Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.     

Towards a programmable magnetic bead microarray in a microfluidic channel

A new hybrid magnetic bead separator that combines an external magnetic field with 175 μm thick current lines buried in the back side of a silicon wafer is presented. A microfluidic channel was etched into the front side of the wafer. The large cross-section of the current lines makes it possible to use larger currents and obtain forces of longer range than from thin current lines at a given power limit. Guiding of magnetic beads in the hybrid magnetic separator and the construction of a programmable microarray of magnetic beads in the microfluidic channel by hydrodynamic focusing is presented.     

Obtaining the bidirectional transmittance distribution function of isotropically scattering materials using an integrating sphere

This paper demonstrates a method to determine the bidirectional transmittance distribution function (BTDF) using an integrating sphere. Information about the sample’s angle-dependent scattering is obtained by making transmittance measurements with the sample at different distances from the integrating sphere. Knowledge about the illuminated area of the sample and the geometry of the sphere port in combination with the measured data combines to a system of equations that includes the angle-dependent transmittance.The resulting system of equations is an ill-posed problem which rarely gives a physical solution. A solvable system is obtained by using Tikhonov regularization on the ill-posed problem. The solution to this system can then be used to obtain the BTDF.Four bulk-scattering samples were characterised using two goniophotometers and the described method to verify the validity of the new method. The agreement shown is excellent for the more diffuse samples. The solution to the low-scattering samples contains unphysical oscillations, but still gives the correct shape of the solution. The origin of the oscillations and why they are more prominent in low-scattering samples are discussed.     

Advanced control and modeling of deposition processes

During the last decade, striking improvements could be achieved for the precise control of deposition processes in optical coating technology. For example, as a consequence of enormous progresses in measure- ment and computer technology, direct optical monitoring in a broad spectral range can be considered as a common tool in many production environments nowadays. Besides the development of the corresponding hardware and measurement channels, advanced approaches for the evaluation of the acquired data and new multiple sensor monitoring strategies moved into the focus of modern research on the way towards de- terministic deposition techniques. In this context, also innovative concepts for the simulation of deposition processes to forecast the result for a specified coating design and automatic online correction algorithms gained of importance to reduce the risk of failure in coating production. The present contribution will be dedicated to selected aspects in this field, especially addressing broad band optical monitoring systems. A short review on examples for existing hardware configurations and software tools will be presented illus- trating the advantages of modern process control techniques. Novel approaches based on the modeling of thin film growth are discussed as an additional strategy to improve the predictability of coating processes.     

On the application of a single‐crystal κ‐diffractometer and a CCD area detector for studies of thin films

A multipurpose six‐axis κ‐diffractometer, together with the brilliance of the ESRF light source and a CCD area detector, has been explored for studying epitaxial relations and crystallinity in thin film systems. The geometrical flexibility of the six‐axis goniometer allows measurement of a large volume in reciprocal space, providing an in‐depth understanding of sample crystal relationships. By a set of examples of LaAlO₃ thin films deposited by the atomic layer deposition technique, the possibilities of the set‐up are presented. A fast panoramic scan provides determination of the crystal orientation matrices, prior to more thorough inspection of single Bragg nodes. Such information, in addition to a broadening analysis of families of single reflections, is shown to correlate well with the crystallinity, crystallite size, strain and epitaxial relationships in the thin films. The proposed set‐up offers fast and easy sample mounting and alignment, along with crucial information on key features of the thin film structures.     

Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem

It may seem odd that Abel, a protagonist of Cauchy's new rigor, spoke of “exceptions” when he criticized Cauchy's theorem on the continuity of sums of continuous functions. However, when interpreted contextually, exceptions appear as both valid and viable entities in the early 19th century. First, Abel's use of the term “exception” and the role of the exception in his binomial paper is documented and analyzed. Second, it is suggested how Abel may have acquainted himself with the exception and his use of it in a process denoted critical revision is discussed. Finally, an interpretation of Abel's exception is given that identifies it as a representative example of a more general transition in the understanding of mathematical objects that took place during the period. With this interpretation, exceptions find their place in a fundamental transition during the early 19th century from a formal approach to analysis toward a more conceptual one.