Geometric Problems Research Articles

Functional calculus and harmonic analysis in geometry

In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This is a succinct survey that hopes to inspire geometers and analysts alike to study these methods so that they can be further developed to be potentially applied to a broader range of questions.

Integrated Teaching of Extreme Geometric Problems in Vietnam

Integrated teaching is a modern, learner-centered trend. Today, Vietnam is strongly innovating from teaching academic knowledge to integrated teaching. Integrated teaching helps learners to develop their competencies. This method makes students to be active and creative. Students have more motivation in learning. Integrated teaching means helps us to find the relationship between these knowledge, methods and other ones. The above things are illustrated by teaching geometric extremes. This is a practical problem in Hau Giang, Vietnam. Teacher gives questions and students answer them. These questions are pedagogic. In addition, the article gives some results of empirical investigation confirming the superiority of integrated teaching compared to traditional teaching. The article proposes a number of research results on the concept of integration and features of integration. Some features of integration are subjective, association, useful combination, expression of introvert and extrovert orientation, creating a unified block, in a certain context. https://doi.org/10.26803/ijlter.18.8.11

Topological sensitivity analysis for identification of voids under Navier’s boundary conditions in linear elasticity

This article is concerned with a geometric inverse problem related to the two-dimensional linear elasticity system. Thereby, voids under Navier’s boundary conditions are reconstructed from the knowledge of partially over-determined boundary data. The proposed approach is based on the so-called energy-like error functional combined with the topological sensitivity method. The topological derivative of the energy-like misfit functional is computed through the topological-shape sensitivity method. Firstly, the shape derivative of the corresponding misfit function is presented briefly from previous work Méjri (2018 J. Inverse Ill-Posed Problems). Then, an explicit solution of the fundamental boundary-value problem in the infinite plane with a circular hole is calculated by the Muskhelishvili formulae. Finally, the asymptotic expansion of the topological gradient is derived explicitly with respect to the nucleation of a void. Numerical tests are performed in order to point out the efficiency of the developed approach.

An Exploration of Teaching for Understanding the Problem for Geometric Proof Development: The Case of Two Secondary School Mathematics Teachers

Understanding the proof problem is necessary in the development of geometric proof in schools. This paper discusses findings from a qualitative case study which explored the mathematical work done by teachers in helping students to understand geometric proof problems. Data were generated from two teachers through video-recorded observations of 30 lessons and tape-recorded post-lesson interviews. Both types of data were analysed using thematic analysis which involved coding of the data relating to the first stage of Polya’s problem-solving framework. The findings suggest that supporting students to understand geometric proof problems involves several tasks, including defining key mathematical terms of the theorem, initiating activities for introducing the theorem and representing the theorem as a statement to be proved. Furthermore, conducting these aspects of mathematical work using empirical exploratory approaches offers students better opportunities to understand the proof problem.

Slender-body theory for plasmonic resonance.

We develop a slender-body theory for plasmonic resonance of slender metallic nanoparticles, focusing on a general class of axisymmetric geometries with locally paraboloidal tips. We adopt a modal approach where one first solves the plasmonic eigenvalue problem, a geometric spectral problem which governs the surface-plasmon modes of the particle; then, the latter modes are used, in conjunction with spectral-decomposition, to analyse localized-surface-plasmon resonance in the quasi-static limit. We show that the permittivity eigenvalues of the axisymmetric modes are strongly singular in the slenderness parameter, implying widely tunable, high-quality-factor, resonances in the near-infrared regime. For that family of modes, we use matched asymptotics to derive an effective eigenvalue problem, a singular non-local Sturm-Liouville problem, where the lumped one-dimensional eigenfunctions represent axial voltage profiles (or charge line densities). We solve the effective eigenvalue problem in closed form for a prolate spheroid and numerically, by expanding the eigenfunctions in Legendre polynomials, for arbitrarily shaped particles. We apply the theory to plane-wave illumination in order to elucidate the excitation of multiple resonances in the case of non-spheroidal particles.

