Triangulating Smooth Submanifolds with Light Scaffolding

We propose an algorithm to sample and mesh a k-submanifold M{\mathcal{M}} of positive reach embedded in \mathbbR^d{\mathbb{R}^{d}} . The algorithm first constructs a crude sample of M{\mathcal{M}} . It then refines the sample according to a prescribed parameter e{\varepsilon} , and builds a mesh that approximates M{\mathcal{M}} . Differently from most algorithms that have been developed for meshing surfaces of \mathbbR ³{\mathbb{R} ^3} , the refinement phase does not rely on a subdivision of \mathbbR ^d{\mathbb{R} ^d} (such as a grid or a triangulation of the sample points) since the size of such scaffoldings depends exponentially on the ambient dimension d. Instead, we only compute local stars consisting of k-dimensional simplices around each sample point. By refining the sample, we can ensure that all stars become coherent leading to a k-dimensional triangulated manifold [^(M)]{\hat{\mathcal{M}}} . The algorithm uses only simple numerical operations. We show that the size of the sample is O(e^-k){O(\varepsilon ^{-k})} and that [^(M)]{\hat{\mathcal{M}}} is a good triangulation of M{\mathcal{M}} . More specifically, we show that M{\mathcal{M}} and [^(M)]{\hat{\mathcal{M}}} are isotopic, that their Hausdorff distance is O(e²){O(\varepsilon ^{2})} and that the maximum angle between their tangent bundles is O(e){O(\varepsilon )} . The asymptotic complexity of the algorithm is T(e) = O(e^-k2-k){T(\varepsilon) = O(\varepsilon ^{-k^2-k})} (for fixed M, d{\mathcal{M}, d} and k).     

Singularly perturbed 1D Cahn–Hilliard equation revisited

We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors ${\{{\mathcal M}_\epsilon\}, \epsilon\geq 0}We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors {M_e}, e 3 0{\{{\mathcal M}_\epsilon\}, \epsilon\geq 0} being the perturbation parameter, such that the map e? M_e{\epsilon \mapsto {\mathcal M}_\epsilon} is H?lder continuous. Besides, the continuity at e = 0{\epsilon=0} is obtained with respect to a metric independent of e.{\epsilon.} Continuity properties of global attractors and inertial manifolds are also examined.     

On exact solutions for quantum particles with spin <Emphasis Type="Italic">S</Emphasis> = 0, 1/2, 1 and de Sitter event horizon

Exact wave solutions for particles with spin 0, 1/2 and 1 in the static coordinates of the de Sitter space–time model are examined in detail. Firstly, for scalar particle, two pairs of linearly independent solutions are specified explicitly: running and standing waves. A known algorithm for calculation of the reflection coefficient R_ej{R_{\epsilon j}} on the background of the de Sitter space–time model is analyzed. It is shown that the determination of R_ej{R_{\epsilon j}} requires an additional constrain on quantum numbers er/ (^h/_2p) c >> j{\epsilon \rho / \hbar c \gg j}, where ρ is a curvature radius. When taken into account of this condition, the R_ej{R_{\epsilon j}} vanishes identically. It is claimed that the calculation of the reflection coefficient R_ej{R_{\epsilon j}} is not required at all because there is no barrier in an effective potential curve on the background of the de Sitter space–time. The same conclusion holds for arbitrary particles with higher spins, it is demonstrated explicitly with the help of exact solutions for electromagnetic and Dirac fields.     

Desingularization of singular Riemannian foliation

Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\mathcal{F}}. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]_n/[^(F)]_n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.     

Global minimization using an Augmented Lagrangian method with variable lower-level constraints

A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the ${\varepsilon_{k}}A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the e_k{\varepsilon_{k}} -global minimization of the Augmented Lagrangian with simple constraints, where e_k ? e{\varepsilon_k \to \varepsilon} . Global convergence to an e{\varepsilon} -global minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.     

A new look at the moduli space of stable hyperelliptic curves

We carry out the log minimal model program for the moduli space ${\bar H_g}We carry out the log minimal model program for the moduli space [`(H)]<sub>g</sub>{\bar H_g} of stable hyperelliptic curves and show that certain log canonical models of [`(H)]_g{\bar H_g} are isomorphic to the proper transform of [`(H)]<sub>g</sub>{\bar H_g} in the corresponding log canonical models of [`(M)]_g{\bar M_g}. For g = 3, we retrieve the compact moduli space [`(B)]<sub>8</sub>{\bar B_{8}} of binary forms as a log canonical model, and obtain a decomposition of the natural map [`(H)]₃ ? [`(B)]<sub>8</sub>{\bar H_3 \to \bar B_{8}} into successive divisorial contractions of the boundary divisors. As a byproduct, we also obtain an isomorphism of [`(B)]₈{\bar B_8} with the GIT quotient of the Chow variety of bicanonically embedded hyperelliptic curves of genus three.     

