the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

adaptive least squares finite element approximations to stokes equations this paper is concerned with adaptive least squares methods for stokes equations based on velocity pressure stress and velocity vorticity pressure formulations to capture the stokes flow region an adaptive algorithm based on mesh redistribution is developed for a nonlinear weighted least squares functional a redistribution approach is considered to generate the optimal grids model problems considered are the flow past a planar channel and a 4 to 1 contraction problems numerical results of model problems illustrating the efficiency of the proposed scheme are presented adaptive least squares methods are used to solve the stokes equations velocity pressure stress and velocity vorticity pressure formulations are used mesh redistribution for planar and 4 to 1 contraction flow problems are developed lower order basis functions in all variables are used in least squares methods hybrid meshes of regular and graded grids are used to reduce the grid effects

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

motion of a set by the curvature of its boundary abstract the study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary often these equations are anisotropic indicating the preferred directions of the crystal structure in the isotropic case this equation is called the curve shortening or the mean curvature flow equation and has been studied by differential geometric tools in general there are no classical solutions to these equations in this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions our approach is closely related to that of osher and sethian chen giga and goto and evans and spruck who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation although we use their results in an essential way we give an intrinsic definition our main results are the existence of a solution large time asymptotics of this solution and its connection to the level set solution of osher and sethian chen giga and goto and evans and spruck in general there is no uniqueness even for classical solutions but we prove a uniqueness result under restrictive assumptions we also construct a class of explicit solutions which are dilations of wulff crystals

the singular set of mean curvature flow with generic singularities a mean curvature flow starting from a closed embedded hypersurface in mathbf r n 1 must develop singularities we show that if the flow has only generic singularities then the space time singular set is contained in finitely many compact embedded n 1 dimensional lipschitz submanifolds plus a set of dimension at most n 2 if the initial hypersurface is mean convex then all singularities are generic and the results apply in mathbf r 3 and mathbf r 4 we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time slice for 2 or 3 convex hypersurfaces in all dimensions the same arguments lead to the same conclusion the flow is completely smooth at almost all times and connected components of the singular set are contained in time slices a key technical point is a strong parabolic reifenberg property that we show in all dimensions and for all flows with only generic singularities we also show that the entire flow clears out very rapidly after a generic singularity these results are essentially optimal

graphs whose positive semi definite matrices have nullity at most two abstract let g v e be a undirected graph containing n vertices and let m g be the set of all hermitian n n matrices m m i j with m i j 0 if i and j are connected by one edge of g with m i j c if i and j are connected by at least two edges with m i j 0 if i j and i and j are not connected by an edge of g and with m i i for i 1 n a real number what is the largest nullity attained by any positive semi definite matrix m m g in this paper we characterize for t 1 and 2 those graphs g for which the maximum nullity is not greater than t

on the possible multiplicities of the eigenvalues of a hermitian matrix whose graph is a tree we consider the general problem of determining which lists of multiplicities for the eigenvalues occur among hermitian matrices the graph of whose off diagonal entries is a given tree several restrictions are cited and a construction strategy is given together these are sufficient to characterize all lists for each tree in two infinite classes the double paths and generalized stars and to tabulate all lists for trees on fewer than nine vertices such tables should be useful for formulating and dispelling general conjectures

positive semidefinite zero forcing numbers of two classes of graphs abstract the positive semidefinite zero forcing number is a variant of the zero forcing number which is an important parameter in the study of minimum rank maximum nullity problems in this paper we consider two classes of graphs matching chain graphs and claw free graphs we first introduce the propagation decomposition of graphs then we use this decomposition to prove a lower bound for the positive semidefinite zero forcing number of a graph we apply this lower bound to find the positive semidefinite zero forcing number of matching chain graphs we show that the positive semidefinite zero forcing number and the zero forcing number agree for matching chain graphs as a consequence we prove the conjecture about the positive semidefinite zero forcing number of the cartesian product of two paths and partially prove the conjecture about the positive semidefinite zero forcing number of the cartesian product of a cycle and a path for claw free graphs we establish a relationship between the positive semidefinite zero forcing number and the zero forcing number we also prove that it is np complete to find the positive semidefinite zero forcing number of line graphs

