Related papers: Unavoidable golden ratio
Some simple facts are proved ruling the Collatz tree and the chains of vertices appearing in it, leading to the reduction of the number of significant elements appearing in the tree. Although the Collatz conjecture remains open, these fact…
Geometrically frustrated solids with non-Euclidean reference metric are ubiquitous in biology and are becoming increasingly relevant in technological applications. Often they acquire a targeted con- figuration of incompatibility through…
Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…
It is argued that the occurrence of disproportionately ("un-natural") large (or small) numbers, as well as deep cancellations, are comparatively natural traits of the way Nature is geared to operate in most complex systems. The idea is…
We survey structures endowed with natural partial orderings and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism order…
The aim of this paper is to present a recursive construction of simple t-designs for arbitrary t. The construction is of purely combinatorial nature and it requires finding solutions for the indices of the ingredient designs that satisfy a…
It is generally believed that major galaxy mergers with mass ratios in the range 1:1-3:1 result in remnants that have properties similar to elliptical galaxies, and minor mergers below 10:1 result in disturbed spiral galaxies. The…
Among the key parameters defining the ISM of galaxies is the fraction of the metals that are locked up in dust: the metals-to-dust ratio. This ratio bears not only on the ISM and its evolution, but particularly on the origin of cosmic dust.…
We identify and explore the intriguing property of resource resonance arising within resource theories of entanglement, coherence and thermodynamics. While the theories considered are reversible asymptotically, the same is generally not…
Let $\Gamma$ be a group of type rotating automorphisms of a building $\cB$ of type $\widetilde A_2$, and suppose that $\Gamma$ acts freely and transitively on the vertex set of $\cB$. The apartments of $\cB$ are tiled by triangles, labelled…
Complex spatiotemporal patterns in nature significantly challenge reductionism-based modern science. The lack of a paradigm beyond reductionism hinders our understanding of the emergence of complexity. The diversity of countless patterns…
We compare the light extinction spectra of elongated gold nanoparticles with different shapes (cylinder, spherocylinder and ellipsoid) and sizes of 10 to 100~nm. We argue that the equivalence of the various moments of mass distribution is…
We perform a systematic study of various versions of massive gravity with and without violation of Lorentz symmetry in arbitrary dimension. These theories are well known to possess very unusual properties, unfamiliar from studies of gauge…
A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire subcritical phase $(\gamma < \sqrt{2d})$ and…
The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures finding optimal solutions by means…
Recently, a new concept, primitive chaos, was proposed, as a concept closely related to the fundamental problems of physics itself such as determinism, causality, free will, predictability, and irreversibility [J. Phys. Soc. Jpn. {\bf 79},…
Granular materials such as sand, powders, and grains are omnipresent in daily life, industrial applications, and earth-science [1]. When unperturbed, they form stable structures that resemble the ones of other amorphous solids like metallic…
The rate at which giant planets accumulate solids and gas is a critical component of planet formation models, yet it is extremely challenging to predict from first principles. Characterizing the heavy element (everything other than hydrogen…
A new angle of view is proposed to find the simple rules dominating complex systems and regular patterns behind random phenomena such as cities. Hierarchy of cities reflects the ubiquitous structure frequently observed in the natural world…
One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…