6,473 research outputs found
Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets
We consider the problem of computing the minimum value of a
polynomial over a compact set , which can be
reformulated as finding a probability measure on minimizing . Lasserre showed that it suffices to consider such measures of the form
, where is a sum-of-squares polynomial and is a given
Borel measure supported on . By bounding the degree of by one gets
a converging hierarchy of upper bounds for . When is
the hypercube , equipped with the Chebyshev measure, the parameters
are known to converge to at a rate in . We
extend this error estimate to a wider class of convex bodies, while also
allowing for a broader class of reference measures, including the Lebesgue
measure. Our analysis applies to simplices, balls and convex bodies that
locally look like a ball. In addition, we show an error estimate in when satisfies a minor geometrical condition, and in when is a convex body, equipped with the Lebesgue measure. This
improves upon the currently best known error estimates in and
for these two respective cases.Comment: 30 pages with 10 figures. Update notes for second version: Added a
new section containing numerical examples that illustrate the theoretical
results -- Fixed minor mistakes/typos -- Improved some notation -- Clarified
certain explanations in the tex
A rigorous model for constraint release in the bulk and the near-wall region
In the present work an attempt is made to build a rigorous theoretical model for the constraint release mechanism found to play an important role in the dynamics of polymer melts. Our goal is a formalism free of adjustable parameters and ''ad-hoc'' assumptions which are inherent to existing theories for constraint release. Our model is capable to describe both thermal and convective constraint release. These processes have the same effect on chains and accordingly can be unified in a single framework. Since polymer chains in the bulk and in the near-wall layer may experience different types of constraint release, the latter case is studied separately. This topic is closely related to the long-standing problem of polymer melt flow instabilities encountered during extrusion. Nowadays it is believed that constraint release plays a crucial role in the dynamics of tethered chains preventing them from being squeezed against the wall. The resulting non-monotonous slip-law is the most probable reason of the so-called spurt instability. \u
Design of a 30 GHz bragg reflector for a Raman FEL
A design of a Bragg reflector for a Raman FEL is described. It is shown that mode conversion occurs whenever the axial wavenumbers of the two modes fulfil the Bragg condition. With a constant ripple of the corrugation it is shown that the reflected radiation also contains higher order modes, assuming that the incident radiation consists only of a TE11 mode. The mode purity can be increased by increasing the length of the reflector at the expense of a smaller reflection bandwidth. A more flexible method is by applying a Hamming window to the corrugation of the reflector. Contributions of other modes to the reflected radiation can in that case be neglected. The reflector will be installed in a Raman laser to be able to compare the amplifier with the oscillator configuration. Therefore some preliminary results are also presented about the start-up of the Raman laser
A universal constitutive model for the interfacial layer between a polymer melt and a solid wall
In a preceeding report we derived the evolution equation for the bond vector probability distribution function (BVPDF) of tethered molecules. It describes the behavior of polymer molecules attached to a solid wall interacting with an adjacent flowing melt of bulk polymer molecules and includes all the major relaxation mechanisms such as constraint release, retraction and convection. The derived equation is quite universal and valid for all flow regimes. In the present paper the developed formalism is further analyzed. We begin our analysis with the simple case of slow flows. Then, as expected, a remarkable reduction of the theory is possible. Later on the more general case is considered. \u
Dynamics of chains grafted on solid wall during polymer melt extrusion
The objective of the present work is the mathematical modeling of the dynamics of polymer molecules grafted on a solid boundary during polymer melt extrusion. This topic is closely related to the long-standing problem of polymer flow instabilities encountered in industry when extruding melts. In order to describe the behavior of the tethered chains, we introduce the bond vector probability distribution function (BVPDF) which appears to be a simple, yet effective mathematical 'tool'. The bond vector, i.e. the tangent vector to a polymer chain depending on the position along the chain and on time, describes the local geometry via its direction and the local stretching of the chain via its length. The BVPDF contains all information about the geometry of the ensemble of chains. Via averaging over the BVPDF we can calculate all interesting macrsocopic quantities, e.g. the thickness of and stress in the layer of tethered molecules. The time dependence of the BVPDF yields the time evolution of the system. We derive the equation of motion for the BVPDF taking into account all important mechanisms, such as reptation and (convective) constraint release. Besides that, we show that all macroscopic quantities of practical interest can be expressed via second order moments of this distribution function. \u
Flow-induced correlation effects within a linear chain in a polymer melt
A framework for a consistent description of the flow-induced correlation effects within a linear polymer chain in a melt is proposed. The formalism shows how correlations between chain segments in the flow can be incorporated into a hierarchy of distribution functions for tangent vectors. The present model allows one to take into account all the major relaxation mechanisms. Special cases of the derived set of equations are shown to yield existing models and shed some light on the connection between them. Consequences of several assumptions widely used in the literature are analyzed within the developed framework
Studies of a Terawatt X-Ray Free-Electron Laser
The possibility of constructing terawatt (TW) x-ray free-electron lasers
(FELs) has been discussed using novel superconducting helical undulators [5].
In this paper, we consider the conditions necessary for achieving powers in
excess of 1 TW in a 1.5 {\AA} FEL using simulations with the MINERVA simulation
code [7]. Steady-state simulations have been conducted using a variety of
undulator and focusing configurations. In particular, strong focusing using
FODO lattices is compared with the natural, weak focusing inherent in helical
undulators. It is found that the most important requirement to reach TW powers
is extreme transverse compression of the electron beam in a strong FODO
lattice. The importance of extreme focusing of the electron beam in the
production of TW power levels means that the undulator is not the prime driver
for a TW FEL, and simulations are also described using planar undulators that
reach near-TW power levels. In addition, TW power levels can be reached using
pure self-amplified spontaneous emission (SASE) or with novel self-seeding
configurations when such extreme focusing of the electron beam is applied.Comment: 10 pages, 12 figure
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