Current Search: Li, Jie (x)
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Title
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APPROXIMATION BY BERNSTEIN POLYNOMIALS AT THE POINT OF DISCONTINUITY.
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Creator
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Liang, Jie, Li, Xin, University of Central Florida
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Abstract / Description
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Chlodovsky showed that if x0 is a point of discontinuity of the first kind of the function f, then the Bernstein polynomials Bn(f; x0) converge to the average of the one-sided limits on the right and on the left of the function f at the point x0. In 2009, Telyakovskii in extended the asymptotic formulas for the deviations of the Bernstein polynomials from the differentiable functions at the first-kind discontinuity points of the highest derivatives of even order and demonstrated the same...
Show moreChlodovsky showed that if x0 is a point of discontinuity of the first kind of the function f, then the Bernstein polynomials Bn(f; x0) converge to the average of the one-sided limits on the right and on the left of the function f at the point x0. In 2009, Telyakovskii in extended the asymptotic formulas for the deviations of the Bernstein polynomials from the differentiable functions at the first-kind discontinuity points of the highest derivatives of even order and demonstrated the same result fails for the odd order case. Then in 2010, Tonkov in found the right formulation and proved the result that was missing in the odd-order case. It turned out that the limit in the odd order case is related to the jump of the highest derivative. The proofs in these two cases look similar but have many subtle differences, so it is desirable to find out if there is a unifying principle for treating both cases. In this thesis, we obtain a unified formulation and proof for the asymptotic results of both Telyakovskii and Tonkov and discuss extension of these results in the case where the highest derivative of the function is only assumed to be bounded at the point under study.
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Date Issued
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2011
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Identifier
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CFH0004099, ucf:44790
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Format
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Document (PDF)
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PURL
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http://purl.flvc.org/ucf/fd/CFH0004099
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Title
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Generation and Characterization of Isolated Attosecond Pulse in the Soft X-ray Regime.
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Creator
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Li, Jie, Chang, Zenghu, Delfyett, Peter, Vanstryland, Eric, Chen, Bo, University of Central Florida
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Abstract / Description
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The observation of any atomic and molecular dynamics requires a probe that has a timescale comparable to the dynamics itself. Ever since the invention of laser, the temporal duration of the laser pulse has been incrementally reduced from several nanoseconds to just attoseconds. Picosecond and femtosecond laser pulses have been widely used to study molecular rotation and vibration. In 2001, the first single isolated attosecond pulse (1 attosecond = 10^-18 seconds.) was demonstrated. Since this...
Show moreThe observation of any atomic and molecular dynamics requires a probe that has a timescale comparable to the dynamics itself. Ever since the invention of laser, the temporal duration of the laser pulse has been incrementally reduced from several nanoseconds to just attoseconds. Picosecond and femtosecond laser pulses have been widely used to study molecular rotation and vibration. In 2001, the first single isolated attosecond pulse (1 attosecond = 10^-18 seconds.) was demonstrated. Since this breakthrough, (")attoscience(") has become a hot topic in ultrafast physics. Attosecond pulses typically have span between EUV to X-ray photon energies and sub-femtosecond pulse duration. It becomes an ideal tool for experimentalists to study ultrafast electron dynamics in atoms, molecules and condensed matter. The conventional scheme for generating attosecond pulses is focusing an intense femtosecond laser pulse into inert gases. The bound electrons are ionized into continuum through tunneling ionization under the strong electrical field. After ionization, the free electron will be accelerated by the laser field away from the parent ion and then recombined with its parent ion and releases its kinetic energy as a photon burst that lasts for a few hundred attoseconds. According to the classical (")three-step model("), high order harmonic will have a higher cutoff photon energy when driven by a longer wavelength laser field. Compared to Ti:sapphire lasers center at a wavelength of 800 nm, an optical parametric amplifier could offer a broad bandwidth at infrared range, which could support few cycle pulses for driving high harmonic generation in the X-ray spectrum range. In this work, an optical parametric chirped-pulse amplification system was developed to deliver CEP-stable 3-mJ, 12-fs pulses centered at 1.7 micron. We implement a chirped-pump technique to phase match the board parametric amplification bandwidth with high conversion efficiency. Using such a laser source, isolated attosecond pulses with photon exceeding 300 eV are achieved by applying the polarization gating technique at 1.7 micron. The intrinsic positive chirp of the attosecond pulses is measured by the attosecond streak camera and retrieved with our PROOF technique. Sn metal filters with negative dispersion were chosen to compensate the intrinsic attochirp. As a result, isolated 53-attosecond soft x-ray pulses are achieved. Such water window attosecond source will be a powerful tool for studying charge distribution/migration in bio-molecules and will bring opportunities to study high field physics or attochemistry.
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Date Issued
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2018
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Identifier
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CFE0007040, ucf:52007
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Format
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Document (PDF)
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PURL
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http://purl.flvc.org/ucf/fd/CFE0007040