Mathematical Issues Research Articles

A Symbolical Approach to Negative Numbers

(ProQuest: ... denotes formulae omitted.)IntroductionStatus QuoIt is a well known fact from the History of Mathematics that negative numbers were not widely accepted by mainstream European mathematicians until the Eighteenth Century, and then only after much controversy (Katz, 2009). By contrast, the arithmetic of rational numbers was known in essentially its modern form to Diophantus in the Third Century (Heath, 1910; Klein, 1968) and the Ancient Egyptians possessed a system of unit fractions more than two thousand years earlier (Chace, 1979). It is therefore unsurprising that negative numbers are regarded as difficult to teach (Hitchcock, 1997; Hefendehl-Hebeker, 1991) or, as the Report of the Cambridge Conference on School Mathematics put it, "Perhaps no area of discussion brought out more viewpoints that the question of how the multiplication of signed numbers should be introduced" (Cambridge, 1963). Comparatively, the properties of rational numbers are greeted with far less trepidation, especially when armed with a suitable physical model. This difference in difficulty is also seen in the order of contemporary school curricula, with negative numbers, regarded as more abstract, generally delayed until secondary school, while fractions, viewed as more concrete, can be taught as early as lower primary.While this may represent the status quo which has existed for many years, mathematics educators are naturally resistant to the idea that certain topics are difficult to teach, or can only be taught to the more able student. Indeed, each new generation of researchers seems to yield a fresh initiative for overcoming the barriers to entry into higher mathematics, such as the New Math in the 60's and 70's (Cambridge, 1963; Southampton, 1961) or the Reform Mathematics in the 80's and 90's (Cockcroft, 1982; Standards, 1989). Most recently, the 2000's and 2010's have seen the development of the Early Algebra movement, which aims to smooth out the abrupt jump from primary school arithmetic to secondary school algebra by selectively introducing certain kinds of algebraic thinking into primary schools (Carraher & Schliemann, 2007). Unlike the New Math and Reform Mathematics movements, both of which achieved mixed success, a key research finding already obtained by Early Algebra research is that young children are capable of a much higher level of abstract thought than previously believed (Mason, 2008). In particular, even children of lower primary school age have been found to be able to work with negative numbers, well before the point at which this topic is normally introduced (Bishop et al., 2011; Wilcox, 2008; Behrend, 2006).Concern over negative numbers is not confined to the Early Algebra movement, however, and the many research articles recently published on this topic show that important issues still remain unresolved (e.g. Almeida & Bruno, 2014; Bishop et al., 2014; Bofferding, 2014; Leong et al., 2014; Whitacre et al., 2012; Selter et al., 2012; Altiparmak & Ozdogan, 2010). All these articles feature classroom based research, where the mathematical content can be assumed fixed, and different teaching approaches are investigated. This article also aims to improve classroom teaching, but the exactly opposite way: it considers different approaches to the mathematics, and existing classroom based research is used to justify that what is proposed is worth trying with children. In other words, rather than searching for a pedagogy which matches the difficulty level of the mathematics, the difficulty level of the mathematics is lowered to match the available pedagogy. That changes to the mathematics should be possible at all is justified in the next section, which highlights unresolved mathematical issues in each of the three strands mentioned in the first paragraph, namely: the historical origins of negative numbers, their mathematical definition, and the physical models used in teaching. …

Pre-service teachers’ awareness of interdisciplinary connections: Mathematics, financial literacy, and social justice issues

ABSTRACTThis article describes a study that investigated pre-service teachers’ awareness of the interdisciplinary connections among mathematics, personal finance, and social justice issues. When pre-service teachers understand these connections, they can prepare their students to be responsible, participatory, and justice-oriented citizens in a global economy. Possessing and articulating this awareness are integral components of providing children and youth with the knowledge and skills to improve their life quality and bring about social change. The 68 participants in the study were pre-service teachers at two public universities in the United States, one in the Midwest and one in the Southeast. The participants at the Midwestern university were enrolled in general education classes, and the participants at the Southeastern university were enrolled in elementary, middle school, and high school mathematics methods classes in the Fall 2010 and Spring 2011 semesters. They completed surveys mid-semester, in which they were prompted to discuss mathematics literacy, financial literacy, and their social applications. The researchers found the majority of the pre-service teachers had narrow or intermediate conceptions of mathematics and financial literacy, discussing the need to understand mathematics and financial terminology, as well as their social connections. However, the participants were largely unable to articulate related social justice issues, seeing mathematics as an isolated discipline.

