Pavel Bělík

Augsburg University · Mathematics Statistics and Computer Science Department

For polynomials some of whose zeros are complex, little is known about the overall convergence properties of Laguerre’s function. This chapter provides an outline of this function viewed as a dynamic system which often studied by many researchers and includes some of the latest research made by the authors. Moreover, the existence of its free criti...

In this work we propose a new algorithm for the computation of statistical equilibrium quantities on a cubic lattice when both an energy and a statistical temperature are involved. We demonstrate that the pivot algorithm used in situations such as protein folding works well for a small range of temperatures near the polymeric case, but it fails in...

In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate syste...

In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate syste...

Self-similarity in tornadic and some non-tornadic supercell flows is studied and power lawsrelating various quantities in such flows are demonstrated. Magnitudes of the exponents in these powerlaws are related to the intensity of the corresponding flow and thus the severity of the supercell storm.The features studied in this paper include the verti...

The stimulation of electromagnetic ion cyclotron (EMIC) waves by a magnetospheric compression is perhaps the closest thing to a controlled experiment that is currently possible in magnetospheric physics, in that one prominent factor that can increase wave growth acts at a well-defined time. We present a detailed analysis of EMIC waves observed in t...

Self-similarity in tornadic and some non-tornadic supercell flows is studied and power laws relating various quantities in such flows are demonstrated. Magnitudes of the exponents in these power laws are related to the intensity of the corresponding flow and thus the severity of the supercell storm. The features studied in this paper include the ve...

Self-similarity in tornadic and some non-tornadic supercell flows is studied and power laws relating various quantities in such flows are demonstrated. Magnitudes of the exponents in these power laws are related to the intensity of the corresponding flow and thus the severity of the supercell storm. The features studied in this paper include the ve...

Processes related to the production of vorticity in the forward and rear flank downdrafts and their interaction with the boundary layer are thought to play a role in tornadogenesis. We argue that an inverse energy cascade is a plausible mechanism for tornadogenesis and tornado maintenance and provides supporting evidence which is both numerical and...

Possible attractor-like flow in tornadic supercells, based on Fujita’s recycling hypothesis.

We describe tornadogenesis and maintenance using the 3-dimensional vortex gas model presented in Chorin (1994) and developed further in Flandoli and Gubinelli (2002). We suggest that high-energy, super-critical vortices in the sense of Benjamin (1962), that have been studied by Fiedler and Rotunno (1986), have negative temperature in the sense of O...

Two-dimensional and three-dimensional vortex gas models are discussed and proposed in this paper as potential models for tornadogenesis and tornado maintenance. The idea of maximization of entropy is utilized which gives rise to negative-temperature systems; in such systems energy is transferred from smaller to larger scales resulting in an inverse...

Previous analyses of Laguerre’s iteration method have provided results on the behavior of this popular method when applied to the polynomials pn(z)=zn-1, n∈N. In this paper, we summarize known analytical results and provide new results. In particular, we study symmetry properties of the Laguerre iteration function and clarify the dynamics of the me...

Recent radar studies indicate power laws for horizontal velocity in terms of distance from the axis of rotation [6] and the fractal nature of the tornado-related vorticity field with respect to the grid size [2]. As the power increases the likelihood of strong tornados appears to increase.

Previous analyses of Laguerre's method have provided results on the convergence and properties of this popular method when applied to the polynomials $p_n(z)=z^n-1$, $n\in\mathbb{N}$. While these analyses appear to provide a fairly complete picture, careful study of the results reveals that more can be said. We provide additional analytical, comput...

A computational study is presented of the training of a martensitic thin film by apply-ing a biaxial loading cycle to the film below the transformation temperature. The Γ-limit of a bulk energy allowing sharp interfaces is discretized by the finite element method. The energy density models the softening of the elastic modulus controlling the low-en...

Recent radar studies indicate power laws for horizontal velocity in terms of distance from the axis of rotation and the fractal nature of the tornado-related vorticity field with respect to the grid size. As the power increases the likelihood of strong tornados appears to increase.

Recent radar studies indicate power laws for horizontal velocity in terms of distance from the axis of rotation and the fractal nature of the tornado-related vorticity field with respect to the grid size, more specifically, natural log of the vorticity and natural log of the grid spacing have a linear relationship, with a constant ratio. As the slo...

