Sequence Of Squares Research Articles

Packing batches of items into a single bin

In λ-packing items are grouped in batches. Items arrive one by one (online) and they are stored in a buffer until either the total volume of stored items is greater than or equal to λ or all items have already arrived. Then items from the buffer are packed offline into a unit capacity bin and the buffer is emptied. We show that any sequence of squares with total area not greater than 1/2 can be 1/4-packed into a single bin (a unit square).

The Conceptual Design of the Octagon at Thessaloniki

Numerous theoretical designs based on the geometrical shape of ad quadratum, sequences of squares or/and octagons have been used to speculate for the existence of archetype patterns in architecture. However, up to now no archaeological evidence of the existence of any such pattern has been documented. The present work examines whether a diagram incised on stone found in the Octagon in Thessaloniki could be a first evidence of a conceptual design in architecture. Two statistically validated models are presented, one to explain the geometrical patterns on the Octagon diagram and another to describe the Octagon’s ground plan. The results are anticipated to contribute not only to the discussion concerning the function of the Octagon monument, but also to the debate concerning the role of type in architecture and the origin of form in central-plan and octagonal buildings.

No evidence for an attentional bias towards implicit temporal regularities

Action and perception are optimized by exploiting temporal regularities, and it has been suggested that the attentional system prioritizes information that contains some form of structure. Indeed, Zhao, Al-Aidroos, and Turk-Browne (Psychological Science, 24(5), 667–677, 2013) found that attention was biased towards the location and low-level visual features of shapes that appeared with a regular order but were irrelevant for the main search task. Here, we investigate whether this bias also holds for irrelevant metrical temporal regularities. In six experiments, participants were asked to perform search tasks. In Experiments 1a–d, sequences of squares, each presented at one of four locations, appeared in between the search trials. Crucially, in one location, the square appeared with a regular rhythm, whereas the timing in the other locations was random. In Experiments 2a and 2b, a sequence of centrally presented colored circles was shown in between the search trials, of which one specific color appeared regularly. We expected that, if attention is automatically biased towards these temporal regularities, reaction times would be faster if the target matches the location (Experiments 1a–d) or color (Experiments 2a–b) of the regular stimulus. However, no reaction time benefit was observed for these targets, suggesting that there was no attentional bias towards the regularity. In addition, we found no evidence for attentional entrainment to the rhythmic stimulus. These results suggest that people do not use implicit rhythmic temporal regularities to guide their attention in the same way as they use order regularities.

Büchi’s Problem in Modular Arithmetic for Arbitrary Quadratic Polynomials

AbstractGiven a prime $p\geqslant 5$ and an integer $s\geqslant 1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^{s}$, such that $f$ is not a square, if a sequence $(f(1),\ldots ,f(N))$ is a sequence of squares, then $N$ is at most $M$. We also provide some explicit formulas for the optimal $M$.

On the properties of k-balancing and k-Lucas-balancing numbers

The k-Lucas-balancing numbers are obtained from a special sequence of squares of k-balancing numbers in a natural form. In this paper, we will study some properties of k-Lucas-balancing numbers and establish relationship between these numbers and k-balancing numbers.

Jacobi sequences of squares of random variables

We express the Jacobi sequences of the square of a real valued random variable with all moments, not necessarily symmetric, as functions of the corresponding sequences of the random variable itself. In the symmetric case, the result is known and, we give a short, purely algebraic proof of it. We apply our result to the square of the Gamma distribution, i.e. the 4th power of the standard Gaussian. The result confirms the conjecture that belongs to the polynomial class, but its principal Jacobi sequence grows like , not as expected.

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.

On monochromatic sums of squares of primes

Let K be a positive integer, {Ai,1≤i≤K} be any partition of the sequence of squares of primes and s(K) be the smallest positive integer such that every sufficiently large integer can be written as the sum of no more than s(K) elements, which belong to one of the sets Ai. In this paper, we prove that s(K)≪ϵK2+ϵ for sufficiently small positive number ϵ and all K≥1.

Optimal bounds for Büchi's problem in modular arithmetic

We study the second order analogue of the problem of finding optimal lower and upper bounds for the length of sequences of squares in arithmetic progression modulo a prime, and some connections with the computational problem of finding a quadratic non-residue modulo a prime. More precisely, we work modulo an integer and our objects of study are those sequences of squares whose the second difference is an invertible constant. The main results of our work is a number of exact formulae that allow to reduce the problem to prime moduli. We also observe several phenomena which are supported by extensive numerical computations.

A chromatic version of Lagrange’s four squares theorem

When the sequence of squares of the natural numbers is coloured with \(K\) colours, \(K\ge 1\) an integer, let \(s(K)\) be the smallest integer such that each sufficiently large integer can be written as the sum of no more than \(s(K)\) squares, all of the same colour. We show that \(s(K) \ll _{\epsilon } K^{2 + \epsilon }\), for any \(\epsilon > 0\) and all \(K \ge 1\), improving on a result of Hegyvari and Hennecart, who obtained \(s(K) \ll (K\log K)^5\).

