General Mathematical Concepts Research Articles

Metaheuristics for variable-size mixed optimization problems: A unified taxonomy and survey

Many real world optimization problems are formulated as mixed-variable optimization problems (MVOPs) which involve both continuous and discrete variables. MVOPs including dimensional variables are characterized by a variable-size search space. Depending on the values of dimensional variables, the number and type of the variables of the problem can vary dynamically. MVOPs and variable-size MVOPs (VMVOPs) are difficult to solve and raise a number of scientific challenges in the design of metaheuristics. Standard metaheuristics have been first designed to address continuous or discrete optimization problems, and are not able to tackle VMVOPs in an efficient way. The development of metaheuristics for solving such problems has attracted the attention of many researchers and is increasingly popular. However, to our knowledge there is no well established taxonomy or comprehensive survey for handling this important family of optimization problems. This paper presents an unified taxonomy for metaheuristic solutions for solving VMVOPs in an attempt to provide a common terminology and classification mechanisms. It provides a general mathematical formulation and concepts of VMVOPs, and identifies the various solving methodologies than can be applied in metaheuristics. The advantages, the weaknesses and the limitations of the presented methodologies are discussed. The proposed taxonomy also allows to identify some open research issues which needs further in-depth investigations.

A New Theoretical Interpretation of Measurement Error and Its Uncertainty

The traditional measurement theory interprets the variance as the dispersion of a measured value, which is actually contrary to a general mathematical concept that the variance of a constant is 0. This paper will fully demonstrate that the variance in measurement theory is actually the evaluation of probability interval of an error instead of the dispersion of a measured value, point out the key point of mistake in the traditional interpretation, and fully interpret a series of changes in conceptual logic and processing method brought about by this new concept.

Experimental investigation and modelling of the curing behaviour of photopolymers

This paper focuses on the experimental investigation and modelling of the crosslinking reaction of photopolymers used in additive manufacturing processes such as digital light processing and stereolithography. Starting with general mathematical concepts for the description of the material behaviour of polymeric materials, the importance of modelling the crosslinking reaction is emphasised. In order to characterise the crosslinking reaction experimentally, photocalorimetric measurements with varying isothermal temperature and light intensity are shown. From the exothermic heat flows measured during the crosslinking reaction, the degree of cure can be determined for each experimental scenario as a function of time. It is shown that the test temperature and light intensity have a significant influence on the crosslinking reaction. A modelling approach for the description of the crosslinking reaction incorporating temperature and light intensity is presented. Moreover, parameter identification and a comparison of the proposed model with the experiments are conducted. The identified model has an excellent match with experimental data, resulting in a least square error smaller than 3 %. The proposed model and identification method opens up several extensions for the modelling of the material behaviour.

A Theoretical Study of Indentation with an Oblate Spheroid Shape

In this study, firstly we introduced general mathematical concepts related to the geometrical form of ellipse and ellipsoid. Secondly, the hardness was calculated when we used an indenter having a geometrical shape of an oblate spheroid; the mathematical expressions of static hardness were determined as functions of the applied load, the radius or the depth of the imprint area. The general static hardness formula was expressed considering the projected and the real area imprint of the oblate spheroid indenter. Thirdly mathematical assumptions were introduced in order to simplify the mirohardness and nanohardness formulas when we used the real imprint area. Finally we did a comparative study between the oblate spheroid and the spherical indenters.

For the Introduction of a Conceptual Perspective in Mathematics: Dedekind, Noether, van der Waerden

,,She [Noether] then appeared as the creator of a new direction in algebra and became the leader, the most consistent and brilliant representative, of a particular mathematical doctrine - of all that is characterized by the term ‚Begriffliche Mathematik‘.“ The aim of this paper is to illuminate this "new direction", which can be characterized as a conceptual [begriffliche] perspective in mathematics, and to comprehend its roots and trace its establishment. Field, ring, ideal, the core concepts of this new direction in mathematical images of knowledge, were conceptualized by Richard Dedekind (1831-1916) within the scope of his number theory research and associated with an understanding of a formation of concepts as a "free creation of the human spirit". They thus stand for an abstract perspective of mathematics in their entirety, described as 'modern algebra' in the 1920s and 1930s, leading to an understanding of mathematics as structural sciences. The establishment of this approach to mathematics, which is based on "general mathematical concepts" [allgemein-mathematische Begriffe], was the success of a cultural movement whose most important protagonists included Emmy Noether (1882-1935) and her pupil Bartel L. van der Waerden (1903-1996). With the use of the term 'conceptual', a perspective is taken in the analysis which allows for developing connections between the thinking of Dedekind, the "working and conceptual methods" [Arbeits- und Auffassungsmethoden] of Noether as well as the methodological approach, represented through the thought space of the Noether School as presented under the term "conceptual world" [Begriffswelt] in the Moderne Algebra of van der Waerden. This essay thus makes a contribution to the history of the introduction of a structural perspective in mathematics, a perspective that is inseparable from the mathematical impact of Noether, her reception of the work of Dedekind and the creative strength of the Noether School.

