Neighboring Squares Research Articles

On a variant of Flory model

We consider a one-dimensional variant of a recently introduced settlement planning problem in which houses can be built on finite portions of the rectangular integer lattice subject to certain requirements on the amount of insolation they receive. In our model, each house occupies a unit square on a 1×n strip, with the restriction that at least one of the neighboring squares must be free. We are interested mostly in situations in which no further building is possible, i.e. in maximal configurations of houses in the strip. We reinterpret the problem as a problem of restricted packing of vertices in a path graph and then apply the transfer matrix method in order to compute the bivariate generating functions for the sequences enumerating all maximal configurations of a given length with respect to the number of houses. This allows us to determine the asymptotic behavior of the enumerating sequences and to compute some interesting statistics. Along the way, we establish close connections between our maximal configurations and several other types of combinatorial objects, including restricted permutations and walks on certain small oriented graphs. We then generalize our results in several directions by considering multi-story houses, by varying the insolation restrictions, and, finally, by considering strips of width 2 and 3. At the end we comment on several possible directions of future research.

Broadband transparency of Babinet complementary metamaterials

According to the Babinet principle, the diffraction pattern from an opaque body is identical to that from a hole of the same shape and size. Intuitively, placing two complementary structures such as an opaque metal body and its transparent counterpart one by one may result in destructive or constructive interference leading to unexpected electromagnetic response. We propose a Babinet principle-based metamaterial made of two complementary metal/hole checkerboards. The unit cell of each layer is either a metal square with 1/4 of 8 neighboring squares, four of which are made of metal, whereas the other four are square holes, or vice versa. Being placed complementary at optimal distance equal to three-unit cell length, the compound bi-layered Babinet structure demonstrates absolute transparency in a very broad frequency range. The observed absolute transparency of the bi-layered Babinet metasurface is the result of the modified multipole interaction of layers with shifted centers of radiation. We demonstrate both theoretically and experimentally absolute transmission of 0 dB for the Babinet metamaterial made of 3 cm sized Cu squares and complementary holes in the broad frequency range from 4.5 to 6.62 GHz in simulations and from 4.6 to 6.4 GHz in the experiment when the distance between two layers is 1.2 cm. Moving layers toward each other leads to blurring of the resonances. The proven concept of simple, reproducible, and scalable design of the Babinet metamaterial paves the way for the fabrication of broadband transparent devices at any frequency, including THz and optical ranges. The main advantage of broadband Babine metamaterials is applications in optical switching, sensing, filtering, and slow light devices.

Investigating geometrical characteristics of collapsed pipes and the changing role of driving factors

Determination of the amount (i.e., area and volume) of soil losses due to erosional landforms, especially collapsed pipes, plays a considerable role in different decision-making approaches. Further, mapping the spatial distribution and predicting the volumetric and areal losses of collapsed pipes (CPs) are essential for supporting ecosystem health. The study was conducted in relation to the area and volume of CPs and their related covariables. It focused on the estimation of soil losses due to collapsed pipes using unmanned aerial vehicle (UAV) images as well as field covariates at the Chatal Watershed, Golestan Province, Iran. A total of 481 soil samples were collected from homogeneous units with an area of approximately 1,410 ha. The potential relationship between the area/volume of collapsed pipes and land use, several topographic attributes (i.e., altitude, slope, and aspect), and soil properties, including soil stability, soil organic matter, clay, silt, and sand contents were analyzed using five distance-based methods (i.e., kernel density (KD), average nearest neighbor (ANN), spatial autocorrelation, hotspot analysis (HSA), and ordinary least square (OLS) analysis. The average nearest neighbor (Ratio = 0.12, Z score = −20.30, p-value < 0.05) and Moran space solidarity (Moran index = 0.258, Z score = 5.50, p-value < 0.05) showed the cluster distribution of area and volume of CPs. Hot spots and cold spots in the southwestern part of the study area were identified using KD and HSA. The relationship between existing independent and dependent variables (area of CPs) using regression analysis of OLS showed that slope and aggregate stability (>2.5 standard deviation) had the highest positive relationship with the dependent variable. Regarding the volume of CPs, land use (especially agricultural lands) had the strongest relationship with the dependent variable. Thus, geometrical characteristics of collapsed pipes can be applied as a quantitative indicator for the identification of hotspot zones (hazardous areas), land use planning, and erosion hazard mitigation. However, more studies are required to measure geometrical characteristics of soil landforms.

