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G. Domokos

Publications and source records attributed to G. Domokos.

At least 19 recordsLinked to original sources
arXiv2024

Decoding planetary surfaces by counting cracks

Planets are often covered with thin cracked shells. From mud films to lithospheres of rock or ice, fracture networks form two-dimensional (2D) tessellations of convex polygons whose geometry encodes their genesis. Here we chart the geometry of 2D fracture mosaics across the solar system, and decode their formative conditions using a new dynamical crack model. We show that mosaics can be projected onto a Symbolic Ternary Diagram, where the relative proportions of ``T'', ``X'' and ``Y'' junctions are uniquely related to contributions from distinct modes of fracture. Most planetary mosaics cluster in a region associated with hierarchical fracture networks, where sequential cracking favors formation of T junctions. Exceptions to this rule may betray the presence of water. Europa's fracture networks stand apart due to the predominance of X junctions; this is a special feature of ice, where healing of cracks by refreezing of water allows new fractures to overprint older ones. Several fracture networks on Mars appear as outliers due the high proportion of Y junctions. These patterns -- previously interpreted as ancient mudcracks and frozen polar terrain, based on geological evidence -- are consistent with the twisting of crack junctions by cyclic volume change. Our findings suggest that counting cracks could aid in the identification of other water-influenced planetary environments.

physics.geo-ph↗
arXiv2021

A characterization of the symmetry groups of mono-monostatic convex bodies

Answering a question of Conway and Guy in a 1968 paper, Lángi in 2021 proved the existence of a monostable polyhedron with $n$-fold rotational symmetry for any $n \geq 3$, and arbitrarily close to a Euclidean ball. In this paper we strengthen this result by characterizing the possible symmetry groups of all mono-monostatic smooth convex bodies and convex polyhedra. Our result also answers a stronger version of the question of Conway and Guy, asked in the above paper of Lángi.

math.MG↗
arXiv2019

On some average properties of convex mosaics

In a convex mosaic in $\mathbb{R} ^d$ we denote the average number of vertices of a cell by $\bar v$ and the average number of cells meeting at a node by $\bar n$. Except for the $d=2$ planar case, there is no known formula prohibiting points in any range of the $[\bar n, \bar v]$ plane (except for the unphysical $\bar n, \bar v < d+1$ strips). Nevertheless, in $d=3$ dimensions if we plot the 28 points corresponding to convex uniform honeycombs, the 28 points corresponding to their duals and the 3 points corresponding to Poisson-Voronoi, Poisson-Delaunay and random hyperplane mosaics, then these points appear to accumulate on a narrow strip of the $[\bar n, \bar v]$ plane. To explore this phenomenon we introduce the harmonic degree $\bar h= \bar n\bar v/(\bar n + \bar v)$ of a $d$-dimensional mosaic. We show that the observed narrow strip on the $[\bar n, \bar v]$ plane corresponds to a narrow range of $\bar h$. We prove that for every $\bar h^{\star} \in (d, 2^{d-1}]$ there exists a convex mosaic with harmonic degree $\bar h^{\star}$ and we conjecture that there exist no $d$-dimensional mosaic outside this range. We also show that the harmonic degree has deeper geometric interpretations. In particular, in case of Euclidean mosaics it is related to the average of the sum of vertex angles and their polars, and in case of 2D mosaics, it is related to the average excess angle.

math.MG↗
arXiv2014

On the monotonicity of spatial critical points evolving under curvature-driven flows

We describe the variation of the number $N(t)$ of spatial critical points of smooth curves (defined as a scalar distance $r$ from a fixed origin $O$) evolving under curvature-driven flows. In the latter, the speed $v$ in the direction of the surface normal may only depend on the curvature $κ$. Under the assumption that only generic saddle-node bifurcations occur, we show that $N(t)$ will decrease if the partial derivative $v_κ$ is positive and increase if it is negative (Theorem 1). Justification for the genericity assumption is provided in Section 5. For surfaces embedded in 3D, the normal speed $v$ under curvature-driven flows may only depend on the principal curvatures $κ, λ$. Here we prove the weaker (stochastic) Theorem 2 under the additional assumption that third-order partial derivatives can be approximated by random variables with zero expected value and covariance. Theorem 2 is a generalization of a result by Kuijper and Florack for the heat equation. We formulate a Conjecture for the case when the reference point coincides with the centre of gravity and we motivate the Conjecture by intermediate results and an example. Since models for collisional abrasion are governed by partial differential equations with $v_κ,v_λ>0$, our results suggest that the decrease of the number of static equilibrium points is characteristic of some natural processes.