on the discrete evss method abstract this paper is a completion of a previous paper written by two of the authors m fortin r guenette r pierre numerical analysis of the modified evss method comput methods appl mech engrg 143 1997 79 95 concerning the devss method which is a discrete variant of the popular elastic viscous split stress evss method introduced by rajagopalan r a brown r c armstrong finite element methods for calculation of steady viscoelastic flow using constitutive equations with a newtonian viscosity j non newtonian fluid mech 36 1990 159 199 within the context of viscoelastic fluids the method uses the rate of deformation tensor as an additional variable in this paper it is shown that a simple modification of the theoretical framework introduced in the first paper m fortin r guenette r pierre numerical analysis of the modified evss method comput methods appl mech engrg 143 1997 79 95 allows the treatment of multi mode constitutive equations of viscoelastic flows as well as that of the velocity gradient introduced as the additional variable instead of the rate of deformation tensor moreover this general framework based on a generalized brezzi babuska theory allows different discretizations for the stress and the additional variable the paper presents a general criteria insuring the stability of the method three particular mixed finite elements satisfying this criteria are studied an error analysis is performed on stokes and viscoelastic flows

on the graph complement conjecture for minimum rank abstract the minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so one of the many unresolved questions on this topic is the so called graph complement conjecture which grew out of a workshop in 2006 this conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement and may be classified as a nordhaus gaddum type problem involving the graph parameter minimum rank the conjectured bound is the order of the graph plus two other variants of the graph complement conjecture are introduced here for the minimum semidefinite rank and the colin de verdiere type parameter we show that if the graph complement conjecture is true for two graphs then it is true for the join of these graphs related results for the graph complement conjecture and the positive semidefinite version for joins of graphs are also established we also report on the use of recent results on partial k trees to establish the graph complement conjecture for graphs of low minimum rank

lower bounds for minimum semidefinite rank from orthogonal removal and chordal supergraphs abstract the minimum semidefinite rank msr of a graph is the minimum rank among positive semidefinite matrices with the given graph the os number is a useful lower bound for msr which arises by considering ordered vertex sets with some connectivity properties in this paper we develop two new interpretations of the os number we first show that os number is also equal to the maximum number of vertices which can be orthogonally removed from a graph under certain nondegeneracy conditions our second interpretation of the os number is as the maximum possible rank of chordal supergraphs who exhibit a notion of connectivity we call isolation preserving these interpretations not only give insight into the os number but also allow us to prove some new results for example we show that msr g g 2 if and only if os g gzsfnc 2

a new lower bound for the positive semidefinite minimum rank of a graph abstract the real positive semidefinite minimum rank of a graph is the minimum rank among all real positive semidefinite matrices that are naturally associated via their zero nonzero pattern to the given graph in this paper we use orthogonal vertex removal and sign patterns to improve the lower bound for the real positive semidefinite minimum rank determined by the os number and the positive semidefinite zero forcing number

on the graph complement conjecture for minimum semidefinite rank in this note we combine a number of recent ideas to give new results on the graph complement conjecture for minimum semidefinite rank

a correction orthogonal representations and connectivity of graphs abstract this note corrects an error in the proof of the main result of the authors paper orthogonal representations and connectivity of graphs which appeared in linear algebra and its applications 114 115 1989 439 454

unitary matrix digraphs and minimum semidefinite rank for an undirected simple graph g the minimum rank among all positive semidefinite matrices with graph g is called the minimum semidefinite rank msr of g in this paper we show that the msr of a given graph may be determined from the msr of a related bipartite graph finding the msr of a given bipartite graph is then shown to be equivalent to determining which digraphs encode the zero nonzero pattern of a unitary matrix we provide an algorithm to construct unitary matrices with a certain pattern and use previous results to give a lower bound for the msr of certain bipartite graphs

lower bounds in minimum rank problems abstract the minimum rank of a graph is the smallest possible rank among all real symmetric matrices with the given graph the minimum semidefinite rank of a graph is the minimum rank among hermitian positive semidefinite matrices with the given graph we explore connections between os sets and a lower bound for minimum rank related to zero forcing sets as well as exhibit graphs for which the difference between the minimum semidefinite rank and these lower bounds can be arbitrarily large

