A New Method In Distribution Theory With A Non-Smooth Framework

In this work, we present a complete treatment of the theory of thick distributions and its asymptotic expansion. We also present several applications of thick distributions in mathematical physics, function spaces, and measure theory. We also discuss regularization using different surfaces. In the last chapter we present some recent applications of distributions in clarifying the moment terms in the heat kernel expansion, and in explaining the relation between the heat kernel expansion and the cylinder kernel expansion

Distributions in spaces with thick points

We present a theory of distributions in a space with a thick point in dimensions n≥2, generalizing the theory of thick distributions in one variable given in Estrada and Fulling (2007) [8]. The higher dimensional situation is quite different from the one dimensional case.We construct topological vector spaces of thick test functions and, by duality, spaces of thick distributions. We study several operations on these distributions, both algebraic and analytic, particularly partial differentiation. We introduce the notion of thick delta functions at the special point, not only of order 0 but of any integral order. We also consider the thick distributions constructed by the Hadamard finite part procedure. We give formulas for the derivatives of important thick distributions, including the finite part of power functions. We obtain the new formula ∂*2Pf(r-1)∂xi∂xj=(3xixj-δijr2)Pf(r-5)+4π(δij-4ninj)δ* for the second order thick derivatives of the finite part of r-1 in R3, where δ* is a thick delta of order 0. © 2013 Elsevier Ltd

Convex Image Segmentation Model Based on Local and Global Intensity Fitting Energy and Split Bregman Method

We propose a convex image segmentation model in a variational level set formulation. Both the local information and the global information are taken into consideration to get better segmentation results. We first propose a globally convex energy functional to combine the local and global intensity fitting terms. The proposed energy functional is then modified by adding an edge detector to force the active contour to the boundary more easily. We then apply the split Bregman method to minimize the proposed energy functional efficiently. By using a weight function that varies with location of the image, the proposed model can balance the weights between the local and global fitting terms dynamically. We have applied the proposed model to synthetic and real images with desirable results. Comparison with other models also demonstrates the accuracy and superiority of the proposed model

Global exponential stabilization of language constrained switched linear discrete-time system based on the s-procedure approach

This paper considers global exponential stabilization (GES) of switched linear discrete-time system under language constraint which is generated by non-deterministic finite state automata. A technique in linear matrix inequalities called S-procedure is employed to provide sufficient conditions of GES which are less conservative than the existing Lyapunov-Metzler condition. Moreover, by revising the construction of Lyapunov matrices and the corresponding switching control policy, a more flexible result is obtained such that stabilization path at each moment might be multiple. Finally, a numerical example is given to illustrate the effectiveness of the proposed results

Hydrodynamic and entropic effects on colloidal diffusion in corrugated channels

In the absence of advection, confined diffusion characterizes transport in many natural and artificial devices, such as ionic channels, zeolites, and nanopores. While extensive theoretical and numerical studies on this subject have produced many important predictions, experimental verifications of the predictions are rare. Here, we experimentally measure colloidal diffusion times in microchannels with periodically varying width and contrast results with predictions from the Fick-Jacobs theory and Brownian dynamics simulation. While the theory and simulation correctly predict the entropic effect of the varying channel width, they fail to account for hydrodynamic effects, which include both an overall decrease and a spatial variation of diffusivity in channels. Neglecting such hydrodynamic effects, the theory and simulation underestimate the mean and standard deviation of first passage times by 40\% in channels with a neck width twice the particle diameter. We further show that the validity of the Fick-Jakobs theory can be restored by reformulating it in terms of the experimentally measured diffusivity. Our work thus demonstrates that hydrodynamic effects play a key role in diffusive transport through narrow channels and should be included in theoretical and numerical models.Comment: 7 pages, 4 figure

Fuzzy Determination of Target Shifting Time and Torque Control of Shifting Phase for Dry Dual Clutch Transmission

