arXiv subjects
Publications and source records attributed to József Vass.
The purpose of this paper is to introduce a curious function of two variables, expressable via the employment of the Lambert W Funtion, which can be generalized to satisfy Euler's Equation of Inviscid Motion over a specific domain, with pressure independent of space variables.
The problem of finding the convex hull of an IFS fractal is relevant in both theoretical and computational settings. Various methods exist that approximate it, but our aim is its exact determination. The finiteness of extremal points is examined a priori from the IFS parameters, revealing some cases when the convex hull problem is solvable. Former results are detailed from the literature, and two new methods are introduced and crystallized for practical applicability -- one more general, the other more efficient. Focal periodicity in the address of extremal points emerges as the central idea.
A discretization scheme is introduced for a set of convection-diffusion equations with a non-linear reaction term, where the convection velocity is constant for each reactant. This constancy allows a transformation to new spatial variables, which ensures the global stability of discretization. Convection-diffusion equations are notorious for their lack of stability, arising from the algebraic interaction of the convection and diffusion terms. Unexpectedly, our implemented numerical algorithm proves to be faster than computing exact solutions derived for a special case, while remaining reasonably accurate, as demonstrated in our runtime and error analysis.
Kinetic rate constants fundamentally characterize the dynamics of the chemical interaction of macromolecules, and thus their study sets a major direction in experimental biochemistry. The estimation of such constants is often challenging, partly due to the noisiness of data, and partly due to the theoretical framework. Novel and qualitatively reasonable methods are presented for the estimation of the rate constants of complex formation and dissociation in Kinetic Capillary Electrophoresis (KCE). This also serves the broader effort to resolve the inverse problem of KCE, where these estimates pose as initial starting points in the non-linear optimization space, along with the asymmetric Gaussian parameters describing the injected plug concentration profiles, which is also hereby estimated. This rate constant estimation method is also compared to an earlier one.
Determining kinetic rate constants is a highly relevant problem in biochemistry, so various methods have been designed to extract them from experimental data. Such methods have two main components: the experimental apparatus and the subsequent analysis, the latter often dependent on mathematical theory. Thus the theoretical approach taken influences the effectiveness of constant determination. A computational inverse problem approach is hereby presented, which does not merely give a single rough approximation of the sought constants, but is inherently capable of determining them from exact signals to arbitrary accuracy. This approach is thus not merely novel, but opens a whole new category of solution approaches in the field, enabled primarily by an efficient direct solver.
The purpose of this paper is to introduce basic concepts that are fundamental in the examination of composite moduli, while avoiding the notoriously difficult problem of prime-factorization. We introduce a new class of numbers, called idempotent numbers, that is unavoidable when researching composite moduli. Among many interesting results, we give generalizations of well-known theorems and definitions, such as the Euler-Fermat Theorem and the concept of primitive roots. We consider the generalization of the equivalence condition for the solvability of a binomial congruence to be the main result of our paper.
A necessary and sufficient condition is provided for the solvability of a binomial congruence with a composite modulus, circumventing its prime factorization. This is a generalization of Euler's Criterion through that of Euler's Theorem, and the concepts of order and primitive roots. Idempotent numbers play a central role in this effort.
Euclidean triangles and IFS fractals seem to be disparate geometrical concepts, unless we consider the Sierpiński gasket, which is a self-similar collection of triangles. The "circumcircle" hints at a direct link, as it can be derived for three-map IFS fractals in general, defined in an Apollonian manner. Following this path, one may discover a broader relationship between polygons and IFS fractals.
Fully Developed Turbulence (FDT) is a theoretical asymptotic phenomenon which can only be approximated experimentally or computationally, so its defining characteristics are hypothetical. It is considered to be a chaotic stationary flow field, with self-similar fractalline features. A number of approximate models exist, often exploiting this self-similarity. The idealized mathematical model of Fractal Potential Flows is hereby presented, and linked philosophically to the phenomenon of FDT on a free surface, based on its experimental characteristics. The model hinges on the recursive iteration of a fluid dynamical transfer operator. The existence of its unique attractor - called the invariant flow - is shown in an appropriate function space, which will serve as our suggested model for the FDT flow field. Its sink singularities are shown to form an IFS fractal, explicitly resolving Mandelbrot's Conjecture. Meanwhile an isometric isomorphism is defined between flows and probability measures, hinting at a wealth of future research. The inverse problem of representing turbulent flow fields with this model is discussed in closing, along with explicit practical considerations for experimental verification and visualization.
IFS fractals - the attractors of Iterated Function Systems - have motivated plenty of research to date, partly due to their simplicity and applicability in various fields, such as the modeling of plants in computer graphics, and the design of fractal antennas. The statement and resolution of the Fractal-Line Intersection Problem is imperative for a more efficient treatment of certain applications. This paper intends to take further steps towards this resolution, building on the literature. For the broad class of hyperdense fractals, a verifiable condition guaranteeing intersection with any line passing through the convex hull of a planar IFS fractal is shown, in general R^d for hyperplanes. The condition also implies a constructive algorithm for finding the points of intersection. Under certain conditions, an infinite number of approximate intersections are guaranteed, if there is at least one. Quantification of the intersection is done via an explicit formula for the invariant measure of IFS.