Calculus students’ quantitative reasoning in the context of solving related rates of change problems

ABSTRACTDespite the increasing amount of research on students’ quantitative reasoning at the secondary level, research on students’ quantitative reasoning at the undergraduate level is scarce. The present study used task-based interviews to examine 16 high-performing undergraduate calculus students’ quantitative reasoning in the context of solving three related rates of change problems-two geometric and one non-geometric. A qualitative analysis of verbal responses and work written by the students when solving the three problems revealed that 15 students created and used diagrams to support their quantitative reasoning in the geometric problems, and that the creation of these diagrams helped the students to solve the two problems. In addition, seven students exhibited lack of facility with the product or quotient rule of differentiation. We argue that the use of diagrams in calculus instruction at the undergraduate level could play an important role in supporting students’ quantitative reasoning in topics that require students to make sense of relationships between quantities such as related rates of change problems. Furthermore, we argue that helping students develop greater facility with rules of differentiation such as the product or quotient rule could increase students’ success in many calculus application problems, including related rates of change problems. Directions for future research are included.

Students’ Spatial Reasoning in Solving Geometrical Transformation Problems

This study describes spatial reasoning of senior high school students in solving geometrical transformation problems. Spatial reasoning consists of three aspects: spatial visualization, mental rotation, and spatial orientation. The approach that is used in this study is descriptive qualitative. Data resource is the test result of reflection, translation, and rotation problems then continued by interview. Collecting data process involves 35 students. They are grouped to three spatial reasoning aspects then selected one respondent to be the most dominant of each aspect. The results of this study are: (1) the students with spatial visualization aspect used drawing strategy and non-spatial strategy in solving geometrical transformation problems. She transformed every vertex of the object and drew assistance lines which connect every vertex of the object to center point; (2) the students with mental rotation aspect used holistic and analytic strategies in solving geometrical transformation problems. Using holistic strategy means imagining the whole of transformational objects to solve easy problems. While using analytic strategy means transforming some components of objects to solve hard problems; (3) the students with spatial orientation didn’t involve mental imagery and she only could determine the position and orientation of the object in solving geometrical transformation problems

Towards a theory of multi-parameter geometrical variational problems: Fibre bundles, differential forms, and Riemannian quasiconvexity

We are concerned with the existence and associated gauge problems of a general class of geometrical minimisation problems, with action integrals defined via differential forms over fibre bundles. We find natural algebraic and analytic conditions which give rise to an associated gauge theory. Moreover, we propose the notion of “Riemannian quasiconvexity” for cost functions whose variables are differential forms on Riemannian manifolds, which extends the classical quasiconvexity condition in the Euclidean settings. The existence of minimisers under the Riemannian quasiconvexity condition has been established. This work may serve as a tentative generalisation of the framework developed in the recent paper [Arch. Ration. Mech. Anal. 234 (2019), pp. 317–349] by Dacorogna–Gangbo.

Multi Objective Geometric Programming with Interval Coefficients: A Parametric Approach

When we talk of optimization in industry we need to pay attention in searching for very powerful and flexible optimization techniques. One of such techniques which has attracted the interest of many researchers in the last few decades is called geometric programming that provides a powerful tool for solving nonlinear problems. As we know in the real world, many applications of geometric programming are engineering design problems. Generally, engineering design problems deal with multi-objective functions, in which their objectives are often in conflicts with each other. This paper considers a solution method when the cost, the constraint coefficients, and the right-hand sides in the multi-objective geometric programming problems are imprecise and represented as interval values. This problem is reduced with the method of weighted sum to a single objective function and further by applying interval-valued function, we solve the problem by geometric programming technique. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering optimization areas. Finally a numerical example is given to illustrate the methodology of solution and efficiency of the present approach.

Flat-band ferromagnetism of SU(N) Hubbard model on Tasaki lattices

We investigate the para-ferro magnetic transition of the repulsive SU(N) Hubbard model on a type of one- and two-dimensional decorated cubic lattices, referred as Tasaki lattices, which feature massive single-particle ground state degeneracy. Under certain restrictions for constructing localized many-particle ground states of flat-band ferromagnetism, the quantum model of strongly correlated electrons is mapped to a classical statistical geometric site-percolation problem, where the nontrivial weights of different configurations must be considered. We prove rigorously the existence of para-ferro transition for the SU(N) Hubbard model on one-dimensional Tasaki lattice and determine the critical density by the transfer-matrix method. In two dimensions, we numerically investigate the phase transition of SU(3), SU(4) and SU(10) Hubbard models by Metropolis Monte Carlo simulation. We find that the critical density exceeds that of standard percolation, and increases with spin degrees of freedom, implying that the effective repulsive interaction becomes stronger for larger N. We further rigorously prove the existence of flat-band ferromagnetism of the SU(N) Hubbard model when the number of particles equals to the degeneracy of the lowest band in the single-particle energy spectrum.

