摘要 :
We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading.We construct a one-to-one correspondence Γ from the set of equivalent well-posed two-point boundary c...
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We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading.We construct a one-to-one correspondence Γ from the set of equivalent well-posed two-point boundary conditions to gl(4, C).Using Γ, we derive eigenconditions for the integral operator KM for each well-posed two-point boundary condition represented by M ∈ gl(4, 8, C).Special features of our eigenconditions include; (1) they isolate the effect of the boundary condition M on Spec KM, (2) they connect Spec KM to Spec Kl,α,k whose structure has been well understood.Using our eigenconditions, we show that, for each nonzero real λ 6∈ Spec Kl,α,k, there exists a real well-posed boundary condition M such that λ ∈ Spec KM.This in particular shows that the integral operators KM arising from well-posed boundary conditions, may not be positive nor contractive in general, as opposed to Kl,α,k.
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We present a mathematical theory of the two-dimensional offset curves from the viewpoint of medial axis transform. We explore the local geometry of the offset curve in relation with the medial axis transform, culminating in the cl...
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We present a mathematical theory of the two-dimensional offset curves from the viewpoint of medial axis transform. We explore the local geometry of the offset curve in relation with the medial axis transform, culminating in the classification of points on the offset curve. We then study the domain decomposition from the viewpoint of offsets, and in particular introduce the concept of monotonic fundamental domain as a device for detecting the correct topology of offsets as well as for stable numerical computation. The monotonic fundamental domains are joined by peaks or valleys of the medial axis transform, or by what we call the critical horizonal section whose algebro-geometric properties are rigorously treated as well.
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摘要 :
We present a mathematical theory of the two-dimensional offset curves from the viewpoint of medial axis transform. We explore the local geometry of the offset curve in relation with the medial axis transform, culminating in the cl...
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We present a mathematical theory of the two-dimensional offset curves from the viewpoint of medial axis transform. We explore the local geometry of the offset curve in relation with the medial axis transform, culminating in the classification of points on the offset curve. We then study the domain decomposition from the viewpoint of offsets, and in particular introduce the concept of monotonic fundamental domain as a device for detecting the correct topology of offsets as well as for stable numerical computation. The monotonic fundamental domains are joined by peaks or valleys of the medial axis transform, or by what we call the critical horizonal section whose algebro-geometric properties are rigorously treated as well.
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Characteristic equations for the whole class of integral operators arising from arbitrary well-posed two-point boundary value problems of finite beam deflection resting on elastic foundation are obtained in terms of $ 4 \ imes 4 $...
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Characteristic equations for the whole class of integral operators arising from arbitrary well-posed two-point boundary value problems of finite beam deflection resting on elastic foundation are obtained in terms of $ 4 \ imes 4 $ matrices in block-diagonal form with explicit $ 2 \ imes 2 $ blocks.
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We consider the static deflection of an infinite beam resting on a nonlinear and non-uniform elastic foundation. The governing equation is a fourth-order nonlinear ordinary differential equation. Using the Green's function for the...
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We consider the static deflection of an infinite beam resting on a nonlinear and non-uniform elastic foundation. The governing equation is a fourth-order nonlinear ordinary differential equation. Using the Green's function for the well-analyzed linear version of the equation, we formulate a new integral equation which is equivalent to the original nonlinear equation. We find a function space on which the corresponding nonlinear integral operator is a contraction, and prove the existence and the uniqueness of the deflection in this function space by using Banach fixed point theorem. 2010 Mathematics Subject Classification: 34A12; 34A34; 45G10; 74K10.
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Given a real valued function f (X, Y), a box region B_0 ⊆R~2 and ε > 0, we want to compute an ε-isotopic polygonal approximation to the restriction of the curves =f~(-1)(0) = {p ∈ K~2 :f (p) = 0) to B_0. We focus on subdivis...
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Given a real valued function f (X, Y), a box region B_0 ⊆R~2 and ε > 0, we want to compute an ε-isotopic polygonal approximation to the restriction of the curves =f~(-1)(0) = {p ∈ K~2 :f (p) = 0) to B_0. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga & Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities.
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We provide a complete proof that there are no eigenvalues of the integral operator $\mathcal{K}_l$ outside the interval $(0,1/k)$. $\mathcal{K}_l$ arises naturally from the deflection problem of a beam with length $2l$ resting hor...
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We provide a complete proof that there are no eigenvalues of the integral operator $\mathcal{K}_l$ outside the interval $(0,1/k)$. $\mathcal{K}_l$ arises naturally from the deflection problem of a beam with length $2l$ resting horizontally on an elastic foundation with spring constant $k$, while some vertical load is applied to the beam.
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Medial axis transform (MAT) is very sensitive to noise, in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. Recently, Cho...
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Medial axis transform (MAT) is very sensitive to noise, in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. Recently, Choiet al. (2002) showed that MAT is stable for a class of 2D domains called weakly injective, if we view this phenomenon with the one-sided Hausdorff distance, rather than with the two-sided Hausdorff distance. In this paper, we extend this result to general 2D domains with natural boundary regularity. We also present explicit bounds for this general one-sided stability of the 2D MAT.
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Despite its usefulness in many applications, the medial axis transform (MAT) is very sensitive to the change of the boundary in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of...
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Despite its usefulness in many applications, the medial axis transform (MAT) is very sensitive to the change of the boundary in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. However, it is known that MATs of 2D domains are stable if we view this phenomenon with the one-sided Hausdorff distance. This result depends on the fact that MATs are stable if the differences between them are measured with the recently introduced hyperbolic Hausdorff distance. In this paper, we extend the result for the one-sided stability of the MAT to a class of 3D domains called weakly injective, which contains many important 3D shapes typically appearing in solid modeling. Especially, the weakly injective 3D domains can have sharp features like corners or edges. In fact, by using the stability of the MAT under the hyperbolic Hausdorff distance, we obtain an explicit bound for the one-sided Hausdorff distance of the MAT of a weakly injective 3D domain with respect to that of a perturbed domain, which is linear with respect to the domain perturbation. We discuss some consequences of this result concerning the computation and the approximation of the MAT of 3D objects.
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摘要 :
Medial axis transform (MAT) is very sensitive to noise, in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. But it turns ...
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Medial axis transform (MAT) is very sensitive to noise, in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. But it turns out that MAT is stable, if we view this phenomenon with the one-sided Hausdorff distance, rather than with the two-sided Hausdorff distance. In this paper, we show that, if the original domain is weakly injective, which means that the MAT of the domain has no end point which is the center of an inscribed circle osculating the boundary at only one point, the one-sided Hausdorff distance of the original domain's MAT with respect to that of the perturbed one is bounded linearly with the Hausdorff distance of the perturbation. We also show by example that the linearity of this bound cannot be achieved for the domains which are not weakly injective. In particular, these results apply to the domains with sharp corners, which were excluded in the past. One consequence of these results is that we can clarify theoretically the notion of extracting "the essential part of the MAT", which is the heart of the existing pruning methods.
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