Showing papers by "Courant Institute of Mathematical Sciences published in 1990"

How to draw a planar graph on a grid

Abstract: Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fary embedding (i.e., straight-line embedding) on the 2n−4 byn−2 grid and provide anO(n) space,O(n logn) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any setF, which can support a Fary embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.

Scalings for tokamak energy confinement

Abstract: On the basis of an analysis of the ITER L-mode energy confinement database, two new scaling expressions for tokamak L-mode energy confinement are proposed, namely a power law scaling and an offset-linear scaling. The analysis indicates that the present multiplicity of scaling expressions for the energy confinement time τE in tokamaks (Goldston, Kaye, Odajima-Shimomura, Rebut-Lallia, etc.) is due both to the lack of variation of a key parameter combination in the database, fs = 0.32 R a−0.75 k0.5 ~ A a0.25k0.5, and to variations in the dependence of τE on the physical parameters among the different tokamaks in the database. By combining multiples of fs and another factor, fq = 1.56 a2 kB/RIp = qeng/3.2, which partially reflects the tokamak to tokamak variation of the dependence of τE on q and therefore implicitly the dependence of τE on Ip and ne, the two proposed confinement scaling expressions can be transformed to forms very close to most of the common scaling expressions. To reduce the multiplicity of the scalings for energy confinement, the database must be improved by adding new data with significant variations in fs, and the physical reasons for the tokamak to tokamak variation of some of the dependences of the energy confinement time on tokamak parameters must be clarified.

Representations of Commonsense Knowledge

Abstract: A central goal of artificial intelligence is to give a computer program commonsense understanding of basic domains such as time, space, simple laws of nature, and simple facts about human minds. Many different systems of representation and inference have been developed for expressing such knowledge and reasoning with it. "Representations of Commonsense Knowledge" is the first thorough study of these techniques. The first three chapters of the book establish a general framework in domain-independent terms, discussing methodology, deductive logics, and theories of plausible inference. Subsequent chapters each deal with representations and inferences in specific domains: quantities, time, space, physics, knowledge and belief, plans and goals, and interactions among agents. The power of these representations in expressing world knowledge and in supporting significant inferences is analyzed using many detailed examples. The discussion includes both representations that have been used in successful AI programs and those that have been developed in purely abstract settings. Representations of Commonsense Knowledge is an essential reference for AI researchers and developers. It can also be used as a textbook in advanced undergraduate or graduate courses; each chapter contains exercises and suggestions for further reading. Readers who have completed the book will be prepared to read original technical papers in the area and to begin their own work in developing useful representations for AI programs.

Combinatorial complexity bounds for arrangements of curves and spheres

Abstract: We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges boundingm cells in an arrangement ofn lines is ?(m2/3n2/3 +n), and that it isO(m2/3n2/3s(n) +n) forn unit-circles, wheres(n) (and laters(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up toO(m3/5n4/5s(n) +n). The same bounds (without thes(n)-terms) hold for the maximum sum of degrees ofm vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees ofm vertices in an arrangement ofn spheres in three dimensions isO(m4/7n9/7s(m, n) +n2), in general, andO(m3/4n3/4s(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances amongm points in three dimensions isO(m3/2s(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.

On the finite volume element method

Abstract: The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH 1 semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h 2). Results on the effects of numerical integration are also included.

Rounding effects of quenched randomness on first-order phase transitions

Abstract: Frozen-in disorder in an otherwise homogeneous system, is modeled by interaction terms with random coefficients, given by independent random variables with a translation-invariant distribution. For such systems, it is proven that ind=2 dimensions there can be no first-order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter. Discontinuities which would amount to a continuous symmetry breaking, in systems which are (stochastically) invariant under the action of a continuous subgroup ofO(N), are suppressed by the randomness in dimensionsd≦4. Specific implications are found in the Random-Field Ising Model, for which we conclude that ind=2 dimensions at all (β,h) the Gibbs state is unique for almost all field configurations, and in the Random-Bond Potts Model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point. The results are explained by the argument of Imry and Ma [1]. The proofs involve the analysis of fluctuations of free energy differences, which are shown (using martingale techniques) to be Gaussian on the suitable scale.

