THE RELATIONAL APPROACH TO THE MATHEMATICAL MODEL OF MEANING OF INFORMATION

A mathematical model of the meaning of the signal is proposed. The meaning is not the signal itself, but its effect on the recipient. Under the action of the signal, the state of the receiver changes, which is the meaning of the signal. The most general mathematical model is the description of the recipient’s state with the help of some mathematical object, and the meaning is modeled by the action of some operator on this object. Various concrete formalisms are considered: abstract automata, matrix representation, algorithms, Markov chains, parameter spaces. The article deals with finite, countable and continuous meanings, reversible and irreversible meanings, ambiguous meanings, decomposition into elementary meanings.

The Representation Theory of Neural Networks

In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical object that we represent using a network quiver. Furthermore, we show that network quivers gently adapt to common neural network concepts such as fully connected layers, convolution operations, residual connections, batch normalization, pooling operations and even randomly wired neural networks. We show that this mathematical representation is by no means an approximation of what neural networks are as it exactly matches reality. This interpretation is algebraic and can be studied with algebraic methods. We also provide a quiver representation model to understand how a neural network creates representations from the data. We show that a neural network saves the data as quiver representations, and maps it to a geometrical space called the moduli space, which is given in terms of the underlying oriented graph of the network, i.e., its quiver. This results as a consequence of our defined objects and of understanding how the neural network computes a prediction in a combinatorial and algebraic way. Overall, representing neural networks through the quiver representation theory leads to 9 consequences and 4 inquiries for future research that we believe are of great interest to better understand what neural networks are and how they work.

MEMBANGUN SIKAP POSITIF UNTUK MENGHINDARI SIKAP PHOBIA MATEMATIKA

Math phobias are anxiety on math skills that happen to most learners. Anxiety is demonstrated by the existence of continuous fear against the object of mathematics and its learning situation. To avoid this, the facilitator must create a situation that allows the learners to build their confidence in math tasks. Math is characterized by abstract ideas in the form of abstract symbols that require delivery within a less abstract language to avoid understanding the idea. A positive attitude towards mathematics appears closely related to the previous experience in both the mathematical object and the classroom situation. Therefore, developing a positive attitude against math phobia requires the content to be delivered in an attractive manner in every learning process. The presentation of mathematical formulas should be adjusted according to the learning basic ability and sometimes accompanied by an inductive approach. As the facilitator of learning, the teacher must seek a learning environment that can encourage learners to behave positively and passionately in learning mathematics.

The Interplay between Individual and Collective Activity: an Analysis of Classroom Discussions about the Sierpinski Triangle

This article uses a cultural-developmental framework to illuminate the interplay between collective and individual activity in the mathematical reasoning displayed in a university Masters level lesson on fractals. During whole class and small group discussions, eleven students, guided by an instructor, engage in inductive reasoning about the area and perimeter of the Sierpinski triangle, a unique mathematical object with zero area and infinite perimeter. As participants conceptualize and communicate about the Sierpinski problem, they unwittingly generate a linguistic register of action word forms (e.g., fencing, zooming) and object word forms (e.g., area, infinity) to serve Sierpinski-linked mathematical reasoning functions, a register that we document in the first analytic section of the article. In the second analytic section, we report a developmental analysis of microgenetic, ontogenetic, and sociogenetic shifts in the word forms constitutive of the register and their varied functions in participants’ activities. In the third analytic section, we provide a cultural analysis of the classroom’s collective practices, practices that enable and constrain participants’ constructions of form-function relations constituting the register. We examine the ways in which participants work to establish a common ground of talk in their communicative exchanges, exchanges supported by classroom norms for public displays of reasoning and active listening to one another’s ideas. We show that it is as participants work to establish a common ground that the register emerges and is reproduced and altered. We conclude by pointing to ways that the analytic framework can be extended to illuminate learning processes in other classroom settings.