Washback Effect of University Entrance exams in Applied Mathematics to Social Sciences

Curricular issues of subject Applied Mathematics to Social Sciences are studied in relation to university entrance exams performed in several Spanish regions between 2009–2014. By using quantitative and qualitative analyses, it has been studied how these exams align with curriculum and how they produce a washback on curriculum and teachers’ work. Additionally, one questionnaire about teachers’ practices has been performed, in order to find out how the exams are influencing teaching methodology development. Main results obtained show that evaluation is producing a bias on the official curriculum, substantially simplifying the specific orientation that should guide applied mathematics. Furthermore, teachers’ practices are influenced by the exams, and they usually approach their teaching methodology to the frequent types of exams. Also, slight differences among the teachers lead to distinguish two behavioral subgroups. Results can also be useful in an international context, because of the importance of standardized exit exams in OECD countries.

Letter to the editor: How I left my comfort zone in pure mathematics

I studied mathematics as my major. When I was a senior, I already felt tired of abstract mathematics. Psychology of education could be better for me, or another field which has its concrete application in real life. However, at the same time, I did not want to move away from mathematics. Then I met the Department of Education, and I started to take some classes related to education to get ready for my teaching carrier. Because with some background in pedagogy, I knew that I would be able to be more helpful to my students in the future. Then I realized that some mathematical issues which my professors probably worked on for hours were quite barren and not useful for my purposes: teaching mathematics, I mean, usable mathematics that is lots of fun!

Broadening the Study of Participation in the Life Sciences: How Critical Theoretical and Mixed-Methodological Approaches Can Enhance Efforts to Broaden Participation.

This research methods Essay details the usefulness of critical theoretical frameworks and critical mixed-methodological approaches for life sciences education research on broadening participation in the life sciences. First, I draw on multidisciplinary research to discuss critical theory and methodologies. Then, I demonstrate the benefits of these approaches for researchers who study diversity and inclusion issues in the life sciences through examples from two critical mixed-methods studies of prominent issues in science, technology, engineering, and mathematics (STEM) participation and recognition. The first study pairs critical discourse analysis of the STEM workforce literature, data, and underlying surveys with quantitative analyses of STEM pathways into the workforce. This example illustrates the necessity of questioning popular models of retention. It also demonstrates the importance of intersecting demographic categories to reveal patterns of experience both within and between groups whose access to and participation in STEM we aim to improve. The second study's critical approach applies research on inequities in prizes awarded by STEM professional societies toward organizational change. This example uses data from the life sciences professional societies to show the importance of placing data within context to broaden participation and understand challenges in creating sustainable change.

Mathematical expression and sampling issues of treatment contrasts: Beyond significance testing and meta-analysis to clinically useful research synthesis

The more two treatments’ outcome distributions overlap, the more ambiguity there is about which would be better for some clients. Effect size and t-statistics ignore this ambiguity by indicating nothing about the contrasted treatments’ outcome ranges, although the wider these are the smaller are these statistics and the more other influences than these given treatments matter for outcomes. Treatment contrast data analysis logically requires valid measurement of all the influences on outcomes. Each influence, measured or not, is somehow sampled in every treatment contrast, and the nature of this sampling affects the contrast’s two outcome distributions. Sampling also affects replications of a treatment contrast, which requires sampling that produces the same statistically expected outcome distributions for each replicate as a logical prerequisite of proper meta-analysis. Because scientific human psychology is most fundamentally about individual persons and cases, rather than aggregations of persons or cases, contrasted treatments’ outcome distributions ought eventually be disaggregated to whatever input dimension gradation configurations collapse their ranges to zero through jointly taking account of every influence on outcomes. Only then are the data about individual persons or cases and so relevant to psychotherapy theory.