Conceptualization and finite element groundwater flow modeling of a flooded underground mine reservoir in the Asturian Coal Basin, Spain

Secure workings in underground coal mines usually require the lowering of water levels via pumping. After mine closure, pump systems are stopped and the water level recovers in a process known as groundwater rebound. In the Asturian Coal Basin, Spain, the closure and flooding of underground coal mines carries associated environmental impacts that ought to be assessed. Among them, those related with groundwater flow and pollutant transport are of main interest. To evaluate the environmental risks during and after mine closure the construction of numerical flow models is suggested, which rely on an appropriate conceptualization of the mine-hydrogeological system. Underground mines of this region are characterized by very steep coal seams due to a folded and faulted geological structure. The resulting geometry translates into a complex network of hundreds of kilometers of mine tunnels and galleries. This is a challenge for the construction of numerical models in issues like geometrical problems, assignation of hydraulic parameters, coupling of physical laws and high computational cost. This paper presents the conceptual and numerical model of two linked mines of the region, Mosquitera and Pumarabule, that recently ended the flooding stage, to be used in assessment of post-closure environmental risks. To implement the geometry of the underground mine, a methodological approach to translate the mining information to the numerical simulator has been developed, that retains the hydraulic behavior of the key underground mine elements. A methodological approach for coupled simulation of mine conduits and porous media flow has been explicitly adapted. The results, in terms of the velocity distribution field inside the mine and possible outflow sites, are useful to provide a basis for later extension of transport models and assessment of hydro chemical impacts on the area.

Faster Approximation Schemes for the Two-Dimensional Knapsack Problem

For geometric optimization problems we often understand the computational complexity on a rough scale, but not very well on a finer scale. One example is the two-dimensional knapsack problem for squares. There is a polynomial time (1+ε)-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1/ε, i.e., Ω ( n 2 2 1/ε ). A double or triple exponential dependence on 1/ε is inherent in how this and other algorithms for other geometric problems work. In this article, we present an efficient PTAS (EPTAS) for knapsack for squares, i.e., a (1+ε)-approximation algorithm with a running time of O ε (1)⋅ n O (1) . In particular, the exponent of n in the running time does not depend on ε at all! Since there can be no fully polynomial time approximation scheme (FPTAS) for the problem (unless P = NP), this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω (2 2 1/ε ) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS, each of these steps needs a running time of Ω ( n 2 2 1/ε ) and we improve both to O ε (1)⋅ n O (1) . We complete our result by giving an algorithm for two-dimensional knapsack for rectangles under (1+ε)-resource augmentation. We improve the previous double-exponential PTAS to an EPTAS and compute even a solution with optimal weight, while the previous PTAS computes only an approximation.

Augmenting Basin-Hopping With Techniques From Unsupervised Machine Learning: Applications in Spectroscopy and Ion Mobility.

Evolutionary algorithms such as the basin-hopping (BH) algorithm have proven to be useful for difficult non-linear optimization problems with multiple modalities and variables. Applications of these algorithms range from characterization of molecular states in statistical physics and molecular biology to geometric packing problems. A key feature of BH is the fact that one can generate a coarse-grained mapping of a potential energy surface (PES) in terms of local minima. These results can then be utilized to gain insights into molecular dynamics and thermodynamic properties. Here we describe how one can employ concepts from unsupervised machine learning to augment BH PES searches to more efficiently identify local minima and the transition states connecting them. Specifically, we introduce the concepts of similarity indices, hierarchical clustering, and multidimensional scaling to the BH methodology. These same machine learning techniques can be used as tools for interpreting and rationalizing experimental results from spectroscopic and ion mobility investigations (e.g., spectral assignment, dynamic collision cross sections). We exemplify this in two case studies: (1) assigning the infrared multiple photon dissociation spectrum of the protonated serine dimer and (2) determining the temperature-dependent collision cross-section of protonated alanine tripeptide.

