree vertex of degree d2 we also show that if t is a tree of order n 2 then 2 t d 2 if v 1 v 2 e d 2 1 d 2 1 2 4 2 if v 1 v 2 e wheree is the set of edges equality holds if t t1 d1 or t t2 d1 where t1 d1 is formed by joining the centres of two copies of k1 d1 1 and t2 d1 is formed by joining the centres of two copies of k1 d1 1 to a new vertex moreover we obtain the lower bounds for the sum of two largest laplacian eigenvalues

a symmetry result for the torsion equation in an unbounded domain in 2d abstract we investigate an overdetermined problem for the torsion equation u 1 with two kinds of free boundary conditions in an unbounded domain and present a symmetry result in the case of 2d the proof is based on the maximum principle in the proof we use the sliding method and some convex arguments

a harnack inequality approach to the regularity of free boundaries part ii flat free boundaries are lipschitz soit u une solution faible d un probleme aux limites libres du type u 0 sur u 0 et sur u 0 et soit le long de la frontiere libre f la relation u v g u x est satisfaite au sens faible alors si f est proche du graphe d une fonction de lipschitz dans la boule unite b 1 de r n f est le graphe d une fonction de lipschitz dans b 1 2

a symmetry result for the torsion equation in an unbounded domain in 3d abstract we investigate an overdetermined problem for the torsion equation u 1 in an unbounded domain in r 3 under some additional conditions we show that the solution is a one dimensional quadratic function and that the boundary of the domain is a pair of parallel planes the proof is based on the maximum principle

a spectral approach to the randic index abstract the higher randic index rt of a simple graph is defined as r t v i 1 v i 2 v i t 1 1 i 1 i 2 i t 1 where i denotes the degree of the vertex vi and v i 1 v i 2 v i t 1 runs over all paths of length t in in this paper we introduce a suitable version of the laplacian of a graph and we formulated r1 in terms of its spectrum moreover bounds on r2 from the eigenvalues either the adjacency matrix or the laplacian matrix of the graph are obtained in the paper

murthy s conjecture on 0 cycles we show that the levine weibel chow group of 0 cycles mathrm ch d a of a reduced affine algebra a of dimension d ge 2 over an algebraically closed field is torsion free among several applications it implies an affirmative solution to an old conjecture of murthy in classical k theory

upper bound on randic energy of some graphs abstract applying the cauchy schwarz inequality we obtain a sharp upper bound on the randic energy of a bipartite graph and of graphs whose adjacency matrix is partitioned into blocks with constant row sum

spectral properties of geometric arithmetic index the concept of geometric arithmetic index was introduced in the chemical graph theory recently but it has shown to be useful one of the main aims of algebraic graph theory is to determine how or whether properties of graphs are reflected in the algebraic properties of some matrices the aim of this paper is to study the geometric arithmetic index ga1 from an algebraic viewpoint since this index is related to the degree of the vertices of the graph our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices moreover using this matrix we define a ga laplacian matrix which determines the geometric arithmetic index of a graph and satisfies properties similar to the ones of the classical laplacian matrix

the one phase free boundary problem for the p laplacian with non constant bernoulli boundary condition our objective here is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a bernoulli type boundary gradient condition and with the p laplacian as the governing operator the main theorems of this paper assert that the exterior and the interior free boundary problem with a bernoulli law i e with a prescribed pressure a x on the free streamline of the flow have convex solutions provided the initial domains are convex the continuous function a x is subject to certain convexity properties in our earlier results we have considered the case of constant a x in the lines of the proof of the main results we also prove the semi continuity up to the boundary of the gradient of the p capacitary potentials in convex rings with c 1 boundaries

existence of classical solutions to a free boundary problem for the p laplace operator i the exterior convex case in this paper we prove under convexity assumptions for the data the existence of classical solutions for a bernoulli type free boundary problem with the p laplacian as the governing operator the method employed here originates from a pioneering work of a beurling where he proves the existence for the harmonic case in the plane though with no geometrical restrictions

the connectivity index of a weighted graph abstract let g be a simple graph and consider the m connectivity index m x g i 1 i n hellip i m 1 1 d i 1 d i m d i m 1 where i 1 i 2 i m 1 runs over all paths of length m in g and di denotes the degree of the vertex i we find upper bounds for m g using the eigenvalues of the laplacian matrix of an associated weighted graph the method provides also lower bounds for 1 g