Some sets of indistinguishability operators as multiresolution families

Multiresolution is a general mathematical concept that allows us to study a property by means of several changes of resolution. From a fixed resolution, a coarser projection can be calculated and then the changes between a finer resolution and a coarser one can be studied. That information can give a good knowledge about the problem under consideration. Also using multiresolution techniques it is possible to present information with a higher or a lower detail, given a way to get the adequate granularity or abstraction for a context.The granularity of a system can be obtained or modeled by the use of indistinguishability operators. In this work the relation between indistinguishability operators and multiresolution theory is studied and several methods to build families of indistinguishability operators with multiresolution capacities are given.

Mathematical Model of Banking Operation

A general mathematical concept is proposed to describe banking operation. To this end, a reference model of banking operation is introduced, which describes the evolution of capital by a Markov process. The ruin probability in the reference model that majorizes the bankruptcy probability of bank in the proposed model is estimated. The value of the initial bank capital such that the bank can operate for infinite time with a sufficiently small ruin probability is indicated.

Entropy—A Universal Concept in Sciences

Entropy represents a universal concept in science suitable for quantifying the uncertainty of a series of random events. We define and describe this notion in an appropriate manner for physicists. We start with a brief recapitulation of the basic concept of the theory probability being useful for the determination of the concept of entropy. The history of how this concept came into its to-day exact form is sketched. We show that the Shannon entropy represents the most adequate measure of the probabilistic uncertainty of a random object. Though the notion of entropy has been introduced in classical thermodynamics as a thermodynamic state variable it relies on concepts studied in the theory of probability and mathematical statistics. We point out that whole formalisms of statistical mechanics can be rewritten in terms of Shannon entropy. The notion “entropy” is differently understood in various science disciplines: in classical physics it represents the thermodynamical state variable; in communication theory it represents the efficiency of transmission of communication; in the theory of general systems the magnitude of the configurational order; in ecology the measure for bio-diversity; in statistics the degree of disorder, etc. All these notions can be mapped on the general mathematical concept of entropy. By means of entropy, the configurational order of complex systems can be exactly quantified. Besides the Shannon entropy, there exists a class of Shannon-like entropies which converge, under certain circumstances, toward Shannon entropy. The Shannon-like entropy is sometimes easier to handle mathematically then Shannon entropy. One of the important Shannon-like entropy is well-known Tsallis entropy. The application of the Shannon and Shannon-like entropies in science is really versatile. Besides the mentioned statistical physics, they play a fundamental role in the quantum information, communication theory, in the description of disorder, etc.

Roughness in modules by using the notion of reference points

A module over a ring is a general mathematical concept for many examples of mathematical objects that can be added to each other and mul- tiplied by scalar numbers. In this paper, we consider a module over a ring as a universe and by using the notion of reference points, we provide local approximations for subsets of the universe.

Mechanical Analysis of Finite Idempotent Relations

We use the technique of interactive theorem proving to develop the theory and an enumeration technique for finite idempotent relations. Starting from a short mathematical characterization of finite idempotents defined and proved in Isabelle/HOL, we derive first an iterative procedure to generate all instances of idempotents over a finite set. From there, we develop a more precise theoretical characterization giving rise to an efficient predicate that can be executed in the programming language ML. Idempotent relations represent a very basic, general mathematical concept but the steps taken to develop their theory with the help of Isabelle/HOL are representative for developing algorithms from a mathematical specification. (Research for this paper has been partly performed in collaboration with J.W. Sanders, University of Oxford and has been supported by grant ARC D/03/20258 of the German Academic Exchange Service (DAAD).)

