an efficient algorithm for testing the generic rigidity of graphs in the plane given a structure made up of n sites connected by b bars the problem of recognizing which subsets of sites form rigid units is not a trivial one because of the non local character of rigidity in central force systems even though this is a very old problem of statics no simple algorithms are available for it so the most usual approach has been to solve the elastic equations which is very time consuming for large systems recently an integer algorithm was proposed for this problem in two dimensions using matching methods from graph theory and laman s theorem for two dimensional graphs the method is relatively simple but its time complexity grows as in the worst case and almost as fast on practical cases so that an improvement is highly desirable in this paper we describe a further elaboration of that procedure which relies upon the description of the system as a collection of rigid bodies connected by bars instead of sites connected by bars sets of rigidly connected objects are replaced by a unique body and this is done recursively as more rigid connections between bodies are discovered at larger scales as a consequence of this rescaling transformation our algorithm has much improved average behaviour even when its worst case complexity remains the time complexity of the body bar algorithm is found to scale as for the randomly diluted triangular lattice while the original site bar version scales as for the same problem

comparison of rigidity and connectivity percolation in two dimensions using a recently developed algorithm for generic rigidity of two dimensional graphs we analyze rigidity and connectivity percolation transitions in two dimensions on lattices of linear size up to l 4096 we compare three different universality classes the generic rigidity class the connectivity class and the generic open quotes braced square net close quotes gbsn we analyze the spanning cluster density p sub infinity the backbone density p sub b and the density of dangling ends p sub d in the generic rigidity gr and connectivity cases the load carrying component of the spanning cluster the backbone is ital fractal at p sub c so that the backbone density behaves as b approximately p minus p sub c sup beta sup prime for p gt p sub c we estimate beta sub gr sup prime 0 25 plus minus 0 02 for generic rigidity and beta sub c sup prime 0 467 plus minus 0 007 for the connectivity case we find the correlation length exponents nu sub gr 1 16 plus minus 0 03 for generic rigidity compared to the exact value for connectivity nu sub c 4 3 in contrast the gbsn undergoes a first order rigidity transition with the backbone density being extensive at p sub c and undergoing a jump discontinuity on reducing ital p across the transition we define a model which tunes continuously between the gbsn and gr classes and show that the gr class is typical copyright ital 1999 italmore the american physical society less

floppy modes and the free energy rigidity and connectivity percolation on bethe lattices we show that the negative of the number of ital floppy modes behaves as a ital free energy for both connectivity and rigidity percolation and we illustrate this result using bethe lattices the rigidity transition on bethe lattices is found to be first order at a bond concentration close to that predicted by maxwell constraint counting we calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster and show how a ital specific heat can be defined as the second derivative of the free energy we demonstrate that the bethe lattice solution is equivalent to that of the random bond model where points are joined randomly with equal probability at all length scales to have a given coordination and then subsequently bonds are randomly removed copyright ital 1999 ital the american physical society

generic rigidity in three dimensional bond bending networks much progress can be made in studying the mechanical stability of frameworks when they are treated as generic which lack any special symmetries this is because testing for rigidity becomes topological in nature rather than geometrical generic rigidity synonymous with graph rigidity depends only on the connectivity of the network making it a simpler problem to deal with in principle a complete combinatorial constraint counting characterization of graph rigidity is given by laman s theorem for two dimensions unfortunately there is no known corresponding theorem for three dimensions herein it is proposed that the theorem of laman generalizes to three dimensions for bar joint networks that have no implied hinge joints particular attention is given to bond bending networks having a truss structure with constraints between nearest and next nearest neighbours that are suitable for modelling many covalent network glasses and macromolecules it is shown that implied hinge joints do not exist in bond bending networks based on the proposition that an all subgraph constraint counting characterization of generic rigidity is recovered in three dimensional bar joint networks having no implied hinge joints an efficient combinatorial algorithm is constructed for bond bending networks complete agreement is found with exact calculations involving diagonalization of dynamical matrices for systems up to degrees of freedom