the construction of weight two arithmetic cohomology let x be a regular noetherian scheme in a letter to soul6 be beilinson conjectured for each non negative integer n the existence of complexes f n x of zariski sheaves on x whose hypercohomology should be in the same relation to algebraic k theory as classical cohomology is to topological ktheory and proposed certain properties that these complexes should satisfy this investigation was continued in l where the conjectured complexes were moved to the a at other points we work only up to 2torsion or up to torsion involving primes dividing the residue field characteristics there seems good reason to expect our complex f 2 x to in fact exactly satisfy all the axioms we now sketch the construction of f 2 x and recall the axioms let a be a regular noetherian ring let w s p e c a t z s p e c a t t t 1 let b bl b 2 b be a finite sequence of exceptional units of a i e b i and

some inverse problems around the tokamak tore supra we consider two inverse problems related to the tokamak textsl tore supra through the study of the magnetostatic equation for the poloidal flux the first one deals with the cauchy issue of recovering in a two dimensional annular domain boundary magnetic values on the inner boundary namely the limiter from available overdetermined data on the outer boundary using tools from complex analysis and properties of genereralized hardy spaces we establish stability and existence properties secondly the inverse problem of recovering the shape of the plasma is addressed thank tools of shape optimization again results about existence and optimality are provided they give rise to a fast algorithm of identification which is applied to several numerical simulations computing good results either for the classical harmonic case or for the data coming from textsl tore supra

on partially and globally overdetermined problems of elliptic type we consider some elliptic pdes with dirichlet and neumann data prescribed on some portion of the boundary of the domain and we obtain rigidity results that give a classication of the solution and of the domain in particular we nd mild conditions under which a partially overdetermined problem is in fact globally overdetermined this enables to use several classical results in order to classify all the domains that admit a solution of suitable general partially overdetermined problems these results may be seen as solutions of suitable inverse problems that is to say given that an overdetermined system possesses a solution we nd the shape of the admissible domains models of these type arise in several areas of mathematical physics and shape optimization

zero cycles and k theory on normal surfaces in this paper we prove a formula conjectured by bloch and srinivas s2 which describes the chow group of zero cycles of a normal quasi projective surface x over a field as an inverse limit of relative chow groups of a desingularisation x relative to multiples of the exceptional divisor we then give several applications of this result a relative version of the famous bloch conjecture on 0 cycles the triviality of the chow group of 0 cycles for any 2 dimensional normal graded q algebra analogue of the bloch beilinson conjecture and the analogue of the roitman theorem for torsion 0 cycles in characteristic p 0 for normal varieties including the case of p torsion

on the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions we prove the existence of convex classical solutions for a general multidimensional multilayer free boundary problem the geometric context of this problem is a nested family of closed convex surfaces except for the innermost and outermost surfaces which are given these surfaces are interpreted as unknown layer interfaces where the layers are the bounded annular domains between them each unknown interface is characterized by a quite general nonlinear equation called a joining condition which relates the first derivatives along the interface of the capacitary potentials in the two adjoining layers as well as the spatial variables a well known special case of this problem involves several stationary immiscible two dimensional flows of ideal fluid related along their interfaces by bernoulli s law

flattening results for elliptic pdes in unbounded domains with applications to overdetermined problems semilinear elliptic pdes are dealt with either in the half space or in overdetermined settings we obtain several new results and also give new proofs of celebrated theorems by exploiting some geometric analysis of level sets a geometric inequality and a pointwise gradient estimate

the normalized incidence energy of a graph abstract the energy of a graph g is the sum of the singular values of its adjacency matrix it is a graph invariant used in mathematical chemistry the normalized incidence energy of the graph g denoted by nie g is defined as the sum of the singular values of its normalized incidence matrix we find some upper and lower bounds and determine the coulson integral formula for nie g based on the integral formula we give a way to compare the normalized incidence energies moreover we show a relation between normalized incidence energy and randic energy

fast numerical methods for bernoulli free boundary problems the numerical solution of the free boundary bernoulli problem is addressed an iterative method based on a level set formulation and boundary element method is proposed issues related to the implementation the accuracy and the generality of the method are discussed the efficiency of the approach is illustrated by numerical results