Contemporary Mathematics Research Articles

Kako su matematičari kao deo istorije matematike predstavljeni u savremenim kineskim udžbenicima matematike?

As part of our efforts to understand how history and culture are manifested in mathematics textbooks, we investigated particularly in this study how mathematicians are represented across three different series of contemporary school mathematics textbooks used in primary and junior secondary schools in China. The study intends to document the ways of representation of mathematicians in the three series of textbooks, compare the similarities and differences, and explore the reasons and implications of the findings concerning mathematics textbook research and development. Our findings show that each series of mathematics textbooks introduced both Chinese and non-Chinese mathematicians in a commendable way, though most of the mathematicians introduced were ancient mathematicians, and all the three series introduced mathematicians each year from Grade 2 to Grade 9. There is also a high level of consistency in the distribution of the introduction of mathematicians in the three series in terms of mathematics contents and the structures of the chapters. In comparison, we found the mathematics textbook series by the PEP presented a more balanced distribution of the introductions of mathematicians in terms of grade levels, ethnic origins and the history timeline. The explanations and implications of the findings are discussed at the end of the paper.

Architectural and structural optimization research of structural forms topologically transformed

In nature, homologues can be seen in structures of common evolutionary origin, which are composed of similar elements in the same zones. Homology shows the relationship of living organisms, which may differ externally, but their structure (bone and muscle) is characterized by significant similarity. The equivalent of homologous transformations in mathematics is homeomorphism, which is one of the basic concepts of the department of contemporary mathematics dealing with the study of the properties of topological geometric figures and solids (properties which do not change even after radical continuous deformation of figures), called topology. Homeomorphism is about transforming figures without tearing them apart and gluing them together, only by stretching, compressing, bending and twisting them, which enables continuity and identity functions to be maintained. Homeomorphic spaces are indistinguishable from the point of view of topology. Homeomorphism of space ensures the preservation of topological invariables, which include a number of features such as closeness, openness, compactness, centrality or cohesion. Optimization searches for solids based on topological transformations, carried out in the scope of architectural and structural criteria can be an important element of analyses in the interdisciplinary design process. On their basis it is possible to make a conscious choice between individual forms inspired by geometry or structures found in nature. The paper presents the results of model analyses of structural forms formed on the basis of topological transformations with determination of their effectiveness in terms of minimum mass. Optimization searches due to the minimum mass are an important element from the design of objects due to the idea of sustainable development.

The Quantum Strategy of Completeness: On the Self-Foundation of Mathematics

Gentzen’s approach by transfinite induction and that of intuitionist Heyting arithmetic to completeness and the self-foundation of mathematics are compared and opposed to the Godel incompleteness results as to Peano arithmetic. Quantum mechanics involves infinity by Hilbert space, but it is finitist as any experimental science. The absence of hidden variables in it interpretable as its completeness should resurrect Hilbert’s finitism at the cost of relevant modification of the latter already hinted by intuitionism and Gentzen’s approaches for completeness. This paper investigates both conditions and philosophical background necessary for that modification. The main conclusion is that the concept of infinity as underlying contemporary mathematics cannot be reduced to a single Peano arithmetic, but to at least two ones independent of each other. Intuitionism, quantum mechanics, and Gentzen’s approaches to completeness an even Hilbert’s finitism can be unified from that viewpoint. Mathematics may found itself by a way of finitism complemented by choice. The concept of information as the quantity of choices underlies that viewpoint. Quantum mechanics interpretable in terms of information and quantum information is inseparable from mathematics and its foundation.

Operations and Actions of Lie Groups on Manifolds

As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the square root of an element (in this case the result is not always in the set). In this paper, we aim to find the operations and actions of Lie groups on manifolds. These actions can be applied to the matrix group and Bi-invariant forms of Lie groups and to generalize the eigenvalues and eigenfunctions of differential operators on Rn. A Lie group is a group as well as differentiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct. It plays an extremely important role in the theory of fiber bundles and also finds vast applications in physics. It represents the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds.