interrogating witnesses for geometric constraint solving classically geometric constraint solvers use graph based methods to decompose systems of geometric constraints these methods have intrinsic limitations which the witness method overcomes a witness is a solution of a variant of the system this paper details the computation of a basis of the vector space of free infinitesimal motions of a typical witness and explains how to use this basis to interrogate the witness for dependence detection the paper shows that the witness method detects all kinds of dependences structural dependences already detectable by graph based methods but also non structural dependences due to known or unknown geometric theorems which are undetectable by graph based methods it also discusses how to decide about the rigidity of a witness and how to decompose it

stepwise positional orientational order and the multicritical multistructural global phase diagram of the s 3 2 ising model from renormalization group theory the spin frac 3 2 ising model with nearest neighbor interactions only is the prototypical system with two different ordering species with concentrations regulated by a chemical potential its global phase diagram obtained in d 3 by renormalization group theory in the migdal kadanoff approximation or equivalently as an exact solution of a d 3 hierarchical lattice with flows subtended by 40 different fixed points presents a very rich structure containing eight different ordered and disordered phases with more than 14 different types of phase diagrams in temperature and chemical potential it exhibits phases with orientational and or positional order it also exhibits quintuple phase transition reentrances universality of critical exponents is conserved across different renormalization group flow basins via redundant fixed points one of the phase diagrams contains a plastic crystal sequence with positional and orientational ordering encountered consecutively as temperature is lowered the global phase diagram also contains double critical points first order and critical lines between two ordered phases critical end points usual and unusual inverted bicritical points tricritical points multiple tetracritical points and zero temperature criticality and bicriticality the four state potts permutation symmetric subspace is contained in this model

riemann hilbert problems for the shapes formed by bodies dissolving melting and eroding in fluid flows the classical stefan problem involves the motion of boundaries during phase transition but this process can be greatly complicated by the presence of a fluid flow here we consider a body undergoing material loss due to either dissolution from molecular diffusion melting from thermodynamic phase change or erosion from fluid mechanical stresses in a fast flowing fluid in each case the task of finding the shape formed by the shrinking body can be posed as a singular riemann hilbert problem a class of exact solutions captures the rounded surfaces formed during dissolution melting as well as the angular features formed during erosion thus unifying these different physical processes under a common framework this study which merges boundary layer theory separated flow theory and riemann hilbert analysis represents a rare instance of an exactly solvable model for high speed fluid flows with free boundaries 2017 wiley periodicals inc

explicit sensor network localization using semidefinite representations and facial reductions the sensor network localization snl problem in embedding dimension r consists of locating the positions of wireless sensors given only the distances between sensors that are within radio range and the positions of a subset of the sensors called anchors current solution techniques relax this problem to a weighted nearest positive semidefinite programming sdp completion problem by using the linear mapping between euclidean distance matrices edm and semidefinite matrices the resulting sdp is solved using primal dual interior point solvers yielding an expensive and inexact solution this relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the sdp cone implying that the slater constraint qualification fails cliques in the graph of the snl problem give rise to this degeneracy in the sdp relaxation in this paper we take advantage of the absence of the slater constraint qualification and derive a technique for the snl problem with exact data that explicitly solves the corresponding rank restricted sdp problem no sdp solvers are used for randomly generated instances we are able to efficiently solve many huge instances of this np hard problem to high accuracy by finding a representation of the minimal face of the sdp cone that contains the sdp matrix representation of the edm the main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the sdp cone that correspond to cliques of the snl problem

a general framework for solving riemann hilbert problems numerically a new numerical framework for the approximation of solutions to matrix valued riemann hilbert problems is developed based on a recent method for the homogeneous painleve ii riemann hilbert problem we demonstrate its effectiveness by computing solutions to other painleve transcendents an implementation in mathematica is made available online

combinatorial models of rigidity and renormalization we first introduce the percolation problems associated with the graph theoretical concepts of k l sparsity and make contact with the physical concepts of ordinary and rigidity percolation we then devise a renormalization transformation for k l percolation problems and investigate its domain of validity in particular we show that it allows an exact solution of k l percolation problems on hierarchical graphs for k l 2k we introduce and solve by renormalization such a model which has the interesting feature of showing both ordinary percolation and rigidity percolation phase transitions depending on the values of the parameters