New Software for Simulation of Electrochemical Reaction Mechanisms of Any Complexity

We introduce here our scientific software KISSA designed for automatic simulation of electrochemical reaction mechanisms of any complexity, including those leading to ECL emission at planar, (hemi)spherical and (hemi)cylindrical working electrodes (KISSA-1D) as well as at microelectrodes such as disk or band (KISSA-2D). Simulations are performed by KISSA using a novel approach [1, 2] developed by the authors which involves conformal coordinate transformations to overcome the numerical difficulties arising from diffusional propagation of species and automatic grid adaptation based on the detection of large kinetic terms to obtain accurate results for both concentrations and electrochemical currents even in the presence of steep concentration gradients. This approach allows for efficient resolution of all physicochemical and mathematical issues connected both with edge effects (nano- and microelectrodes) and considerable disparity of time and length scales of the underlying processes arising due to quickly changing electrode potential and/or extremely rapid homogeneous chemical reactions for any complex mechanisms. The software offers a possibility to take into account spontaneous natural convection which limits the extent of the diffusion layer and leads to the establishment of a steady state current even at macroelectrodes. Furthermore, this enables setting correct and thermodynamically consistent initial conditions within the stagnant layer when the system is pre-equilibrated at a given rest potential for a finite period of time [3]. The software also allows simulating complex adsorption dynamics, i.e., kinetics and chemical reactions involving surface-bound species [4, 5]. The programs offer convenient entry of a reaction mechanism and parameters and provide access to the computed dynamic spatial distributions of all important quantities such as currents and current densities, concentration distributions of all reactants, surface coverage of adsorbed species [4, 5] as well as ECL intensity [6] (if applicable) through a graphical user interface [7]. [1] C. Amatore, O.V. Klymenko, I. Svir, Electrochem. Commun. 12 (2010) 1170. [2] O.V. Klymenko, I. Svir, A. Oleinick, C. Amatore, ChemPhysChem 13 (2012) 845. [3] C. Amatore, O.V. Klymenko, I. Svir, Anal. Chem. 84 (2012) 2792. [4] O.V. Klymenko, I. Svir, C. Amatore, J. Electroanal. Chem. 688 (2013) 320. [5] O.V. Klymenko, I. Svir, C. Amatore, Molecular Physics, 112 (2014) 1273. [6] O.V. Klymenko, I. Svir, C. Amatore, ChemPhysChem, 14 (2013) 2237. [7] http://kissagroup.com/index.php/bibliography

A REVIEW - CAPTCHA AS GRAPHICAL PASSWORDS—A NEW SECURITY PRIMITIVE BASED ON HARD AI PROBLEMS

Most of the safety primeval square measure supported mathematical issues. This analysis goals to check existing parole and to style a brand new improved graphical parole pattern. Captcha as a graphical parole. during this paper, we tend to discuss a brand new security primeval supported exhausting computer science issues, a innovative of graphical parole systems created on dominant of Captcha technology, what we are saying Captcha as graphical passwords (CaRP). CaRP is each a Captcha and a graphical parole pattern. With the mix of CAPTCHA and graphical parole addresses a like on-line estimation attacks, relay attacks, combination of with dual-view technology, and shoulder-surfing attacks. If the parole is in search nominative then CaRP parole are often found solely risk by automatic on-line estimation attack.

Entangled Harmonic Oscillators and Space-Time Entanglement

The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lorentz-covariant picture of the bound state, which requires both space and time separations between two constituent particles. These space and time variables become entangled as the bound state moves with a relativistic speed. It is shown also that our inability to measure the time-separation variable leads to an entanglement entropy together with a rise in the temperature of the bound state. As was noted by Paul A. M. Dirac in 1963, the system of two oscillators contains the symmetries of the O ( 3 , 2 ) de Sitter group containing two O ( 3 , 1 ) Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the S p ( 4 ) group, which serves as the basic language for two-mode squeezed states. Since the S p ( 4 ) symmetry contains both rotations and squeezes, one interesting case is the combination of rotation and squeeze, resulting in a shear. While the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified.

Recent research on geometry education: an ICME-13 survey team report

This survey on the theme of Geometry Education (including new technologies) focuses chiefly on the time span since 2008. Based on our review of the research literature published during this time span (in refereed journal articles, conference proceedings and edited books), we have jointly identified seven major threads of contributions that span from the early years of learning (pre-school and primary school) through to post-compulsory education and to the issue of mathematics teacher education for geometry. These threads are as follows: developments and trends in the use of theories; advances in the understanding of visuo spatial reasoning; the use and role of diagrams and gestures; advances in the understanding of the role of digital technologies; advances in the understanding of the teaching and learning of definitions; advances in the understanding of the teaching and learning of the proving process; and, moving beyond traditional Euclidean approaches. Within each theme, we identify relevant research and also offer commentary on future directions.