Error estimate of the P 1 nonconforming finite element method for the penalized unsteady Navier-Stokes equations

We consider a finite element method for the penalty formulation of the time dependent Navier-Stokes equations. Usually the improper choice of the finite element space will lead that the error estimate (inversely) depends on the penalty parameter e{\epsilon}. We use the classical P ₁ nonconforming finite element space for the spatial discretization. Optimal e{\epsilon}-uniform error estimations for both velocity and pressure are obtained.     

Pursuit–Evasion Games with incomplete information in discrete time

Pursuit–Evasion Games (in discrete time) are stochastic games with nonnegative daily payoffs, with the final payoff being the cumulative sum of payoffs during the game. We show that such games admit a value even in the presence of incomplete information and that this value is uniform, i.e. there are e{\epsilon}-optimal strategies for both players that are e{\epsilon}-optimal in any long enough prefix of the game. We give an example to demonstrate that nonnegativity is essential and expand the results to Leavable Games.     

The singular limit dynamics of the phase-field equations

We consider the phase-field equations subject to Dirichlet boundary conditions. We construct families of exponential attractors and inertial manifolds which are continuous at any parameter of perturbation ${\epsilon >0 }${\epsilon >0 } including the singular limit case e = 0{\epsilon=0}. Besides, the continuity at e = 0{\epsilon=0} is obtained with respect to a metric independent of e{\epsilon}. Continuity properties of the global attractors are also examined.     

Minimal rotational hypersurfaces in Minkowski (<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">β</Emphasis>)-space

In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal rotational hypersurfaces generated by plane curves rotating around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski (α, β)-space (\mathbbVⁿ⁺¹,[(F_b)\tilde]){(\mathbb{V}^{n+1},\tilde{F_b})} , where \mathbbVⁿ⁺¹{\mathbb{V}^{n+1}} is an (n+1)-dimensional real vector space, [(F_b)\tilde]=[(a)\tilde]f([(b)\tilde]/[(a)\tilde]), [(a)\tilde]{\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}} is the Euclidean metric, [(b)\tilde]{\tilde{\beta}} is a one form of constant length b:=||[(b)\tilde]||_[(a)\tilde], [(b)\tilde]^\sharp{b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}} is the dual vector of [(b)\tilde]{\tilde{\beta}} with respect to [(a)\tilde]{\tilde{\alpha}} . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski Randers 3-space (\mathbbV³,[(a)\tilde]+[(b)\tilde]){(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})} .     

Goodness in the enumeration and singleton degrees

We investigate and extend the notion of a good approximation with respect to the enumeration ${({\mathcal D}_{\rm e})}We investigate and extend the notion of a good approximation with respect to the enumeration (D_e){({\mathcal D}_{\rm e})} and singleton (D_s){({\mathcal D}_{\rm s})} degrees. We refine two results by Griffith, on the inversion of the jump of sets with a good approximation, and we consider the relation between the double jump and index sets, in the context of enumeration reducibility. We study partial order embeddings i_s{\iota_s} and [^(i)]_s{\hat{\iota}_s} of, respectively, D_e{{\mathcal D}_{\rm e}} and D_T{{\mathcal D}_{\rm T}} (the Turing degrees) into D_s{{\mathcal D}_{\rm s}} , and we show that the image of D_T{{\mathcal D}_{\rm T}} under [^(i)]_s{\hat{\iota}_s} is precisely the class of retraceable singleton degrees. We define the notion of a good enumeration, or singleton, degree to be the property of containing the set of good stages of some good approximation, and we show that i_s{\iota_s} preserves the latter, as also other naturally arising properties such as that of totality or of being G⁰_n{\Gamma^0_n} , for G ? {S,P,D}{\Gamma \in \{\Sigma,\Pi,\Delta\}} and n > 0. We prove that the good enumeration and singleton degrees are immune and that the good S⁰₂{\Sigma^0_2} singleton degrees are hyperimmune. Finally we show that, for singleton degrees a _s < b _s such that b _s is good, any countable partial order can be embedded in the interval (a _s, b _s).     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces [`(T)]<sub>g,n</sub>{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces T<sub>g,n</sub>{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike T<sub>g,n</sub>{\mathcal{T}_{g,n}}, the space [`(T)]_g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.     

Gaussian mean curvature flow

In the Euclidean Space \mathbb Rⁿ⁺¹{\mathbb {R}^{n+1}} with a density e^{e\frac12 n m2 |x|2}, (e = ±1){e^{\varepsilon \frac12 n \mu^2 |x|^2},} {(\varepsilon =\pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ \varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere) depending on the relation between the bound of its principal curvatures and μ.     

Closed-Range Composition Operators on {\mathbb{A}^2} and the Bloch Space

For any analytic self-map j{\varphi} of {z : |z| <  1} we give four separate conditions, each of which is necessary and sufficient for the composition operator C_j{C_{\varphi}} to be closed-range on the Bloch space B{\mathcal{B}} . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if C_j{C_{\varphi}} is closed-range on the Bergman space \mathbbA²{\mathbb{A}^2} , then it is closed-range on B{\mathcal{B}} , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem.     

