In the realm of mathematics education, geometry problems assume a pivotal role by fostering abstract thinking, establishing a connection between theory and practice, and offering a tangible portrayal ...of reality. This study focuses on comprehending problem-solving methodologies by observing the eye movements of 45 primary and multi-year grammar school pupils, aged 11 to 14, as they tackled pictorial geometry problems without computation. The utilization of eye-tracking technology, specifically the OGAMA tool, was essential in unveiling the nuanced strategies employed by students. Visual attention metrics were determined through fixations on predefined areas of interest, identified using the ScanGraph tool. Through an analysis of eye movements, participants were categorized into three distinct groups based on their problem-solving strategies. This categorization facilitated an exploration of the correlation between the chosen strategy and the success rate in solving geometry problems without computational aids. The findings underscore the imperative for continued investigation into strategies for solving geometry problems without computation. Additionally, the research aims to broaden its scope by delving into the metacognitive strategies applied in solving imaginative geometric tasks.
It responds to a real problem with the objective: to offer sustained considerations in didactic activities to favor learning of geometric problems in the teaching process of mathematics grade twelve. ...Masters thesis is generalized, with activities that allow: guide of action in organization and direction of the process, solidity and durability of knowledge; development of integral general culture, logical thinking and preparation for life. Indicators favored comparing results before and after the proposal, verifying feasibility and feasibility, by positively transforming students in compliance with Mathematics objectives in twelve, and Mathematics and Mother Language Mathematics Programs, in solving problems.
Se responde a un problema real con el objetivo: ofrecer consideraciones sustentadas en actividades didácticas para favorecer aprendizaje de problemas geométricos en el proceso enseñanza aprendizaje de Matemática grado doce y Matemática Básica en la Educación superior. Se generaliza tesis de Maestría, con actividades que permiten: guía de acción en organización y dirección del proceso, solidez y perdurabilidad de conocimientos; desarrollo de cultura general integral, pensamiento lógico y preparación para la vida. Indicadores favorecieron comparar resultados antes y después de la propuesta, verificándose factibilidad y viabilidad, al transformarse positivamente los estudiantes en cumplimiento de objetivos de Matemática en doce y Matemática Básica en la educación superior, Programas directores de Matemática y Lengua materna, en la resolución de problemas.
It is known that the critical points of the distance function induced by a dense sample P of a submanifold Σ of ℝn are distributed into two groups, one lying close to Σ itself, called the shallow, ...and the other close to medial axis of Σ, called deep critical points. We prove that under (uniform) sampling assumption, the union of stable manifolds of the shallow critical points have the same homotopy type as Σ itself and the union of the stable manifolds of the deep critical points have the homotopy type of the complement of Σ. The separation of critical points under uniform sampling entails a separation in terms of distance of critical points to the sample. This means that if a given sample is dense enough with respect to two or more submanifolds of ℝn, the homotopy types of all such submanifolds together with those of their complements are captured as unions of stable manifolds of shallow versus those of deep critical points, in a filtration of the flow complex based on the distance of critical points to the sample. This results in an algorithm for homotopic manifold reconstruction when the target dimension is unknown.
A method for solving an inverse geometric problem is presented by reconstructing the unknown subsurface cavity geometry with the boundary element method (BEM) and a genetic algorithm in combination ...with the Nelder-Mead non-linear simplex optimization method. The heat conduction problem is solved by the BEM which calculates the difference between the measured temperature at the exposed surface and the computed temperature under the current update of the unknown subsurface flaws and cavities. In a first step, clusters of singularities are utilized to solve the inverse problem and to identify the location of the centroid(s) of the subsurface cavity(ies)/flaw(s). In a second step, the reconstruction of the estimated cavity(ies)/flaw(s) geometry(ies) is accomplished by utilizing an anchored grid pattern upon which cubic spline knots are restricted to move in the search for the unknown geometry. The solution is achieved using a genetic algorithm accelerated with the Nelder-Mead non-linear simplex method. The automated algorithm successfully reconstructs single and multiple subsurface cavities within two dimensional mediums. The cavity detection was enhanced by applying multiple boundary condition sets (MBCS) to the same geometry. This extra information supplied on the boundary made the subsurface cavity easily detectable despite its low heat signature effect on the boundaries.