Convergence to a-stable Lévy motion for chaotic billiards with several cusps at flat points

We consider billiards with several possibly non-isometric and asymmetric cusps at flat points; the case of a single symmetric cusp was studied previously in Zhang (2017 Dynamical Systems, Ergodic Theory, and Probability: in Memory of Kolya Chernov (Contemporary Mathematics vol 698) (Providence, RI: American Mathematical Society) pp 287–316) and Jung and Zhang (2018 Ann. Henri Poincaré 19 3815–53). In particular, we show that properly normalized Birkhoff sums of Hölder observables, with respect to the billiard map, converge in Skorokhod’s M1-topology to an -stable Lévy motion, where depends on the ‘curvature’ of the flattest points and the skewness parameter depends on the values of the observable at those same points. Previously, Jung and Zhang (2018 Ann. Henri Poincaré 19 3815–53) proved convergence of the one-point marginals to totally skewed -stable distributions for a symmetric cusp. The limits we prove here are stronger, since they are in the functional sense, but also allow for more varied behaviour due to the presence of multiple cusps. In particular, the general limits we obtain allow for any skewness parameter, as opposed to just the totally skewed cases. We also show that convergence in the stronger J1-topology is not possible.

Strongly graded vertex algebras generated by vertex Lie algebras

We construct three families of vertex algebras along with their modules from appropriate vertex Lie algebras, using the constructions in [Vertex Lie algebra, vertex Poisson algebras and vertex algebras, in Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory[Formula: see text] Proceedings of an International Conference at University of Virginia[Formula: see text] May 2000, in Contemporary Mathematics, Vol. 297 (American Mathematical Society, 2002), pp. 69–96] by Dong, Li and Mason. These vertex algebras are strongly graded vertex algebras introduced in [Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, in Conformal Field Theories and Tensor Categories[Formula: see text] Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, eds. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2 (Springer, New York, 2014), pp. 169–248] by Huang, Lepowsky and Zhang in their logarithmic tensor category theory and can also be realized as vertex algebras associated to certain well-known infinite dimensional Lie algebras. We classify irreducible [Formula: see text]-gradable weak modules for these vertex algebras by determining their Zhu’s algebras. We find examples of strongly graded generalized modules for these vertex algebras that satisfy the [Formula: see text]-cofiniteness condition introduced in [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009] by the second author. In particular, by a result of the second author [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009, 26 pp.], the convergence and extension property for products and iterates of logarithmic intertwining operators in [Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011); arXiv:1110.1929 ] among such strongly graded generalized modules is verified.

MATHEMATICAL RIGOR AND PROOF

AbstractMathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely, the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it.

Differential heterogenesis and the emergence of semiotic function

AbstractIn this study, we analyse the notion of “differential heterogenesis” proposed by Deleuze and Guattari on a morphogenetic perspective. We propose a mathematical framework to envisage the emergence of singular forms from the assemblages of heterogeneous operators. In opposition to the kind of differential calculus that is usually adopted in mathematical-physical modelling, which tends to assume a homogeneous differential equation applied to an entire homogeneous region, heterogenesis allows differential constraints of qualitatively different kinds in different points of space and time. These constraints can then change in time, opening the possibility for new kinds of differential dynamics and the emergence of distinct entities and forms. Formally, we show that operators with different phase spaces can be assembled on the basis of a result of Rothschild & Stein (1976. Hypoelliptic differential operators and nilpotent groups. Acta Mathematica 137. 247–320). Furthermore, operators with different dynamics can be assembled by means of a partition of the unit.After stating the concept of differential heterogenesis in terms of contemporary mathematics, we show that this construction sheds light on the constitution of the semiotic function. In fact, both the Merleau-Pontian and the Deleuzian approaches share a common conceptualisation of the semiotic function and its emergence in terms of amorphodynamics of heterogeneous assemblages with a divergent actualisation. This divergent actualisation allows the co-constitution of various expression and content planes. Finally, we show that the divergent actualisation can be interpreted as the directions of principal eigenvectors of the actualized flow.

Humanities-as-Technique: New Images of Knowledge and Ontological Construction

The sociocultural landscape of contemporaneity can be represented as a porous inhomogeneous space consisting of border zones. These zones (exchange zones), understood by P. Galison (1999, 137-60) as “local coordination of beliefs and actions,” can be deployed due to already existing conditions, for example, a project may arise at a university. However, such zones do not create sustainable social configurations and infrastructures associated with new areas of knowledge. In this article, we expand the concept of trading zones and call them zones requiring local ontologies that arise on the border between different knowledge and imagination systems and, in turn, require their own infrastructure. This approach supports the view on humanities not as “pre-packaged” knowledge (Daston 2009, 807), but as special “technique” forming local ontologies. For a humanities-as-technique to emerge, it is necessary that two or more systems of knowledge coincide in time and form an intermediate zone, on the basis of which some local ontology can unfold. Thus, for example, when combining contemporary philosophy and contemporary mathematics (topology), new images of knowledge emerge. Humanities-as-technique are considered as the basis for the establishment of trading zones, which are understood as local ontologies. We also propose a system of concepts that can be used to describe and construct these trading zones on the territory of the humanities in a techno-oriented cultural environment.