Statistical ensemble error bounds for homogenized microheterogeneous solids

Typically, in order to characterize the homogenized effective macroscopic response of new materials possessing random heterogeneous microstructure, a relation between averages $ {\left\langle {\user2{ \sigma }} \right\rangle }_{\Omega } = \mathbb{E}^{*} :{\left\langle {\user2{ \epsilon }} \right\rangle }_{\Omega } $ {\left\langle {\user2{ \sigma }} \right\rangle }_{\Omega } = \mathbb{E}^{*} :{\left\langle {\user2{ \epsilon }} \right\rangle }_{\Omega } is sought, where $ {\left\langle \cdot \right\rangle }_{\Omega } {\mathop = \limits^{{\text{def}}} }\tfrac{1} {{|\Omega |}}{\int_\Omega { \cdot \,d} }\Omega , $ {\left\langle \cdot \right\rangle }_{\Omega } {\mathop = \limits^{{\text{def}}} }\tfrac{1} {{|\Omega |}}{\int_\Omega { \cdot \,d} }\Omega , and where \varvecs\varvec{\sigma} and \varvece\varvec{\epsilon} are the stress and strain tensor fields within a statistically representative volume element (SRVE) of volume |$ \mathbb{E}^{*} , $ \mathbb{E}^{*} , is known as the effective property, and is the elasticity tensor used in usual macroscale analyses. In order to generate homogenized responses computationally, a series of detailed boundary value representations resolving the heterogeneous microstructure, posed over the SRVEs domain, must be solved. This requires an enormous numerical effort that can overwhelm most computational facilities. A natural way of generating an approximation to the SRVEs response is by first computing the response of smaller (subrepresentative) samples, each with a different random realization of the microstructural type under investigation, and then to ensemble average the results afterwards. Compared to a direct simulation of an SRVE, testing many small samples is a computationally inexpensive process since the number of floating point operations is greatly reduced, as well as the fact that the samples responses can be computed trivially in parallel. However, there is an inherent error in this process. Clearly the populations ensemble average is not the SRVEs response. However, as shown in this work, the moments on the distribution of the population can be used to generate rigorous upper and lower error bounds on the quality of the ensemble-generated response. Two-sided bounds are given on the SRVE response in terms of the ensemble average, its standard deviation and its skewness.     

Fully developed laminar slip and no-slip flow in rough microtubes

The effect of surface roughness on developed laminar flow in microtubes is investigated. The tube boundary is defined by r=R[1+e sin( lq)]{r=R\left[{1+\varepsilon\, {\rm sin}\left( {\lambda \theta }\right)}\right]}, with R representing the reference radius and e{\varepsilon} and λ the roughness parameters. The momentum equation is solved using Fourier–Galerkin–Tau method with slip at the boundary. A novel semi-analytical method is developed to predict friction factor and pressure drop in corrugated rough microtubes for continuum flow and slip flow that are not restricted to small values of el{\varepsilon \lambda } . The analytical solution collapses onto the perturbation solution ofDuan and Muzychka (J. Fluids Eng., 130:031102, 2008) for small enough values of el{\varepsilon \lambda } .     

Relative universality and universality obtained by adding constants

A variety ${\mathbb{V}}${\mathbb{V}} is var-relatively universal if it contains a subvariety \mathbbW{\mathbb{W}} such that the class of all homomorphisms that do not factorize through any algebra in \mathbbW{\mathbb{W}} is algebraically universal. And \mathbbV{\mathbb{V}} has an algebraically universal α-expansion a\mathbbV{\alpha\mathbb{V}} if adding α nullary operations to all algebras in \mathbbV{\mathbb{V}} gives rise to a class a\mathbbV{\alpha\mathbb{V}} of algebras that is algebraically universal. The first two authors have conjectured that any varrelative universal variety \mathbbV{\mathbb{V}} has an algebraically universal α-expansion a\mathbbV{\alpha\mathbb{V}} . This note contains a more general result that proves this conjecture.     

Length spectra and the Teichmüller metric for surfaces with boundary

We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call “ε ₀-relative e{\epsilon}-thick parts”, and whose definition depends on the choice of some positive constants ε ₀ and e{\epsilon}. Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families of arcs.     

Balance in Stable Categories

We study when the stable category ${\mathcal A}/\langle{\mathcal T}\rangleWe study when the stable category A/áT?{\mathcal A}/\langle{\mathcal T}\rangle of an abelian category A{\mathcal A} modulo a full additive subcategory T{\mathcal T} is balanced and, in case T{\mathcal T} is functorially finite in A{\mathcal A}, we study a weak version of balance for A/áT?{\mathcal A}/\langle{\mathcal T}\rangle. Precise necessary and sufficient conditions are given in case T{\mathcal T} is either a Serre class or a class consisting of projective objects. The results in this second case apply very neatly to (generalizations of) hereditary abelian categories.