Brute Force Computations and Reference Solutions

Self-scaled conjugate gradient training algorithms

This paper reports from a case study with teachers at two schools in Norway participating in developmental projects aiming for inquiry communities in mathematics teaching and learning. In the reported case study, the teachers participated in one of the developmental projects focusing on implementation and use of computer software in mathematics teaching. I study teachers’ initial orchestration of dynamic geometry software (DGS) in mathematics teaching at lower secondary school. By utilising the notion of ‘instrumental orchestration’ from the theoretical perspective known as the ‘instrumental approach’ (Drijvers et al., in Educ Stud Math 75:213–234, 2010; Trouche, in Int J Comput Math Learn 9:281–307, 2004), I examine how teachers in their initial teaching with DGS empower students’ mathematics learning with the DGS software. According to this perspective, it involves teachers’ orchestration of two interrelated processes instrumentation and instrumentalisation. Analytical findings indicate that a difference in teachers’ empowerment is evident and consequences for students’ opportunities to engage with mathematics represented by the DGS are presented.

Computational realization of period analysis for monitoring cancer patient survival.

Progressive collapse is defined as the spread of an initial local failure of a critical load bearing structural element to the remaining structure, ultimately leading to the collapse of the entire building or of a large portion of it. After the partial collapse of the Ronan Point apartment tower in 1968, when a localised gas explosion ultimately led to the collapse of the whole corner of the building, progressive collapse firstly drew public attention due to its catastrophic consequences on human casualties and economic losses. Other recent collapse events, such as those of the Murrah Federal Building in 1995, Sampoong department store in 1995 and World Trade Centre towers in 2001, have further highlighted the importance of this topic and made engineering communities aware of the necessity to prevent the occurrence of progressive collapse in building structures. In the past decades, considerable research has been devoted to investigate progressive collapse of building structures. These research typically followed an alternate load path approach. This approach allows an initial failure to occur, while the remaining structure needs to seek alternate load paths to dissipate damage and prevent progressive collapse. In practice, the progressive collapse resistance of building structures is generally evaluated by notionally removing one or more load bearing structural elements, representing the initial failure. Despite all research efforts made, they have predominately focused on frame systems. Relevant investigations on reinforced concrete (RC) flat plate structures, a commonly used structural system, are still scarce. The flat plate system contains slabs, of uniform thickness, directly supported by columns without using beams, drop panels and column capitals. This provides structures with increased storey height, architectural convenience and significant savings in construction costs. Nevertheless, the slab-column connection is prone to punching shear failure, which may propagate to the vicinity of connections, potentially leading to progressive collapse. The present study aims at filling some of the current knowledge gap with comprehensive experimental, analytical and numerical investigations on RC flat plate structures. The experimental program consists of four quasi-static large-deformation tests performed on three 1/3-scale, 2×2-bay RC flat plate substructures subjected to critical column removal scenarios. The test specimens included one specimen tested under two consecutive corner column removal scenarios, one specimen under an edge column removal and one specimen under the scenario of concurrently removing both an edge and an interior column. For the loading scheme, a multi-point loading system was specially designed to simulate an increasing uniformly distributed load (UDL). The loading points of this system were equally positioned on the bays adjacent to the removed column(s) to apply the load until failure of the slab. Throughout each test, the structural behaviour of the slab was monitored by extensive apparatuses mounted to the slab both internally and externally. Failure and post-failure behaviours, failure modes, and collapse resisting mechanisms were observed and analysed. Additionally, a complement of analytical studies to the experimental tests, based on the observed yield lines, was carried out to estimate the flexural capacity of the test specimens. To numerically investigate the progressive collapse behaviour of RC flat plate structures, a set of 3D nonlinear finite element modelling approaches, using LS-DYNA software, was established. A total of five numerical models were explicitly created based on the four experimental tests in present research and one from the literature. The proposed modelling approach was basically composed of appropriate definitions of element types, bond-slip relationships, loadings and boundary conditions, and more importantly development of the material models for concrete and reinforcements and calibration of failure criteria for element erosion. A quasi-static loading scheme in displacement control in the numerical models was adopted, the same as that in the conducted tests, through explicitly modelling the multi-point loading system in order to replicate the true test conditions. The numerical models were respectively validated by comparing the load displacement responses, failure modes, crack patterns, and displacement and strain results against the experimental ones, which confirmed the appropriateness and reliability of the proposed models. The validated modelling techniques could be applied to further perform parametric studies of the progressive collapse resistance of RC flat plate substructures and prototype structures. In summary, the significant and original contribution of this research is to (1) obtain a solid understanding of the collapse behaviour of RC flat plate structures under critical column removal scenarios, (2) examine the analytical solution using the yield line theory to predict the flexural capacity of RC flat plate structures, (3) establish a set of numerical modelling techniques which can reasonably replicate the experimental process and provide valuable references of numerically analysing progressive collapse of RC flat plate structures, and (4) give design insights into Australian and international Standards in mitigating progressive collapse of RC flat plate structures.

