Fibers of tropicalization

We use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the Zariski topology. For subvarieties of tori over fields of generalized power series, points in each tropical fiber are obtained “constructively” using Kedlaya’s transfinite version of Newton’s method.     

On the Jacobian of the Harmonic Moment Map

In this paper we represent harmonic moments in the language of transfinite functions, that is projective limits of polynomials in infinitely many variables. We obtain also an explicit formula for the Jacobian of a generalized harmonic moment map. Submitted: June 29, 2008. Accepted: July 15, 2008.     

复杂外形设计中的C^1,C^2样条插值曲面

根据几种复杂外形设计的特点，木文构造了三角形域上Ｓ１２样条插值曲面，三角形域上的Ｃ２超限插值曲面，矩形参数域上Ｃ２超限插值曲面和任意四边形域上双三次Ｃ１，Ｃ２样条插值曲面，给出了一类有效的边界条件确定方法．同时，算法皆已应用到人体外形描述和飞机外形设计中．     

Transfinite diameter notions in ℂ N and integrals of Vandermonde determinants

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in ? ^N . An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann. 337 (2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in ? ^N . Our results extend to certain weights and measures defined on cones in ? ^N .     

A whitehead theorem for long towers of spaces

We show that one can construct the universalR-homology isomorphismK →E _RX of Bousfield [1] by a transfinite iteration of an elementary homology correction map. This correction map is essentially the same as the one used classically to define Adams spectral sequence. This yields a topological characterization of the class of local spaces as the smallests containingK(A, n)’s and closed under homotopy inverse limit. This research was partially supported by the National Science Foundation, Grant # MCS76-08795, and by the U.S.-Israel Bi-National Science Foundation Grant # 680.     

Interpolation transfinie sur le triangle,le tétraèdre et le pentaèdre. Application à la création de maillages et à la condition de Dirichlet

The Lagrange's interpolation L(ν) of a function ν on a polyhedron K is defined from the values of the function at points p_i of the boundary Γ(K) and ℒ(ν)(p_i) = ν(p_i) at these points. The transfinite interpolation is defined in the same way, but on the whole boundary Γ(K). Three new transfinite interpolations of degree one on the triangle, tetrahedron, and pentahedron are added to the Gordon and Hall's transfinite interpolations on the quadrangle and hexaedron. All these interpolations leave the polynomials of degree ≤1 invariant. The applications to the generation of regular meshes and the possibility of exactly taking into account the Dirichlet boundary condition, show the usefulness of these transfinite interpolations.     

Random walks on finitely structured transfinite networks

A general theory for random walks on transfinite networks whose ranks are arbitrary natural numbers is established herein. In such networks, nodes of higher ranks connect together transfinite networks of lower ranks. The probabilities for transitions through such nodes are obtained as extensions of the Nash-Williams rule for random walks on ordinary infinite networks. The analysis is based on the theory of transfinite electrical networks, but it requires that the transfinite network have a structure that generalizes local-finiteness for ordinary infinite networks. The shorting together of nodes of different ranks are allowed; this complicates transitions through such nodes but provides a considerably more general theory. It is shown that, with respect to any finite set of nodes of any ranks, a transfinite random walk can be represented by an irreducible reversible Makov chain, whose state space is that set of nodes.This work was supported by the National Science Foundation under the grants DMS-9200738 and MIP-9200748.     

Limit spaces and transfinite types

 We give a characterisation of an extension of the Kleene-Kreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types. Received: 24 August 1999 / Published online: 12 July 2002     

Supersymmetry, transfinite neural networks, hyperbolic manifolds, quantum gravity and the Higgs

Connections between the transfinite extension of neural networks and hyperbolic manifolds are pointed out. The found relations are subsequently used to elucidate a certain quantum gravity equation that is reinterpreted via Spinor varieties. Finally it is shown how a definite picture of ^(∞) spacetime may be obtained from a transfinite extension of a certain four dimensional hyperbolic manifold based on Coxeter 120-cell polytop and how this helps in understanding the need of postulating the existence of not only one but three Higgs particles.     

Holomorphic complex differential equations: An elementary proof that the time- p map is holomorphic

We give an elementary proof that, in its domain of definition, the time-p map of a scalar, autonomous holomorphic, complex differential equation, is itself holomorphic. This result is used by Sverdlove [1] when considering limit cycles in complex holomorphic differential equations. However no proof or reference for the result is given in [1]. Although this result must be well established, a proof does not appear to be readily accessible in the reference literature.     

Understanding the Small Object Argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an “algebraic” refinement of the small object argument, cast in terms of Grandis and Tholen’s natural weak factorisation systems, which rectifies each of these three deficiencies.     