method of riemann surfaces in the study of supercavitating flow around two hydrofoils in a channel abstract in the framework of the tulin model of supercavitating flow the problem of reconstructing the free surface of a channel and the shapes of the cavities behind two hydrofoils placed in an ideal fluid is solved in closed form the conformal map that transforms a parametric plane with three cuts along the real axis into the triple connected flow domain is found by quadratures the use of the theory of riemann surfaces the schottky doubles enables the non linear model problem to be reduced to two separate riemann hilbert problems on a hyperelliptic surface of genus two the solution to the first problem is a rational function with certain zeros and poles on a riemann surface the second problem is solved in terms of singular integrals with the weierstrass kernel the essential singularities of the solution at the infinite points of the surface due to a pole of the kernel are removed by solving a real analogue of the jacobi inversion problem on the surface the unknown parameters of the conformal map are recovered from a system of certain algebraic and transcendental equations

motion of a yawed supercavitating wedge beneath a free surface the problem that is studied concerns a yawed wedge under a free surface moving at a uniform speed the model involves a trailing cavity whose boundary is a dividing streamline through the vertex of the wedge the cavity closure mechanism is described according to the tulin terent ev single spiral vortex model a closed form solution to the governing nonlinear boundary value problem is found by the method of conformal mappings the doubly connected flow domain is treated as the image by this map of the exterior of two slits in a parametric plane the mapping function is constructed through the solution to two boundary value problems of the theory of analytic functions the hilbert problem for two slits in a plane and the riemann hilbert problem on an elliptic surface numerical results for the shape of the cavity and the free surface the yaw angle the drag and lift coefficients and the circulation are reported

single spiral vortex model for a cavitating elastic curvilinear foil a two dimensional nonlinear inverse fluid structure interaction problem for a curvilinear elastic hydrofoil is considered a cavity formed behind the foil is modeled according to the single spiral vortex model by tulin the fluid structure problem is decoupled by the method of successive approximations for the cavitation problem the foil is modeled as a polygon the method of conformal mappings and the riemann hilbert problem are employed at this stage the classical detachment mechanism for a smooth arc is satisfied for the polygon approximately the deformation of the smooth foil is described by the governing equations of the thin shell theory with the clamped clamped boundary conditions the loading acting on the middle surface of the foil is prescribed as the difference between the fluid and vapor pressure computed in the fluid problem numerical results include those for the cavity profile the drag coefficient the pressure distribution the speed and the displacements of the elastic foil

a generalized traction integral equation for stokes flow with applications to near wall particle mobility and viscous erosion abstract a double layer integral equation for the surface tractions on a body moving in a viscous fluid is derived which allows for the incorporation of a background flow and or the presence of a plane wall the lorentz reciprocal theorem is used to link the surface tractions on the body to integrals involving the background velocity and stress fields on an imaginary bounding sphere or hemisphere for wall bounded flows the derivation requires the velocity and stress fields associated with numerous fundamental singularity solutions which we provide for free space and wall bounded domains two sample applications of the method are discussed we study the tractions on an ellipsoid moving near a plane wall which provides a more detailed understanding of the well studied glancing and reversing trajectories in the context of particle sedimentation and the erosion of bodies by a viscous flow in which the surface is ablated at a rate proportional to the local viscous shear stress simulations and analytical estimates suggest that a spherical body in a uniform flow first reduces nearly but not exactly to the drag minimizing profile and then vanishes in finite time the shape dynamics of an eroding body in a shear flow and near a wall are also investigated stagnation points on the body surface lead generically to the formation of cusps whose number depends on the flow configuration and or the presence of nearby boundaries

conditions for unique graph realizations the graph realization problem is that of computing the relative locations of a set of vertices placed in euclidean space relying only upon some set of inter vertex distance measurements this paper is concerned with the closely related problem of determining whether or not a graph has a unique realization both these problems are np hard but the proofs rely upon special combinations of edge lengths if one assumes the vertex locations are unrelated then the uniqueness question can be approached from a purely graph theoretic angle that ignores edge lengths this paper identifies three necessary graph theoretic conditions for a graph to have a unique realization in any dimension efficient sequential and nc algorithms are presented for each condition although these algorithms have very different flavors in different dimensions