math.AP↗
arXiv2014

How river rocks round: resolving the shape-size paradox

River-bed sediments display two universal downstream trends: fining, in which particle size decreases; and rounding, where pebble shapes evolve toward ellipsoids. Rounding is known to result from transport-induced abrasion; however many researchers argue that the contribution of abrasion to downstream fining is negligible. This presents a paradox: downstream shape change indicates substantial abrasion, while size change apparently rules it out. Here we use laboratory experiments and numerical modeling to show quantitatively that pebble abrasion is a curvature-driven flow problem. As a consequence, abrasion occurs in two well-separated phases: first, pebble edges rapidly round without any change in axis dimensions until the shape becomes entirely convex; and second, axis dimensions are then slowly reduced while the particle remains convex. Explicit study of pebble shape evolution helps resolve the shape-size paradox by reconciling discrepancies between laboratory and field studies, and enhances our ability to decipher the transport history of a river rock.

physics.geo-ph↗
arXiv2013

Geometrical and physical models of abrasion

We extend the geometrical theory presented in [5] for collisional and frictional particle abrasion to include an independent physical equation for the evolution of mass and volume. We introduce volume weight functions as multipliers of the geometric equations and use these mutipliers to enforce physical volume evolution in the unified equations. The latter predict, in accordance with Sternberg's Law, exponential decay for volume evolution. We describe both the PDE versions, which are generalisations of Bloore's equations and their heuristic ODE approximations, called the box equations. The latter are suitable for tracking the collective abrasion of large particle populations. The mutual abrasion of identical particles, called the self-dual ows, play a key role in explaining geological scenarios. We give stability criteria for the self-dual ows in terms of the parameters of the physical volume evolution models and show that under reasonable assumptions these criteria can be met by physical systems. We also study a natural generalisation, the unidirectional Bloore equation, covering the case of unidirectional abrasion. We have previously shown that his equation admits travelling front solutions with circular profiles. More generally, in three dimensions, they are so-called linear or special Weingarten surfaces.

physics.geo-ph↗
arXiv2013

The robustness of equilibria on convex solids

We examine the minimal magnitude of perturbations necessary to change the number $N$ of static equilibrium points of a convex solid $K$. We call the normalized volume of the minimally necessary truncation robustness and we seek shapes with maximal robustness for fixed values of $N$. While the upward robustness (referring to the increase of $N$) of smooth, homogeneous convex solids is known to be zero, little is known about their downward robustness. The difficulty of the latter problem is related to the coupling (via integrals) between the geometry of the hull $\bd K$ and the location of the center of gravity $G$. Here we first investigate two simpler, decoupled problems by examining truncations of $\bd K$ with $G$ fixed, and displacements of $G$ with $\bd K$ fixed, leading to the concept of external \rm and internal \rm robustness, respectively. In dimension 2, we find that for any fixed number $N=2S$, the convex solids with both maximal external and maximal internal robustness are regular $S$-gons. Based on this result we conjecture that regular polygons have maximal downward robustness also in the original, coupled problem. We also show that in the decoupled problems, 3-dimensional regular polyhedra have maximal internal robustness, however, only under additional constraints. Finally, we prove results for the full problem in case of 3 dimensional solids. These results appear to explain why monostatic pebbles (with either one stable, or one unstable point of equilibrium) are found so rarely in Nature.

math.MG↗
arXiv2012

Circular, stationary profiles emerging in unidirectional abrasion

We describe a PDE model of bedrock abrasion by impact of moving particles and show that by assuming unidirectional impacts the modification of a geometrical PDE due to Bloore exhibits circular arcs as solitary profiles. We demonstrate the existence and stability of these stationary, travelling shapes by numerical experiments based on finite difference approximations. Our simulations show that, depending on initial profile shape and other parameters, these circular profiles may evolve via long transients which, in a geological setting, may appear as non-circular stationary profiles.

physics.geo-ph↗
arXiv2012

The evolution of pebble size and shape in space and time

We propose a mathematical model which suggests that the two main geological observations about shingle beaches, i.e. the emergence of predominant pebble size ratios and strong segregation by size are interrelated. Our model is a based on a system of ODEs called the box equations, describing the evolution of pebble ratios. We derive these ODEs as a heuristic approximation of Bloore's PDE describing collisional abrasion. While representing a radical simplification of the latter, our system admits the inclusion of additional terms related to frictional abrasion. We show that nontrivial attractors (corresponding to predominant pebble size ratios) only exist in the presence of friction. By interpreting our equations as a Markov process, we illustrate by direct simulation that these attractors may only stabilized by the ongoing segregation process.

