Set Of Squares Research Articles

Expansion, random walks and sieving in $$S{L_2}({\mathbb{F}_p}[t])$$

We construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of \(S{L_2}({\mathbb{F}_p}[t])\) modulo certain square-free ideals. We describe some applications of our results to simple random walks on such subgroups, specifically giving bounds on the rate of escape of such walks from algebraic subvarieties, the set of squares and the set of elements with reducible characteristic polynomial in \(S{L_2}({\mathbb{F}_p}[t])\).

Online Square-into-Square Packing

In 1967, Moon and Moser proved a tight bound on the critical density of squares in squares: any set of squares with a total area of at most 1/2 can be packed into a unit square, which is tight. The proof requires full knowledge of the set, as the algorithmic solution consists in sorting the objects by decreasing size, and packing them greedily into shelves. Since then, the online version of the problem has remained open; the best upper bound is still 1/2, while the currently best lower bound is 1/3, due to Han et al. (Theory Comput Syst 43(1):38---55, 2008). In this paper, we present a new lower bound of 11/32, based on a dynamic shelf allocation scheme, which may be interesting in itself. We also give results for the closely related problem in which the size of the square container is not fixed, but must be dynamically increased in order to accommodate online sequences of objects. For this variant, we establish an upper bound of 3/7 for the critical density, and a lower bound of 1/8. When aiming for accommodating an online sequence of squares, this corresponds to a $$2.82{\ldots }$$2.82ź-competitive method for minimizing the required container size, and a lower bound of $$1.33{\ldots }$$1.33ź for the achievable factor.

An Improved Upper Bound on the Growth Constant of Polyominoes

Polyominoes are edge-connected sets of squares on the square lattice. The symbol λ usually denotes the growth constant of A(n), the sequence that enumerates polyominoes. In this paper we prove that λ≤4.5685 by analyzing the growth constant of a sequence B(n), for which B(n)≥A(n) for any value of n∈N. The recursive formula for B(n) is based on the representation of a polyomino as the assembly of a pair of smaller polyominoes and a code that describes the assembly. Then, an upper bound on the growth constant of B(n) is derived by a careful analysis of this assembly.

Porosity dependency of an optimized stent design for an intracranial aneurysm.

Optimal design of stents for a cerebral aneurysm is desired for efficient flow reduction in the aneurysm. In this study, we aimed to optimize stent design at several porosities, estimate the influence of stent design on aneurysm flow, and evaluate the ability of stents to reduce flow. Stent models were constructed as sets of squares or rectangles in the necks of a two-dimensional (2D) and realistic aneurysm. Then, automated optimization was performed using a combination of simulated annealing and lattice Boltzmann flow simulation. By simulated annealing, stents were gradually modified to reduce the average velocity in an aneurysm. As a result of optimization, stents of all porosities demonstrated an inhomogeneous distribution with dense struts in the inflow area. Flow reduction was increased compared with the initial stent. Under the condition of high porosity, flow reduction by the stent drastically increased as porosity decreased. Under low porosity, the increase of velocity reduction was moderate even as porosity decreased. Optimization can enhance flow reduction by stents. However, the increase in reduction associated with decreasing porosity is moderate under lower-porosity conditions. This threshold may help in the choice of stent porosity for each specific aneurysm.

Directed enzymatic activation of 1-D DNA tiles.

The tile assembly model is a Turing universal model of self-assembly where a set of square shaped tiles with programmable sticky sides undergo coordinated self-assembly to form arbitrary shapes, thereby computing arbitrary functions. Activatable tiles are a theoretical extension to the Tile assembly model that enhances its robustness by protecting the sticky sides of tiles until a tile is partially incorporated into a growing assembly. In this article, we experimentally demonstrate a simplified version of the Activatable tile assembly model. In particular, we demonstrate the simultaneous assembly of protected DNA tiles where a set of inert tiles are activated via a DNA polymerase to undergo linear assembly. We then demonstrate stepwise activated assembly where a set of inert tiles are activated sequentially one after another as a result of attachment to a growing 1-D assembly. We hope that these results will pave the way for more sophisticated demonstrations of activated assemblies.