Mean-Field Critical Behaviour for Percolation in High Dimensions

Abstract: The triangle condition for percolation states that\(\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)} \) is finite at the critical point, where τ(x, y) is the probability that the sitesx andy are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on thed-dimensional hypercubic lattice, ifd is sufficiently large, and (ii) in more than six dimensions for a class of “spread-out” models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values\((\gamma = \beta = 1,\delta = \Delta _t = 2, t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2)\) and that the percolation density is continuous at the critical point. We also prove thatv2 in (i) and (ii), wherev2 is the critical exponent for the correlation length.

Second variation and stabilities of minimal lagrangian submanifolds in Kähler manifolds

Abstract: First, we derive a new second variation formula which holds for any minimal lagrangian submanifold in K~ihler manifolds. Using this, we prove that any minimal lagrangian submanifold in Kfihler manifolds with nonpositive Chern class is stable. We also prove that in Kfihler manifolds with positive Chern class, any stable minimal lagrangian submanifold L satisfies that H~(L, ~) = {0} . Next, we introduce the notion of hamiltonian stability (either local or global) of minimal lagrangian submanifolds in Einstein-Kfihler manifolds. Using our second variation formula, we find a general criterion for the local hamiltonian stability for minimal lagrangian submanifolds in Einstein-K/ihler manifolds. This criterion is satisfied for all real forms which are Einstein, of irreducible Hermitian symmetric spaces, e.g, the cannonical RP" c CP". It is also satisfied by the Clifford torus T" c CP".

On characterizing the set of possible effective tensors of composites: The variational method and the translation method

Abstract: A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization-problem simplifies the derivation of the Hashin-Shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non-local projection operator, T1. An important class of bounds, namely trace bounds on the effective tensors of two-component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two-components are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou. An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet-type variational principles and bounds for the effective tensors of general non-selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.

A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equation

Abstract: A velocity pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations

Nonlinear wave equations, formation of singularities

Abstract: Equations in one space variable Blow-up in higher dimensions Longtime existence for solutions of nonlinear wave equations with small initial data Appendix I. Uniqueness for nonlinear wave equations Appendix II. Klainerman's inequality.

Numerical implementation of a variational method for electrical impedance tomography

Abstract: A variational method for computing electrical conductivity distributions from boundary measurements was proposed by Kohn and Vogelius (1987). The authors explore the numerical performance of that technique. A version of Newton's method is used for the minimisation, and synthetic data for the boundary measurements. The variational method is found to be generally stable and robust, reproducing the locations and shapes of conducting objects well, provided that smooth boundary data are used. Early termination appears to have a desirable smoothing effect upon the reconstruction. Contrary to the suggestion of Kohn and Vogelius, the method is not enhanced by allowing the conductivity to be anisotropic.

Convergence of the point vortex method for the 2-D euler equations

Abstract: We prove consistency, stability and convergence of the point vortex approximation to the 2-D incompressible Euler equations with smooth solutions. We first show that the discretization error is second-order accurate. Then we show that the method is stable in l p norm. Consequently the method converge in l p norm for all time. The convergence is also illustrated by a numerical experiment

Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system

Abstract: Partial differential equations that conserve energy can often be written as infinite-dimensional Hamiltonian systems of the following general form: du/dt=JE'(u), where J:X * →X is a symplectic matrix (i.e., JJ * =−1) and E: X→R is a C 2 functional defined on some Hilbert space X

Asymptotic properties of linear field equations in minkowski space

Abstract: The aim of this paper is to derive the uniform asymptotic behavior of solutions to linear field equations in Minkowski space, based on geometric consideration and generalized energy estimates. Our method relies on a systematic use of the invariance properties of the field equations with respect to the conformal group of the Minkowski space and thus departs substantially from the classical method of analyzing the fundamental solution

On the slowness of phase boundary motion in one space dimension

Abstract: We study the limiting behavior of the solution of u t −e 2 u xx +u 3 −u=0, a

Some existence results for fully nonlinear elliptic equations of monge‐ampère type

Abstract: We study the existence and uniqueness of solutions of Monge-Ampere-type equations. This type of equations has been studied extensively by Caffarelli, Nirenberg, Spruck and many others. (See [5] through [8] and the references therein.) We present some existence and uniqueness results for this type of equations on compact Riemannian manifolds with non-negative sectional curvature. We have also generalized some results in [7].