Case Study Topography Integrated Urban Center Concept Project on Coastal Line of Kartal/?stanbul

The concept of creating a Topography integrated urban center is to create an urban center that integrated with the city. The first step of the Concept is to settle by using the natural elevation in the land and to create volumes compatible with land by raising together with the elevation. While the passenger circulation at starting elevation is moved into a project with the cave-style volume settled in land elevation. The new area of the square to be defined in the center of the building is intended to form an area combining the neighboring squares Kartal Square and Freedom Square, as well as contributing to the silhouette of Kartal from the sea with the location of the square and building. The project is a central complex design that deals with various urban problems thanks to professionals, local people of Kartal, and clubs which established with the municipality in a comprehensive way to search for solutions to be organized urban workshops and conferences.&#x0D; Key Words&#x0D; Urban Center, Topography integrated, Sustainable Building, Blending with Topography, Walkable Green Roofs, Serving to People, 7/24 Living Place.

How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares.

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like 2, is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers (specifically, the radicands of radical expressions) as natural numbers. Strategy self-reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task-specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.

Minesweeper with Limited Moves

We consider the problem of playing Minesweeper with a limited number of moves: Given a partially revealed board, a number of available clicks k, and a target probability p, can we win with probability p. We win if we do not click on a mine, and, after our sequence of at most k clicks (which reveal information about the neighboring squares) can correctly identify the placement of all mines. We make the assumption, that, at all times, all placements of mines consistent with the currently revealed squares are equiprobable. Our main results are that the problem is PSPACE-complete, and it remains PSPACE-complete when p is a constant, in particular when p = 1. When k = 0 (i.e., we are not allowed to click anywhere), the problem is PP-complete in general, but co-NP-complete when p is a constant, and in particular when p = 1.

Temporal Asymmetry in Dark–Bright Processing Initiates Propagating Activity across Primary Visual Cortex

Differences between visual pathways representing darks and lights have been shown to affect spatial resolution and detection timing. Both psychophysical and physiological studies suggest an underlying retinal origin with amplification in primary visual cortex (V1). Here we show that temporal asymmetries in the processing of darks and lights create motion in terms of propagating activity across V1. Exploiting the high spatiotemporal resolution of voltage-sensitive dye imaging, we captured population responses to abrupt local changes of luminance in cat V1. For stimulation we used two neighboring small squares presented on either bright or dark backgrounds. When a single square changed from dark to bright or vice versa, we found coherent population activity emerging at the respective retinal input locations. However, faster rising and decay times were obtained for the bright to dark than the dark to bright changes. When the two squares changed luminance simultaneously in opposite polarities, we detected a propagating wave front of activity that originated at the cortical location representing the darkened square and rapidly expanded toward the region representing the brightened location. Thus, simultaneous input led to sequential activation across cortical retinotopy. Importantly, this effect was independent of the squares' contrast with the background. We suggest imbalance in dark-bright processing as a driving force in the generation of wave-like activity. Such propagation may convey motion signals and influence perception of shape whenever abrupt shifts in visual objects or gaze cause counterchange of luminance at high-contrast borders. An elementary process in vision is the detection of darks and lights through the retina via ON and OFF channels. Psychophysical and physiological studies suggest that differences between these channels affect spatial resolution and detection thresholds. Here we show that temporal asymmetries in the processing of darks and lights create motion signals across visual cortex. Using two neighboring squares, which simultaneously counterchanged luminance, we discovered propagating activity that was strictly drawn out from cortical regions representing the darkened location. Thus, a synchronous stimulus event translated into sequential wave-like brain activation. Such propagation may convey motion signals accessible in higher brain areas, whenever abrupt shifts in visual objects or gaze cause counterchange of luminance at high-contrast borders.

When crowding of crowding leads to uncrowding

In crowding, target discrimination is impaired by flanking elements. What happens if a target is neighbored by flankers A and in addition by further flankers B? According to pooling models, crowding is the consequence of averaging features of target and flankers. Hence, pooling models predict strong crowding when the target is flanked by two flankers. We determined offset discrimination thresholds for verniers at 9° of eccentricity. When the vernier was embedded in a square, thresholds increased compared to the unflanked threshold - a classical crowding effect. Surprisingly, when adding more squares, thresholds did not increase, but crowding almost vanished. Hence, crowding of crowding leads to uncrowding. We propose that this un-crowding effect can be explained in terms of grouping. Grouping between the vernier and the central square leads to crowding by the Gestalt principle of common region. Grouping of the central and the neighboring squares by the Gestalt principle of similarity, leads to ungrouping of the vernier, and to un-crowding.

More results on overlapping squares

Three recent papers (Fan et al., 2006; Simpson, 2007; Kopylova and Smyth, 2012) [5,11,8] have considered in complementary ways the combinatorial consequences of assuming that three squares overlap in a string. In this paper we provide a unifying framework for these results: we show that in 12 of 14 subcases that arise the postulated occurrence of three neighboring squares forces a breakdown into highly periodic behavior, thus essentially trivial and easily recognizable. In particular, we provide a proof of Subcase 4 for the first time, and we simplify and refine the previously established results for Subcases 11-14.

