Related papers: Cyclic Oritatami Systems Cannot Fold Infinite Frac…
Accurately predicting experimentally realizable 3D molecular crystal structures from their 2D chemical graphs is a long-standing open challenge in computational chemistry called crystal structure prediction (CSP). Efficiently solving this…
In many communication contexts, the capabilities of the involved actors cannot be known beforehand, whether it is a cell, a plant, an insect, or even a life form unknown to Earth. Regardless of the recipient, the message space and time…
The importance of nonlinearities in material constitutive relations has long been appreciated in the continuum mechanics of macroscopic rods. Although the moment (torque) response to bending is almost universally linear for small deflection…
The Path Contraction and Cycle Contraction problems take as input an undirected graph $G$ with $n$ vertices, $m$ edges and an integer $k$ and determine whether one can obtain a path or a cycle, respectively, by performing at most $k$ edge…
A comprehensive model of natural language processing in the brain must accommodate four components: representations, operations, structures and encoding. It further requires a principled account of how these components mechanistically, and…
This paper gives a (polynomial time) algorithm to decide whether a given Discrete Self-Similar Fractal Shape can be assembled in the aTAM model.In the positive case, the construction relies on a Self-Assembling System in the aTAM which…
We develop a formalism for the calculation of the frequency band structure of a phononic crystal consisting of non-overlapping elastic spheres, characterized by Lam\'e coefficients which may be complex and frequency dependent, arranged…
Criticality is traditionally regarded as an unstable, fine-tuned fixed point of the renormalization group. We introduce an iterative bicolored percolation process in two dimensions and show that it can both preserve criticality and…
To help understand the underlying mechanisms of neural networks (NNs), several groups have, in recent years, studied the number of linear regions $\ell$ of piecewise linear functions generated by deep neural networks (DNN). In particular,…
A random overlap structure (ROSt) is a measure on pairs (X,Q) where X is a locally finite sequence in the real line with a maximum and Q a positive semidefinite matrix of overlaps intrinsic to the particles X. Such a measure is said to be…
We have proposed the construction of optical quantum computer (OQC) on regular domain structure (RDS) crystal. By using RDS crystal, we can perform all the logical operations on one RDS crystal. Moreover, RDS crystals are parctically…
In this paper we consider the problem of RNA folding with pseudoknots. We use a graphical representation in which the secondary structures are described by planar diagrams. Pseudoknots are identified as non-planar diagrams. We analyze the…
Consider an oriented curve $\Gamma$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol R^3$. This can be expressed as the image of an…
All constructive methods employed in modern mathematics produce only countable sets, even when designed to transcend countability. We show that any constructive argument for uncountability -- excluding diagonalization techniques --…
Fractal geometry, defined by self-similar patterns across scales, is crucial for understanding natural structures. This work addresses the fractal inverse problem, which involves extracting fractal codes from images to explain these…
New tilings of certain subsets of $\mathbb{R}^{M}$ are studied, tilings associated with fractal blow-ups of certain similitude iterated function systems (IFS). For each such IFS with attractor satisfying the open set condition, our…
We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible $0$-form symmetries. These symmetries can arise as…
We provide a general framework to construct fractal interpolation surfaces (FISs) for a prescribed countably infinite data set on a rectangular grid. Using this as a crucial tool, we obtain a parameterized family of bivariate fractal…
Our understanding of vibrations in solids currently rests on concepts and techniques designed for crystals and explicitly relying on periodicity, hence inapplicable to amorphous materials. As a consequence, no established framework enables…
For self-similar fractals, the Minkowski content and fractal curvature have been introduced as a suitable limit of the geometric characteristics of its parallel sets, i.e., of uniformly thin coatings of the fractal. For some self-conformal…