Oriented matroids—combinatorial structures underlying loop quantum gravity

We analyze combinatorial structures which play a central role in determining spectral properties of the volume operator (Ashtekar A and Lewandowski J 1998 Adv. Theor. Math. Phys. 1 388) in loop quantum gravity (LQG). These structures encode geometrical information of the embedding of arbitrary valence vertices of a graph in three-dimensional Riemannian space and can be represented by sign strings containing relative orientations of embedded edges. We demonstrate that these signature factors are a special representation of the general mathematical concept of an oriented matroid (Ziegler G M 1998 Electron. J. Comb.; Björner A et al 1999 Oriented Matroids (Cambridge: Cambridge University Press)). Moreover, we show that oriented matroids can also be used to describe the topology (connectedness) of directed graphs. Hence, the mathematical methods developed for oriented matroids can be applied to the difficult combinatorics of embedded graphs underlying the construction of LQG. As a first application we revisit the analysis of Brunnemann and Rideout (2008 Class. Quantum Grav. 25 065001 and 065002), and find that enumeration of all possible sign configurations used there is equivalent to enumerating all realizable oriented matroids of rank 3 (Ziegler G M 1998 Electron. J. Comb.; Björner A et al 1999 Oriented Matroids (Cambridge: Cambridge University Press)), and thus can be greatly simplified. We find that for 7-valent vertices having no coplanar triples of edge tangents, the smallest non-zero eigenvalue of the volume spectrum does not grow as one increases the maximum spin jmax at the vertex, for any orientation of the edge tangents. This indicates that, in contrast to the area operator, considering large jmax does not necessarily imply large volume eigenvalues. In addition we give an outlook to possible starting points for rewriting the combinatorics of LQG in terms of oriented matroids.

Colliding galaxies, rotating neutron stars and merging black holes—visualizing high dimensional datasets on arbitrary meshes

Visualization of datasets stemming from diverse sources is challenged by the large variety of substantial differences in topology, geometry and nature of the associated data fields. Since there is no standard on how to formulate and treat data for scientific visualization, algorithms are frequently implemented in a highly domain-specific way. Here, we explore the potential of point-wise rendering as a generic way to represent single or multiple fields instantaneously on arbitrary mesh types. This approach is discussed within the terminology of fiber bundles as a general mathematical concept to model scalar-, vector- and tensorfields given on topological spaces (with manifolds as a particular case). We give application examples based on datasets originating from astrophysics and show first results of a tensor field visualization of a recently produced complex dataset of colliding black holes in their final orbit. We finally propose a data layout representing the mathematical concept of a ‘field’ generic enough to handle all cases involved.

Model reduction using the Vorobyev moment problem

Given a nonsingular complex matrix $A\in{\mathbb C}^{N\times N}$ and complex vectors v and w of length N, one may wish to estimate the quadratic form w * A − 1 v, where w * denotes the conjugate transpose of w. This problem appears in many applications, and Gene Golub was the key figure in its investigations for decades. He focused mainly on the case A Hermitian positive definite (HPD) and emphasized the relationship of the algebraically formulated problems with classical topics in analysis - moments, orthogonal polynomials and quadrature. The essence of his view can be found in his contribution Matrix Computations and the Theory of Moments, given at the International Congress of Mathematicians in Zurich in 1994. As in many other areas, Gene Golub has inspired a long list of coauthors for work on the problem, and our contribution can also be seen as a consequence of his lasting inspiration. In this paper we will consider a general mathematical concept of matching moments model reduction, which as well as its use in many other applications, is the basis for the development of various approaches for estimation of the quadratic form above. The idea of model reduction via matching moments is well known and widely used in approximation of dynamical systems, but it goes back to Stieltjes, with some preceding work done by Chebyshev and Heine. The algebraic moment matching problem can for A HPD be formulated as a variant of the Stieltjes moment problem, and can be solved using Gauss-Christoffel quadrature. Using the operator moment problem suggested by Vorobyev, we will generalize model reduction based on matching moments to the non-Hermitian case in a straightforward way. Unlike in the model reduction literature, the presented proofs follow directly from the construction of the Vorobyev moment problem.