Synchronization Protocols and Implementation Issues in Wireless Sensor Networks: A Review

Time synchronization in wireless sensor networks (WSNs) is a topic that has been attracting the research community in the last decade. Most performance evaluations of the proposed solutions have been limited to theoretical analysis and simulation. They consequently ignored several practical aspects, e.g., packet handling jitters, clock drifting, packet loss, and mote limitations, which affect real implementation on sensor motes. Authors of some pragmatic solutions followed empirical approaches for the evaluation, where the proposed solutions have been implemented on real motes and evaluated in testbed experiments. This paper gives an insight on issues related to the implementation of synchronization protocols in WSN. The challenges related to WSN environment are presented; the importance of real implementation and testbed evaluation are motivated by some experiments we conducted. The most relevant implementations of the literature are then reviewed, discussed, and qualitatively compared. While there are several survey papers that present and compare the protocols from the conception perspectives, as well as others that deal with mathematical and signal processing issues of the estimators, a survey on practical aspects related to the implementation is missing. To our knowledge, this paper is the first one that takes into account the practical aspect of existing solutions.

Einstein׳s physical strategy, energy conservation, symmetries, and stability: “But Grossmann & I believed that the conservation laws were not satisfied”

Recent work on the history of General Relativity by Renn et al. shows that Einstein found his field equations partly by a physical strategy including the Newtonian limit, the electromagnetic analogy, and energy conservation. Such themes are similar to those later used by particle physicists. How do Einstein׳s physical strategy and the particle physics derivations compare? What energy–momentum complex(es) did he use and why? Did Einstein tie conservation to symmetries, and if so, to which? How did his work relate to emerging knowledge (1911–1914) of the canonical energy–momentum tensor and its translation-induced conservation? After initially using energy–momentum tensors hand-crafted from the gravitational field equations, Einstein used an identity from his assumed linear coordinate covariance xμ′=Mνμxν to relate it to the canonical tensor. Usually he avoided using matter Euler–Lagrange equations and so was not well positioned to use or reinvent the Herglotz–Mie–Born understanding that the canonical tensor was conserved due to translation symmetries, a result with roots in Lagrange, Hamilton and Jacobi. Whereas Mie and Born were concerned about the canonical tensor׳s asymmetry, Einstein did not need to worry because his Entwurf Lagrangian is modeled not so much on Maxwell׳s theory (which avoids negative-energies but gets an asymmetric canonical tensor as a result) as on a scalar theory (the Newtonian limit). Einstein׳s theory thus has a symmetric canonical energy–momentum tensor. But as a result, it also has 3 negative-energy field degrees of freedom (later called “ghosts” in particle physics). Thus the Entwurf theory fails a 1920s–1930s a priori particle physics stability test with antecedents in Lagrange׳s and Dirichlet׳s stability work; one might anticipate possible gravitational instability.This critique of the Entwurf theory can be compared with Einstein׳s 1915 critique of his Entwurf theory for not admitting rotating coordinates and not getting Mercury׳s perihelion right. One can live with absolute rotation but cannot live with instability.Particle physics also can be useful in the historiography of gravity and space–time, both in assessing the growth of objective knowledge and in suggesting novel lines of inquiry to see whether and how Einstein faced the substantially mathematical issues later encountered in particle physics. This topic can be a useful case study in the history of science on recently reconsidered questions of presentism, whiggism and the like. Future work will show how the history of General Relativity, especially Noether׳s work, sheds light on particle physics.

System Engineering Approach of Diabetes Treatment

The paper gives a short system engineering review on the treatment of diabetes covering control engineering and mathematical modeling issues. The Artificial Pancreas (AP) concept is discussed together with the Intensive Care (ICU) idea for automatic treatment of diabetic patients. This highly interdisciplinary concept involves beyond medical sciences, control engineering and biomedical engineering knowledge. Having already developed continuous glucose sensor devices and highly performant insulin pumps, the automatic control concept could be an efficient solution for millions of people living with diabetes for accurate metabolic conditions management.

Commentary on the statistical properties of noise and its implication on general linear models in functional near-infrared spectroscopy.

