Pullback Operation Research Articles

A pullback operation on a class of currents

A pullback operation on a class of currents

Eliminating Thurston obstructions and controlling dynamics on curves

AbstractEvery Thurston map $f\colon S^2\rightarrow S^2$ on a $2$ -sphere $S^2$ induces a pull-back operation on Jordan curves $\alpha \subset S^2\smallsetminus {P_f}$ , where ${P_f}$ is the postcritical set of f. Here the isotopy class $[f^{-1}(\alpha )]$ (relative to ${P_f}$ ) only depends on the isotopy class $[\alpha ]$ . We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the map f can be seen as a fixed point of the pull-back operation. We show that if a Thurston map f with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying $2$ -sphere and construct a new Thurston map $\widehat f$ for which this obstruction is eliminated. We prove that no other obstruction arises and so $\widehat f$ is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.

Canonical subsheaves of torsionfree semistable sheaves

Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety defined over an algebraically closed field. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ also admits a filtration of subbundles for which each successive quotient is a stable vector bundle whose slope is $\mu (F)$. We also prove that $F$ has a unique maximal reflexive subsheaf $W$ such that $F/W$ is torsionfree and $W$ admits a filtration of subsheaves for which each successive quotient is a stable reflexive sheaf whose slope is $\mu (F)$. We show that these canonical subsheaves behave well with respect to the pullback operation by étale Galois covering maps. Given a separable finite surjective map $\phi \colon Y \longrightarrow X$ between normal projective varieties, we give a criterion for the induced homomorphism of étale fundamental groups $\phi _*\colon \pi ^{\textrm {et}}_{1}(Y) \longrightarrow \pi ^{\textrm {et}}_{1}(X)$ to be surjective. The criterion in question is expressed in terms of the above mentioned unique maximal locally free subsheaf associated to the direct image $\phi _*{\mathcal O}_Y$.

An Improved Model to Calculate Pullback Force of Trenchless Horizontal Directional Drilling Pipeline

In the pullback operation of horizontal directional drilling pipeline crossing, the existing calculation and prediction models of pullback force are relatively simple. Each pullback force maze greatly simplifies the wellbore trajectory and fails to make a detailed analysis of the pipeline stress and external resistance when the pipeline is pulled back in each characteristic trajectory area. The factors considered are relatively simple. Therefore, it is necessary to improve the calculation method of pullback force. This paper aims to establish an improved model, enhancing the earth pressure calculation method of unloading arch and winch calculation method, and carries out an example calculation of the improved formula. Therefore, it is necessary to study the pullback process of horizontal directional drilling pipeline. Firstly, this paper analyzes the calculation method of pullback force in horizontal directional drilling; studies the calculation formula and principle of common pullback force through examples; obtains the advantages, disadvantages, and applicable scope of different formulas; and improves the calculation model of pullback force and step resistance. The numerical simulation of the step crossing process is carried out, and the variation law of local stress and strain of the pipeline and relevant conclusions are obtained. The results show that the estimates of the winch calculation method are close to the actual pullback load of the project. The earth pressure calculation method of the unloading arch and the winch calculation method are improved, and a more stable and reliable calculation formula is obtained, which provides more valuable calculation data for the actual project. In the process of pullback, the pipeline will encounter step resistance after passing through the soft and hard staggered stratum, which will suddenly increase the increment of pipeline pullback force and lead to engineering accidents. If the pullback load suddenly increases and then decreases, it may encounter similar pipeline collision accidents. At the same time, emergency measures can be taken to prevent the crossing accident and ensure the safe pullback of the pipeline.

A note on smooth forms on analytic spaces

We prove that any smooth mapping between reduced analytic spaces induces a natural pullback operation on smooth differential forms.

Automatic correction of the initial rotation angle error improves 3D reconstruction in endoscopic airway optical coherence tomography.

Endoscopic airway optical coherence tomography (OCT) is an advanced imaging modality capable of capturing the internal anatomy and geometry of the airway. Due to fiber-optic catheter bending and friction, the rotation speed of the endoscopic probe is usually non-uniform: at each B-scan image, the initial rotation angle of the probe is easily misaligned with that of the previous slices. During the pullback operation, this initial rotation angle error (IRAE) will be accumulated and will result in distortion and deformation of the reconstructed 3D airway structure. Previous attempts to correct this error were mainly manual corrections, which are time-consuming and suffered from observer variation. In this paper, we present a method to correct the IRAE for anatomically improved visualization of the airway. Our method derived the rotation angular difference of adjacent B-scans by measuring their contour similarity and then tracks the IRAE by formulating its continuous drift as a graph-based problem. The algorithm was tested on a simulated airway contour dataset, and also on experimental datasets acquired by two different long range endoscopic airway OCT platforms. Effective and smooth compensation of the frame-by-frame initial angle difference was achieved. Our method has real-time capability and thus has the potential to improve clinical imaging efficiency.