We consider a modification of the fluid flow model for a swirling vortex developed by J. Serrin, where velocity decreases as the reciprocal of the distance from the vortex axis. Recent studies, based on radar data of selected severe weather events, indicate that the angular momentum in a tornado may not be constant with the radius, and thus suggest...

We give an analysis of the stability and displacement error for linear and circular atomistic chains in the plane when the atomistic energy is approximated by the Cauchy-Born continuum energy and by the quasi-nonlocal atomistic-to-continuum coupling energy. We consider atomistic energies that include Lennard-Jones type nearest neighbor and next nea...

We describe an asymptotic model for the behavior of PET-like heat-shrinkable thin films that includes both membrane and bending energies when the thickness of the film is positive. We compare the model to Koiter’s shell model and to models in which a membrane energy or a bending energy are obtained by Γ-convergence techniques. We also provide compu...

This paper investigates the partial differential equation for the evolving distribution of prostate-specific antigen (PSA) levels following radiotherapy. We also present results on the behavior of moments for the evolving distribution of PSA levels and estimate the probability of long-term treatment success and failure related to values of treatmen...

A finite element approximation of the thin film limit for a sharp interface bulk energy for martensitic crystals is given. The energy density models the softening of the elastic modulus controlling the low-energy path from the cubic to the tetragonal lattice, the loss of stability of the tetragonal phase at high temperatures and the loss of stabili...

We give results for the -limit of a scaled elastic energy of a film as the thickness h > 0 converges to zero. The elastic energy density models materials with multiple phases or variants and is thus non-convex. The model includes an interfacial energy that allows sharp interfaces between the phases and variants and is proportional to the total vari...

Resistivity logs, as directly used for the determination of Water Saturation profiles, have always been of focal interest for the oil industry; it's clear that the quality of these measurements, currently used in the net pay and hydrocarbon-in-place determination, must be very high. As a consequence, more accurate and flexible resistivity tools hav...

We develop a free energy density to model a structural first-order phase transformation from a high-temperature cubic phase to a low-temperature tetragonal phase. The free energy density models the softening of the elastic modulus controlling the low-energy path from the cubic to the tetragonal lattice, the loss of stability of the tetragonal phase...

We develop a computational model for the martensitic first-order structural phase transformation in a single crystal thin film, and we use this model to study the effect of spatial compositional fluctuation, spatial temporal noise, and the loss of stability of the metastable phase at temperatures sufficiently far from the transformation temperature...

We study the approximation properties of piecewise constant functions with respect to triangular and rectangular finite elements in a metric defined on functions of bounded variation. We apply our results to a thin film model for martensitic crystals and to the approximation of deformations with microstructure. 1.

We rigorously derive a thin film limit for martensitic crystals that utilizes the total variation of the deformation gradient to model the energy on surfaces separating regions of di#erent variants. We find that the deformation for an infinitesimally thin film minimizes a two-dimensional energy.

We propose a computational model for a stress-induced martensitic phase transformation of a single-crystal thin film by indentation and its reverse transformation to austenite by heating. Our model utilizes a surface energy that allows sharp interfaces with finite energy and a penalty that forces the film to lie above the indenter and undergo a str...

We rigorously derive a thin film limit for martensitic crystals that utilizes the total variation of the deformation gradient to model the energy on surfaces separating regions of different variants. We find that the deformation for an infinitesimally thin film minimizes a two-dimensional energy.

Thesis (Ph. D.)--University of Minnesota, 2000. Includes bibliographical references (leaves 111-116)

In classical continuum mechanics a state of pure shear is defined as one for which there is some orthonormal basis relative to which the normal components of the Cauchy stress tensor vanish. An equivalent characterization is that the trace of the Cauchy stress tensor must vanish. We give an elementary but complete discussion of this fundamental the...

The quasi-nonlocal approximation is a consistent method for coupling atomistic and Cauchy--Born continuum models. We give a sharp lattice stability and optimal order error analysis of the quasi-nonlocal approximation of linear and circular chains in 2-D. Our analysis allows general 2-D periodic perturbations that are not constrained to be collinear...