Functional near-infrared spectroscopy (fNIRS) is a noninvasive neuroimaging technique that uses low levels of light to measure changes in cerebral blood oxygenation levels. In the majority of NIRS functional brain studies, analysis of this data is based on a statistical comparison of hemodynamic levels between a baseline and task or between multiple task conditions by means of a linear regression model: the so-called general linear model. Although these methods are similar to their implementation in other fields, particularly for functional magnetic resonance imaging, the specific application of these methods in fNIRS research differs in several key ways related to the sources of noise and artifacts unique to fNIRS. In this brief communication, we discuss the application of linear regression models in fNIRS and the modifications needed to generalize these models in order to deal with structured (colored) noise due to systemic physiology and noise heteroscedasticity due to motion artifacts. The objective of this work is to present an overview of these noise properties in the context of the linear model as it applies to fNIRS data. This work is aimed at explaining these mathematical issues to the general fNIRS experimental researcher but is not intended to be a complete mathematical treatment of these concepts.

Max Euwe's Set-Theoretic Observations on the Game of Chess — Introductory Notes

This is a brief introduction to the Classic on Constructive Game Theory by Max Euwe (18), translated and republished in this issue of New Mathematics and Natural Computation. The task and aim of the Introduction is simply to provide a context, and some historical notes, particularly on the difference between Brouwer's approach to Set Theory — an Intuitionistic-Constructive approach and that of Zermelo.

RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

The following article has been included in a multiple retraction: Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract In 2015 SAGE were made aware of concerns regarding the Special Issue of Mathematics & Mechanics of Solids on Advances in Canonical Duality Theory, guest-edited by Professor David Gao. At the request of the Guest Editor, the Special Issue has been retracted, due to conflict of interest regarding Professor Gao’s role as Guest Editor and co-author on a number of submitted papers. In addition the peer review process was less rigorous than the journal requires. The Guest Editor takes full responsibility for the retraction. The following articles that were due to appear in the Special Issue have therefore been retracted: Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization in complex systems; David Y Gao, Ning Ruan, and Vittorio Latorre. http://mms.sagepub.com/content/early/2015/02/24/1081286514566533.abstract Canonical Dual Approach for Contact Mechanics Problems with Friction; Vittorio Latorre, Simone Sagratella, David Y Gao. http://mms.sagepub.com/content/early/2015/01/20/1081286514566534.abstract Canonical Duality Theory for Solving Non-Monotone Variational Inequality Problems; Guoshan Liu, David Y Gao, Shouyang Wang. http://mms.sagepub.com/content/early/2015/02/04/1081286514566535.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part I; Shu-Cherng Fang, David Y Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wen-Xun Xing. http://mms.sagepub.com/content/early/2015/02/24/1081286514566704.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material; David Y Gao and E. Hajilarov. http://mms.sagepub.com/content/early/2015/07/06/1081286515591084.abstract Triality Theory and Complete Post-buckling Solutions of Large Deformed Beam by Canonical Dual Finite Element Method; Kun Cai, David Y Gao, Qinghua Qin. http://mms.sagepub.com/content/early/2015/06/28/1081286515591085.abstract Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory; Yi Chen, David Y Gao. http://mms.sagepub.com/content/early/2015/04/08/1081286515577122.abstract Unified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstract A Framework of Canonical Dual Algorithms for Global Optimization; Xiaojun Zhou, David Y Gao, Chunhua Yang. http://mms.sagepub.com/content/early/2015/07/22/1081286515592190.abstract Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems; Ning Ruan, David Y Gao. http://mms.sagepub.com/content/early/2015/07/08/1081286515591087.abstract Global Optimization Solutions to a Class of Non-convex Quadratic Minimization Problems with Quadratic Constraints; Yubo Yuan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591086.abstract On Minimal Distance between Two Non-Convex Surfaces; Daniel Morales-Silva, David Y Gao. http://mms.sagepub.com/content/early/2015/07/27/1081286515592949.abstract The Editor-in-Chief and SAGE strive to uphold the very highest standards of publication ethics and are committed to supporting the high standards of integrity of Mathematics & Mechanics of Solids. Authors, reviewers, editors and interested readers are encouraged to consult SAGE’s ethics statements and the Committee on Publication Ethics (COPE) website for guidelines on publication ethics.

