Sides Of Rectangle Research Articles

Studies of Auxetic Structures Assembled from Rotating Rectangles.

The subject of the work is analysis, which presents a renowned auxetic structure based on so-called rotating polygons, which has been subject to modification. This modification entails introducing pivot points on unit cell surfaces near rectangle corners. This innovative system reveals previously unexplored correlations between Poisson's ratio, the ratio of rectangle side lengths, pivot point placement, and structural opening. Formulas have been derived using geometric relationships to compute the structure's linear dimensions and Poisson's ratio. The obtained findings suggest that Poisson's ratio is intricately tied to the structure's opening degree, varying as the structure undergoes stretching. Notably, there are critical parameter limits beyond which Poisson's ratio turns positive, leading to the loss of auxetic properties. For elongated rectangles, extremely high negative Poisson's ratio values are obtained, but only for small opening angles, while with further stretching, the structure loses its auxetic properties. This observed trend is consistent across a broad category of structures comprised of rotating rectangles.

Аналог задачи Дезина для уравнения параболо-гиперболического типа с условиями периодичности

В прямоугольной области рассматривается неоднородное уравнение смешанного параболо-гиперболического типа второго порядка. Для данного уравнения исследуется аналог задачи А. А. Дезина, который заключается в отыскании решения уравнения, удовлетворяющего внутренне-краевому условию, связывающему значение искомой функции на линии изменения типа уравнения со значением нормальной производной на границе в области гиперболичности, и неоднородным нелокальным краевым условиям периодичности. Приводится подстановка, позволяющая свести задачу к эквивалентной и, не теряя общности, ограничиться исследованием задачи с однородными условиями для неоднородного уравнения. Доказаны теоремы единственности и существования решения задачи, решение выписано в явном виде. Решение поставленной задачи ищется в виде суммы ряда Фурье по ортонормированной системе собственных функций соответствующей одномерной спектральной задачи. Установлен критерий единственности решения задачи. Для случая, когда нарушен критерий единственности, приведен пример нетривиального решения однородной задачи и получено необходимое и достаточное условие существования решения неоднородной задачи. При обосновании существования решения возникает проблема малых знаменателей в сумме ряда относительно соотношения сторон прямоугольника в гиперболической части области. Получена оценка отделенности знаменателя от нуля при некоторых условиях относительно параметров задачи, которая при определенных условиях на заданные функции позволяет доказать абсолютную и равномерную сходимость как формально построенного решения, так и соответствующих производных, входящих в уравнение.

Global exact optimization for covering a rectangle with 6 circles

We address the problem of covering a rectangle with six identical circles, whose radius is to be minimized. We focus on open cases from Melissen and Schuur (Discrete Appl Math 99:149–156, 2000). Depending on the rectangle side lengths, different configurations of the circles, corresponding to the different ways they are placed, yield the optimal covering. We prove the optimality of the two configurations corresponding to open cases. For the first one, we propose a mathematical mixed-integer nonlinear optimization formulation, that allows one to compute global optimal solutions. For the second one, we provide an analytical expression of the optimal radius as a function of one of the rectangle side lengths. All open cases are thus closed for the optimal covering of a rectangle with six circles.

Robust visual line-following navigation system for humanoid robots

This paper implements a novel line-following system for humanoid robots. Camera embedded on the robot’s head captures the image and then extracts the line using a high-speed and high-accuracy rectangular search method. This method divides the search location into three sides of rectangle and performs image convolution by edge detection matrix. The extracted line is used to calculate relative parameters, including forward velocity, lateral velocity and angular velocity that drive line-following walking. A proposed path curvature estimation method generates the forward velocity and guidance reference point of the robot. A classical PID controller and a PID controller with angle compensation are then used to set the lateral velocity and angular velocity of the robot, improving the performance in tracking a curved line. Line-following experiments for various shapes were conducted using humanoid robot NAO. Experimental results demonstrate the robot can follow different line shapes with the tracking error remaining at a low level. This is a significant improvement from existing biped robot visual navigation systems.

