topology of pseudoknotted homopolymers we consider the folding of a self avoiding homopolymer on a lattice with saturating hydrogen bond interactions our goal is to numerically evaluate the statistical distribution of the topological genus of pseudoknotted configurations the genus has been recently proposed for classifying pseudoknots and their topological complexity in the context of rna folding we compare our results on the distribution of the genus of pseudoknots with the theoretical predictions of an existing combinatorial model for an infinitely flexible and stretchable homopolymer we thus obtain that steric and geometric constraints considerably limit the topological complexity of pseudoknotted configurations as it occurs for instance in real rna molecules we also analyze the scaling properties at large homopolymer length and the genus distributions above and below the critical temperature between the swollen phase and the compact globule phase both in two and three dimensions

a theoretical and numerical analysis of the measured equation of invariance the measured equation of invariance mei has been previously introduced to efficiently and accurately handle the boundary truncation for finite methods the present authors give a theoretical analysis that provides several important insights into the capabilities of the mei from the numerical study they can explain why the mei works better than one would expect both the theoretical and the numerical analyses demonstrate that the accuracy of the solution is dependent on the electrical size of the geometry as well as the distance between the mesh boundary and the geometry from the analysis the authors propose a new set of metrons that is less sensitive to the excitation than the previously proposed sinusoidal metrons

kp hierarchy for hodge integrals abstract starting from the elsv formula we derive a number of new equations on the generating functions for hodge integrals over the moduli space of complex curves this gives a new simple and uniform treatment of certain known results on hodge integrals like witten s conjecture virasoro constrains faber s g conjecture etc among other results we show that a properly arranged generating function for hodge integrals satisfies the equations of the kp hierarchy

an exact sublinear algorithm for the max flow vertex disjoint paths and communication problems on random graphs this paper describes a randomized algorithm for solving the maximum flow maximum cut problem on connected random graphs the algorithm is very fast it does not look up most vertices in the graph another feature of this algorithm is that it almost surely provides along with an optimal solution a proof of optimality of the solution in addition the algorithm s solution is by construction a collection of vertex disjoint paths which is maximum under a restriction on the graph s density an optimal solution to the np hard communication problem is provided as well that is finding a maximum collection of vertex disjoint paths between sender receiver pairs of terminals the algorithm lends itself to a sublogarithmic parallel and distributed implementation its effectiveness is demonstrated via extensive empirical study

moduli spaces and macromolecules techniques from moduli spaces are applied to biological macromolecules the first main result provides new a priori constraints on protein geometry discovered empirically and confirmed computationally the second main result identifies up to homotopy the natural moduli space of several interacting rna molecules with the riemann moduli space of a surface with several boundary components in each fixed genus applications to rna folding prediction are discussed the mathematical and biological frameworks are surveyed and presented from first principles

choice of the perfectly matched layer boundary condition for frequency domain maxwell s equations solvers we show that the performance of frequency domain solvers of maxwell s equations is greatly affected by the kind of the perfectly matched layer pml used in particular we demonstrate that using the stretched coordinate pml sc pml results in significantly faster convergence speed than using the uniaxial pml upml such a difference in convergence behavior is explained by an analysis of the condition number of the coefficient matrices additionally we develop a diagonal preconditioning scheme that significantly improves solver performance when upml is used

fatgraph models of proteins we introduce a new model of proteins that extends and enhances the traditional graphical representation by associating a combinatorial object called a fatgraph to any protein based upon its intrinsic geometry fatgraphs can easily be stored and manipulated as triples of permutations and these methods are therefore amenable to fast computer implementation applications include the refinement of structural protein classifications and the prediction of geometric and other properties of proteins from their chemical structures 2010 wiley periodicals inc

the decorated teichmuller space of punctured surfaces a principal 5 bundle over the usual teichmuller space of ans times punctured surface is introduced the bundle is mapping class group equivariant and admits an invariant foliation several coordinatizations of the total space of the bundle are developed there is furthermore a natural cell decomposition of the bundle finally we compute the coordinate action of the mapping class group on the total space the total space is found to have a rich equivariant geometric structure we sketch some connections with arithmetic groups diophantine approximations and certain problems in plane euclidean geometry furthermore these investigations lead to an explicit scheme of integration over the moduli spaces and to the construction of a universal teichmuller space which we hope will provide a formalism for understanding some connections between the teichmuller theory the kp hierarchy and the virasoro algebra these latter applications are pursued elsewhere