A Quick Turn of Foot: Rigid Foot-Ground Contact Models for Human Motion Prediction.

Computer simulation can be used to predict human walking motions as a tool of basic science, device design, and for surgical planning. One the challenges of predicting human walking is accurately synthesizing both the movements and ground forces of the stance foot. Though the foot is commonly modeled as a viscoelastic element, rigid foot-ground contact models offer some advantages: fitting is reduced to a geometric problem, and the numerical stiffness of the equations of motion is similar in both swing and stance. In this work, we evaluate two rigid-foot ground contact models: the ellipse-foot (a single-segment foot), and the double-circle foot (a two-segment foot). To evaluate the foot models we use three different comparisons to experimental data: first we compare how accurately the kinematics of the ankle frame fit those of the model when it is forced to track the measured center-of-pressure (CoP) kinematics; second, we compare how each foot affects how accuracy of a sagittal plane gait model that tracks a subjects walking motion; and third, we assess how each model affects a walking motion prediction. For the prediction problem we consider a unique cost function that includes terms related to both muscular effort and foot-ground impacts. Although the ellipse-foot is superior to the double-circle foot in terms of fit and the accuracy of the tracking OCP solution, the predictive simulation reveals that the ellipse-foot is capable of producing large force transients due to its geometry: when the ankle quickly traverses its u-shaped trajectory, the body is accelerated the body upwards, and large ground forces result. In contrast, the two-segment double-circle foot produces ground forces that are of a similar magnitude to the experimental subject because the additional forefoot segment plastically contacts the ground, arresting its motion, similar to a human foot.

Asymptotic Spreading for General Heterogeneous Fisher-KPP Type Equations

In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations: ∂ t u − ∑ i , j = 1 N a i , j ( t , x ) ∂ i j u − ∑ i = 1 N q i ( t , x ) ∂ i u = f ( t , x , u ) . \begin{equation*} \partial _{t} u - \sum _{i,j=1}^N a_{i,j}(t,x)\partial _{ij}u-\sum _{i=1}^N q_i(t,x)\partial _i u=f(t,x,u). \end{equation*} These are concerned with the dynamics of the solution starting from initial data with compact support. The nonlinearity f f is of Fisher-KPP type, and admits 0 0 as an unstable steady state and 1 1 as a globally attractive one (or, more generally, admits entire solutions p ± ( t , x ) p^\pm (t,x) , where p − p^- is unstable and p + p^+ is globally attractive). Here, the coefficients a i , j , q i , f a_{i,j}, q_i, f are only assumed to be uniformly elliptic, continuous and bounded in ( t , x ) (t,x) . To describe the spreading dynamics, we construct two non-empty star-shaped compact sets S _ ⊂ S ¯ ⊂ R N \underline {\mathcal {S}}\subset \overline {\mathcal {S}} \subset \mathbb {R}^N such that for all compact set K ⊂ i n t ( S _ ) K\subset \mathrm {int}(\underline {\mathcal {S}}) (resp. all closed set F ⊂ R N ∖ S ¯ F\subset \mathbb {R}^N\backslash \overline {\mathcal {S}} ), one has lim t → + ∞ sup x ∈ t K | u ( t , x ) − 1 | = 0 \lim _{t\to +\infty } \sup _{x\in tK} |u(t,x)-1| = 0 (resp. lim t → + ∞ sup x ∈ t F | u ( t , x ) | = 0 \lim _{t\to +\infty } \sup _{x\in tF} |u(t,x)| =0 ). The characterizations of these sets involve two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. In particular, it allows us to show that S ¯ = S _ \overline {\mathcal {S}}=\underline {\mathcal {S}} and to establish an exact asymptotic speed of propagation in various frameworks. These include: almost periodic, asymptotically almost periodic, uniquely ergodic, slowly varying, radially periodic and random stationary ergodic equations. In dimension N N , if the coefficients converge in radial segments, again we show that S ¯ = S _ \overline {\mathcal {S}}=\underline {\mathcal {S}} and this set is characterized using some geometric optics minimization problem. Lastly, we construct an explicit example of non-convex expansion sets.