physics.geo-ph↗
arXiv2012

Spacetime interpretation of Torsion in Prismatic Bodies

A non-linear theory for the plastic deformation of prismatic bodies is constructed which interpolates between Prandtl's linear soap-film approximation and Nádai's sand-pile model . Geometrically Prandtl's soap film and Nádai's wavefront are unified into a single smooth surface of constant mean curvature in three-dimensional Minkowski spacetime. Although the theory is non-linear, a general solution may be given in terms of a freely specifiable holomorphic function of a single complex variable.

gr-qc↗
arXiv2009

Formation of sharp edges and planar areas of asteroids by polyhedral abrasion

While the number of asteroids with known shapes has drastically increased over the past few years, little is known on the the time-evolution of shapes and the underlying physical processes. Here we propose an averaged abrasion model based on micro-collisons, accounting for asteroids not necessarily evolving toward regular spheroids, rather (depending on the fall-back rate of ejecta) following an alternative path, thus confirming photometry-derived features, e.g. existence of large, relatively flat areas separated by edges. We show that our model is realistic, since the bulk of the collisions falls into this category.

astro-ph.EP↗
arXiv2005

TeV String Theories, Mini Black Holes and Trans-GZK Cosmic Rays

We review the proposal that trans-GZK cosmic ray interactions are caused by neutrino primaries. The primaries cause excitations of strings and give rise to extensive air showers (EAS) resembling EAS induced by nuclei. We also show that in ``low scale'' string models (of characteristic energy about 70TeV) the excited string and the mini black hole pictures are equivalent.

hep-ph↗
arXiv2004

Strings, Black Holes and the Extreme Energy Cosmic Rays

In a large class of models the string excitation and black hole pictures invoked as an explanation of trans-GZK cosmic ray events are equivalent. Single particle inclusive distributions are thermal at the Hagedorn temperature. The hadron multiplicities are reminiscent of multiplicities in heavy nucleus initiated interactions.

hep-ph↗
arXiv2003

What Can we learn from Cosmic Rays?

Ultra high energy cosmic rays (UHECR) pose a problem either for particle physics or for astrophysics (or for both) by the unexpectedly high number of cosmic ray showers observed with energy above about 5x10^{19}eV, the Greisen-Zatsepin-Kuzmin (GZK) cutoff. Our emphasis is on those possible solutions of the puzzle which assume that ultra high energy neutrinos travel cosmic distances. We present, in detail, a model which is based on a low energy (50 to 100 TeV) transition to a higher than four dimensional string regime. Neutrino-quark cross sections grow exponentially close to the threshold of this new scale because of the fast increase of the density of string states and effectively acquire hadronic strength.

hep-ph↗
arXiv2003

Countintg Extra Dimensions: Magnetic Cherenkov Radiation from High Energy Neutrinos

In theories which require a space of dimension d>4, there is a natural mechanism of suppressing neutrino masses: while Standard Model fields are confined to a 3-brane, right handed neutrinos live in the bulk. Due to Kaluza-Klein excitations, the effective magnetic moments of neutrinos are enhanced. The effective magnetic moment is a monotonically growing function of the energy of the neutrino: consequently, high energy neutrinos can emit observable amounts of magnetic Cherenkov radiation. By observing the energy dependence of the magnetic Cherenkov radiation, one may be able to determine the number of compactified dimensions.

hep-ph↗
arXiv2002

Low Scale String Unification and the Highest Energy Cosmic Rays

String unification at a scale of a few tens of TeV explains the existence of cosmic ray interactions beyond the Greisen-Zatsepin-Kuzmin (GZK) cutoff. Trans-GZK cosmic rays are neutrinos which can penetrate the cosmic microwave background. In interactions with atmospheric nuclei they have sufficient energy for exciting string modes. We present a model for the description of such interactions and discuss the properties of the resulting extensive air showers. Presently available data on trans-GZK cosmic rays suggest a string scale around 80 TeV.

hep-ph↗
arXiv2001

Particle Candidates of Ultrahigh Energy Cosmic Rays

We discuss candidates for trans-GZK cosmic rays observed in a variety of detectors. Three types of primaries are represented among the abstracts submitted to this meeting: neutrin os causing a Z-burst, protons arising from the decay of ultra-heavy metastable particles and neutrinos within the framework of low scale string-like models of unification. We attempt to evaluate the relative merits of these schemes. No definite conclusion can be reached at this time. However, we point out that some schemes are more credible/predictive than others. Data to be gathered by the Pierre Auger observatories as well as orbiting detectors (OWL, Airwatch...) should be able to decide between the various schemes.

hep-ph↗