Students’ Mathematical Literacy in Solving PISA Problems Based on Keirsey Personality Theory

This research is descriptive-qualitative research. The purpose is to describe students’ mathematical literacy in solving PISA on space and shape content based on Keirsey personality theory. The subjects are four junior high school students grade eight with guardian, artisan, rational or idealist personality. Data collecting methods used test and interview. Data of Keirsey Personality test, PISA test, and interview were analysed. Profile of mathematical literacy of each subject are described as follows. In formulating, guardian subject identified mathematical aspects are formula of rectangle area and sides length; significant variables are terms/conditions in problem and formula of ever encountered question; translated into mathematical language those are measurement and arithmetic operations. In employing, he devised and implemented strategies using ease of calculation on area-subtraction principle; declared truth of result but the reason was less correct; didn’t use and switch between different representations. In interpreting, he declared result as area of house floor; declared reasonableness according measurement estimation.In formulating, artisan subject identified mathematical aspects are plane and sides length; significant variables are solution procedure on both of daily problem and ever encountered question; translated into mathematical language those are measurement, variables, and arithmetic operations as well as symbol representation. In employing, he devised and implemented strategies using two design comparison; declared truth of result without reason; used symbol representation only. In interpreting, he expressed result as floor area of house; declared reasonableness according measurement estimation.In formulating, rational subject identified mathematical aspects are scale and sides length; significant variables are solution strategy on ever encountered question; translated into mathematical language those are measurement, variable, arithmetic operation as well as symbol and graphic representation. In employing, he devised and implemented strategies using additional plane forming on area-subtraction principle; declared truth of result according calculation process; used and switched between symbol and graphic representation. In interpreting, he declared result as house area within terrace and wall; declared reasonableness according measurement estimation.In formulating, idealist subject identified mathematical aspects are sides length; significant variables are terms/condition in problem; translated into mathematical language those are measurement, variables, arithmetic operations as well as symbol and graphic representation. In employing, he devised and implemented strategies using trial and error and two design in process of finding solutions; declared truth of result according the use of two design of solution; used and switched between symbol and graphic representation. In interpreting, he declared result as floor area of house; declared reasonableness according measurement estimation.

Weighted random-geometric and random-rectangular graphs: spectral and eigenfunction properties of the adjacency matrix

Within a random-matrix-theory approach, we use the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length $\ell$ to study spectral and eigenfunction properties (of adjacency matrices) of weighted random--geometric and random--rectangular graphs. A random--geometric graph (RGG) considers a set of vertices uniformly and independently distributed on the unit square, while for a random--rectangular graph (RRG) the embedding geometry is a rectangle. The RRG model depends on three parameters: The rectangle side lengths $a$ and $1/a$, the connection radius $r$, and the number of vertices $N$. We then study in detail the case $a=1$ which corresponds to weighted RGGs and explore weighted RRGs characterized by $a\sim 1$, i.e.~two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when $a\gg1$. In general we look for the scaling properties of $P(s)$ and $\ell$ as a function of $a$, $r$ and $N$. We find that the ratio $r/N^\gamma$, with $\gamma(a)\approx -1/2$, fixes the properties of both RGGs and RRGs. Moreover, when $a\ge 10$ we show that spectral and eigenfunction properties of weighted RRGs are universal for the fixed ratio $r/{\cal C}N^\gamma$, with ${\cal C}\approx a$.

The inverse problem for the Lavrent’ev–Bitsadze equation connected with the search of elements in the right-hand side

We consider an equation of mixed elliptic-hyperbolic type, whose right-hand side represents a product of two one-dimensional functions. We establish a criterion for the unique solvability of this equation and construct its solution as a sum of series on the set of its eigenfunctions. Under certain constraints imposed on the ratio of the rectangle sides, on boundary functions, and on known multipliers in the right-hand side of the equation, we obtain estimates separating small denominators that appear in coefficients of constructed series from zero.