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

weighted least squares approximationsto nonlinear hyperbolic equations some approximate methods for solving nonlinear hyperbolic equations are presented and analyzed the methods consist of discretizing with respect to time and solving the resulting hyperbolic equation for fixed time by least squares finite element methods stability of appropriate weighted least squares approximations is provided uniform and time varying optimal grids based on the mesh redistribution approach are considered numerical results for the inviscid burgers equation are also presented

issues related to least squares finite element methods for the stokes equations least squares finite element methods have become increasingly popular for the approximate solution of first order systems of partial differential equations here after a brief review of some existing theories a number of issues connected with the use of such methods for the velocity vorticity pressure formulation of the stokes equations in two dimensions in realistic settings are studied through a series of computational experiments finite element spaces that are not covered by existing theories are considered included in these are piecewise linear approximations for the velocity mixed boundary conditions which are also not covered by existing theories are also considered as is enhancing mass conservation next problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved a conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh dependent weights in the least squares functional almost always improves the accuracy of the approximations concluding remarks concerning three dimensional problems the nonlinear navier stokes equations and the conditioning of the discrete systems are provided

a nonlinear weighted least squares finite element method for stokes equations the paper concerns a nonlinear weighted least squares finite element method for the solutions of the incompressible stokes equations based on the application of the least squares minimization principle to an equivalent first order velocity pressure stress system model problem considered is the flow in a planar channel the least squares functional involves the l 2 norms of the residuals of each equation multiplied by a nonlinear weighting function and mesh dependent weights using linear approximations for all variables by properly adjusting the importance of the mass conservation equation and a carefully chosen nonlinear weighting function the least squares solutions exhibit optimal l 2 norm error convergence in all unknowns numerical solutions of the flow pass through a 4 to 1 contraction channel will also be considered

linearly independent vertices and minimum semidefinite rank we study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs we present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph and conjecture that the resulting graph invariant is equal to minimum semidefinite rank rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs we also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex

singularities of mean convex level set flow in general ambient manifolds abstract we prove two new estimates for the level set flow of mean convex domains in riemannian manifolds our estimates give control exponential in time for the infimum of the mean curvature and the ratio between the norm of the second fundamental form and the mean curvature in particular the estimates remove a stumbling block that has been left after the work of white 16 17 20 and haslhofer kleiner 9 and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension

the size of the singular set in mean curvature flow of mean convex sets in this paper we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point it is most convenient to describe the results in terms of the level set flow also called biggest flow i2 of chen giga goto cgg and evans spruck es under the level set flow any closed set k in r generates a one parameter family of closed sets ft k t 0 with f0 k k if the boundary of k is a smooth compact hypersurface then so is the boundary of ft k for t in some interval 0 and for such t s the evolution coincides with motion by mean curvature as defined classically by partial differential equations and differential geometry as in h1 however if k is compact then the boundary of ft k will necessarily become singular at some finite time our goal is to show that the singular sets are necessarily quite small this we do provided the initial set k is compact and mean convex in the sense that ft k interior k for all t 0 in case m k is a smooth hypersurface we have the following equivalent characterization of mean convexity k is mean convex if and only if the mean curvature of m with respect to the inward unit normal is everywhere non negative given a compact set k let k be the region in spacetime swept out by the ft k k x t r r t 0 x ft k a point x x t in the boundary of k is called a regular point if 1 x has a neighborhood in which k is a smooth manifold with boundary and 2 the tangent plane to k at x is not horizontal i e is not r 0 note that if x x t is a regular point then in a neighborhood of x ft k is a smooth manifold with boundary in r a point x x t in k with t 0 that is not a regular point is called a singular point

positive semidefinite zero forcing abstract the positive semidefinite zero forcing number z g of a graph g was introduced in barioli et al 2010 4 we establish a variety of properties of z g any vertex of g can be in a minimum positive semidefinite zero forcing set this is not true for standard zero forcing the graph parameters tw g tree width z g and z g standard zero forcing number all satisfy the graph complement conjecture see barioli et al 2012 3 graphs having extreme values of the positive semidefinite zero forcing number are characterized the effect of various graph operations on positive semidefinite zero forcing number and connections with other graph parameters are studied

