Lipschitzian retracts and curves as invariant sets

By an invariant set in a metric space we mean a non-empty compact set K such that K = ⋃ _i=1 ⁿ T _i (K) for some contractions T ₁, …, T _n of the space. We prove that, under not too restrictive conditions, the union of finitely many invariant sets is an invariant set. Hence we establish collage theorems for non-affine invariant sets in terms of Lipschitzian retracts. We show that any rectifiable curve is an invariant set though there is a simple arc which is not an invariant set.      

Direct measurement of oxygen extraction with fMRI using 6% CO2 inhalation

The blood-oxygenation-level-dependent (BOLD) signal is an indirect hemodynamic signal that is sensitive to cerebral blood flow (CBF), cerebral blood volume (CBV) and cerebral metabolic rate of oxygen. Therefore, the BOLD signal amplitude and dynamics cannot be interpreted unambiguously without additional physiological measurements, and thus, there remains a need for a functional magnetic resonance imaging (fMRI) signal, which is more closely related to the underlying neuronal activity. In this study, we measured CBF with continuous arterial spin labeling, CBV with an exogenous contrast agent and BOLD combined with intracortical electrophysiological recording in the primary visual cortex of the anesthetized monkey. During inhalation of 6% CO2, it was observed that CBF and CBV are not further increased by a visual stimulus, although baseline CBF for 6% CO2 is below the maximal value of CBF. In contrast, the electrophysiological response to the stimulation was found to be preserved during hypercapnia. As a consequence, the simultaneously measured BOLD signal responds negatively to a visual stimulation for 6% CO2 inhalation in the same voxels responding positively during normocapnia. These observations suggest that the fMRI response to a sensory stimulus for 6% CO2 inhalation occurs in the absence of a hemodynamic response, and it therefore directly reflects oxygen extraction into the tissue.     

Spin-orbit Ab initio investigation of the photodissociation of dibromomethane in the gas and solution phases

A clear and reliable theoretical investigation on dibromomethane (CH(2)Br(2)) photodissociation is desired. The calculation must consider: (i) relativistic effects; (ii) the potential energy curves (PECs) of spin-orbit coupling states; (iii) geometry optimization by the method with both static and dynamic electron correlations; (iv) solvent effects on the photodissociation in the solution. All these have been considered in this study by state-of-the-art quantum chemical calculations. The experimentally observed photodissociation in the gas phase with products of spin-orbit-coupled states, Br((2)P(3/2)) and Br*((2)P(1/2)), was assigned by multi-state second order multiconfigurational perturbation theory in conjunction with spin-orbit interaction through complete active space state interaction (MS-CASPT2/CASSI-SO) PECs. The mechanisms of the experimentally observed photodissociation and photoisomerization in solvent were elucidated by the MS-CASPT2/CASSI-SO method combined with polarized continuum model of the solvent.     

Asymptotic behaviors of the Lorenz curve and Gini index in sampling from a length-biased distribution

In this work, we consider the nonparametric estimators of the Lorenz curve and Gini index based on a sample from the corresponding length-biased distribution. We show that this estimators are strongly consistent for the associated Lorenz curve and Gini index. Strong Gaussian approximations for the associated Lorenz process are established under appropriate assumptions. We apply the strong Gaussian approximation technique to obtain a functional law for the iterated logarithm for the Lorenz curve. Also, we obtain an asymptotic normality for the corresponding Gini index.     

A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

Algorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G¹ Hermite data; however, one could also obtain higher order algorithms.     

Conditions for coincidence of two cubic Bézier curves

This paper presents a necessary and sufficient condition for judging whether two cubic Bézier curves are coincident: two cubic Bézier curves whose control points are not collinear are coincident if and only if their corresponding control points are coincident or one curve is the reversal of the other curve. However, this is not true for degree higher than 3. This paper provides a set of counterexamples of degree 4.     

On the conditions for the coincidence of two cubic Bézier curves

In a recent article, Wang et al. [2] derive a necessary and sufficient condition for the coincidence of two cubic Bézier curves with non-collinear control points. The condition reads that their control points must be either coincident or in reverse order. We point out that this uniqueness of the control points for polynomial cubics is a straightforward consequence of a previous and more general result of Barry and Patterson, namely the uniqueness of the control points for rational Bézier curves. Moreover, this uniqueness applies to properly parameterized polynomial curves of arbitrary degree.     

The blow up analysis of the general curve shortening flow

It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.     

Local structure of bifurcation curves for nonlinear Sturm-Liouville problems

We consider the nonlinear bifurcation problem arising in population dynamics and nonlinear Schrödinger equation:

     

Embeddings of -dimensional separable metric spaces into the product of Sierpinski curves

We give a short proof of the following fact: the set of embeddings of any -dimensional separable metric space into a certain -dimensional subset of the -product of Sierpinski curves is residual in .