A dynamic adaptive eigenfracture method for failure in brittle materials

We present a dynamic adaptive eigenfracture algorithm for the study of crack propagation in brittle materials. The original eigenfracture method is an variational approximation of the Griffith-type fracture with convergence as the element size vanishes. To improve its convergence rate, we extend the eigenfracture approach by using the effective energy release rate as the adaptive refinement criteria. Elements are subdivided based on nested refinement method with a pull-back operation. The dynamic adaptive eigenfracture method is validated by comparing to three-point bending tests of concrete and pure-tension tests of the material PMMA. The predicted crack angle in concrete and the simulation results of fracture propagation speed in PMMA show excellent agreement with experimental measurements. Even through a relatively coarse mesh employed in the simulations, the dynamic adaptive eigenfracture method can address the problem of mesh biased crack path.

A Lagrangian formulation for interacting particles on a deformable medium

Many problems in softmatter and membrane biophysics, such as finding the equilibrium configurations of protein clusters on cell-membranes, the highly defect ridden structure of Gag polyproteins in immature HIV capsids, and the unusual fluid-like state of Archaeal viruses can be understood as systems of interacting particles (typically representing proteins or protein capsomers) on deformable surfaces. The coupled interactions between the particles and the underlying elastic medium to which they are constrained pose significant computational challenges. Existing methods often employ expensive constraints to anchor particles to the surface or artificially restrict the particles movement yielding spurious equilibrium states. In this work, a new method is presented where the particles positions are parameterized in the reference configuration. This pull-back operation allows the particles to move freely on the surface without the need of additional constraints. Furthermore, the mapping necessary to describe the particles positions in a discretized setting, e.g., in a finite element mesh, can be done on the reference undeformed configuration at a reduced computational cost. The proposed method is applied to particles moving on manifolds in one and two dimensions to study its convergence characteristics and error when compared to derived analytical and semi-analytical solutions. Finally, the newly developed method is applied to generalized Tammes problems of packing interacting point particles on a deformable shell.

A global perspective to connections on principal 2-bundles

Abstract For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation along Morita equivalences between Lie groupoids. Using this notion, we define connections on principal 2-bundles as Lie 2-algebra-valued 1-forms on the total space Lie groupoid of the 2-bundle, satisfying a condition in complete analogy to connections on ordinary principal bundles. We carefully treat various notions of curvature, and prove a classification result by the non-abelian differential cohomology of Breen–Messing. This provides a consistent, global perspective to higher gauge theory.

ELECTROMAGNETISM IN THE STANDARD MODEL

This paper examines particularly the U (1)×SU(2) of the Standard Model by focusing on the electromagnetic interaction term in the Lagrangian, which appears to be the electromagnetic waves of real photons, not the virtue photons as the force carriers of Maxwell Equations; while E and B are sym- metric in the former, they are not in the latter, for E is a radial force and B is a sideways force. We give a review of the pull-back operation of a tensor for transformation of frames. We also derive Special Relativity from Maxwell Equations and we find that the vector potential A is a built-in mechanism of any electromagnetic wave to prevent it from exerting electric forces; i.e., A is responsible for photons being of no electric charges. Additionally, we show that Special Relativity determines the Lorentz gauge uniquely and that gauge transformations can be presented as frame transformations by an application of cohomology with differential forms as cast in our model of a combined spacetime 4-manifold of the Universe.

Linear connections for reproducing kernels on vector bundles

We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism of the input kernel. The aforementioned correspondence turns out to be a canonical functor between categories of kernels and linear connections. A number of examples of linear connections including the ones associated to classical kernels, homogeneous reproducing kernels and kernels occurring in the dilation theory for completely positive maps are given, together with their covariant derivatives.

Globally Optimal Estimation of Nonrigid Image Distortion

Image alignment in the presence of non-rigid distortions is a challenging task. Typically, this involves estimating the parameters of a dense deformation field that warps a distorted image back to its undistorted template. Generative approaches based on parameter optimization such as Lucas-Kanade can get trapped within local minima. On the other hand, discriminative approaches like nearest-neighbor require a large number of training samples that grows exponentially with respect to the dimension of the parameter space, and polynomially with the desired accuracy 1/?. In this work, we develop a novel data-driven iterative algorithm that combines the best of both generative and discriminative approaches. For this, we introduce the notion of a pull-back operation that enables us to predict the parameters of the test image using training samples that are not in its neighborhood (not ?-close) in the parameter space. We prove that our algorithm converges to the global optimum using a significantly lower number of training samples that grows only logarithmically with the desired accuracy. We analyze the behavior of our algorithm extensively using synthetic data and demonstrate successful results on experiments with complex deformations due to water and clothing.