The dirichlet problem for an equation of mixed elliptic-hyperbolic type with variable potential

We study the Dirichlet problem for an equation of mixed elliptic-hyperbolic type with variable potential in the rectangular area. We establish a criterion of the uniqueness of a solution to this problem. The uniqueness of a solution is proved on the basis of the completeness of systems of functions corresponding to one-dimensional spectral problem. Solutionwas constructed as a sumof series on the systems of eigenfunctions. The existence is proved under certain conditions on the ratio of the rectangle sides of hyperbolic part of the equation, upon the boundary functions and function of potential.

Random rectangular graphs.

A generalization of the random geometric graph (RGG) model is proposed by considering a set of points uniformly and independently distributed on a rectangle of unit area instead of on a unit square [0,1](2). The topological properties of the random rectangular graphs (RRGs) generated by this model are then studied as a function of the rectangle sides lengths a and b=1/a, and the radius r used to connect the nodes. When a=1 we recover the RGG, and when a→∞ the very elongated rectangle generated resembles a one-dimensional RGG. We obtain here analytical expressions for the average degree, degree distribution, connectivity, average path length, and clustering coefficient for RRG. These results provide evidence that show that most of these properties depend on the connection radius and the side length of the rectangle, usually in a monotonic way. The clustering coefficient, however, increases when the square is transformed into a slightly elongated rectangle, and after this maximum it decays with the increase of the elongation of the rectangle. We support all our findings by computational simulations that show the goodness of the theoretical models proposed for RRGs.

Error Estimates of the Finite-Difference Scheme for a One-Dimensional Parabolic Equation with Allowance for the Effect of Initial and Boundary Conditions

Error estimates are obtained for the finite-difference scheme in solving a one-dimensional heat equation. These estimates take into account the effect of initial and boundary conditions. The order of accuracy in a spatiotemporal rectangle is proved to be higher near the rectangle bottom and along the rectangle side on which the Dirichlet boundary condition is specified.

1D nanostructured Na7V4(P2O7)4(PO4) as high-potential and superior-performance cathode material for sodium-ion batteries.

Tailoring materials into nanostructure offers unprecedented opportunities in the utilization of their functional properties. High-purity Na7V4(P2O7)4(PO4) with 1D nanostructure is prepared as a cathode material for rechargeable Na-ion batteries. An efficient synthetic approach is developed by carefully controlling the crystal growth in the molten sodium phosphate. Based on the XRD, XPS, TG, and morphological characterization, a molten-salt assisted mechanism for nanoarchitecture formation is revealed. The prepared Na7V4(P2O7)4(PO4) nanorod has rectangle sides and preferential [001] growth orientation. GITT evaluation indicates that the sodium de/intercalation of Na7V4(P2O7)4(PO4) nanorod involves V(3+)/V(4+) redox reaction and Na5V(3.5+)4(P2O7)4(PO4) as intermediate phase, which results in two pairs of potential plateaus at the equilibrium potentials of 3.8713 V (V(3+)/V(3.5+)) and 3.8879 V (V(3.5+)/V(4+)), respectively. The unique nanoarchitecture of the phase-pure Na7V4(P2O7)4(PO4) facilitates its reversible sodium de/intercalation, which is beneficial to the high-rate capability and the cycling stability. The Na7V4(P2O7)4(PO4) cathode delivers 80% of the capacity (obtained at C/20) at the 10 C rate and 95% of the initial capacity after 200 cycles. Therefore, it is feasible to design and fabricate an advanced rechargeable sodium-ion battery by employment of 1D nanostructured Na7V4(P2O7)4(PO4) as the cathode material.

Experimental research on the plastic upsetting between fixed walls of thin strips in the form of a rectangle and sector

This paper describes experiments and analysis on the upsetting between approaching rigid plates of a thin plastic sample, limited fixed walls and originally shaped rectangle or sector. Fixed wall located along the long rectangle sides and along the radial sector boundaries. It describes the definition of experimental laws, which allows to choose the corresponding mathematical model.

