About the Collatz conjecture

This paper refers to the Collatz conjecture. The origin and the formalization of the Collatz problem are presented in the first section, named “Introduction”. In the second section, entitled “Properties of the Collatz function”, we treat mainly the bijectivity of the Collatz function. Using the obtained results, we construct a (set of) binary tree(s) which “simulate(s)”– in a way that will be specified – the computations of the values of the Collatz function. In the third section, we give an “efficient” algorithm for computing the number of iterations (recursive calls) of the Collatz function. A comparison between our algorithm and the standard one is also presented, the first being at least 2.25 “faster” (3.00 in medium). Finally, we describe a class of natural numbers for which the conjecture is true. Received 28 April 1997 / 10 June 1997     

Comparison of a new algorithm with the supervised classifications

In this study, a new classification algorithm in which only the selected pixels have been attempted to be classified (selected pixels classification: SPC) has been introduced and compared with the well known supervised classification methods such as maximum likelihood, minimum distance, nearest neighbour and condensed nearest neighbour. To examine the algorithm, Landsat Thematic Mapper (TM) data have been used to classify the crop cover in the selected region. It is clearly demonstrated that the SPC method has the higher accuracy with comparable CPU times.     

Multifractal characterization of unworn hydrogel contact lens surfaces

Purpose: The aim of this work was to estimate the multifractal spectra of 3D surface roughness for unworn hydrogel contact lenses (CL), obtained with atomic force microscopy (AFM) analysis. Materials and methods: Contact lenses made from vifilcon A (Focus® Monthly Toric Visitint® model, CIBA Vision Corp.) were investigated. CL surface roughness was studied by AFM in tapping‐mode™, in an aqueous environment, on square areas ranging from 1 to 4 μm². A detailed methodology for CL surface multifractal characterization, which may be applied for AFM data, was presented. Results: The CL surface roughness revealed multifractal geometry at various magnifications. The generalized dimension D_q and the singularity spectrum f(α) provided quantitative values that characterize the local scale properties of CL surface geometry at nanometer scale. Conclusions: Multifractal analysis provides different yet complementary information to that offered by traditional surface statistical parameters. POLYM. ENG. SCI., 54:1066–1080, 2014. © 2013 Society of Plastics Engineers     

An exponential approach for the system of nonlinear delay integro-differential equations describing biological species living together

Neural Computing and Applications - In this paper, we consider a system of nonlinear delay integro-differential equations with convolution kernels, which arises in biology. This problem...     

Investigating the effect of production process of ball mill refiner on some physical quality parameters of compound chocolate: response surface methodology approach

Chocolate compound was produced using ball mill refiner, and the effect of agitator shaft speed and refining time on the physical quality parameters (particle size, colour and steady‐state rheology) of compound chocolate was determined using response surface methodology. The shaft speed and refining time range were selected between 40–60 r.p.m. and 10–30 min, respectively. Determination coefficient of the models established for particle size, Newtonian viscosity and colour parameters (brightness, chroma and hue angle) were found to be very close to unity. Increasing shaft speed and time induced a reduction in particle size and an increase in viscosity of the samples. Temperature sweep test was also performed, and the obtained data were successfully fitted to Arrhenius equation to calculate the corresponding parameters representing temperature dependency of the compounds. The results highlighted that the establishment of such models can provide essential information in terms of optimisation of production processes regarding usage purpose of the compound chocolate.     

Lamellar eutectic growth with anisotropic interphase boundaries: Experimental study using the rotating directional solidification method

We report on an experimental study of the effects of interphase boundary anisotropy on eutectic microstructures using a new methodology called rotating directional solidification (RDS), which consists of rotating a thin sample with respect to a fixed unidirectional thermal gradient. The systems used are thin, large eutectic grains of the CBr₄–C₂Cl₆ and In–In₂Bi lamellar eutectic alloys. The shape of the observed RDS lamellar trajectories turns out to be a reproducible eutectic-grain-dependent feature, in agreement with the theoretical prediction that these trajectories are approximately homothetic to the Wulff form of the interphase boundary in the sample plane. We show that different modes of lamellar growth, ranging from quasi-isotropic to (crystallographically) locked, exist in different eutectic grains of the two alloys studied. A detailed characterisation of these modes is given, with particular attention to the as-yet poorly understood aspects of locked lamellar growth.     

Cyclic distillation technology – a mini‐review

Process intensification in distillation systems has received much attention during past decades, with the aim of increasing both energy and separation efficiency. Various techniques, such as internal heat‐integrated distillation, membrane distillation, rotating packed bed, dividing‐wall columns and reactive distillation were studied and reported in the literature. All these techniques employ the conventional continuous counter‐current contact of vapor and liquid phases. Cyclic distillation technology is based on an alternative operating mode using separate phase movement which leads to key practical advantages in both chemical and biochemical processes. This article provides a mini‐review of cyclic distillation technology. The topics covered include the working principle, design and control methods, main benefits and limitations as well as current industrial applications. Cyclic distillation can be rather easily implemented in existing columns by simply changing the internals and the operating mode, thus bringing new life to old distillation towers by significantly increasing the column throughput, reducing the energy requirements and offering better separation performance. © 2016 Society of Chemical Industry