Enhancing teaching and learning of fluid mechanics with interactive computational modelling

Contemporary science, technology, engineering and mathematics (STEM) increasingly require advanced knowledge about mathematical physics models and methods of scientific computation. In STEM educational environments this entails the necessity to implement pedagogical curricula and methodologies that epistemologically balance the inclusion of interactive engagement sequences of computational modelling activities, and offer students opportunities to develop meaningful knowledge of physics, mathematics and scientific computation, as well as of the specific STEM concepts and processes. Our approach in this context is based on interactive engagement activities built around computational modelling experiments implemented in the Modellus environment that span the range of different kinds of modelling, from exploratory to expressive modelling. In this paper we describe research concerning a sequence of activities about hydrostatic pressure forces and torques, a theme of an introductory fluid mechanics course for undergraduate university engineering students having only elementary knowledge of secondary education physics and mathematics and no significant prior knowledge about scientific computation. We analyse student’s perceptions about the activities and the effects generated on the learning process. Using a Likert scale questionnaire, we show that students reacted positively to the Modellus based activities, considering them helpful in the learning process of the mathematics and physics of fluid mechanics and for their overall professional training. The results also show that students considered Modellus a useful software for computational activities that help the learning of mathematical physics models, sufficiently easy to learn and user-friendly. Based on the analysis of student’s work content, we show that students were able to construct and explore the proposed mathematical physics models and simulations, and establish meaningful and operationally reified relations with the relevant hydrostatic phenomena. We also show that the computational modelling activities were effective in resolving several difficulties persisting after theoretical lectures and problem-solving paper and pen activities.

($${{\mathbf {t}}},{{\mathbf {q}}}$$)-Deformed Q-Systems, DAHA and Quantum Toroidal Algebras via Generalized Macdonald Operators

We introduce the natural (t, q)-deformation of the Q-system algebra in type A. The q-Whittaker limit $$t\rightarrow \infty $$ gives the quantum Q-system algebra of Di Francesco and Kedem (Lett Math Phys 107(2):301–341, [DFK17]), a deformation of the Groethendieck ring of finite dimensional Yangian modules, compatible with graded tensor products (Hatayama et al. in: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Volume 248 of Contemporary Mathematics, Amer. Math. Soc., Providence, [HKO+99]; Feigin and Loktev in: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Volume 194 of Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, [FL99]; Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, [DFK14]). We show that the (q, t)-deformed algebra is isomorphic to the spherical double affine Hecke algebra of type $${\mathfrak {gl}}_N$$ . Moreover, we describe the kernel of the surjective homomorphism from the quantum toroidal algebra (Miki in J Math Phys 48(12):123520, [Mik07]) and the elliptic Hall algebra (Schiffmann and Vasserot in Compos Math 147(1):188–234, [SV11]) to this new algebra. It is generated by (q, t)-determinants, new objects which are a deformation of the quantum determinant associated with the quantum Q-system. The functional representation of the algebra is generated by generalized Macdonald operators, obtained from the usual Macdonald operators by the $$SL_2({\mathbb {Z}})$$ -action on the spherical Double Affine Hecke Algebra. The generating function for generalized Macdonald operators acts by plethysms on the space of symmetric functions. We give the relation to the plethystic operators from Macdonald theory of Bergeron et al. (J Comb 7(4):671–714, [BGLX16]) in the limit $$N\rightarrow \infty $$ . Thus, the (q, t)-deformation of the Q-system cluster algebra leads directly to Macdonald theory.