a fast chebyshev method for simulating flexible wing propulsion abstract we develop a highly efficient numerical method to simulate small amplitude flapping propulsion by a flexible wing in a nearly inviscid fluid we allow the wing s elastic modulus and mass density to vary arbitrarily with an eye towards optimizing these distributions for propulsive performance the method to determine the wing kinematics is based on chebyshev collocation of the 1d beam equation as coupled to the surrounding 2d fluid flow through small amplitude analysis of the euler equations with trailing edge vortex shedding the complete hydrodynamics can be represented by a nonlocal operator that acts on the 1d wing kinematics a class of semi analytical solutions permits fast evaluation of this operator with o n log n operations where n is the number of collocation points on the wing this is in contrast to the minimum o n 2 cost of a direct 2d fluid solver the coupled wing fluid problem is thus recast as a pde with nonlocal operator which we solve using a preconditioned iterative method these techniques yield a solver of near optimal complexity o n log n allowing one to rapidly search the infinite dimensional parameter space of all possible material distributions and even perform optimization over this space

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

an algorithm for two dimensional rigidity percolation many important macroscopic properties of materials depend upon the number of microscopic degrees of freedom the task of counting the number of such degrees of freedom can be computationally very expensive we describe a new approach for this calculation which is appropriate for two dimensional glass like networks building upon recent work in graph rigidity this purely combinatorial algorithm is formulated in terms of a simplepebble game it has allowed for the first studies of the rigidity transition ingenericnetworks which are models of amorphous materials and glasses in the context of generic rigidity percolation we show how to calculate the number of internal degrees of freedom identify all rigid clusters and locate the overconstrained regions for a network ofnsites the pebble game has a worst case performance ofo n2 in our applications its performance scaled asn1 15at the rigidity transition while away from the transition region it grew linearly

global stability and decay for the classical stefan problem the classical one phase stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water this is accom plished by solving the heat equation on a time dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free boundary we establish a global in time stability result for nearly spherical geometries and small tempera tures using a novel hybrid methodology which combines energy estimates decay estimates and hopf type inequalities 1 1 the problem formulation we consider the problem of global existence and asymptotic stability of classical solutions to the classical stefan problem describing the evolving free boundary between the liquid and solid phases the temperature of the liquid p t x and the a priori unknown moving phase boundary t must satisfy the following system of equations

nonlinear steepest descent and numerical solution of riemann hilbert problems the effective and efficient numerical solution of riemann hilbert problems has been demonstrated in recent work with the aid of ideas from the method of nonlinear steepest descent for riemann hilbert problems the resulting numerical methods have been shown in practice to retain accuracy as values of certain parameters become arbitrarily large remarkably this numerical approach does not require knowledge of local parametrices rather the deformed contour is scaled near stationary points at a specific rate the primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved to do so we first construct a general theoretical framework for the numerical solution of riemann hilbert problems second we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes in particular we prove sufficient conditions for numerical methods to retain accuracy finally we compute solutions to the homogeneous painleve ii equation and the modified korteweg de vries equation to explicitly demonstrate the practical validity of the theory 2014 wiley periodicals inc

compact optimal quadratic spline collocation methods for the helmholtz equation quadratic spline collocation methods are formulated for the numerical solution of the helmholtz equation in the unit square subject to non homogeneous dirichlet neumann and mixed boundary conditions and also periodic boundary conditions the methods are constructed so that they are a of optimal accuracy and b compact that is the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems using fast fourier transforms the computational cost of such an algorithm is o n 2logn on an nxn uniform partition of the unit square the results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena in particular it is shown that the methods are fourth order accurate at the nodes of the partition

double cavity flow past a wedge a mathematical model of supercavitating flow past a wedge with sides of arbitrary length is proposed the flow branches at a point on the lower side of the wedge at the vertex of the wedge and at the ends of the wedge the flow breaks away forming a nose bubble and a trailing cavity the closure mechanism is described by the tulin single spiral vortex model the flow domain is mapped into a parametric plane cut along a unit segment the conformal mapping function is reconstructed through the exact solution of two riemann hilbert problems on a genus zero riemann surface to complete the solution one needs to determine five real parameters from a certain system of transcendental equations numerical results are presented for the case when a wedge can rotate about the vertex in the flow domain in this case the flow branches at the vertex and the number of the parameters to be determined is 3