On the General Erdős-Turán Conjecture

The general Erdős-Turán conjecture states that if A is an infinite, strictly increasing sequence of natural numbers whose general term satisfies an≤cn2, for some constant c>0 and for all n, then the number of representations functions of A is unbounded. Here, we introduce the function ψ(n), giving the minimum of the maximal number of representations of a finite sequence A={ak:1≤k≤n} of n natural numbers satisfying ak≤k2 for all k. We show that ψ(n) is an increasing function of n and that the general Erdős-Turán conjecture is equivalent to limn→∞ψ(n)=∞. We also compute some values of ψ(n). We further introduce and study the notion of capacity, which is related to the ψ function by the fact that limn→∞ψ(n) is the capacity of the set of squares of positive integers, but which is also of intrinsic interest.

Approximation of Spatio-Temporal Random Processes Using Tensor Decomposition

AbstractA new representation of spatio-temporal random processes is proposed in this work. In practical applications, such processes are used to model velocity fields, temperature distributions, response of vibrating systems, to name a few. Finding an efficient representation for any random process leads to encapsulation of information which makes it more convenient for a practical implementations, for instance, in a computational mechanics problem. For a single-parameter process such as spatial or temporal process, the eigenvalue decomposition of the covariance matrix leads to the well-known Karhunen-Loève (KL) decomposition. However, for multiparameter processes such as a spatio-temporal process, the covariance function itself can be defined in multiple ways. Here the process is assumed to be measured at a finite set of spatial locations and a finite number of time instants. Then the spatial covariance matrix at different time instants are considered to define the covariance of the process. This set of square, symmetric, positive semi-definite matrices is then represented as a third-order tensor. A suitable decomposition of this tensor can identify the dominant components of the process, and these components are then used to define a closed-form representation of the process. The procedure is analogous to the KL decomposition for a single-parameter process, however, the decompositions and interpretations vary significantly. The tensor decompositions are successfully applied on (i) a heat conduction problem, (ii) a vibration problem, and (iii) a covariance function taken from the literature that was fitted to model a measured wind velocity data. It is observed that the proposed representation provides an efficient approximation to some processes. Furthermore, a comparison with KL decomposition showed that the proposed method is computationally cheaper than the KL, both in terms of computer memory and execution time.

Factor-set of binary matrices and Fibonacci numbers

The article discusses the set of square n×n binary matrices with the same number of 1’s in each row and each column. An equivalence relation on this set is introduced. Each binary matrix is represented using ordered n-tuples of natural numbers. We are looking for a formula which calculates the number of elements of each factor-set by the introduced equivalence relation. We show a relationship between some particular values of the parameters and the Fibonacci sequence.

Packing cubes into a cube is NP-complete in the strong sense

While the problem of packing two-dimensional squares into a square, in which a set of squares is packed into a big square, has been proved to be NP-complete, the computational complexity of the d-dimensional ( $$ d\ge 3 $$ d ? 3 ) problems of packing hypercubes into a hypercube remains an open question (Acta Inf 41(9):595---606, 2005; Theor Comput Sci 410(44):4504---4532, 2009). In this paper, the authors show that the three-dimensional problem version of packing cubes into a cube is NP-complete in the strong sense.

P-Adic dynamics and angle doubling

We consider the squaring map over the p-adic numbers for an odd prime p, and study its symbolic dynamics on the unit circle in ℤp, the p-adic integers. When the map is restricted to the set of squares, we show an equivalence to angle doubling (mod 1) for rational angles. For primes p ≡ 3 (mod 4), this map may be represented as a unitary permutation matrix of the type used in quantum phase estimation.