Polynomial time ultrapowers and the consistency of circuit lower bounds

A polynomial time ultrapower is a structure given by the set of polynomial time computable functions modulo some ultrafilter. They model the universal theory $$\forall \mathsf {PV}$$ of all polynomial time functions. Generalizing a theorem of Hirschfeld (Israel J Math 20(2):111–126, 1975), we show that every countable model of $$\forall \mathsf {PV}$$ is isomorphic to an existentially closed substructure of a polynomial time ultrapower. Moreover, one can take a substructure of a special form, namely a limit polynomial time ultrapower in the classical sense of Keisler (in: Bergelson, V., Blass, A., Di Nasso, M., Jin, R. (eds.) Ultrafilters across mathematics, contemporary mathematics vol 530, pp 163–179. AMS, New York, 1963). Using a polynomial time ultrapower over a nonstandard Herbrand saturated model of $$\forall \mathsf {PV}$$ we show that $$\forall \mathsf {PV}$$ is consistent with a formal statement of a polynomial size circuit lower bound for a polynomial time computable function. This improves upon a recent result of Krajicek and Oliveira (Logical methods in computer science 13 (1:4), 2017).

Artifacts, Agency and Classroom Activity: Materialist Perspectives on Mathematics Education Technology

This article analyzes an episode of classroom mathematics activity mediated by graphing technology from 3 different theoretical perspectives. An important line of research in the learning sciences focuses on graphs as inscriptions, foregrounding learners’ interactions with and around the material properties of graphical displays. Several recent studies in mathematics education have emphasized the utility of an instrumental genesis approach for conceptualizing mathematics learning and teaching with graphing calculators and other tools. A third perspective, influential in science and technology studies but less widely used in education research, calls for attention to material agency, considering the ways both learners and tools might be conceptualized as actors in a classroom activity system. This article explores the potential of this alternate approach for extending insights into technology-supported mathematics teaching and learning. It is argued that a material agency account can complement and enrich theoretical frameworks for analyzing technology-mediated classroom activity. Collectively, drawing these perspectives together to examine a classroom episode in which a digital device played a pivotal role provides a framework for considering the complexities and the tradeoffs involved in cultivating proficient tool use and conceptual understanding in contemporary mathematics classrooms.

Figurative Numbers Contribution in Perceiving the Legality in Numerous Strings Tasks and Long-term Memory of Numerous Data

Contemporary mathematics teaching is mostly reduced to the application of algebraic formulas and algebraic procedures. The visual-logical approach in solving mathematical tasks is very little represented in teaching mathematics. Such practice should be changed since visualization is of great importance in the process of learning and understanding mathematics as well as in solving mathematical tasks. This paper suggests the possibility of developing students’ ability to perceive lawfulness among numbers by introducing figurative numbers in mathematics teaching. Considering the visual presentation of figurate numbers and obvious rules among their members, students find them interesting and easy for understanding. They can also be a very good paradigms for many tasks with numerous arrays. The research carried out in this paper has shown that figurative numbers contribute to a visual-logical approach in solving tasks with numerous arrays and provide long-term storage of numerous data.

Şükrü Sayan’ın “Kemmiyyât-ı Mevhûmenin Sûret-i İrâesine Dair Yeni bir Nazariyye” Adlı Makalesi

<p class="ql-align-justify">Mathematician Şükrü Sayan (d. 1943) appended an article entitled “On a novel theory concerning the designation of imaginary quantities” to the end of his analytic geometry textbook Hendese-i Tahlîliyye (Vol.1, Istanbul, 1915), which is held to be the most proficient publication on the subject of the period, in Turkish. Although Sayan proposed to expose “a novel theory” the article, in fact, does not introduce a novel mathematical concept or facilitate algebraic calculations. Sayan’s article is primarily an explication of a notational modification, and represents yet another outcome of the Ottoman mathematicians’ ambition to come up with new work, rather than original and fundamental contributions to the field. Şükrü Sayan’s article is reviewed and assessed within the contexts of contemporary mathematics and analytic geometry.&nbsp;

A tribute to Juliusz Schauder

An analysis of the mathematical contributions of Juliusz Schauder through the writings of contemporary mathematicians, and in particular Leray, Hadamard, Banach, Randolph, Federer, Lorentz, Ladyzhenskaya, Ural'tseva, Orlicz, G\aa rding and Pietsch.