Rice and Rice‐Shapiro Theorems for transfinite correction grammars

Hay and, then, Johnson extended the classic Rice and Rice‐Shapiro Theorems for computably enumerable sets, to analogs for all the higher levels in the finite Ershov Hierarchy. The present paper extends their work (with some motivations presented) to analogs in the transfinite Ershov Hierarchy. Some of the transfinite cases are done for all transfinite notations in Kleene's important system of notations, $\mathcal {O}$. Other cases are done for all transfinite notations in a very natural, proper subsystem $\mathcal {O}_{\mathrm{Cantor}}$ of $\mathcal {O}$, where $\mathcal {O}_{\mathrm{Cantor}}$ has at least one notation for each constructive ordinal. In these latter cases it is open as to what happens for the entire set of transfinite notations in $(\mathcal {O} -\mathcal {O}_{\mathrm{Cantor}})$.     

On transfinite extension of asymptotic dimension

We prove that a transfinite extension of the asymptotic dimension asind is trivial. We introduce a transfinite extension of the asymptotic dimension asdim and give an example of a metric proper space which has transfinite infinite dimension.     

The Finite Cell Method for large deformation analysis

The Finite Cell Method (FCM) combines the fictitious domain approach with high-order finite elements and adaptive integration. For linear elastic problems with smooth solution, FCM has been shown to achieve exponential rates of convergence in energy norm, while its structured cell grid guarantees simple mesh generation irrespective of the geometric complexity involved. In this contribution, the FCM idea is combined with standard finite element technology for the solution of geometrically nonlinear problems. In particular, a modified FCM formulation is introduced, which resets the deformed configuration of the fictitious domain to the deformation-free reference configuration after each Newton iteration. Numerical experiments show that this intervention allows for stable nonlinear FCM analysis with very small values of the penalty parameter, while the accuracy of the geometrically nonlinear solution within the physical domain remains unaffected. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On computing Poincaré map by Hénon method

The trajectory of the autonomous chaotic system deviates from the original path leading to a deformation in its attractor while calculating Poincaré map using the method presented by Hénon [Hénon M. Physica D 1982;5:412]. Also, the Poincaré map obtained is found to be the Poincaré map of deformed attractor instead of the original attractor. In order to overcome these drawbacks, this method is slightly modified by introducing an important change in the existing algorithm. Then it is shown that the modified Hénon method calculates the Poincaré map of the original attractor and it does not affect the system dynamics (attractor). The modified method is illustrated by means of the Lorenz and Chua systems.     

Holomorphic complex differential equations: An elementary proof that the time-p map is holomorphic

We give an elementary proof that, in its domain of definition, the time-p map of a scalar, autonomous holomorphic, complex differential equation, is itself holomorphic. This result is used by Sverdlove [1] when considering limit cycles in complex holomorphic differential equations. However no proof or reference for the result is given in [1]. Although this result must be well established, a proof does not appear to be readily accessible in the reference literature.     

Geometry of the Moment Map for Representations of Quivers

We study the moment map associated to the cotangent bundle of the space of representations of a quiver, determining when it is flat, and giving a stratification of its Marsden–Weinstein reductions. In order to do this we determine the possible dimension vectors of simple representations of deformed preprojective algebras. In an appendix we use deformed preprojective algebras to give a simple proof of much of Kac's Theorem on representations of quivers in characteristic zero.     

Finite energy and totally geodesic maps from locally symmetric spaces of finite volume

LetM=G/K be a locally symmetric space of finite volume and rank 2. We show that any map fromM of weighted finite energy in the sense of Saper can be deformed into a finite energy map. As a consequence such maps can be deformed into totally geodesic ones, and a geometric generalization of Margulis' superrigidity theorem is obtained.     

Terrain map resolution

Studies of land combat processes usually deal with interactions between combatants on (and above) specified terrain. When this involves modeling through computer simulation (or even tracking units in a field test over actual terrain), the terrain is usually represented as a digital terrain map for computational purposes. With a digital terrain map, the terrain surface is represented as a simple surface (often a step function) over a two-dimensional domain covered by a tesselation of squares or hexagons. As the size of the underlying units (let us assume squares) in the domain is reduced, the resolution of the map increases, but the computational cost of preparing, storing and using the map increases exponentially. Thus, there is a question of how high the resolution needs to be for a given application. We study this question from the point of view of accuracy of intervisibility calculations as a function of terrain resolution.The approach involves computation of theoretical total intervisibility with varying levels of terrain resolution over simple “pseudo terrains” having assumed mathematical forms. It is found that intervisibility approximations based on digital terrain rapidly approach the theoretical limiting intervisibility as resolution is increased. Relationships between intervisibility surfaces and current military doctrine concerning location of defending forces at the “military crest” are discussed.     

The existence of harmonic maps from Finsler manifolds to Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler man-ifold and study the characterisation of harmonic maps,in the spirit of lshihara.Using heatequation method we show that any map from a compact Finsler manifold M to a com-pact Riemannian manifold with non-positive sectional curvature can be deformed into aharmonic map which has minimum energy in its homotopy class.