Two-patch model of spatial niche segregation

Spatial niche segregation between two habitats and the related adaptive dynamics are investigated. Independent population regulations operate in the two patches by a single resource in each. The populations migrate between the habitats with a constant rate. In line with a general mathematical concept published elsewhere, the niche of a species is described by the measures of the two-way interactions between the species and the resources. Increasing migration rate tends to equalize the population sizes between the habitats and diminish the dependence of the niches on the environmental tolerances of the species. In line with the expectations, when two species coexist, their realized niches are more segregated than their fundamental ones. We demonstrate that robust coexistence requires sufficient niche segregation. That is, the parameter range that allows coexistence of the two species shrinks to nil when the niche-differences between the species disappear. In turn, niche segregation requires separation of tolerances and sufficiently low migration rate. For the evolutionary study we assume a continuous, clonally inherited character that has different optima at the two patches. Evolution of this trait may end up in an intermediate “generalist” optimum, or it can branch and leads to a dimorphic population. The condition of the latter outcome is in line with the conditions that allow niche segregation: the patches have to be sufficiently different and the migration has to be sufficiently low.

On the divergence of line search methods

We discuss the convergence of line search methods for minimization. We explain how Newton's method and the BFGS method can fail even if the restrictions of the objective function to the search lines are strictly convex functions, the level sets of the objective functions are compact, the line searches are exact and the Wolfe conditions are satisfied. This explanation illustrates a new way to combine general mathematical concepts and symbolic computation to analyze the convergence of line search methods. It also illustrate the limitations of the asymptotic analysis of the iterates of nonlinear programming algorithms.

On the divergence of line search methods

We discuss the convergence of line search methods for minimization. We explain how Newton's method and the BFGS method can fail even if the restrictions of the objective function to the search lines are strictly convex functions, the level sets of the objective functions are compact, the line searches are exact and the Wolfe conditions are satisfied. This explanation illustrates a new way to combine general mathematical concepts and symbolic computation to analyze the convergence of line search methods. It also illustrate the limitations of the asymptotic analysis of the iterates of nonlinear programming algorithms.

On the intimate relationship between the principle of duality and the Hough transform

This paper shows the relationship between the principle of duality and the Hough transform. We show that the definition of the Hough transform actually corresponds to an application of the principle of duality. In this interpretation, the general mathematical concepts of figures and high–dimensional coordinates in projective space are re–expressed in terms of shapes and mappings used for shape extraction in Euclidean space. So far, some works have demonstrated the equivalence between the definition of several concepts used in pattern matching and the Hough transform. These works have related the Hough transform to particular developments of diverse concepts including template matching, the Radon transform, maximum likelihood estimation and robust statistics. The correspondence between the Hough transform definition and the principle of duality here developed suggests a more general definition of the Hough transform with a deeper meaning in pure geometry. This definition introduces the formalism of projective geometry to shape extraction and analysis and thus the ideas, properties and geometric relationships in the projective space can have an interpretation for the development of pattern matching techniques. We discuss the analogy between the definition of a figure in geometry and the notion of a shape in the Hough transform, and how this relationship and the generalization of the principle of duality to high–dimensional spaces, defined by space coordinates, are related to the extensions of the Hough transform. Parametric forms are used to develop a dual analytic expression of general forms. The intimate relationship between the Hough transform and the principle of duality increases our understanding of the dual nature of pattern matching which can now benefit from established results in projective geometry.

Response: On Claims That Answer the Wrong Questions

Response: On Claims That Answer the Wrong Questions

General mathematical concept of compensation in sports science with quantitative analysis in the case of sprinting performance

AbstractIn many of the known sports disciplines, especially in athletics, the criterion which determines the positions of the competitors is a simple physical value, mostly a time or a distance, and the athlete with the minimum or maximum, respectively, takes the first place. Moreover, sports science explains this criterion by a set of the so‐called basic abilities. Compensation means the balance of the inferiority of such a basic ability by the superiority of another one.In the following paper, a general abstract concept to analyse compensation in a quantitative way is presented first. It can be applied to any discipline with a measurable criterion, if, in addition, the performance can be described by a kinematic function.Second, the proceeding is worked out in detail for the short sprint based on the modelling of the velocity function of a sprinter and the indicators for his basic abilities in [6]. The two aspects of compensation, namely the set of indicators which belong to the same criterion and the improvement of the criterion caused by the improvement of a basic ability, are discussed in a quantitative way. They turn out to be describable by the mathematical structures of ‘surfaces of constant pure running time’ and certain gradients, respectively.

The Unfathomable Influence of Kolmogorov

A. N. Kolmogorov contributed much to mathematical statistics, both through his own statistical works and through the researches of his students and disciples in statistics. But the case will be made that he influenced statistics still more through his general mathematical concepts and approaches. This paper gives some examples of unexpected applications of Kolmogorov's ideas in topics of nonparametric statistics.