spaces of rna secondary structures abstract we prove two topological theorems in physical chemistry namely we introduce a hybrid of transverse and tangential measures on train tracks to prove sphericity of various simplicial complexes which arise from certain idealized models of physical chemistry these complexes are at once identified with thurston s space of projective geodesic laminations on an ideal polygon and with the analogue of a compactification described elsewhere of the moduli space of a punctured riemann surface the physical structures we study are various sub collections of the set of all possible planar chemical bonds among the sites of a linear macromolecule each such collection we consider has a natural partial ordering and the geometric realizations of appropriate posets are shown to be topological spheres such a topological statement encodes a wealth of combinatorial data as we briefly discuss in fact our primary motivation here is to study secondary structures on rna this imposes the further restriction that there can be at most one base pair supported at a given site of underlying linear macromolecule and imposing this restriction leads to the class of binary macromolecules our main results here assert the sphericity of certain topological spaces of both arbitrary and binary macromolecules and it is the latter which we hope may have applications to rna our techniques are largely elaborations of elementary topological techniques from techmuller theory and the theory of train tracks

a hybrid finite element method for 3 d scattering using nodal and edge elements a hybrid finite element method for three dimensional scattering is presented and numerical examples shown this approach which couples finite element discretization with the method of moments is particularly well suited for monostatic radar cross section calculations the method is based on a scalar and vector potential formulation of maxwell s equations the use of nodal elements and a highly efficient body of revolution implementation of the method of moments combined nodal and edge elements are employed to accurately model fields around corners and edges a curvature based sampling criterion is derived and shown to ensure accurate answers for highly curved scatterers numerical results and cray computer timings are compared with published results for an edge element code using radiation boundary conditions

coupling finite element and integral equation solutions using decoupled boundary meshes electromagnetic scattering a method is outlined for calculating scattered fields from inhomogeneous penetrable objects using a coupled finite element integral equation solution the finite element equation can efficiently model fields in penetrable and inhomogeneous regions while the integral equation exactly models fields on the finite element mesh boundary and in the exterior region by decoupling the interior finite element and exterior integral equation meshes considerable flexibility is found in both the number of field expansion points as well as their density only the nonmetal portions of the object need be modeled using a finite element expansion exterior perfect conducting surfaces are modeled using an integral equation with a single unknown field since e sub tan is identically zero on these surfaces numerical convergence accuracy and stability at interior resonant frequencies are studied in detail

modeling of electromagnetic absorption scattering problems using hp adaptive finite elements a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for affine elements is presented the proposed method is illustrated and verified with a series of 2d experiments including elements with curved boundaries and nonhomogeneous media

probabilistic analysis of network flow algorithms this paper is concerned with the design and probabilistic analysis of algorithms for the maximum flow problem and capacitated transportation problems these algorithms run in linear time and under certain assumptions about the probability distribution of edge capacities obtain an optimal solution with high probability the design of our algorithms is based on the following general method which we call the mimicking method for solving problems in which some of the input data are deterministic and some are random with a known distribution 1 replace each random variable in the problem by its expectation this gives a deterministic problem instance that has a special form making it particularly easy to solve 2 solve the resulting deterministic problem instance 3 taking into account the actual values of the random variables mimic the solution of the deterministic instance to obtain a near optimal solution to the original problem 4 fine tune this suboptimal solution to obtain an optimal solution we present linear time algorithms to compute a feasible flow in directed and undirected capacitated transportation problem instances the algorithms are shown to be successful with high probability when the probability distribution of the input data satisfies certain assumptions we also consider the maximum flow problem with multiple sources and sinks we show that with high probability the minimum cut isolates either the sources or the sinks and we give a linear time algorithm that produces a maximum flow with high probability