an adaptively refined least squares finite element method for generalized newtonian fluid flows using the carreau model we implemented an adaptively refined least squares finite element approach for the navier stokes equations that govern generalized newtonian fluid flows using the carreau model to capture the flow region we developed an adaptive mesh refinement approach based on the least squares method the generated refined grids agree well with the physical attributes of the flows we also proved that the least squares approximation converges to the linearized versions solutions of the carreau model at the best possible rate model problems considered in the study are the flow past a planar channel and 4 to 1 contraction problems we presented the numerical results of the model problems revealing the efficiency of the proposed scheme and investigated the physical parameter effects

efficient stress velocity least squares finite element formulations for the incompressible navier stokes equations abstract in this contribution three different mixed least squares finite element methods lsfems are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies the deliberations are made for the incompressible navier stokes equations the generally known first order div grad system in terms of the total stress velocity and pressure svp formulation is the basis for two further div grad least squares formulations in terms of total stress and velocity sv whereby both formulations are derived from different fountainheads the extended sv formulation is an enrichment of the known stress velocity formulation proposed by cai et al 2004 the second sv formulation is based on the substitution of the pressure in the svp formulation thus it is based on a discontinuous pressure interpolation advantage of the sv formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom nevertheless the approximation quality has to last the demand of the established least squares formulations the drawback of a poor mass conservation is well investigated for the lsfem and known to overcome by using high order interpolations for all solution variables therefore the main attention of this contribution is focused on accuracy while aiming for high efficiency which is intrinsic for the fundamental idea in here in the light of efficient lsfem solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h type adaptive mesh refinement

on the minimum rank of the join of graphs and decomposable graphs abstract for a given undirected graph g the minimum rank of g is defined to be the smallest possible rank over all real symmetric matrices a whose i j th entry is nonzero whenever i j and i j is an edge in g in this work we consider joins and unions of graphs and characterize the minimum rank of such graphs in the case of balanced inertia several consequences are provided for decomposable graphs also known as cographs

the inertia set of the join of graphs abstract let g v e be a graph with v 1 2 n denote by s g the set of all real symmetric n n matrices a a i j with a i j 0 i j if and only if ij is an edge of g denote by i g the set of all pairs p q of natural numbers such that there exists a matrix a s g with at most p positive and q negative eigenvalues we show that if g is the join of g 1 and g 2 then i g 1 1 i g 1 k 1 i g 2 k 1 1 1 further we show that if g is a graph with s isolated vertices then i g k 1 i g i k 1 s where g denotes the graph obtained from g be removing all isolated vertices and we give a combinatorial characterization of graphs g with 1 1 i g we use these results to determine i g for every complete multipartite graph g

erratum to partial regularity at the first singular time for hypersurfaces evolving by mean curvature the purpose of this note is to inform the reader of a major error in the original article namely in the proof of theorem 1 8 i would like to point out that under the additional assumption of embeddedness of the solution the dimension estimates for k convex hypersurfaces in corollary 1 7 have been established by white 2 for k n and in 1 for all 1 k n i therefore believe that this statement is also correct in the case of immersed solutions but the proof may require different techniques to ours i am also confident that the statement of theorem 1 1 is correct for immersed hypersurfaces the error in the proof of theorem 1 8 can be found in inequality 11 on page 13 of our paper the centre of the ball should be y j j rather than y j the resulting ball may not contain the unit ball around the origin therefore the validity of inequality 13 and all subsequent statements in the proof are unclear since the proof of theorem 1 1 uses theorem 1 8 the validity of theorem 1 1 is also unclear the same holds for corollary 1 2 remark 1 3 as well as corollaries 1 4 and 1 5 the curvature integral estimate for k convex hypersurfaces with 1 k n 1 is however correct corollary 1 7 relies on corollary 1 2 though so its validity is again not clear i thank my colleague dr theodora bourni for pointing out this error to me

inscribed radius estimates for inverse curvature flow in sphere and hyperbolic space abstract in this paper we study a fully nonlinear inverse curvature flow in sphere and hyperbolic space and prove a non collapsing property for this flow using maximum principle precisely when ambient space is sphere we show that upon some conditions on speed function the radius of the largest touching interior ball is bounded below by a multiple of the reciprocal of the speed where ambient space is hyperbolic we obtain an upper bound of the curvature of the largest touching interior ball for star shaped inverse mean curvature flow