Robust estimation of fractional models in the frequency domain using set membership methods

In this paper, the usual definition of Grünwald–Letnikov fractional derivative is first extended to interval derivatives in order to deal with uncertainties in the differentiation orders. The Laplace transform of interval derivatives is computed and its monotonicity is studied in the frequency domain. Next, the main objectives of this paper are presented as the implementation of three methods for set membership parameters estimation of fractional differentiation models based on complex frequency data. The first one uses a rectangular inclusion function with rectangle sides corresponding to real and imaginary parts of the complex frequency response; the second one uses a polar inclusion function and the gain/phase representation; the third one uses a circular inclusion function with disk representation. Each inclusion function introduces pessimism differently. It is shown that all three approaches are complementary and that the results can be merged to obtain a smaller feasible solution set. The proposed methods can be applied to estimate parameters of certain/uncertain linear time variant/invariant systems.

A DOCUMENT RECTIFICATION APPROACH DEALING WITH BOTH PERSPECTIVE DISTORTION AND WARPING BASED ON TEXT FLOW CURVE FITTING

In this paper, we propose a coordinate transform model and document rectification process for book dewarping. This model can handle both perspective distortion and book surface warping problems. The main idea is to find the boundary of a region, which is a rectangle on a flat surface before warping and using the correspondence between the warped boundary and the original rectangle sides to map each pixel in the warped region into the rectangle. On a book page, two text lines and the left and right boundaries of a text region form a rectangle before warping. Therefore, we detect two text flow curves and text region boundaries on the left and right sides on a warped image and use the two curves and two line segments to do the transformation. The experimental results illustrate the effectiveness of our proposed book dewarping approach.

Transient and asymptotic dispersion in confined sphere packings with cylindrical and non-cylindrical conduit geometries

We study the time and length scales of hydrodynamic dispersion in confined monodisperse sphere packings as a function of the conduit geometry. By a modified Jodrey-Tory algorithm, we generated packings at a bed porosity (interstitial void fraction) of ε=0.40 in conduits with circular, rectangular, or semicircular cross section of area 100πd(p)(2) (where d(p) is the sphere diameter) and dimensions of about 20d(p) (cylinder diameter) by 6553.6d(p) (length), 25d(p) by 12.5d(p) (rectangle sides) by 8192d(p) or 14.1d(p) (radius of semicircle) by 8192d(p), respectively. The fluid-flow velocity field in the generated packings was calculated by the lattice Boltzmann method for Péclet numbers of up to 500, and convective-diffusive mass transport of 4×10(6) inert tracers was modelled with a random-walk particle-tracking technique. We present lateral porosity and velocity distributions for all packings and monitor the time evolution of longitudinal dispersion up to the asymptotic (long-time) limit. The characteristic length scales for asymptotic behaviour are explained from the symmetry of each conduit's velocity field. Finally, we quantify the influence of the confinement and of a specific conduit geometry on the velocity dependence of the asymptotic dispersion coefficients.

그래디언트 라돈변환을 이용한 만화영상의 외곽 경계사각형 검출

최근, 인터넷에 다양한 컨텐츠가 있으며, 특히 만화영상은 인기 있는 디지털 컨텐츠 중의 하나이다. 이러한 만화영상은 대부분 스캐너에 의해 스캔되며, 크기, 기울기, 경계선 부분의 여백 등이 정규화되어 있지 않다. 이와 같은 정규화 과정은 만화영상 분석에서 매우 중요한 단계로, 만화영상의 외곽 경계사각형을 검출하는 방법으로 정규화를 수행할 수 있다. 본 논문에서는 그래디언트 라돈 변환을 사용하여 만화영상에서 외곽의 경계사각형을 검출하는 방법을 제안한다. 만화영상에서 경계 사각형의 변의 후보인 선분을 검출하기 위하여 그래디언트 정보를 이용한 라돈 변환을 적용하고, 후보 선분과 지역 히스토그램을 이용하여 최종 외곽의 경계사각형 검출하였다. 제안 방법으로 만화 영상에서 효과적으로 외곽의 경계사각형을 검출함을 실험으로 보였다. Today, there is a wide variety of digital contents on the Internet. Especially, comic images are one of popular digital contents. Most of them are scanned from comic books by digital scanners, but they were not normalized in sense of their size, skew and boundary margin. The normalization is very important step in comic image analysis. It can be achieved by finding out-boundary rectangles in comic images. In this paper, we propose a method for detecting the out-boundary rectangle using the gradient Radon transform in comic images. We applied the Radon transform using image gradients to extract line segments which are the out-boundary rectangle sides' candidates in comic images. The final out-boundary rectangle can be detected by local histogram and the candidate line segments. Experimental results show that our proposed method effectively detect the out-boundary rectangle in comic images.