generic rigidity percolation in two dimensions we study rigidity percolation for random central force networks on the bondand site diluted generic triangular lattice here each site location is randomly displaced from the perfect lattice removing any special symmetries using the pebble game algorithm the total number of floppy modes are counted exactly and exhibit a cusp singularity in the second derivative at the transition from a rigid to a floppy structure the critical thresholds for bond and site dilution are found to be 0 66020 ifmmode pm else textpm fi 0 0003 and 0 69755 ifmmode pm else textpm fi 0 0003 respectively the network is decomposed into unique rigid clusters and we apply the usual percolation scaling theory from finite size scaling we find that the generic rigidity percolation transition is second order but in a different universality class from connectivity percolation with the exponents ensuremath alpha 0 48 ifmmode pm else textpm fi 0 05 ensuremath beta 0 175 ifmmode pm else textpm fi 0 02 and ensuremath nu 1 21 ifmmode pm else textpm fi 0 06 the fractal dimension of the spanning rigid clusters and the spanning stressed regions at the critical threshold are found to be mathit d mathit f 1 86 ifmmode pm else textpm fi 0 02 and mathit d mathit bb 1 80 ifmmode pm else textpm fi 0 03 respectively textcopyright 1996 the american physical society

exact reconstruction of euclidean distance geometry problem using low rank matrix completion the euclidean distance geometry problem arises in a wide variety of applications from determining molecular conformations in computational chemistry to localization in sensor networks when the distance information is incomplete the problem can be formulated as a nuclear norm minimization problem in this paper this minimization program is recast as a matrix completion problem of a low rank r gram matrix with respect to a suitable basis the well known restricted isometry property cannot be satisfied in this scenario instead a dual basis approach is introduced to theoretically analyze the reconstruction problem if the gram matrix satisfies certain coherence conditions with parameter nu the main result shows that the underlying configuration of n points can be recovered with very high probability from o nr nu log 2 n uniformly random samples computationally simple and fast algorithms are designed to solve the euclidean distance geometry problem numerical tests on different 3 d data and protein molecules validate the effectiveness and efficiency of the proposed algorithms

ideal free streamline flow over a curved obstacle in the classical two dimensional model the description of helmholtz s kirchhoff flow is a problem of complex analysis which can be solved analytically only for a few simple bodies or polygonal contours using the schwarz christoffel map this paper presents a practical method for computing flows over arbitrary obstacles whose boundaries may be piecewise smooth curves while the impinging flow may be an unbounded flow a jet or a semi infinite stream i e the ocean

minimal and redundant bearing rigidity conditions and applications this article studies the notions of minimal and 1 redundant bearing rigidity a necessary and sufficient condition for the numbers of edges in a graph of n n geq 3 vertices to be minimally bearing rigid mbr in mathbb r d d geq 2 is proposed if 3 leq n leq d 1 a graph is mbr if and only if it is the cycle graph in case n d 1 a generically bearing rigid graph is minimal if it has precisely 1 lfloor frac n 2 d 1 rfloor times d text mod n 2 d 1 text sgn text mod n 2 d 1 edges then several conditions for 1 redundant bearing rigidity are derived based on the mathematical conditions some algorithms for generating generically minimally and 1 redundantly bearing rigid graphs are given furthermore two applications of the new notions to optimal network design and formation merging are also reported

an efficient undeniable group oriented signature scheme three main methods with favorable security and efficiency can be considered in the design of cryptosystems these methods are integer factorization systems of which rsa is the best known example discrete logarithm systems such as dsa and elliptic curve cryptosystems ecc math comput 48 1987 203 advances in cryptology crypto 85 springer verlag 1986 p 417 the elliptic curve cryptosystem has been applied in this study because of its low computational cost and short key size which reduces the overheads in a limited hardware environment thus the elliptic curve constitutes a suitable environment for the cryptosystems this study improves the undeniable signature scheme proceedings of the eleventh national conference on information security 2001 p 331 by reducing its time complexity such an improvement makes the proposed scheme more efficient and secure