Effective Summation and Interpolation of Series by Self-Similar Root Approximants

We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by a number of examples. The accuracy of the method is not worse, and in many cases better, than that of Pade approximants, when the latter can be defined.Comment: Latex file, 18 page

Scale-free fractal interpolation

An iterated function system that defines a fractal interpolation function, where ordinate scaling is replaced by a nonlinear contraction, is investigated here. In such a manner, fractal interpolation functions associated with Matkowski contractions for finite as well as infinite (countable) sets of data are obtained. Furthermore, we construct an extension of the concept of α-fractal interpolation functions, herein called R-fractal interpolation functions, related to a finite as well as to a countable iterated function system and provide approximation properties of the R-fractal functions. Moreover, we obtain smooth R-fractal interpolation functions and provide results that ensure the existence of differentiable R-fractal interpolation functions both for the finite and the infinite (countable) cases

Influence of external flows on crystal growth: numerical investigation

We use a combined phase-field/lattice-Boltzmann scheme [D. Medvedev, K. Kassner, Phys. Rev. E {\bf 72}, 056703 (2005)] to simulate non-facetted crystal growth from an undercooled melt in external flows. Selected growth parameters are determined numerically. For growth patterns at moderate to high undercooling and relatively large anisotropy, the values of the tip radius and selection parameter plotted as a function of the Peclet number fall approximately on single curves. Hence, it may be argued that a parallel flow changes the selected tip radius and growth velocity solely by modifying (increasing) the Peclet number. This has interesting implications for the availability of current selection theories as predictors of growth characteristics under flow. At smaller anisotropy, a modification of the morphology diagram in the plane undercooling versus anisotropy is observed. The transition line from dendrites to doublons is shifted in favour of dendritic patterns, which become faster than doublons as the flow speed is increased, thus rendering the basin of attraction of dendritic structures larger. For small anisotropy and Prandtl number, we find oscillations of the tip velocity in the presence of flow. On increasing the fluid viscosity or decreasing the flow velocity, we observe a reduction in the amplitude of these oscillations.Comment: 10 pages, 7 figures, accepted for Physical Review E; size of some images had to be substantially reduced in comparison to original, resulting in low qualit

Towards a More Well-Founded Cosmology

First, this paper broaches the definition of science and the epistemic yield of tenets and approaches: phenomenological (descriptive only), well-founded (solid first principles, conducive to deep understanding), provisional (falsifiable if universal, verifiable if existential), and imaginary (fictitious entities or processes, conducive to empirically unsupported beliefs). The Big-Bang pardigm and the {\Lambda}CDM "concordance model" involve such beliefs: the emanation of the universe out of a non-physical stage, cosmic inflation (invented ad hoc), {\Lambda} (fictitious energy), and exotic dark matter. They fail in the confidence check that is required in empirical science. They also face a problem in delimiting what expands from what does not. In the more well-founded cosmology that emerges, energy is conserved, the universe is persistent (not transient) and the 'perfect cosmological principle' holds. Waves and other perturbations that propagate at c (the escape velocity from the universe) expand exponentially with distance. This dilatation results from gravitation. The cosmic web of galaxies does not expand. Potential {\Phi} varies as -H/(cz) instead of -1/r. Inertial forces arise from gravitational interaction with the rest of the universe (not with space). They are increased where the universe appears blueshifted and decreased more than proportionately at very low accelerations. A cut-off acceleration a0 = 0.168 cH is deduced. This explains the successful description of galaxy rotation curves by MoND. A fully elaborated physical theory is still pending. The recycling of energy via a cosmic ocean filled with photons (the CMB), neutrinos and gravitons, and wider implications for science, are briefly discussed

Analytic traveling-wave solutions of the Kardar-Parisi-Zhang interface growing equation with different kind of noise terms

The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the traveling-wave Ansatz is analyzed. As a new feature additional analytic terms are added. From the mathematical point of view, these can be considered as various noise distribution functions. Six different cases were investigated among others Gaussian, Lorentzian, white or even pink noise. Analytic solutions are evaluated and analyzed for all cases. All results are expressible with various special functions Mathieu, Bessel, Airy or Whittaker functions showing a very rich mathematical structure with some common general characteristics. This study is the continuation of our former work, where the same physical phenomena was investigated with the self-similar Ansatz. The differences and similarities among the various solutions are enlightened.Comment: 14 pages,14 figures. arXiv admin note: text overlap with arXiv:1904.0183

Arbitrary topology meshes in geometric design and vector graphics

Meshes are a powerful means to represent objects and shapes both in 2D and 3D, but the techniques based on meshes can only be used in certain regular settings and restrict their usage. Meshes with an arbitrary topology have many interesting applications in geometric design and (vector) graphics, and can give designers more freedom in designing complex objects. In the first part of the thesis we look at how these meshes can be used in computer aided design to represent objects that consist of multiple regular meshes that are constructed together. Then we extend the B-spline surface technique from the regular setting to work on extraordinary regions in meshes so that multisided B-spline patches are created. In addition, we show how to render multisided objects efficiently, through using the GPU and tessellation. In the second part of the thesis we look at how the gradient mesh vector graphics primitives can be combined with procedural noise functions to create expressive but sparsely defined vector graphic images. We also look at how the gradient mesh can be extended to arbitrary topology variants. Here, we compare existing work with two new formulations of a polygonal gradient mesh. Finally we show how we can turn any image into a vector graphics image in an efficient manner. This vectorisation process automatically extracts important image features and constructs a mesh around it. This automatic pipeline is very efficient and even facilitates interactive image vectorisation

Rapid model-guided design of organ-scale synthetic vasculature for biomanufacturing

Our ability to produce human-scale bio-manufactured organs is critically limited by the need for vascularization and perfusion. For tissues of variable size and shape, including arbitrarily complex geometries, designing and printing vasculature capable of adequate perfusion has posed a major hurdle. Here, we introduce a model-driven design pipeline combining accelerated optimization methods for fast synthetic vascular tree generation and computational hemodynamics models. We demonstrate rapid generation, simulation, and 3D printing of synthetic vasculature in complex geometries, from small tissue constructs to organ scale networks. We introduce key algorithmic advances that all together accelerate synthetic vascular generation by more than 230-fold compared to standard methods and enable their use in arbitrarily complex shapes through localized implicit functions. Furthermore, we provide techniques for joining vascular trees into watertight networks suitable for hemodynamic CFD and 3D fabrication. We demonstrate that organ-scale vascular network models can be generated in silico within minutes and can be used to perfuse engineered and anatomic models including a bioreactor, annulus, bi-ventricular heart, and gyrus. We further show that this flexible pipeline can be applied to two common modes of bioprinting with free-form reversible embedding of suspended hydrogels and writing into soft matter. Our synthetic vascular tree generation pipeline enables rapid, scalable vascular model generation and fluid analysis for bio-manufactured tissues necessary for future scale up and production.Comment: 58 pages (19 main and 39 supplement pages), 4 main figures, 9 supplement figure