on the formulation of hybrid finite element and boundary integral methods for 3 d scattering this paper studies in detail a variety of formulations for the hybrid finite element and boundary integral fe bi method for three dimensional 3 d electromagnetic scattering by inhomogeneous objects it is shown that the efficiency and accuracy of the fe bi method depends highly on the formulation and discretization of the boundary integral equation bie used a simple analysis of the matrix condition number identifies the efficiency of the different fe bi formulations and an analysis of weighting functions shows that the traditional fe bi formulations cannot produce accurate solutions a new formulation is then proposed and numerical results show that the resulting solution has a good efficiency and accuracy and is completely immune to the problem of interior resonance finally the multilevel fast multipole algorithm mlfma is employed to significantly reduce the memory requirement and computational complexity of the proposed fe bi method

enumeration of rna structures by matrix models we enumerate the number of rna contact structures according to their genus i e the topological character of their pseudoknots by using a recently proposed matrix model formulation for the rna folding problem we obtain exact results for the simple case of an rna molecule with an infinitely flexible backbone in which any arbitrary pair of bases is allowed we analyze the distribution of the genus of pseudoknots as a function of the total number of nucleotides along the phosphate sugar backbone

theoretical modeling of cavity backed patch antennas using a hybrid technique a hybrid technique that combines the method of moments mom and the finite element method fem to analyze cavity backed patch antennas is presented this technique features the use of fem in solving the electromagnetic field distribution in the cavity and the use of mom in solving integral equations outside the cavity the results of mom and fem are combined through the continuity conditions on the boundary of the cavity due to the flexibility of fem complex cavities filled with inhomogeneous media can be analyzed by this technique the results obtained by this hybrid technique are compared to the finite difference time domain fdtd results and good agreement is found

finite type invariants and fatgraphs abstract we define an invariant g m of pairs m g where m is a 3 manifold obtained by surgery on some framed link in the cylinder i is a connected surface with at least one boundary component and g is a fatgraph spine of in effect g is the composition with the n maps of le murakami ohtsuki of the link invariant of andersen mattes reshetikhin computed relative to choices determined by the fatgraph g this provides a basic connection between 2d geometry and 3d quantum topology for each fixed g this invariant is shown to be universal for homology cylinders i e g establishes an isomorphism from an appropriate vector space h of homology cylinders to a certain algebra of jacobi diagrams via composition g g 1 for any pair of fatgraph spines g g of we derive a representation of the ptolemy groupoid i e the combinatorial model for the fundamental path groupoid of teichmuller space as a group of automorphisms of this algebra the space h comes equipped with a geometrically natural product induced by stacking cylinders on top of one another and furthermore supports related operations which arise by gluing a homology handlebody to one end of a cylinder or to another homology handlebody we compute how g interacts with all three operations explicitly in terms of natural products on jacobi diagrams and certain diagrammatic constants our main result gives an explicit extension of the lmo invariant of 3 manifolds to the ptolemy groupoid in terms of these operations and this groupoid extension nearly fits the paradigm of a tqft we finally re derive the morita penner cocycle representing the first johnson homomorphism using a variant generalization of g

an amplitude finite element formulation for electromagnetic radiation and scattering electromagnetic radiation and scattering in an exterior domain within the context of a finite element method has traditionally involved imposing a suitable absorbing boundary condition on the truncation boundary of the numerical domain to inhibit reflection from it in this work based on the wilcox asymptotic expansion of the electric far field we propose an amplitude formulation within the framework of the nodal finite element method whereby the highly oscillatory radial part of the field is separated out a priori so that the standard lagrange interpolation functions that are used have to capture a relatively gently varying function since these elements can be used in the immediate vicinity of the radiator or scatterer with few exceptions which we enumerate it is more effective compared to methods that impose absorbing boundary conditions at the truncation boundary especially for high frequency problems the proposed method is based on the standard galerkin finite element formulation and uses standard lagrange interpolation functions standard gaussian quadrature and the same degrees of freedom as a conventional formulation we show the effectiveness of the proposed formulation on a wide variety of radiation and scattering problems involving both conducting and dielectric bodies and involving both convex and non convex domains with sharp corners

phase accuracy comparisons and improved far field estimates for 3 d edge elements on tetrahedral meshes abstract edge element methods have proved very effective for 3 d electromagnetic computations and are widely used on unstructured meshes however the accuracy of standard edge elements can be criticised because of their low order this paper analyses discrete dispersion relations together with numerical propagation accuracy to determine the effect of tetrahedral shape on the phase accuracy of standard 3 d edge element approximations in comparison to other methods scattering computations for the sphere obtained with edge elements are compared with results obtained with vertex elements and a new formulation of the far field integral approximations for use with edge elements is shown to give improved cross sections over conventional formulations