Let be a set of points located in a rectangle and is a point that is not in . This article describes the design, implementation, and experimentation of different algorithms to solve the following ...two problems: (i) Maximum Empty Rectangle (MER), which consists in finding an empty rectangle with a maximum area contained in R and does not contain any point from and (ii) Query Maximum Empty Rectangle (QMER), which consists in finding the rectangle with the same restrictions given for the MER problem but must also contain . It is assumed that both problems have insufficient main memory to store all the objects in set . According to the literature, both problems are very practical in fields such as data mining and Geographic Information Systems (GIS). Specifically, the present study proposes two algorithms that assume that is stored in secondary memory (mainly disk) and that it is impossible to store it completely in main memory. The first algorithm solves the QMER problem and consists of decreasing the size of S by using dominance areas and then processing the points that are not eliminated using an algorithm proposed by Orlowski (1990). The second algorithm solves the MER problem and consists of dividing R into four subrectangles that generate four subsets of similar size; these are processed using an algorithm proposed in Edmons et al. (2003), and finally the partial solutions are combined to obtain a global solution. For the purpose of verifying algorithm efficiency, results are shown for a series of experiments that consider synthetic and real data.
In this paper we discuss the complexity and approximability of the minimum corridor connection problem where, given a rectilinear decomposition of a rectilinear polygon into “rooms”, one has to find ...the minimum length tree along the edges of the decomposition such that every room is incident to a vertex of the tree. We show that the problem is strongly NP-hard and give a subexponential time exact algorithm. For the special case when the room connectivity graph is
k-outerplanar the algorithm running time becomes cubic. We develop a polynomial time approximation scheme for the case when all rooms are fat and have nearly the same size. When rooms are fat but are of varying size we give a polynomial time constant factor approximation algorithm.
Creative thinking was characterized by the emergence of new ideas which was an important skill for students. Lateral thinking was one way to grow new ideas. Generally, related research was carried ...out to see the thinking process and its level only with the results of the majority of low levels (especially in geometry problems), so further research was needed to find out what were the obstacles. This study aims to find mathematical lateral thinking barriers based on brain dominance. This research was a qualitative research with a case study approach. The instruments were a brain dominance test, creativity questions, and geometry questions for lateral thinking. Triangulation of data collection techniques was carried out To ensure data validity. Obtained two main subjects LCS and RCS. Analysis of subjects' answers, interview and observations transcripts were carried out by coding the inhibiting factors then analyzed by Toulmin's arguments analysis. We findings that left-brain dominant student experience ontogenic barriers in finding something new caused bounded by concepts and right-brain dominant student experience epistemological barriers in finding different ways cause bounded by context. Further research was needed with students who have high mathematical abilities and creativity to find out how to overcome lateral thinking obstacles.
On Euclidean vehicle routing with allocation Remy, Jan; Spöhel, Reto; Weißl, Andreas
Computational geometry : theory and applications,
05/2010, Letnik:
43, Številka:
4
Journal Article
Recenzirano
Odprti dostop
The (Euclidean)
Vehicle Routing Allocation Problem (VRAP) is a generalization of Euclidean TSP. We do not require that all points lie on the salesman tour. However, points that do not lie on the tour ...are allocated, i.e., they are directly connected to the nearest tour point, paying a higher (per-unit) cost. More formally, the input is a set of
n points
P
⊂
R
d
and functions
α
:
P
→
0
,
∞
)
and
β
:
P
→
1
,
∞
)
. We wish to compute a subset
T
⊆
P
and a salesman tour
π through
T such that the total length of the tour plus the total allocation cost is minimum. The allocation cost for a single point
p
∈
P
∖
T
is
α
(
p
)
+
β
(
p
)
⋅
d
(
p
,
q
)
, where
q
∈
T
is the nearest point on the tour. We give a PTAS with complexity
O
(
n
log
d
+
3
n
)
for this problem. Moreover, we propose an
O
(
n
polylog
(
n
)
)
-time PTAS for the Steiner variant of this problem. This dramatically improves a recent result of Armon et al. A. Armon, A. Avidor, O. Schwartz, Cooperative TSP, in: Proceedings of the 14th Annual European Symposium on Algorithms, 2006, pp. 40–51.
In this paper, we consider the problem of detection of subsurface cavities by the method of infrared computerised axial tomography (IR-CAT). The location and size of interior cavities are determined ...using temperature and heat flux measurements at the surface of a heat conductor. The problem has numerous applications in non-destructive testing of materials. A real coded genetic algorithm (GA) is employed to optimise the object function measuring the goodness of fitness to the data. The solution methodology developed has a general character and can be applied for various cavities or inclusions detection problems.
In this paper we consider the identification of the geometric structure of the boundary of the solution domain for the three-dimensional Laplace equation. We investigate the determination of the ...shape of an unknown portion of the boundary of a solution domain from Cauchy data on the remaining portion of the boundary. This problem arises in the study of quantitative non-destructive evaluation of corrosion in materials in which boundary measurements of currents and voltages are used to determine the material loss caused by corrosion. The domain identification problem is considered as a variational problem to minimize a defect functional, which utilises some additional data on certain known parts of the boundary. A sequential quadratic programming (SQP) optimization algorithm is used in order to minimise the objective functional. The unknown boundary is parameterized using B-splines. The Laplace equation is discretised using the method of fundamental solutions (MFS). Numerical results are presented and discussed for several test examples.
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