Maxi-HDD Pull Loads for Nonlevel Grade for Polyethylene Pipe

ASTM F1962, “Standard Guide for Use of Maxi-Horizontal Directional Drilling for Placement of Polyethylene Pipe or Conduit Under Obstacles, Including River Crossings,” provides a procedure for estimating pull loads and stresses on polyethylene pipe primarily as a function of the drilled path and buoyant weight of the pipe in the drilled hole. The original equations initially were developed assuming a level grade or that the horizontal directional drilling (HDD) entry and exit elevations were the same. In practice, however, it is not uncommon that there would be a finite grade or difference in elevation between the entry and exit points of the drilling operation. It is, therefore, of interest to extend the equations and methodology of ASTM F1962 to address the installation of polyethylene pipe for the case of a nonlevel grade. The primary source of drag corresponds to the frictional drag associated with the buoyant weight of the polyethylene pipe, which buoyancy also tends to pull the pipe upwards, opposite gravity, along sloped portions of the bore path. The effects on the pull load depend on the degree to which the drilled hole is free to drain slurry from the lowest access (entry or exit) point during the pullback operation, and the possible deployment of antibuoyancy measures (i.e., water ballast added to pipe interior). This paper addresses the installation of the pipe in the absence of antibuoyancy measures (i.e., no ballast). A subsequent paper will address the case with added ballast. This paper provides a better understanding of the factors influencing the pull loads in the maxi-HDD installation of pipelines.

Computerized planning of prostate cryosurgery with pullback operation

This study is a part of an ongoing project to develop computerized tools for cryosurgery planning. Prostate cryosurgery is frequently performed as a two-stage process in which the first-stage targets are frozen with the cryoprobes at a greater insertion depth, while the second stage of freezing is performed after some cryoprobes are pulled back to new locations, this process being known as the pullback procedure. This paper proposes a method to generate computationally a preferred cryoprobe layout for the pullback operation, using a previously proposed bubble-packing method for cryosurgery planning. For this purpose, additional constraints are imposed to align packed bubbles along the same insertion path between the different stages, and to vary the number of bubbles at each stage. The method is demonstrated on 3D patient models from four individuals, using cryoprobes with a diameter of 1.5 mm and an active length of 15 mm. Results are verified with bioheat transfer simulations, and compared with the non-pullback operations using cryoprobes with longer (25 mm) active lengths. When compared with results of the non-pullback procedure, results with pullback suggest some improvement in freezing in the apex region of the prostate, with a slight increase in overall freezing damage to surrounding healthy tissues.

Computerized planning of prostate cryosurgery with pullback operation

This study is a part of an ongoing project to develop computerized tools for cryosurgery planning. Prostate cryosurgery is frequently performed as a two-stage process in which the first-stage targets are frozen with the cryoprobes at a greater insertion depth, while the second stage of freezing is performed after some cryoprobes are pulled back to new locations, this process being known as the pullback procedure. This paper proposes a method to generate computationally a preferred cryoprobe layout for the pullback operation, using a previously proposed bubble-packing method for cryosurgery planning. For this purpose, additional constraints are imposed to align packed bubbles along the same insertion path between the different stages, and to vary the number of bubbles at each stage. The method is demonstrated on 3D patient models from four individuals, using cryoprobes with a diameter of 1.5 mm and an active length of 15 mm. Results are verified with bioheat transfer simulations, and compared with the non-pullback operations using cryoprobes with longer (25 mm) active lengths. When compared with results of the non-pullback procedure, results with pullback suggest some improvement in freezing in the apex region of the prostate, with a slight increase in overall freezing damage to surrounding healthy tissues.

Vorticity and symplecticity in Lagrangian fluid dynamics

The relationship between potential vorticity (PV) and the symplectic form is explored, for the shallow-water equations governing Lagrangian particle paths. Starting with the symplectic form, the PV is found by the pullback operation to the reference space. At first sight, the encoding of PV in the symplectic form appears to be independent of the particle relabelling symmetry. The analysis is carried a step further in two ways. Using the ‘conservation of symplecticity’ as a starting point, the fluxes of symplecticity arise as differential forms, and a complete pull back of the flux forms leads to a geometric description of PV conservation. Secondly, symmetry methods are used to give a rigorous connection between particle relabelling, symplecticity and PV conservation. Generalizations of these issues to semi-geostrophic flow and three-dimensional Lagrangian fluid flows, and the connection with Ertel’s theorem are also discussed.

Transformation of the solutions of the Maxwell equations on a Lorentzian manifold by means of a certain class of diffeomorphisms

Poincaré invariance of the Maxwell equations in Minkowski space enables one to generate in a usual way new solutions (corresponding to new sources) from a given one by means of the pullback operation. A method is proposed generalizing this to a wider class of diffeomorphisms f:M → M of a Lorentzian manifold (M,g). It results in the simple explicit formulas for obtaining the new field Ff and its source jf from a given pair (F,j). Some applications, e.g., a uniformly rotating or radially pulsating source, are presented.

The Pull-Back Operation in Physical Geodesy And The Behavior of Plumb Lines

The Pull-Back Operation in Physical Geodesy And The Behavior of Plumb Lines