8 research outputs found

    Escher's Problem and Numerical Sequences

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    Counting problems lead naturally to integer sequences. For example if one asks for the number of subsets of an nn-set, the answer is 2n2^n, or the integer sequence 1, 2, 4, 8, ldots1,~2,~4,~8,~ldots. Conversely, given an integer sequence, or part of it, one may ask if there is an associated counting problem. There might be several different counting problems that produce the same integer sequence. To illustrate the nature of mathematical research involving integer sequences, we will consider Escher's counting problem and a generalization, as well as counting problems associated with the Catalan numbers, and the Collatz conjecture. We will also discuss the purpose of the On-Line-Encyclopedia of Integer Sequences

    High-sensitivity detection of trace gases using dynamic photoacoustic spectroscopy

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    Lincoln Laboratory of Massachusetts Institute of Technology has developed a technique known as dynamic photoacoustic spectroscopy (DPAS) that could enable remote detection of trace gases via a field-portable laser-based system. A fielded DPAS system has the potential to enable rapid, early warning of airborne chemical threats. DPAS is a new form of photoacoustic spectroscopy that relies on a laser beam swept at the speed of sound to amplify an otherwise weak photoacoustic signal. We experimentally determine the sensitivity of this technique using trace quantities of SF[subscript 6] gas. A clutter-limited sensitivity of ~100  ppt is estimated for an integration path of 0.43 m. Additionally, detection at ranges over 5 m using two different detection modalities is demonstrated: a parabolic microphone and a laser vibrometer. Its utility in detecting ammonia emanating from solid samples in an ambient environment is also demonstrated.United States. Dept. of the Air Force (Contract FA8721-05-C-0002

    • ROTHSCHILD ET AL. Recent Trends in Optical Lithography Recent Trends in Optical Lithography

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    ■ The fast-paced evolution of optical lithography has been a key enabler in the dramatic size reduction of semiconductor devices and circuits over the last three decades. Various methods have been devised to pattern at dimensions smaller than the wavelength used in the process. In addition, the patterning wavelength itself has been reduced and will continue to decrease in the future. As a result, it is expected that optical lithography will remain the technology of choice in lithography for at least another decade. Lincoln Laboratory has played a seminal role in the progress of optical lithography; it pioneered 193-nm lithography, which is used in advanced production, and 157-nm lithography, which is under active development. Lincoln Laboratory also initiated exploration of liquidimmersion lithography and studied the feasibility of 121-nm lithography. Many of the challenges related to practical implementation of short-wavelength optical lithography are materials-related, including engineering of new materials, improving on existing materials, and optimizing their photochemistry. This article examines the technical issues facing optical lithography and Lincoln Laboratory’s contributions toward their resolution. Optical lithography, the technology of patterning, has enabled semiconductor devices to progressively shrink since the inception of integrated circuits more than three decades ago. Throughout the 1980s and 1990s, the trend of miniaturization continued unabated and even accelerated. Current semiconductor devices are being mass produced with 130-nm dense features; by 2007 these devices will have 65-nm dense features. Optical lithography has been, and will remain for the foreseeable future, the critical technology that makes this trend possible. (To learn the fundamentals of optical lithography, see the sidebar entitled “Optical Lithograph
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