According to the recent development of the Computational Mechanics, the future development and competitive ability of finite element analysis software seems to be related with the implementation of complicated physical, mechanical and geometric models of solids and fluids. These are coupled models, problems that include physical and geometric nonlinearities, the models or boundary-value problems with small physical or geometric parameters. Thin-walled solids, deformation with large strains and shape distortions, problems coupling solids and fluids supply well- known examples. The modeling of composite materials is another and quite important example nowadays. It begins with solving so- called cell problems and leads to modeling deforming and damaging of composite structural elements as well as to technological problems simulation. The latter type of problems is the problem of a resin with short fibers flow into a mold of complex shape. Another example concerns the process of a resin with long fibers polymerization in a mold followed by the problem of laminate warping. Porous ground and fractured rock are not composites in the commonly used meaning of the term. However, Compositional Mechanics methods are used for their analysis. It is reasonable to mention rather complicated problem of fluid filtration in a porous media experiencing large strains. It seems that a multi scale approach is the most general technique of composite mechanics. It results in so-called local problems in the representative volume element. This paper shows finite-element implementations of local problems developed by the author. Mechanical models and computational algorithms were implemented as home-made computer code. The code has been thoroughly tested and can be used together with FIDESES finite element analysis software as third party package. I may be noted that the developed numerical simulations were elaborated during long term cooperation with the Technical University of Berlin, Dr. Mirtsch GmbH and famous French tire maker Michelin. Further development of the package can be associated with the use of a multi scale approach aiming composite structural elements deformation and progressive damaging modeling, resin with short fibers flow simulation as well as numerical simulation of laminate production process.

The non-stationary gas flow equation in homogeneous and heterogeneous media is considered. Areas of this type are often encountered when considering gas production processes from a gas-bearing reservoir. The heterogeneous structure of the reservoir can strongly affect the extraction processes. For the numerical solution of the problem under consideration, we construct an approximation of the equations on a coarse grid using the numerical averaging method. The method allows us to solve the problem using less computational power and in a shorter time. A numerical comparison of the results of solving the model problem is carried out in a two-dimensional domain for the cases of linear and nonlinear variants of the equations, as well as in a homogeneous and heterogeneous medium. The finite element solution on a fine mesh was taken as a reference solution. The computational realization was performed using the FEniCS library. The construction of the geometric computational domain was performed in the program Gmsh. Paraview and the Matplotlib library in Python were used for data visualization.

Explaining the effect of geometric habits of mind on mathematics courses can affect how to realize the conceptual teaching. Within this scope the aim of the study is to determine the relationship between mathematics preservice teacher’s geometric habits of mind and mathematics achievement. Therefore, this study is carried out with 30 senior preservice teacher in a state university in Turkey. In this study, the relationship between preservice teacher’s geometric habits of mind achievements tests’ scores and mathematics courses’ scores are analyzed. As a result of the study, positive and statistically significant relationship is found between preservice teacher’s geometric habits of mind scores and some mathematics courses such that Geometry, Analytical Geometry 2 and Mathematical Software. It is mean that preservice teachers use geometric habits of mind while encountered problems.

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: Several block canonical forms for describing a class of multivariable control systems have been developed in this project. Based on these canonical structures, some algebraic and geometric theories have been developed for analysis and design of multivariable control systems. Several matrix-valued functions have been defined and computational algorithms have been established for computing these matrix-valued functions. These matrix-valued functions and associated computational algorithms have been utilized for analysis and design of general large-scale continuous-time and discrete-time systems. Based on these research results, twenty-two papers have been published in the referred journals. Keywords: Control theory; Linear systems; Matrix algebra; Abstracts.

The use of a mixed biometric model for breeding evaluation of small cattle has been discussed in the article. This model of breeding evaluation involves a large number of matrix operations. At the same time, the volumes of the formed matrices are directly proportional to the number of animals in the evaluated sample as well as to the number of their off spring. An algorithm for generating matrices of estimated effects that have a large dimension has been presented in the paper. This task is the most time-consuming when using a mixed biometric model. Currently, there are the large number of mathematical packages that provide ample opportunities for performing calculations. A special place in this series is occupied by the integrated mathematical package MATLAB has been designed specifically for performing matrix operations. The authors rely on the use of this package in their work. At the same time the algorithm presented in this paper has the property of universality and can be applied by users in any other software product. Since the matrices of the estimated effects consist of zeros and ones we propose the two-step procedure for forming these matrices. At the first stage, a zero matrix of the required dimension is created. At the second stage, in accordance with the data on the number of evaluated animals, the number of herds for which off spring are distributed, the number and affiliation of evaluated animals to genetic groups, the elements of the matrix are determined, in which zeros are replaced by ones. The advantage of the proposed algorithm is its versatility, and the representation of the algorithm in the form of a block diagram will allow you to design it as a separate proceduresubroutine.

