A remarkable sequence related to π\pi and 2\sqrt{2}

We prove that five ways to define entry A086377 in the On-Line Encyclopedia of Integer Sequences do lead to the same integer sequence

Abstract numeration systems on bounded languages and multiplication by a constant

A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with at least three letters, we show that multiplication by an integer $\lambda\ge2$ does not preserve $S$-recognizability, meaning that there always exists a $S$-recognizable set $X$ such that $\lambda X$ is not $S$-recognizable. The main tool is a bijection between the representation of an integer over a bounded language and its decomposition as a sum of binomial coefficients with certain properties, the so-called combinatorial numeration system

A versatile maskless microscope projection photolithography system and its application in light-directed fabrication of DNA microarrays

We present a maskless microscope projection lithography system (MPLS), in which photomasks have been replaced by a Digital Micromirror Device type spatial light modulator (DMD, Texas Instruments). Employing video projector technology high resolution patterns, designed as bitmap images on the computer, are displayed using a micromirror array consisting of about 786000 tiny individually addressable tilting mirrors. The DMD, which is located in the image plane of an infinity corrected microscope, is projected onto a substrate placed in the focal plane of the microscope objective. With a 5x(0.25 NA) Fluar microscope objective, a fivefold reduction of the image to a total size of 9 mm2 and a minimum feature size of 3.5 microns is achieved. Our system can be used in the visible range as well as in the near UV (with a light intensity of up to 76 mW/cm2 around the 365 nm Hg-line). We developed an inexpensive and simple method to enable exact focusing and controlling of the image quality of the projected patterns. Our MPLS has originally been designed for the light-directed in situ synthesis of DNA microarrays. One requirement is a high UV intensity to keep the fabrication process reasonably short. Another demand is a sufficient contrast ratio over small distances (of about 5 microns). This is necessary to achieve a high density of features (i.e. separated sites on the substrate at which different DNA sequences are synthesized in parallel fashion) while at the same time the number of stray light induced DNA sequence errors is kept reasonably small. We demonstrate the performance of the apparatus in light-directed DNA chip synthesis and discuss its advantages and limitations.Comment: 12 pages, 9 figures, journal articl

The Gargasian (Middle Aptian) of La Marcouline section at Cassis-La Bédoule (SE France): Stable isotope record and orbital cyclicity

International audienceBulk rock stable isotope analysis of La Marcouline section (Cassis-La Bédoule area, SE France) revealed a general trend of decreasing δ13C and δ18O values from the bottom to the top of the section. The decrease in δ13C values reflects a global trend in Middle Aptian times, namely a return to pre-excursion values of δ13C values following a major, positive excursion in the Early Aptian, which is a reflection of the Oceanic Anoxic Event (OAE) 1a. Power spectra of the bedding rhythms suggest that precessional and long eccentricity cycles of the Milankovitch band controlled the deposition of marl-limestone bundles. This can be interpreted as the result of a forcing by insolation at low latitudes which resulted in a monsoon-controlled precipitation pattern, that in turn induced the deposition of clay-rich beds. Conversely, limestone beds were formed in periods of dryer climate. Higher sea-surface productivity during wet periods may have been caused by an increase in continental runoff and a consequent enhancement in the delivery of nutrients to epicontinental basins. An orbital cyclicity in the sedimentary patterns in La Marcouline section provides the opportunity to calibrate the duration of the well-established G. ferreolensis foraminiferal Zone with that of orbital chronology. The G. ferreolensis Zone at Cassis-La Bédoule spans 33 precessional cycles and its duration is thus estimated to be approximately 760 ka. This length of time is significantly shorter than the estimates of published timescales for this zone and thus may be an argument for proposing that the Gargasian substage is significantly shorter than its currently accepted range

A precise bare simulation approach to the minimization of some distances. Foundations

In information theory -- as well as in the adjacent fields of statistics, machine learning, artificial intelligence, signal processing and pattern recognition -- many flexibilizations of the omnipresent Kullback-Leibler information distance (relative entropy) and of the closely related Shannon entropy have become frequently used tools. To tackle corresponding constrained minimization (respectively maximization) problems by a newly developed dimension-free bare (pure) simulation method, is the main goal of this paper. Almost no assumptions (like convexity) on the set of constraints are needed, within our discrete setup of arbitrary dimension, and our method is precise (i.e., converges in the limit). As a side effect, we also derive an innovative way of constructing new useful distances/divergences. To illustrate the core of our approach, we present numerous examples. The potential for widespread applicability is indicated, too; in particular, we deliver many recent references for uses of the involved distances/divergences and entropies in various different research fields (which may also serve as an interdisciplinary interface)