Fuzzy Logic: Modern Mathematics and Myopic Aesthetics In Martin Chuzzlewit

Martin Chuzzlewit is a blurry novel. Where Dickens typically invites his audience to picture precisely what he sees (whether in life or only in his imagination), in this book he keeps us at a troubling distance, rendering textual imagery—characters, places, events—in language that tends to obscure rather than clarify. I interpret the blurriness of this text through key innovations in Victorian mathematics that give rise to the fuzzy logic of the twentieth century. A spirit of algebraic abstraction pervaded nineteenth-century mathematics. I look here to the emergence of many-valued truth systems in (algebraized) British logic, which suggested that a vague term might still be considered an articulate one. I propose that Dickens was experimenting with the bounds of language—and the parameters of the real—in ways similar to contemporary mathematicians, and that vagueness in his writing anticipates perspectival shifts towards abstract aesthetics in the decades that followed. Furthermore, I argue that Dickens feigns shortsightedness in this book to manifest an uncomfortable distance between novel and reader, as an expression of the estrangement he felt on his 1842 American tour, and as a warning to audiences in America and England alike that ego makes strangers of us all.

On Bounds for Quantum Error Correcting Codes over EJ-Integers

There are some differences between quantum and classical error corrections [Nielsen M.A., and I.L. Chuang, “Quantum Computation and Quantum Information”, Cambridge University Press, Cambridge, 2002.]. Hence, these differences should be considered when a new procedure is performed. In our recent study, we construct new quantum error correcting codes over different mathematical structures. The classical codes over Eisenstein-Jacobi(EJ) integers are mentioned in [Huber, K., “Codes over Eisenstein-Jacobi integers”, Contemporary Mathematics 168 (1994), 165.]. There is an efficient algorithm for the encoding and decoding procedures of these codes [Huber, K., “Codes over Eisenstein-Jacobi integers”, Contemporary Mathematics 168 (1994), 165.]. For coding over two-dimensional signal spaces like QAM signals, block codes over these integers p = 7, 13, 19, 31, 37, 43, 61, … can be useful [Dong, X., C.B. Soh, E. Gunawan and L. Tang, “Groups of Algebraic Integers used for Coding QAM Signals”, Information Theory, IEEE 44 (1998), 1848–1860.]. Thus, in this study, we introduce quantum error correcting codes over EJ-integers. This type of quantum codes may lead to codes with some new and good parameters.

A disability studies in mathematics education review of intellectual disabilities: Directions for future inquiry and practice

We employ a Disability Studies in Mathematics Education perspective to explore current research involving students with intellectual disabilities. Such perspective affords a critical and social-cultural angle into the construction of disability in mathematics education. Results of our exploration suggest that current research on students with intellectual disabilities largely focused on finding deficits and neglected students’ capacity for abstract thought. Moreover, while socio-cultural or socio-political (e.g., equity-based) foci are prevalent in contemporary mathematics education, such foci were largely absent in research involving students with intellectual disabilities. Yet, we identify features of several articles that can inform future socio-political mathematics education research related to individuals with intellectual disabilities. These features are critical for the continual advancement of new knowledge, equitable practices, and global, inclusive education agenda.

Paolo Mancosu.*Abstraction and Infinity

Paolo Mancosu’s Abstraction and Infinity is an extended — and, most importantly, historically informed and inflected — investigation into abstraction generally, and abstraction principles in particular. An abstraction principle is a formula of the form: ... The two best known abstraction principles are Basic Law V: ... ... Abstraction principles are central to both Gottlob Frege’s original logicist attempt to reduce all of mathematics to logic (where |${\sf BLV}$| was, for Frege, a logical truth; see [Frege, 1980; 2013]) and more recent neo-logicist attempts to achieve something similar (where other abstraction principles, including |${\sf HP}$|⁠, serve as implicit definitions of mathematical concepts; see [Wright, 1983; Linnebo, 2018]). Of course, Frege’s project collapsed upon the discovery that |${\sf BLV}$| was inconsistent, and neo-logicists continue to struggle to extend the success of |${\sf HP}$| to other mathematical domains. Mancosu’s investigation of abstraction principles does not focus primarily on either the historical project of reconstructing Frege’s logicism in detail or the philosophical project of demonstrating that most or all of contemporary mathematics can be reconstructed in terms of neo-logicistically acceptable abstraction principles. Rather, as the title of the book suggests, Mancosu focuses instead on the connections between mathematical and philosophical work on abstraction and parallel developments in the mathematics and philosophy of the infinite. Nevertheless, the historical narrative that Mancosu constructs, and the philosophical puzzles he raises along the way, have deep implications for both the historical reconstruction of Frege’s original project and more recent attempts to construct Frege-inspired foundations for mathematics.