Nontrivial solutions to a checkerboard problem

The squares of an m n checkerboard are alternately colored black and red. It has been shown that for every pair m;n of positive integers, it is possible to place coins on some of the squares of the checkerboard (at most one coin per square) in such a way that for every two squares of the same color the numbers of coins on neighboring squares are of the same parity, while for every two squares of different colors the numbers of coins on neighboring squares are of opposite parity. All solutions to this problem have been what is referred to as trivial solutions, namely, for either black or red, no coins are placed on any square of that color. A nontrivial solution then requires at least one coin to be placed on a square of each color. For some pairs m;n of positive integers, however, nontrivial solutions do not exist. All pairs m;n of positive integers are determined for which there is a nontrivial solution.

Irredundance and domination in kings graphs

Each king on an n×n chessboard is said to attack its own square and its neighboring squares, i.e., the nine or fewer squares within one move of the king. A set of kings is said to form an irredundant set if each attacks a square attacked by no other king in the set. We prove that the maximum size of an irredundant set of kings is bounded between (n−1)2/3 and n2/3, and that the minimum size of a maximal irredundant set of kings is bounded between n2/9 and ⌊(n+2)/3⌋2, where the latter upper and lower bounds are in fact equal when n≡0(mod3). Results are given for related domination and independence problems.

Instantaneous spin correlations inLa2CuO4

We have carried out a neutron scattering study of the instantaneous spin-spin correlations in La2CuO4 (T_N = 325 K) over the temperature range 337 K to 824 K. Incident neutron energies varying from 14.7 meV to 115 meV have been employed in order to guarantee that the energy integration is carried out properly. The results so-obtained for the spin correlation length as a function of temperature when expressed in reduced units agree quantitatively both with previous results for the two dimensional (2D) tetragonal material Sr2CuO2Cl2 and with quantum Monte Carlo results for the nearest neighbor square lattice S=1/2 Heisenberg model. All of the experimental and numerical results for the correlation length are well described without any adjustable parameters by the behavior predicted for the quantum non-linear sigma model in the low temperature renormalized classical regime. The amplitude, on the other hand, deviates subtly from the predicted low temperature behavior. These results are discussed in the context of recent theory for the 2D quantum Heisenberg model.

Models of the Hofstadter-type

Spectra and eigenfunctions of discrete Hamiltonians are computed using algebraic, analytic, and numerical tools. In particular, we consider the Hofstadter and the Second Neighbor Square Lattice model, the Triangular Lattice model in an inhomogenous magnetic field, the Doubly-discrete Quantum Pendulum, and the Honeycomb model. Qualitative properties of the spectra are related to symmetries. Semiclassical analysis in the algebraic setting for the Doubly-discrete Quantum Pendulum is shown to match numerical results well. The connection to integrable models is mentioned.

Stable spins in the zero temperature spinodal decomposition of 2D Potts models

We present the results of zero temperature Monte Carlo simulations of the q-state Potts model on a square lattice with either four or eight neighbors, and for the triangular lattice with six neighbors. In agreement with previous works, we observe that the domain growth process gets blocked for the nearest-neighbor square lattice when q is large enough, whereas for the eight neighbor square lattice and for the triangular lattice no blocking is observed. Our simulations indicate that the number of spins which never flipped from the beginning of the simulation up to time t follows a power law as a function of the energy, even in the case of blocking. The exponent of this power law varies from less than sol1 2 for the Ising case (1 q = 2) to 2 for q → ∞ and seems to be universal. The effect of blocking on this exponent is invisible at least up to q = 7.

Crossover from Coulomb-blockade to ohmic conduction in small tunnel junctions

We discuss the suppression of Coulomb effects in the low temperature conductanceg(T) of small tunnel junctions with increasing dissipation or bare conductanceg. The conductance is expressed in terms of the spin correlation fuction of a classical one dimensionalXY-model with ferromagnetic nearest neighbor plus inverse square interaction. It is shown that at low temperatures the conductance vanishes asymptotically likeT2 instead of exponentially. A Coulomb gap in the sense of a thermally activated contribution tog(T) is present only for bare conductances smaller thangc∼1. A simple model for the spin correlation functions is compared with experiments.

On forcing functions in Kauffman's random Boolean networks

The phase transition between frozen and chaotic behavior in Kauffman's cellular automata on a nearest neighbor square lattice does not agree with the percolation threshold of the forcing functions.