RETRACTED: Global solutions to spherically constrained quadratic minimization via canonical duality theory

The following article has been included in a multiple retraction: Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory; Yi Chen, David Y Gao. http://mms.sagepub.com/content/early/2015/04/08/1081286515577122.abstract In 2015 SAGE were made aware of concerns regarding the Special Issue of Mathematics & Mechanics of Solids on Advances in Canonical Duality Theory, guest-edited by Professor David Gao. At the request of the Guest Editor, the Special Issue has been retracted, due to conflict of interest regarding Professor Gao’s role as Guest Editor and co-author on a number of submitted papers. In addition the peer review process was less rigorous than the journal requires. The Guest Editor takes full responsibility for the retraction. The following articles that were due to appear in the Special Issue have therefore been retracted: Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization in complex systems; David Y Gao, Ning Ruan, and Vittorio Latorre. http://mms.sagepub.com/content/early/2015/02/24/1081286514566533.abstract Canonical Dual Approach for Contact Mechanics Problems with Friction; Vittorio Latorre, Simone Sagratella, David Y Gao. http://mms.sagepub.com/content/early/2015/01/20/1081286514566534.abstract Canonical Duality Theory for Solving Non-Monotone Variational Inequality Problems; Guoshan Liu, David Y Gao, Shouyang Wang. http://mms.sagepub.com/content/early/2015/02/04/1081286514566535.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part I; Shu-Cherng Fang, David Y Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wen-Xun Xing. http://mms.sagepub.com/content/early/2015/02/24/1081286514566704.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material; David Y Gao and E. Hajilarov. http://mms.sagepub.com/content/early/2015/07/06/1081286515591084.abstract Triality Theory and Complete Post-buckling Solutions of Large Deformed Beam by Canonical Dual Finite Element Method; Kun Cai, David Y Gao, Qinghua Qin. http://mms.sagepub.com/content/early/2015/06/28/1081286515591085.abstract Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory; Yi Chen, David Y Gao. http://mms.sagepub.com/content/early/2015/04/08/1081286515577122.abstract Unified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstract A Framework of Canonical Dual Algorithms for Global Optimization; Xiaojun Zhou, David Y Gao, Chunhua Yang. http://mms.sagepub.com/content/early/2015/07/22/1081286515592190.abstract Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems; Ning Ruan, David Y Gao. http://mms.sagepub.com/content/early/2015/07/08/1081286515591087.abstract Global Optimization Solutions to a Class of Non-convex Quadratic Minimization Problems with Quadratic Constraints; Yubo Yuan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591086.abstract On Minimal Distance between Two Non-Convex Surfaces; Daniel Morales-Silva, David Y Gao. http://mms.sagepub.com/content/early/2015/07/27/1081286515592949.abstract The Editor-in-Chief and SAGE strive to uphold the very highest standards of publication ethics and are committed to supporting the high standards of integrity of Mathematics & Mechanics of Solids. Authors, reviewers, editors and interested readers are encouraged to consult SAGE’s ethics statements and the Committee on Publication Ethics (COPE) website for guidelines on publication ethics.

Lorentz transformation from an elementary point of view

Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a $G$-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.

Irrational images – the visualization of abstract mathematical terms

In this article we would like to draw attention to the cognitive potential hidden in an image and in the art which employs it. We will focus on the visualization of basic mathematical objects i.e. irrational numbers. Our starting point will be the easy and intuitive case of the square root of two, as it is observed in the diagonal of a square. Next we will move over to the golden ratio hidden in a regular pentagon. With the visualization of the irrational number ϕ Normal 0 21 false false false PL ZH-CN X-NONE we will use a looped, endless animation. Finally, we will have a closer look at the famous number π Normal 0 21 false false false PL ZH-CN X-NONE and we will suggest an attempt at its clearly visual representation. In the last section of the article we will consider the possibility of indicating rational and irrational real numbers represented by dimensionless points on a straight line. We will also try to present a straight line on a flat surface which - as we know has length - but has no width. The above issues will enable us to see the extent to which mathematics may be inspirational for art, as well as how art may familiarize us with mathematical issues and explain them.

Second contribution to the ENE critique: Reply to Davidson, part 1

ABSTRACTOn first encounter, the ergodic/nonergodic (ENE) approach has apparent plausibility. Although concerned by some of its problems for many years, it was only after more concentrated reflection on both its parts and their combinations that I became aware of its manifold deficiencies, some of which I outlined in my previous critique (O’Donnell, 2014). In this paper, facilitated by Davidson’s (2015b) rejoinder, these criticisms are deepened, broadened, and strengthened. Because the debate deals with fundamental matters in several disciplines, a considerable amount of investigation, unpacking, and logical dissection is required to clarify the argumentation beneath the compressed and seemingly smooth surface of the ENE position. For this reason, my reply is divided into two parts. This contribution primarily examines the central role of framing in ENE arguments, and clarifies the various misunderstandings and misrepresentations to which it leads. The subsequent contribution provides more detailed discussion of mathematical, stochastic, and methodological issues.