rigorous auxiliary variable based implementation of a second order abc for the vector fem the finite element method fem is commonly used for electromagnetic radiation and scattering analysis when an infinite free space exterior domain needs to be incorporated into the method a radiation boundary condition must be enforced an approach which has received considerable attention is to employ approximate conditions known as absorbing boundary conditions abcs that preserve the sparsity of the original fem system upon discretization in the case of time harmonic analysis based on the vector wave equation in three dimensions the symmetric spherical bayliss turkel type abcs of first and second orders are well established the second order version is expected to be more accurate however when using the standard curl conforming approach to fem discretization an implementation difficulty is encountered relating to successive derivatives being required of the nonconforming field components this issue is addressed here by introducing a scheme where the nonconforming first order derivatives are projected onto a suitably conforming auxiliary field of which another derivative can then be taken instead additional computational costs are minimal and the scheme retains the symmetry of the standard formulation numerical results demonstrate the superior performance of the rigorously implemented second order abc over its first order counterpart

discontinuous galerkin inspired framework for vector generalized finite element methods discontinuous galerkin methods have been extensively used in simulations of various electromagnetic problems in both time and frequency domains in this paper we exploit features of these methods to provide a framework that can be used to overcome a fundamental bottleneck in using as well as augmenting the capabilities of vector generalized finite element methods vgfem this approach permits the use of vgfem for analysis of piecewise inhomogeneous domains without defining additional constraints on basis functions and allows hybridization with other finite element formulations interior penalty discontinuous galerkin method ip dgm is used to enable communication among partitioned domains we will present several results that demonstrate applications of these hybrid methods as well as their convergence characteristics

steepest descent calculation of rna pseudoknots we enumerate possible topologies of pseudoknots in single stranded rna molecules we use a steepest descent approximation in the large n matrix field theory and a feynman diagram formalism to describe the resulting pseudoknot structure

implementation of the second order abc in the feti dpem method for 3d em problems the electromagnetic dual primal finite element tearing and interconnecting feti dpem method originally equipped with the first order absorbing boundary condition abc is combined with the second order abc implemented using auxiliary variables to enhance the accuracy of electromagnetic scattering and radiation simulations when the second order abc surface is partitioned by domain decomposition it is necessary to enforce an extra continuity condition on the auxiliary variables representing a continuous surface divergence for the electric field at the junctions of the subdomain interfaces and the truncation surface a simple approach to enforce such a continuity is proposed without altering the overall framework of the feti dpem method numerical examples are provided to demonstrate the accuracy improvement due to the use of the second order abc

nodal based finite element modeling of maxwell s equations weak forms are derived for maxwell s equations which are suitable for implementation on conventional c sup 0 elements with scalar bases the governing equations are expressed in terms of general vector and scalar potentials for the electric field intensity vector gauge theory is invoked to close the system and dictates the continuity requirements for the potentials at material interfaces as well as the blend of boundary conditions at exterior boundaries two specific gauges are presented both of which lead to helmholtz weak forms which are parasite free and enjoy simple physically meaningful boundary conditions a general and numerically efficient procedure for enforcing the jump discontinuities on the normal components of vector fields at dielectric interfaces and boundary conditions on curved surfaces is also given

continuous potential maxwell solutions on nodal based finite elements a nodal based finite element approach for computing electric fields in heterogeneous media is presented the primary calculation is formulated in terms of continuous potentials so that no special care is required on element assembly at dielectric interfaces the resulting galerkin weak form matrices exhibit the special helmholtz structure which guarantees the absence of parasitic solutions in driven problems with physically well posed boundary conditions the enhanced sparsity of the helmholtz form mitigates the extra coupling effort associated with introduction of a fourth degree of freedom relative to direct e solution e can be extracted from the computed potentials as a postprocessing step either at nodal positions or element centroids solutions obtained with this approach for several benchmark and practical problems are shown to be parasite free and essentially indistinguishable from previously reported direct e computations