Gravity Inversion of Talwani Model using Very Fast Simulated Annealing

Global approaches for estimating geophysical model parameters have been proposed by several authors, including their application for gravity interpretation, which is currently limited to simple and fixed geometrical problems. This paper proposes implementation of Very Fast Simulated Annealing (VFSA) in two-dimensional gravity interpretation problems, which are still rarely addressed. The modeling domain was divided into smaller sub-domains and gravity anomaly calculation was carried out based on the Talwani formulation. To improve the uniqueness of the solution of under-determined problems, specific constraints were added in addition to the assumed known symmetry axes. The inversion of VFSA was tested on synthetic data generated by simple models and on previously published real data to evaluate the applicability of the proposed approach to the interpretation of field data.

THE ANALYSIS OF DIFFICULTY OF STUDENT’S GEOMETRY PROBLEM SOLVING BASED ON VAN HIELE’S THINKING THEORY WITH MACROMEDIA FLASH

This study aims to describe the level of students' geometry thinking based on Van Hiele's thinking theory, the level of students' geometric problem solving abilities, and analyzing students' difficulties in solving geometric problems. This type of research is qualitative descriptive research. The researcher took six research subjects consisting of low, medium, and high level geometry problem solving abilities to be interviewed and analyzed the types of difficulties experienced. The results of this study indicate that there are 26 students reaching level 0 (visualization), 21 students reach level 1 (analysis), 13 students reach level 2 (informal deduction), 6 students reach level 3 (deduction) and no one can reach level 4 (rigor). Furthermore, there are 13 students who have a low level of problem solving ability, 7 students have a medium level of problem solving ability, and 6 students have a high level of problem solving ability. The difficulties experienced by the research subject are described in each problem.

Process variation-aware gate sizing with fuzzy geometric programming

Limitations of manufacturing process in nanometer dimensions have led to large discrepancies between the circuit design and the manufactured chip known as process variation. In this paper, a new gate sizing method based on fuzzy geometric optimization approach is introduced for power minimization of digital circuits subjected to timing constraints under process variations. In the proposed geometric optimization approach, an accurate gate delay model is applied in which the delay variation is represented by fuzzy numbers; therefore, the subsequent optimization problem is a fuzzy geometric programing problem. In order to efficiently solve the fuzzy geometric programing problem, firstly, the problem is converted into a fuzzy linear programming problem and then, using the expected value and Mellin transform, it is converted into a non-fuzzy problem which is solved by traditional solution techniques. Experimental results show that, the proposed technique achieve more power reduction compared to a fuzzy linear programming method.

Special Issue: Highlights From ASME CIE 2018

Special Issue: Highlights From ASME CIE 2018

Number Sequence Prediction Problems for Evaluating Computational Powers of Neural Networks

Inspired by number series tests to measure human intelligence, we suggest number sequence prediction tasks to assess neural network models’ computational powers for solving algorithmic problems. We define the complexity and difficulty of a number sequence prediction task with the structure of the smallest automaton that can generate the sequence. We suggest two types of number sequence prediction problems: the number-level and the digit-level problems. The number-level problems format sequences as 2-dimensional grids of digits and the digit-level problems provide a single digit input per a time step. The complexity of a number-level sequence prediction can be defined with the depth of an equivalent combinatorial logic, and the complexity of a digit-level sequence prediction can be defined with an equivalent state automaton for the generation rule. Experiments with number-level sequences suggest that CNN models are capable of learning the compound operations of sequence generation rules, but the depths of the compound operations are limited. For the digitlevel problems, simple GRU and LSTM models can solve some problems with the complexity of finite state automata. Memory augmented models such as Stack-RNN, Attention, and Neural Turing Machines can solve the reverse-order task which has the complexity of simple pushdown automaton. However, all of above cannot solve general Fibonacci, Arithmetic or Geometric sequence generation problems that represent the complexity of queue automata or Turing machines. The results show that our number sequence prediction problems effectively evaluate machine learning models’ computational capabilities.