An effective algorithm to compute Mandelbrot sets in parameter planes

Agraïments: The second author is partially supported by the Polish NCN grant decision DEC-2012/06/M/ST1/00168 and the MDM-2014-445 Maria de Maeztu.In 2000 McMullen proved that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algo- rithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps

An effective algorithm to compute Mandelbrot sets in parameter planes.

McMullen in 2000 proved that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algorithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps

Yang-Lee and Fisher Zeros of Multisite Interaction Ising Models on the Cayley-type Lattices

A general analytical formula for recurrence relations of multisite interaction Ising models in an external magnetic field on the Cayley-type lattices is derived. Using the theory of complex analytical dynamics on the Riemann sphere, a numerical algorithm to obtain Yang-Lee and Fisher zeros of the models is developed. It is shown that the sets of Yang-Lee and Fisher zeros are almost always fractals, that could be associated with Mandelbrot-like sets on the complex magnetic field and temperature planes respectively.Comment: 9 pages, 3 figures; with minor correction

Discovering Regularity in Point Clouds of Urban Scenes

Despite the apparent chaos of the urban environment, cities are actually replete with regularity. From the grid of streets laid out over the earth, to the lattice of windows thrown up into the sky, periodic regularity abounds in the urban scene. Just as salient, though less uniform, are the self-similar branching patterns of trees and vegetation that line streets and fill parks. We propose novel methods for discovering these regularities in 3D range scans acquired by a time-of-flight laser sensor. The applications of this regularity information are broad, and we present two original algorithms. The first exploits the efficiency of the Fourier transform for the real-time detection of periodicity in building facades. Periodic regularity is discovered online by doing a plane sweep across the scene and analyzing the frequency space of each column in the sweep. The simplicity and online nature of this algorithm allow it to be embedded in scanner hardware, making periodicity detection a built-in feature of future 3D cameras. We demonstrate the usefulness of periodicity in view registration, compression, segmentation, and facade reconstruction. The second algorithm leverages the hierarchical decomposition and locality in space of the wavelet transform to find stochastic parameters for procedural models that succinctly describe vegetation. These procedural models facilitate the generation of virtual worlds for architecture, gaming, and augmented reality. The self-similarity of vegetation can be inferred using multi-resolution analysis to discover the underlying branching patterns. We present a unified framework of these tools, enabling the modeling, transmission, and compression of high-resolution, accurate, and immersive 3D images

A theoretical reflection on smart shape modeling

This paper presents, as far as the authors are aware, a complete and extended new taxonomy of shape specification modeling techniques and a characterization of shape design systems, all based on the relationship of users’ knowledge to the modeling system they use to generate shapes. In-depth knowledge of this relationship is not usually revealed in the regular university training courses such as bachelor’s, master’s and continuing education. For this reason, we believe that it is necessary to modify the learning process, offering a more global vision of all the currently existing techniques and extending training in those related to algorithmic modeling techniques. We consider the latter to be the most powerful current techniques for modeling complex shapes that cannot be modeled with the usual techniques known to date. Therefore, the most complete training should include everything from the usual geometry to textual programming. This would take us a step further along the way to more powerful design environments. The proposed taxonomy could serve as a guideline to help improve the learning process of students and designers in a complex environment with increasingly powerful requirements and tools. The term “smart” is widely used nowadays, e.g. smart phones, smart cars, smart homes, smart cities... and similar terms such as “smart shape modeling”. Nowadays, the term smart is applied from a marketing point of view, whenever an innovation is used to solve a complex problem. This is the case for what is currently called smart shape modeling. However, in the future; this concept should mean a much better design environment than today. The smart future requires better trained and skilled engineers, architects, designers or technical students. This means that they must be prepared to be able to contribute to the creation of new knowledge, to the use of innovations to solve complex problems of form, and to the extraction of the relevant pieces of intelligence from the growing volume of knowledge and technologies accessible today. Our taxonomy is presented from the point of view of methods that are possibly furthest away from what is considered today as “intelligent shape modeling” to the limit of what is achievable today and which the authors call “Generic Shape Algorithm”. Finally, we discuss the characteristics that a shape modeling system must have to be truly “intelligent”: it must be “proactive” in applying innovative ideas to achieve a solution to a complex problem

Bayesian semiparametric long memory models for discretized event data

We introduce a new class of semiparametric latent variable models for long memory discretized event data. The proposed methodology is motivated by a study of bird vocalizations in the Amazon rain forest; the timings of vocalizations exhibit self-similarity and long range dependence ruling out models based on Poisson processes. The proposed class of FRActional Probit (FRAP) models is based on thresholding of a latent process consisting of an additive expansion of a smooth Gaussian process with a fractional Brownian motion. We develop a Bayesian approach to inference using Markov chain Monte Carlo, and show good performance in simulation studies. Applying the methods to the Amazon bird vocalization data, we find substantial evidence for self-similarity and non-Markovian/Poisson dynamics. To accommodate the bird vocalization data, in which there are many different species of birds exhibiting their own vocalization dynamics, a hierarchical expansion of FRAP is provided in Supplementary Materials

A new approach to upscaling fracture network models while preserving geostatistical and geomechanical characteristics

A new approach to upscaling two-dimensional fracture network models is proposed for preserving geostatistical and geomechanical characteristics of a smaller-scale “source” fracture pattern. First, the scaling properties of an outcrop system are examined in terms of spatial organization, lengths, connectivity, and normal/shear displacements using fractal geometry and power law relations. The fracture pattern is observed to be nonfractal with the fractal dimension D ≈ 2, while its length distribution tends to follow a power law with the exponent 2 < a < 3. To introduce a realistic distribution of fracture aperture and shear displacement, a geomechanical model using the combined finite-discrete element method captures the response of a fractured rock sample with a domain size L = 2 m under in situ stresses. Next, a novel scheme accommodating discrete-time random walks in recursive self-referencing lattices is developed to nucleate and propagate fractures together with their stress- and scale-dependent attributes into larger domains of up to 54 m × 54 m. The advantages of this approach include preserving the nonplanarity of natural cracks, capturing the existence of long fractures, retaining the realism of variable apertures, and respecting the stress dependency of displacement-length correlations. Hydraulic behavior of multiscale growth realizations is modeled by single-phase flow simulation, where distinct permeability scaling trends are observed for different geomechanical scenarios. A transition zone is identified where flow structure shifts from extremely channeled to distributed as the network scale increases. The results of this paper have implications for upscaling network characteristics for reservoir simulation