Cumulative quantum mechanics (CQM). Part I: Prerequisites and fundamentals of CQM

There is proposed a formulation of the fundamentals of cumulative quantum mechanics (CQM) that allows one to describe the resonance cos-waves with the ψ n function of an electron (ψ n (r) ∼ cos(k n r)/r k ) unlimited (with k ≠ 0) in the resonator center in hollow quantum resonators with any type of symmetry (plane—k = 0, spherical—k = 1, and cylindrical—k = 0.5). Irregular in the center of a resonator, the cos solutions are regularized in the resonator center by the geometric normalization coefficient corresponding to the symmetry type and being χ(r) = 2 k π1/2 r k at k ≠ 0 (if k = 0, then χ = 1). The stratification of the probability of finding the particle in the quantum resonator volume is similarly determined by the energy of a particle or a full set of squares of the corresponding quantum numbers ((n − 1/2)2 for cos-waves and n 2 for sin-waves) for any type of the resonator symmetry. An analytical CQM model of the polarization resonance electron capture (dynamic localization due to the self-formation of a potential barrier cumulating this electron into a molecule) is proposed. When there occurs the polarization capture of an electron by the allotropic hollow forms of carbon (fullerenes and nanotubes), the electron energy E n > 0. The problem on the Vysikaylo polarization effect of the first type (or the problem on polarization cumulation of the de Broglie waves of electrons with characteristic dimension of ∼ 1 nm) is reduced to the problem of G.A. Gamov: “a quantum particle in a box with a potential barrier on its boundary.” The energy spectrum of states localized by the barrier, E n > 0 (a metastable IQ particle is a partially open quantum dot, line, or pit), as in the case of E n < 0 (a stable FQ particle is a closed quantum dot, line, or pit) is determined by the effective internal dimensions of a box (R + r ind) with the polarization forces effectively acting at a distance r ind from the polarizable molecule. The CQM allows one to describe with E n > 0 both the limited cumulation of ψn(r)-functions for de Broglie-Fresnel generalized interference and the unlimited cumulation of ψn(r)-functions to the center of a quantum resonator with Vysikaylo-de Broglie-Fraunhofer generalized interference in hollow polarizable spherically or cylindrically symmetric quantum resonators for de Broglie electron waves. In the framework of the CQM, there have been analytically calculated the eigen quantum pairs: ψ n (r)-functions, respectively stratified profiles of the probability of a particle’s location in the resonator cavity, W n (r) and E n > 0, the eigen energies of the electrons localized in the quantum resonator (C60 and C70, etc.) by polarization forces. It is proved that, alongside with the classical energy spectrum for asymmetric ψ n -functions (sin-waves) with E n ∼ n 2 for hollow quantum resonators, there exist quantum resonances for symmetric ψ n -functions (cos-waves) with E n ∼ (n − 1/2)2, which can be realized in the experiments.

Hilbert cubes in progression-free sets and in the set of squares

We present a new method to give upper bounds on the dimension of Hilbert cubes in certain sets. As a special case we improve Hegyvari and Sarkozy’s upper bound O((logN)1/3) for the maximal dimension of a Hilbert cube in the set of squares in [1,N] to O((log logN)2).

A set of squares without arithmetic progressions

There is a subset of the first N squares which has > cN/ √ log log N elements and contains no three-term arithmetic progression.

Clipping Distortion Performance of Nonsquare $M$-QAM OFDM Systems on Nonlinear Time-Variant Channels

Orthogonal frequency-division multiplexing (OFDM) systems usually make use of a set of square M quadrature-amplitude modulation ( M-QAM) constellations to obtain a good tradeoff between throughput and symbol-error robustness. However, the switch to the next constellation increases the number of bits per modulation symbol by two. The introduction of nonsquare M -QAM constellations in such systems brings extra advantages such as smoother transition among bit rates and a reduction of the peak-to-average ratio of the OFDM signal. Therefore, this paper unfolds analytical expressions to evaluate the symbol-error performance of nonsquare M-QAM OFDM on nonlinear time-variant additive white Gaussian noise channels, taking clipping distortion into account. Cross and overlaid M-QAM are considered. Analytical performances are evaluated and compared with computational simulations, which show good agreement.

Particle Filter with State Permutations for Solving Image Jigsaw Puzzles.