Based on the independently developed five-speed dry dual clutch transmission (DDCT), the paper proposes the torque coordinating control strategy between engine and two clutches, which obtains engine speed and clutch transferred torque in the shifting process, adequately reflecting the driver intention and improving the shifting quality. Five-degree-of-freedom (DOF) shifting dynamics model of DDCT with single intermediate shaft is firstly established according to its physical characteristics. Then the quantitative control objectives of the shifting process are presented. The fuzzy decision of shifting time and the model-based torque coordinating control strategy are proposed and also verified by simulating under different driving intentions in up-/downshifting processes with the DCT model established on the MATLAB/Simulink. Simulation results validate that the shifting control algorithm proposed in this paper can not only meet the shifting quality requirements, but also adapt to the various shifting intentions, having a strong robustness

The Fourier transform of thick distributions

We first construct a space $\mathcal{W}\left( \mathbb{R}_{\text{c}} ^{n}\right)$ whose elements are test functions defined in $\mathbb{R} _{\text{c}}^{n}=\mathbb{R}^{n}\cup\left\{ \mathbf{\infty}\right\} ,$ the one point compactification of $\mathbb{R}^{n},$ that have a thick expansion at infinity of special logarithmic type, and its dual space $\mathcal{W}^{\prime }\left( \mathbb{R}_{\text{c}}^{n}\right) ,$ the space of $sl-$thick distributions. We show that there is a canonical projection of $\mathcal{W} ^{\prime}\left( \mathbb{R}_{\text{c}}^{n}\right)$ onto $\mathcal{S} ^{\prime}\left( \mathbb{R}^{n}\right) .$ We study several $sl-$thick distributions and consider operations in $\mathcal{W}^{\prime}\left( \mathbb{R}_{\text{c}}^{n}\right) .$ We define and study the Fourier transform of thick test functions of $\mathcal{S}_{\ast}\left( \mathbb{R}^{n}\right)$ and thick tempered distributions of $\mathcal{S}_{\ast}^{\prime}\left( \mathbb{R}^{n}\right) .$ We construct isomorphisms $$\mathcal{F}_{\ast}:\mathcal{S}_{\ast}^{\prime}\left( \mathbb{R}^{n}\right) \longrightarrow\mathcal{W}^{\prime}\left( \mathbb{R}_{\text{c}}^{n}\right) \,,$$ $$\mathcal{F}^{\ast}:\mathcal{W}^{\prime}\left( \mathbb{R}_{\text{c}} ^{n}\right) \longrightarrow\mathcal{S}_{\ast}^{\prime}\left( \mathbb{R} ^{n}\right) \,,$$ that extend the Fourier transform of tempered distributions, namely, $\Pi\mathcal{F}_{\ast}=\mathcal{F}\Pi$ and $\Pi\mathcal{F}^{\ast} =\mathcal{F}\Pi,$ where $\Pi$ are the canonical projections of $\mathcal{S} _{\ast}^{\prime}\left( \mathbb{R}^{n}\right)$ or $\mathcal{W}^{\prime }\left( \mathbb{R}_{\text{c}}^{n}\right)$ onto $\mathcal{S}^{\prime}\left( \mathbb{R}^{n}\right) .$ We determine the Fourier transform of several finite part regularizations and of general thick delta functions.Comment: 21 page

Hexapartite steering based on a four-wave-mixing process with a spatially structured pump

Multipartite Einstein-Podolsky-Rosen (EPR) steering has been widely studied, for realizing safer quantum communication. The steering properties of six spatially separated beams from the four-wave-mixing process with a spatially structured pump are investigated. Behaviors of all (1+i)/(i+1)-mode (i=1,2,3) steerings are understandable, if the role of the corresponding relative interaction strengths are taken into account. Moreover, stronger collective multipartite steerings including five modes also can be obtained in our scheme, which has potential applications in ultra-secure multiuser quantum networks when the issue of trust is critical. By further discussing about all monogamy relations, it is noticed that the type-IV monogamy relations, which are naturally included in our model, are conditionally satisfied. Matrix representation is used to express the steerings for the first time, which is very useful to understand the monogomy relations intuitively. Different steering properties obtained in this compact phase-insensitive scheme have potential applications for different kinds of quantum communication tasks