The Proximal Point Algorithm Revisited

In this paper, we consider the proximal point algorithm for the problem of finding zeros of any given maximal monotone operator in an infinite-dimensional Hilbert space. For the usual distance between the origin and the operator's value at each iterate, we put forth a new idea to achieve a new result on the speed at which the distance sequence tends to zero globally, provided that the problem's solution set is nonempty and the sequence of squares of the regularization parameters is nonsummable. We show that it is comparable to a classical result of Brezis and Lions in general and becomes better whenever the proximal point algorithm does converge strongly. Furthermore, we also reveal its similarity to Guler's classical results in the context of convex minimization in the sense of strictly convex quadratic functions, and we discuss an application to an ∈-approximation solution of the problem above.

Online Square Packing with Gravity

We analyze the problem of packing squares in an online fashion: Given a semi-infinite strip of width 1 and an unknown sequence of squares of side length in [0,1] that arrive from above, one at a time. The objective is to pack these items as they arrive, minimizing the resulting height. Just like in the classical game of Tetris, each square must be moved along a collision-free path to its final destination. In addition, we account for gravity in both motion (squares must never move up) and position (any final destination must be supported from below). A similar problem has been considered before; the best previous result is by Azar and Epstein, who gave a 4-competitive algorithm in a setting without gravity (i.e., with the possibility of letting squares “hang in the air”) based on ideas of shelf packing: Squares are assigned to different horizontal levels, allowing an analysis that is reminiscent of some bin-packing arguments. We apply a geometric analysis to establish a competitive factor of 3.5 for the bottom-left heuristic and present a $\frac{34}{13} \approx 2.6154$ -competitive algorithm.

First Digits of Squares and Cubes

SummaryBenford's Law is the observation that in many lists of numbers that arise in the real world, for d = 1, 2, …, 9, the likelihood that the first digit is d is log10 ((d + 1)/d). A similar phenomenon is known for essentially all geometric sequences ark, namely the relative frequencies RN(d) of the first digit d of the first N terms in such a sequence tend to log10((d + 1)/d). In this note, it is shown that the analogous statements for the sequence of squares and the sequence of cubes do not hold. However, in these cases, interesting subsequences of RN(d) do converge, but not to log10((d + 1)/d).

Generalized Thue-Morse sequences of squares

We consider compact group generalizations T(n) of the Thue-Morse sequence and prove that the subsequence T(n 2) is uniformly distributed with respect to a measure gv that is absolutely continuous with respect to the Haar measure. The proof is based on a proper generalization of the Fourier based method of Mauduit and Rivat in their study of the sum-of-digits function of squares to group representations.

Probabilistic constructions of B 2[g] sequences

We use the probabilistic method to prove that for any positive integer g there exists an infinite B 2[g] sequence $$ \mathcal{A} $$ = {a k } such that a k ≤ k 2+1/g (log k)1/g+o(1) as k→∞. The exponent 2+1/g improves the previous one, 2 + 2/g, obtained by Erdős and Renyi in 1960. We obtain a similar result for B 2[g] sequences of squares.

On-line Packing Squares into n Unit Squares

If $ n \geq 3$, then any sequence of squares of side lengths not greater than $1$ whose total area does not exceed $ { 1 \over 4}(n+1)$ can be on-line packed into $n$ unit squares.

Translative packing of a square with sequences of squares

Let $S$ be a square and let $S'$ be a square of unit area with a diagonal parallel to a side of $S$. Any (finite or infinite) sequence of homothetic copies of $S$ whose total area does not exceed $ { 4 \over 9}$ can be packed translatively into $S'$.

Parallel packing and covering of an equilateral triangle with sequences of squares

An equilateral triangle Te of sides 1 can be parallel covered with any sequence of squares whose total area is not smaller than 1:5. Moreover, any sequence of squares whose total area does not exceed \( \frac{3} {4}(2 - \sqrt 3 ) \) (2 − √3) can be parallel packed into Te.

The pebbling threshold of the square of cliques

Given an initial configuration of pebbles on a graph, one can move pebbles in pairs along edges, at the cost of one of the pebbles moved, with the objective of reaching a specified target vertex. The pebbling number of a graph is the minimum number of pebbles so that every configuration of that many pebbles can reach any chosen target. The pebbling threshold of a sequence of graphs is roughly the number of pebbles so that almost every (resp. almost no) configuration of asymptotically more (resp. fewer) pebbles can reach any chosen target. In this paper we find the pebbling threshold of the sequence of squares of cliques, improving upon an earlier result of Boyle and verifying an important instance of a probabilistic version of Graham's product conjecture.

Online Removable Square Packing

The online removable square packing problem is a two-dimen-sional version of the online removable Knapsack problem. For a sequence of squares with side length at most 1, we are requested to pack a subset of them into a unit square bin in an online fashion where the online player can decide whether to take the current square or not and which squares currently in the unit square to remove. The goal is to maximize the total packed area. Our results include: (i) No online algorithm can achieve a better competitive ratio than $(\sqrt{5}+3)/2\approx 2.618$. (ii) The matching upper bound is achieved by a relatively simple online algorithm if repacking is allowed. (iii) Without repacking, we can achieve an upper bound of 3 by using the concept of bricks of Januszewski and Lassak. (iv) The offline version of the problem admits a PTAS.