Quantitative Steinitz's theorems with applications to multifingered grasping

Abstract: We prove the following quantitative form of a classical theorem of Steintiz: Letm be sufficiently large. If the convex hull of a subsetS of Euclideand-space contains a unit ball centered on the origin, then there is a subset ofS with at mostm points whose convex hull contains a solid ball also centered on the origin and havingresidual radius $$1 - 3d\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ The casem=2d was first considered by Baranyet al. [1]. We also show an upper bound on the achievable radius: the residual radius must be less than $$1 - \frac{1}{{17}}\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ These results have applications in the problem of computing the so-calledclosure grasps by anm-fingered robot hand. The above quantitative form of Steinitz's theorem gives a notion ofefficiency for closure grasps. The theorem also gives rise to some new problems in computational geometry. We present some efficient algorithms for these problems, especially in the two-dimensional case.

Grad ø perturbations of massless Gaussian fields

Abstract: We investigate weak perturbations of the continuum massless Gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity.

Fractional balanced reduction: model reduction via fractional representation

Abstract: A method for the model reduction of finite-dimensional, linear, time-invariant (FDLTI) plants is proposed which uses normalized fractional representations is proposed. The method, dubbed fractional balanced reduction, applies balance and truncate to a special representation of the graph operator of the plant. This operation yields the graph operator of a reduced order plant. The method has such properties as existence of an a priori error bound in the graph metric and preservation of sector containment. Coupling fractional representations with principal component analysis gives a model reduction method that is capable of producing, in a numerically stable way, a good reduced order model using the whole full order model. Sector properties are also preserved-these are useful for deciding stability when nonlinearities are in the loop. >

The complexity and construction of many faces in arrangements of lines and of segments

Abstract: We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m2/3??n2/3+2?+n) for any?>0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of thesem faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and its expected time complexity isO(m2/3??n2/3+2? logn+n logn logm). If instead of lines we have an arrangement ofn line segments, then the maximum number of edges ofm faces isO(m2/3??n2/3+2?+n? (n) logm) for any?>0, where?(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and takes expected timeO(m2/3??n2/3+2? log+n?(n) log2n logm).

On the upper critical dimension of lattice trees and lattice animals

Abstract: We give a rigorous proof of mean-field critical behavior for the susceptibility (γ=1/2) and the correlation length (v=1/4) for models of lattice trees and lattice animals in two cases: (i) for the usual model with trees or animals constructed from nearest-neighbor bonds, in sufficiently high dimensions, and (ii) for a class of “spread-out” or long-range models in which trees and animals are constructed from bonds of various lengths, above eight dimensions This provides further evidence that for these models the upper critical dimension is equal to eight The proof involves obtaining an infrared bound and showing that a certain “square diagram” is finite at the critical point, and uses an expansion related to the lace expansion for the self-avoiding walk

Fluctuations of extensive functions of quenched random couplings

Abstract: An extensive quantity is a family of functionsΨv of random parameters, indexed by the finite regionsV (subsets of ℤd) over whichΨv are additive up to corrections satisfying the boundary estimate stated below. It is shown that unless the randomness is nonessential, in the sense that limΨv/|V| has a unique value in the absolute (i.e., not just probabilistic) sense, the variance of such a quantity grows as the volume ofV. Of particular interest is the free energy of a system with random couplings; for suchΨv bounds are derived also for the generating functionE(etΨ). In a separate application, variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment.

Triangles in space or building (and analyzing) castles in the air

Abstract: We show that the total combinatorial complexity of all non-convex cells in an arrangement ofn (possibly intersecting) triangles in 3-space isO(n 7/3 logn) and that this bound is almost tight in the worst case. Our bound significantly improves a previous nearly cubic bound of Pach and Sharir. We also present a (nearly) worst-case optimal randomized algorithm for calculating a single cell of the arrangement and an alternative less efficient, but still subcubic algorithm for calculating all non-convex cells, analyze some special cases of the problem where improved bounds (and faster algorithms) can be obtained, and describe applications of our results to translational motion planning for polyhedra in 3-space.