embedded minimal tori in s3 and the lawson conjecture we show that any embedded minimal torus in s 3 is congruent to the clifford torus this answers a question posed by h b lawson jr in 1970

on mass conservation in least squares methods we compare three least squares finite element reformulations of the stokes equations paying particular attention to mass conservation the first problem we approximate has a simple analytical solution over a convex region even for this simple problem without special treatment of the conservation of mass term very poor numerical solutions may result sufficiently weighting this term leads to a dramatic improvement in the results over a range of test problems

decoupled algorithm for solving phan thien tanner viscoelastic fluid by finite element method in this paper we present a decoupled algorithm for the viscoelastic fluid flow obeying the phan thien tanner ptt differential model the method consists to solve alternatively a stokes like problem by weighted least squares wls finite element method and the constitutive equation by streamline upwind petrov galerkin supg method a priori error estimate for the wls supg finite element method is derived the existence and uniqueness of the approximate solution are obtained using fixed point techniques we show that the mapping of the decoupled algorithm is locally contracting the planar channel flow problem is used to illustrate our theoretical results we encounter no limit of the weissenberg number for reasonable values of e a dimensionless material parameter in our calculations

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

zero forcing number grundy domination number and their variants abstract this paper presents strong connections between four variants of the zero forcing number and four variants of the grundy domination number these connections bridge the domination problem and the minimum rank problem providing a linear algebra approach to the domination problem we show that several grundy domination type parameters are bounded above by minimum rank type parameters we also give a method to calculate the l grundy domination number by the grundy total domination number giving some linear algebra bounds for the l grundy domination number

least squares p r finite element methods for incompressible non newtonian flows a least squares mixed p type finite element method is developed for numerical solution of incompressible non newtonian flows hierarchical piecewise polynomials are introduced as element basis functions while singularities and boundary layers are treated by a combination of mesh redistribution and polynomial refinement scaling of the original differential equations is found to be important for the least squares minimization process we discuss both nonlinear algebraic and differential constitutive models and present numerical examples to illustrate benefits and shortcomings of the present approach

the nature of singularities in mean curvature flow of mean convex sets let k be a compact subset of r or more generally of an n 1 dimensional riemannian manifold we suppose that k is mean convex if the boundary of k is smooth and connected this means that the mean curvature of k is everywhere nonnegative with respect to the inward unit normal and is not identically 0 more generally it means that ft k is contained in the interior of k for t 0 where ft k is the set obtained by letting k evolve for time t under the level set mean curvature flow as k evolves it traces out a closed set k of spacetime k x t r r x ft k also there is associated to k a brakke flow m t 7 mt of rectifiable varifolds we call the pair m k a mean convex flow let x x t be a point in spacetime with t 0 suppose xi ti is a sequence of points converging to x and i is a sequence of positive numbers tending to infinity translate the pair m and k in spacetime by y 7 y xi ti and then dilate parabolically by y 7 iy i to get new flows mi and ki the sequence mi ki is called a blow up sequence at x general compactness theorems guarantee that this sequence will have subsequential limits a subsequential limit m k is called a limit flow here m t 7 m t

vector representations of graphs and distinguishing quantum product states with one way locc abstract distinguishing sets of quantum states shared by two parties using only local operations and classical communication measurements is a fundamental topic in quantum communication and quantum information theory we introduce a graph theoretic approach based on the theory of vector representations of graphs to the core problem of distinguishing product states with one way locc we establish a number of results that show how distinguishing such states can be framed in terms of properties of the underlying graphs associated with a set of vector product states we also present a number of illustrative examples

orthogonal representations minimum rank and graph complements orthogonal representations are used to show that complements of certain sparse graphs have positive semidefinite minimum rank at most 4 this bound applies to the complement of a 2 tree and to the complement of a unicyclic graph hence for such graphs the sum of the minimum rank of the graph and the minimum rank of its complement is at most two more than the order of the graph the minimum rank of the complement of a 2 tree is determined exactly