Symmetry and models of single-walled TiO2 nanotubes with rectangular morphology

Abstract The formalism of line symmetry groups for one-periodic (1D) nanostructures with rotohelical symmetry has been applied for symmetry analysis of single-walled titania nanotubes (SW TiO2 NTs) formed by rolling up the stoichiometric two-periodic (2D) slabs of anatase structure. Either six- or twelve-layer (101) slabs have been cut from TiO2 crystal in a stable anatase phase. After structural optimization, the latter keeps the centered rectangular symmetry of initial slab slightly compressed along a direction coincided with large sides of elemental rectangles. We have considered two sets of SW TiO2 NTs with optimized six- and twelve-layer structures, which possess chiralities (−n, n) and (n, n) of anatase nanotubes. To analyze the structural and electronic properties of titania slabs and nanotubes, we have performed their ab initio LCAO calculations, using the hybrid Hartree-Fock/Kohn-Sham exchange-correlation functional PBE0. The band gaps (Δɛ gap) and strain energies (E strain) of six-layer nanotubes have been computed and analyzed as functions of NT diameter (D NT). As to models of 12-layer SW TiO2 NTs of both chiralities, their optimization results in structural exfoliation, i.e., the multi-walled structure should be rather formed in nanotubes with such a number of atomic layers.

Cue combination and the effect of horizontal disparity and perspective on stereoacuity

Relative depth judgments of vertical lines based on horizontal disparity deteriorate enormously when the lines form part of closed configurations (Westheimer, 1979). In studies showing this effect, perspective was not manipulated and thus produced inconsistency between horizontal disparity and perspective. We show that stereoacuity improves dramatically when perspective and horizontal disparity are made consistent. Observers appear to use unhelpful perspective cues in judging the relative depth of the vertical sides of rectangles in a way not incompatible with a form of cue weighting. However, 95% confidence intervals for the weights derived for cues usually exceed the a-priori [0-1] range.

電子デバイスの導電性樹脂による実装プロセスの検討

Electric devices with semiconductors are applied to all apparatus including substation equipment, transport machines, home electronics, and cellular phones. Power modules deal large current, and high frequency/optical modules control GHz band signals. As a result, these semiconductors have more than 100 times heat density of memory or MPU chips. Pb-rich high temperature solder and expensive Au-rich solder are applied to these modules, however, thermal stress might be a problem not only for long-term reliability but also for the initial characteristics. The authors studied the assembly of these electric devices using conductive adhesive as a substitute bonding material. We proved that atmospheric aluminum oxides caused electric resistance and that power chips with long rectangle sides over 10 mm have a much larger thermal resistance than theoretical values. We found that it is effective to scratch and remove these oxides through transferred adhesive on aluminum electrodes and to diebond them onto the solder projection previously formed on the die pads.

Persistence of strong correlations in the energy spectrum of a separable Hamiltonian system: The rectangular box

The variance of the number of levels in an energy interval around a level with large quantum numbers (semiclassical quantization) is studied for a particle in a rectangular box. Sampling involves changing the ratio of the rectangle's sides while keeping the area constant. For sufficiently narrow intervals, one finds the usual linear growth with the width of the interval. For wider intervals, the variance undergoes large, nondecaying oscillations around what is expected to be the saturation value. These oscillations can be explained as a superposition of just a few harmonics that correspond to the shortest periodic orbits in the rectangle. The analytical and numerical results are in excellent agreement.