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The comprehension of mathematical notions can profit from the personal background of a population of students, drawing upon their former knowledge in another field. In particular, the embodiment of mathematics provides us with a bridge to transfer competencies from music to intuitive understanding of non-trivial geometries and topologies. In this study, we analyse an experiment in which music composition students are introduced to the Klein bottle for the first time. Through the Ontological and Semiotic Approach (OSA), a conceptual framework in mathematics education, we evaluate the evolution from an initial intuition of the Klein bottle to a formal understanding of the object via musical renditions and verbal presentations to both neophyte and specialist audiences in mathematics. [ABSTRACT FROM AUTHOR]

Advances in medical imaging and artificial intelligence have led to improvements in diagnosis and non-invasive patient examination accuracy. The use of the fundamental method for Magnetic resonance imaging (MRI) brain scans as a screening tool has increased in recent years. Numerous studies have proposed a variety of feature extraction methods to classify the abnormal growths in MRI scans. Recently, the MRI texture analysis and the use of deep features have resulted in remarkable performance improvements in the classification and diagnosis of challenging pathologies, like brain tumors. This study proposes employing a handcrafted model based on Weierstrass elliptic function (WEF) and deep feature based on DenseNet-201 to classify brain tumors in MRI images. By calculating the energy of each individual pixel, the Weierstrass coefficients of the WEF are used to capture high frequency image details of the brain image. The WEF mode works to extract the nonlinear patterns in MRI images based on the probability of each pixel. While the dense connectivity of DenseNet-201’s architecture allows to learn features at multiple scales and abstraction levels. These features are passed to support vector machines classifier, which classifies the brain tumor. The results of classification accuracy achieved is 98.55% for combined features of WEF with trained DenseNet-201. Findings on the brain tumor segmentation dataset indicated that the proposed method performed better than alternative techniques for classifying brain tumors.

It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using 1 -based infinite numbers is applied to measure the set A (where the number 1 is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable or it has the cardinality of the continuum, the 1 -based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of A are obtained. Both estimates are expressed in 1 -based numbers. [ABSTRACT FROM AUTHOR]

Mathematical 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. [ABSTRACT FROM AUTHOR]

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its "architecture" or "unity." This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. [ABSTRACT FROM AUTHOR]

This paper shows an application of plausible reasoning methods (mainly, specialization and analogous) in mathematical modeling. Our attention is how a practitioner to determine analogously a more balanced scientific model to assist the desire during solving the entire problem. Taking interval samples modeling as a problem, we exemplify (with consideration paid to the motivation and course of discovering) how to discover, based on the classical corresponding methods, three linear regression models and two linear-like interpolation models relying on n-variable interval input-1-variable interval output samples. The rationality of these recommended models are proved, and applications of them are illuminated in detail by examples. Strategies to model further interval samples towards satisfactory are also exposed. [ABSTRACT FROM AUTHOR]

We describe in the form of a dialogue a development of various reflections on the combinatorics of set partitions; among the topics we pursue are the number of ways of partitioning a finite set into a fixed number d of subsets of odd or even size, into a parts of odd and b parts of even size, and into d parts, each of which has a size in a certain congruence class modulo some natural number m . To this end, pattern guessing, recursion and induction, combinatorial interpretation and generating functions are employed. The participants of the dialogue represent different perspectives on and approaches to mathematics. [ABSTRACT FROM AUTHOR]

In a wide range of fields, the word "modular" is used to describe complex systems that can be decomposed into smaller systems with limited interactions between them. This essay argues that mathematical knowledge can fruitfully be understood as having a modular structure and explores the ways in which modularity in mathematics is epistemically advantageous. [ABSTRACT FROM AUTHOR]

Inspired by the Ryōan-ji garden in Kyoto, we formulate and solve some problems on the arrangement of two sets of points in the plane. We imagine a Zen Master and a Zen Monk that want to create a dry garden: they will discover the importance of convexity and collinearity, as well as incidence and betweenness. We present these problems with a didactic exposition and we suggest some generalizations which require other mathematical tools. [ABSTRACT FROM AUTHOR]

The Einstein initial foundations of general relativity have to do with his great intuition and they are not clear as it is for special relativity. As has been widely emphasized, for example, in the book of Ohanian and Ruffini, the very name of the theory indicates a misconception. Despite this, the high school textbooks (at least the Italian ones) and books of scientific divulgation introduce the Einstein gravitational theory still following the initial approach leading, in our opinion, to misinterpretations. A careful student, for example, immediately asks: it is not true that the Earth rotates because I can consider it at rest thanks to general relativity theory. The relativity of motion is trivial in mathematics while it has deep meaning in physics and it is not sufficiently analyzed. Similarly, the arbitrary choice of the origin and the perfect equivalence between all coordinate systems are mathematical properties satisfied by general relativity and students can confuse it with physical equivalence between reference systems that have a deeper meaning and the Einstein theory does not verify it. Gravitational force does not exist and this is the real core of Einsteinian revolution. As happens in Newtonian physics and Special Relativity, also in general relativity the relative motions are the geodesic motions with the difference that spacetime can be curved and geodesics may not be a straight line. Similarly, in general relativity forces cause non-geodesic motion. Geodesic and non-geodesic are tensorial properties and for this reason they are absolute. [ABSTRACT FROM AUTHOR]

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two particular sequences, a new characterization of this number will appear. In that process, an infinite sum of iterated radicals is obtained. Based on that result, this article will then proceed to analyse that sum. The conditions under which the infinite sum yields an integer will be inspected, thereby calculating the value of the sum. After that, a method is established to develop some algebraic irrationals as a continued fraction. Finally, the results will be applied to an infinite difference of iterated radicals. [ABSTRACT FROM AUTHOR]