a nonlinear weighted least squares finite element method for the carreau yasuda non newtonian model abstract we study a nonlinear weighted least squares finite element method for the navier stokes equations governing non newtonian fluid flows by using the carreau yasuda model the carreau yasuda model is used to describe the shear thinning behavior of blood we prove that the least squares approximation converges to linearized solutions of the non newtonian model at the optimal rate by using continuous piecewise linear finite element spaces for all variables and by appropriately adjusting the nonlinear weighting function we obtain optimal l 2 norm error convergence rates in all variables numerical results are given for a carreau fluid in the 4 to 1 contraction problem revealing the shear thinning behavior the physical parameter effects are also investigated

a local regularity theorem for mean curvature flow this paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the gaussian densities are close to 1

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

a weighted least squares finite element method for phan thien tanner viscoelastic fluid abstract in this paper we present the weighted least squares finite element approximations to the steady phan thien tanner ptt viscoelastic fluid we approximate the nonlinear viscoelastic model by linearizing the constitutive equation and apply a least squares method to solve the linearized system the least squares functional involves the residuals of each equation in l 2 norm multiplied by mesh dependent weights the method is tested on a planar channel flow problem numerical results exhibit the convergence rates of the proposed scheme

a collocation penalty least squares finite element formulation for incompressible flows abstract we present a penalty least squares finite element formulation for incompressible flows the pressure degree of freedom is eliminated using the classical penalty approach and the least squares model is formed in terms of velocity vorticity and dilatation an iterative penalization is implemented which allows the use of low penalty parameters and retains a manageable conditioning number of the global coefficient matrix the h convergence is verified using the exact solution of kovasznay s flow numerical results are presented for a number of benchmark problems e g steady flow past a large circular cylinder in a channel transient flow over a backward facing step unsteady flow past a circular cylinder and buoyancy driven natural convection in a square cavity for all numerical examples the effect of penalty parameter on the accuracy is investigated thoroughly and it is concluded that the present model produces accurate results for low penalty parameters 10 100 these problems are also solved on coarse meshes to show that good mass conservation is achieved

a fully nonlinear flow for two convex hypersurfaces in riemannian manifolds we consider a one parameter family of closed embedded hypersurfaces moving with normal velocity g kappa big sum i j frac 1 lambda i lambda j 2 kappa big 1 where lambda 1 le cdots le lambda n denote the curvature eigenvalues and kappa is a nonnegative constant this defines a fully nonlinear parabolic equation provided that lambda 1 lambda 2 2 kappa in contrast to mean curvature flow this flow preserves the condition lambda 1 lambda 2 2 kappa in a general ambient manifold our main goal in this paper is to extend the surgery algorithm of huisken sinestrari to this fully nonlinear flow this is the first construction of this kind for a fully nonlinear flow as a corollary we show that a compact riemannian manifold satisfying overline r 1313 overline r 2323 ge 2 kappa 2 with non empty boundary satisfying lambda 1 lambda 2 2 kappa is diffeomorphic to a 1 handlebody the main technical advance is the pointwise curvature derivative estimate the proof of this estimate requires a new argument as the existing techniques for mean curvature flow due to huisken sinestrari haslhofer kleiner and brian white cannot be generalized to the fully nonlinear setting to establish this estimate we employ an induction on scales argument this relies on a combination of several ingredients including the almost convexity estimate the inscribed radius estimate as well as a regularity result for radial graphs we expect that this technique will be useful in other situations as well

orthogonal representations and connectivity of graphs abstract it is proved that a graph on n nodes is k connected if and only if its nodes can be represented by real vectors in dimension n k such that a nonadjacent nodes are represented by orthogonal vectors and b any n k of them are linearly independent we show that the closure of the set of all representations with properties a and b is irreducible as an algebraic variety and study the question of irreducibility of the variety of all representations with property a