We deal with an image jigsaw puzzle problem, which is defined as reconstructing an image from a set of square and non-overlapping image patches. It is known that a general instance of this problem is NP-complete, and it is also challenging for humans, since in the considered setting the original image is not given. Recently a graphical model has been proposed to solve this and related problems. The target label probability function is then maximized using loopy belief propagation. We also formulate the problem as maximizing a label probability function and use exactly the same pairwise potentials. Our main contribution is a novel inference approach in the sampling framework of Particle Filter (PF). Usually in the PF framework it is assumed that the observations arrive sequentially, e.g., the observations are naturally ordered by their time stamps in the tracking scenario. Based on this assumption, the posterior density over the corresponding hidden states is estimated. In the jigsaw puzzle problem all observations (puzzle pieces) are given at once without any particular order. Therefore, we relax the assumption of having ordered observations and extend the PF framework to estimate the posterior density by exploring different orders of observations and selecting the most informative permutations of observations. This significantly broadens the scope of applications of the PF inference. Our experimental results demonstrate that the proposed inference framework significantly outperforms the loopy belief propagation in solving the image jigsaw puzzle problem. In particular, the extended PF inference triples the accuracy of the label assignment compared to that using loopy belief propagation.

On packing squares into a rectangle

We prove that every set of squares with total area 1 can be packed into a rectangle of area at most 2867 / 2048 = 1.399 … . This improves on the previous best bound of 1.53. Also, our proof yields a linear time algorithm for finding such a packing.

Bruegel's Statistical Signature

![Figure][1] Authentic Bruegel / Imitation Bruegel CREDIT: MET Beauty may be in the eye of the beholder, but a painting's authenticity may be encoded in patterns a computer can detect. To show that, mathematician Daniel Rockmore and colleagues at Dartmouth College applied a technique from neuroscience called sparse coding to analyze digitized gray-scale images of 16th century paintings by Pieter Bruegel the Elder and imitations of his work. They fed tiny square patches of the images through an algorithm to modify and “train” a different set of squares that started out randomly shaded. The trained squares could then be superimposed on each other to reconstruct any patch from a painting. Crucially, the algorithm patterned the squares so that a few would suffice to reproduce a patch from a real Bruegel. If the team trained the patterns using any seven of eight real Bruegels, then, on average, the number needed to recreate a random patch from the eighth was smaller than the number needed to fit a patch from any of five fakes (with one exception). So the scheme distinguishes real Bruegels from fakes, the team reported last week online in the Proceedings of the National Academy of Sciences . The technique would be “just one tool” to help determine a painting's authenticity, Rockmore cautions. James Coddington, chief conservator at the Museum of Modern Art in New York City, says its real utility may lie in comparing the works of one or several artists. “Ask the art historians—is it telling us something that we already knew, or is it giving us new food for thought?” [1]: pending:yes

The harmonic oscillator in quantum mechanics: A third way

Courses on undergraduate quantum mechanics usually focus on solutions of the Schrödinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced, only academic examples are used, such as the matrix representation of Dirac’s raising and lowering operators or the angular momentum operators. We introduce some of the same one-dimensional examples as matrix diagonalization problems, with a basis that consists of the infinite set of square well eigenfunctions. Undergraduate students are well equipped to handle such problems in familiar contexts. We pay special attention to the one-dimensional harmonic oscillator. This paper should equip students to obtain the low lying bound states of any one-dimensional short range potential.

An $L^1$ ergodic theorem for sparse random subsequences

We prove an L^1 subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of universally L^1-good sequences nearly as sparse as the set of squares. In the process, we prove that a certain deterministic condition implies a weak maximal inequality for a sequence of \ell^1 convolution operators.

A group structure on squares

We show that there is an abelian group structure on the orbit set of squares of unimodular rows of length n over a commutative ring of stable dimension d when d = 2 n - 3, n odd and also an abelian group structure on the orbit set of fourth powers of unimodular rows of length n over a commutative ring of stable dimension d when d = 2n - 3, n even.