Mathematical models of competitive processes in the economy using known universal models that describe the behavior of counterparties in the market are found in this article. The object of research is to build mathematical models of competitive processes in a pair – secondhand dealer, and to process and convert the economic parameters that are necessary for the models. The methods of research use analysing and solving equations for the known and improved universal models that describe the behaviour of contractors in the market. Mathematical package for construction of schedules of dependences is used. The goals are to construct and research the modified model on the basis of mathematical model by Lotka- Volterra and its further development, and to create software product for statistical processing the economic information. Results – the mathematical model manufacturer – secondhand dealer is created, its modified version is received, and with the help of mathematical package Mathcad, researches of models were done. Prospects of the further improvement of models are revealed. The software product is developed, allowing to process the statistical economic information. On the basis of mathematical model by Lotka-Volterra and its further development a mathematical model of several producers and one middleman is created. Its modified version is studied using mathematical package Mathcad. The nonstable behavior of the counterparties is found. For this model some prospects for further improvements is identified.

The best spectral conjugate gradient algorithm by (Birgin, E. and Martínez, J.M., 2001, A spectral conjugate gradient method for unconstrained optimization. Applied Mathematics and Optimization, 43, 117–128). which is mainly a scaled variant of (Perry, J.M., 1977, A class of Conjugate gradient algorithms with a two step varaiable metric memory, Discussion Paper 269, Center for Mathematical Studies in Economics and Management Science, Northwestern University), is modified in such a way as to overcome the lack of positive definiteness of the matrix defining the search direction. This modification is based on the quasi-Newton BFGS updating formula. The computational scheme is embedded into the restart philosophy of Beale–Powell. The parameter scaling the gradient is selected as spectral gradient or in an anticipative way by means of a formula using the function values in two successive points. In very mild conditions it is shown that, for strongly convex functions, the algorithm is global convergent. Computational results and performance profiles for a set consisting of 700 unconstrained optimization problems show that this new scaled nonlinear conjugate gradient algorithm substantially outperforms known conjugate gradient methods including: the spectral conjugate gradient SCG by Birgin and Martínez, the scaled Fletcher and Reeves, the Polak and Ribière algorithms and the CONMIN by (Shanno, D.F. and Phua, K.H., 1976, Algorithm 500, Minimization of unconstrained multivariate functions. ACM Transactions on Mathematical Software, 2, 87–94).

The stability analysis of slopes is crucial for designing appropriate slope stabilisation methods based on the physical and engineering properties of the soil in the slope area. Slope stability is typically evaluated using the factor of safety (FOS) determined through the Limit Equilibrium Method (LEM). However, the Finite Element Method (FEM) has gained significant attention in geotechnical engineering for conducting comprehensive numerical slope stability analyses. By employing mathematical models and computational algorithms, engineers can simulate slope behaviour and predict stability more accurately and efficiently than traditional methods. Thus, this paper aims to analyse the FOS of slopes and identify the critical slip surface using the LEM and FEM performed by computer software. Two slope profiles were analysed using GEO5 for LEM and PLAXIS 2D for FEM. The findings were compared and discussed. The results show that the FEM generally yields higher FOS values, while the LEM provides slightly lower values. The shapes and locations of the critical slip circles are almost identical for each analysis and method. Overall, this comparative study is crucial in ensuring the validity of the performed analysis by utilising computer software to expedite the analysis and design of the suitable slope stabilisation method.

This paper has dealt with biophysically accurate, or plausible, excitation systems. These are obtained from experiments, and so are complicated, often of high order, and are continually being updated by new experimental results. This is especially true for the excitation equations that represent cardiac tissue. Biophysically relevant problems require quantitatively accurate answers: What happens at what values of what parameter? Thus, numerical methods and results are important, and there is a strong tendency for both the model and the method of analysis to be given as computer algorithms; a current model for cardiac membrane dynamics is a computer package, and the use of path tracking procedures such as AUTO or PATH is becoming common. Perhaps the best way to proceed in the investigation of complicated and exotic dynamics in models of cardiac tissue would be to combine model formulation from voltage clamp data (the derivation of the excitation equations) with bifurcation, and even perhaps singularity theory analysis of dynamical systems, into an expert system.

A traditional approach to developing multivariate statistical theory is algebraic. Sets of observations are represented by matrices, linear combinations are formed from these matrices by multiplying them by coefficient matrices, and useful statistics are found by imposing various criteria of optimization on these combinations. Matrix algebra is the vehicle for these calculations. A second approach is computational. Since many users find that they do not need to know the mathematical basis of the techniques as long as they have a way to transform data into results, the computation can be done by a package of computer programs that somebody else has written. An approach from this perspective emphasizes how the computer packages are used, and is usually coupled with rules that allow one to extract the most important numbers from the output and interpret them. Useful as both approaches are--particularly when combined--they can overlook an important aspect of multivariate analysis. To apply it correctly, one needs a way to conceptualize the multivariate relationships that exist among variables. This book is designed to help the reader develop a way of thinking about multivariate statistics, as well as to understand in a broader and more intuitive sense what the procedures do and how their results are interpreted. Presenting important procedures of multivariate statistical theory geometrically, the author hopes that this emphasis on the geometry will give the reader a coherent picture into which all the multivariate techniques fit.