probabilistic analysis of the capacitated transportation problem we consider the capacitated transportation problem defined by sets of supplies ai i i demands bj j j and capacities cij i i j j assuming that the capacities are random variables we prove asymptotic conditions on the supplies and demands which assure that a feasible solution exists almost surely the proof is constructive and supplies an algorithm whose running time is o i j we then apply the results to the maximum flow problem

a study of discretization error in the finite element approximation of wave solutions a dispersion analysis is used to study the errors caused by the spatial discretization of the finite element method for the two dimensional scalar helmholtz equation it is shown that the error can be determined analytically for a uniform mesh of infinite extent numerical results are presented to show the effects of several parameters on the error these parameters are the nodal density the electrical size of the mesh the direction of propagation of the incident wave the type of element and the type of boundary condition

the finite element modeling of three dimensional electromagnetic fields using edge and nodal elements an efficient and accurate finite element method is presented for computing transient as well as time harmonic electromagnetic fields in three dimensional configurations containing arbitrarily inhomogeneous media that may be anisotropic to obtain accurate results with an optimum computational efficiency both consistently linear edge and consistently linear nodal elements are used for approximating the spatial distribution of the field compared with earlier work the formulation is generalized by adding a method for explicitly modeling the normal continuity along interfaces that are free of surface charge in addition the conditions for efficiently solving time harmonic problems using a code designed for solving transient problems are discussed a general and simple method for implementing arbitrary inhomogeneous absorbing boundary conditions for modeling arbitrary incident fields is introduced

influence of particle shape in the additive manufacturing process for ceramics abstract additive manufacturing am of ceramics through a vat photopolymerization vp process is a promising technique due to the high intrinsic resolution and the evanescence of stresses introduced during the layerwise additive manufacturing process compared to the regular vat photopolymerization process the addition of ceramic powder increases physical complexity hence large scale adoption and optimization of am for ceramics requires a thorough understanding of the underlying physics this work focuses on the light scattering introduced through a complex interplay between optical properties of different constituents and its effect on the photopolymerization the effect of the shape of the ceramic particle is investigated for isolated and multiple particle cases using a parametric analysis with regular convex polygons although the analysis of two dimensional isolated particle shapes indicates a preference for one shape over the other this difference vanishes in a multiple scattering situation a three dimensional analysis furthermore shows that a two dimensional approximation of the three dimensional scattering problem is relevant and can be used to further the process understanding of ceramic vp

edge based finite elements and vector abcs applied to 3 d scattering an edge based finite element formulation with vector absorbing boundary conditions is presented for scattering by composite structures having boundaries satisfying impedance and or transition conditions remarkably accurate results are obtained by placing the mesh a small fraction of a wavelength away from the scatterer

enumeration of chord diagrams on many intervals and their non orientable analogs two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here the real segments may contain additional bivalent vertices in one model but not in the other in the former case we record in a generating function the number of fatgraph boundary cycles containing a fixed number of bivalent vertices while in the latter we instead record the number of boundary cycles of each fixed length second order non linear algebraic partial differential equations are derived which are satisfied by these generating functions in each case giving efficient enumerative schemes moreover these generating functions provide multi parameter families of solutions to the kp hierarchy for each model there is furthermore a non orientable analog and each such model likewise has its own associated differential equation the enumerative problems we solve are interpreted in terms of certain polygon gluings as specific applications we discuss models of several interacting rna molecules we also study a matrix integral which computes numbers of chord diagrams in both orientable and non orientable cases in the model with bivalent vertices and the large n limit is computed using techniques of free probability

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

character expansion methods for matrix models of dually weighted graphs we consider generalized one matrix models in which external fields allow control over the coordination numbers on both the original and dual lattices we rederive in a simple fashion a character expansion formula for these models originally due to itzykson and di francesco and then demonstrate how to take the largen limit of this expansion the relationship to the usual matrix model resolvent is elucidated our methods give as a by product an extremely simple derivation of the migdal integral equation describing the largen limit of the itzykson zuber formula we illustrate and check our methods by analysing a number of models solvable by traditional means we then proceed to solve a new model a sum over planar graphys possessing even coordination numbers on both the original and the dual lattice we conclude by formulating the equations for the case of arbitrary sets of even self dual coupling constants this opens the way for studying the deep problems of phase transitions from random to flat lattices january 1995