bounds for minimum semidefinite rank from superpositions and cutsets abstract the real complex minimum semidefinite rank of a graph is the minimum rank among all real symmetric complex hermitian positive semidefinite matrices that are naturally associated via their zero nonzero pattern to the given graph in this paper we give an upper bound on the minimum semidefinite rank of a graph when the graph is modified from the superposition of two graphs by cancelling some number of edges we also provide a lower bound for the minimum semidefinite rank of a graph determined by a given cutset when the complement of the cutset is a star forest these lower and upper bounds coincide and we can compute the minimum semidefinite rank in terms of smaller graphs this result encompasses the previously known case in which the cut set has order two or smaller next we provide results for when the cut set has order three using these results we provide an example where the positive semidefinite zero forcing number is strictly greater than the maximum positive semidefinite nullity

the minimum semidefinite rank of a triangle free graph abstract we employ a result of moshe rosenfeld to show that the minimum semidefinite rank of a triangle free graph with no isolated vertex must be at least half the number of its vertices we define a rosenfeld graph to be such a graph that achieves equality in this bound and we explore the structure of these special graphs their structure turns out to be intimately connected with the zero nonzero patterns of the unitary matrices finally we suggest an exploration of the connection between the girth of a graph and its minimum semidefinite rank and provide a conjecture in this direction

zero forcing parameters and minimum rank problems the zero forcing number z g which is the minimum number of vertices in a zero forcing set of a graph g is used to study the maximum nullity minimum rank of the family of symmetric matrices described by g it is shown that for a connected graph of order at least two no vertex is in every zero forcing set the positive semidefinite zero forcing number z g is introduced and shown to be equal to g os g where os g is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank the positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs an example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex hermitian positive semidefinite minimum rank is presented

motion of level sets by mean curvature ii we give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface

undirected graphs of hermitian matrices that admit only two distinct eigenvalues abstract we consider the problem of determining those undirected n vertex graphs with a corresponding hermitian matrix that admits only two distinct eigenvalues with multiplicities k and n k after giving some general algebraic characterizations of these dual multiplicity graphs we then prove two major graph theoretic necessary conditions on such graphs construction techniques are then developed and these lead to a characterization of dual multiplicity graphs for which the lesser multiplicity is two

on the complexity of the positive semidefinite zero forcing number abstract the positive semidefinite zero forcing number of a graph is a graph parameter that arises from a non traditional type of graph colouring and is related to a more conventional version of zero forcing we establish a relation between the zero forcing and the fast mixed searching which implies some np completeness results for the zero forcing problem relationships between positive semidefinite zero forcing sets and clique coverings are well understood for chordal graphs building upon constructions associated with optimal tree covers and forest covers we present a linear time algorithm for computing the positive semidefinite zero forcing number of chordal graphs we also prove that it is np complete to determine whether a graph has a positive semidefinite zero forcing set with an additional property

least squares methods for the velocity pressure stress formulation of the stokes equations we formulate and study finite element methods for the solution of the incompressible stokes equations based on the application of least squares minimization principle to an equivalent first order velocity pressure stress system our least squares functional involves the l norms of the residuals of each equation multiplied by a mesh dependent weight each weight is determined according to the agmon douglis nirenberg index of the corresponding equation as a result the approximating spaces are not subject to the lbb condition and conforming discretizations are possible with merely continuous finite element spaces moreover the resulting discrete problems involve only symmetric positive definite systems of linear equations i e assembly of the discretization matrix is not required even at the element level we prove that the least squares approximadons converge to the solutions of the stokes problem at the best possible rate and then present some numerical examples illustrating our theoretical results among other things these numerical examples indicate that the method is not optimal without the weights in the least squares functional

a nonlinear weighted least squares finite element method for the oldroyd b viscoelastic flow this paper studies a nonlinear weighted least squares finite element method for the steady oldroyd b viscoelastic flow our least squares functional involves the l 2 norm of the residuals of each equation multiplied by a proper weight the weights include mass conservation constant mesh dependent weight and a nonlinear weighting function we prove that the least squares approximation converges to the solutions of the linearized versions of the viscoelastic fluid model at the best possible rate and then present the planar channel flow problem illustrating our theoretical results results of the least squares approach indicate that with carefully chosen nonlinear weighting functions and linear basis functions the numerical solution exhibits a second order convergence rate for velocity and viscous stress and superlinear convergence rate in polymeric stress and pressure the method is applied to the 4 to 1 planar contraction problem the effects of weissenberg numbers are also presented in the work