Epistemic Complexity of the Mathematical Object “Integral”

The literature in mathematics education identifies a traditional formal mechanistic-type paradigm in Integral Calculus teaching which is focused on the content to be taught but not on how to teach it. Resorting to the history of the genesis of knowledge makes it possible to identify variables in the mathematical content of the curriculum that have a positive influence on the appropriation of the notions and procedures of calculus, enabling a particularised way of teaching. Objective: The objective of this research was to characterise the anthology of the integral seen from the epistemic complexity that composes it based on historiography. Design: The modelling of epistemic complexity for the definite integral was considered, based on the theoretical construct “epistemic configuration”. Analysis and results: Formalising this complexity revealed logical keys and epistemological elements in the process of the theoretical constitution that reflected epistemological ruptures which, in the organisation of the information, gave rise to three periods for the integral. The characterisation of this complexity and the connection of its components were used to design a process of teaching the integral that was applied to three groups of university students. The implementation showed that a paradigm shift in the teaching process is possible, allowing students to develop mathematical competencies.

Deep Neural Network Concepts for Classification using Convolutional Neural Network: A Systematic Review and Evaluation

In recent years, artificial intelligence (AI) has piqued the curiosity of researchers. Convolutional Neural Networks (CNN) is a deep learning (DL) approach commonly utilized to solve problems. In standard machine learning tasks, biologically inspired computational models surpass prior types of artificial intelligence by a considerable margin. The Convolutional Neural Network (CNN) is one of the most stunning types of ANN architecture. The goal of this research is to provide information and expertise on many areas of CNN. Understanding the concepts, benefits, and limitations of CNN is critical for maximizing its potential to improve image categorization performance. This article has integrated the usage of a mathematical object called covering arrays to construct the set of ideal parameters for neural network design due to the complexity of the tuning process for the correct selection of the parameters used for this form of neural network.

SpinX: Time-resolved 3D Analysis of Mitotic Spindle Dynamics using Deep Learning Techniques and Mathematical Modelling

Time-lapse microscopy movies have transformed the study of subcellular dynamics. However, manual analysis of movies can introduce bias and variability, obscuring important insights. While automation can overcome such limitations, spatial and temporal discontinuities in time-lapse movies render methods such as object segmentation and tracking difficult. Here we present SpinX, a framework for reconstructing gaps between successive frames by combining Deep Learning and mathematical object modelling. By incorporating expert feedback through selective annotations, SpinX identifies subcellular structures, despite confounding neighbour-cell information, non-uniform illumination and variable marker intensities. The automation and continuity introduced allows precise 3-Dimensional tracking and analysis of spindle movements with respect to the cell cortex for the first time. We demonstrate the utility of SpinX using distinct spindle markers and drug treatments. In summary, SpinX provides an exciting opportunity to study spindle dynamics in a sophisticated way, creating a framework for step changes in studies using time-lapse microscopy.

Exceptional complex structures and the hypermultiplet moduli of 5d Minkowski compactifications of M-theory

We present a detailed study of a new mathematical object in E6(6)ℝ+ generalised geometry called an ‘exceptional complex structure’ (ECS). It is the extension of a conventional complex structure to one that includes all the degrees of freedom of M-theory or type IIB supergravity in six or five dimensions, and as such characterises, in part, the geometry of generic supersymmetric compactifications to five-dimensional Minkowkski space. We define an ECS as an integrable U*(6) × ℝ+ structure and show it is equivalent to a particular form of involutive subbundle of the complexified generalised tangent bundle L1 ⊂ Eℂ. We also define a refinement, an SU*(6) structure, and show that its integrability requires in addition a vanishing moment map on the space of structures. We are able to classify all possible ECSs, showing that they are characterised by two numbers denoted ‘type’ and ‘class’. We then use the deformation theory of ECS to find the moduli of any SU*(6) structure. We relate these structures to the geometry of generic minimally supersymmetric flux backgrounds of M-theory of the form ℝ4,1 × M, where the SU*(6) moduli correspond to the hypermultiplet moduli in the lower-dimensional theory. Such geometries are of class zero or one. The former are equivalent to a choice of (non-metric-compatible) conventional SL(3, ℂ) structure and strikingly have the same space of hypermultiplet moduli as the fluxless Calabi-Yau case.

The Brain and the New Foundations of Mathematics

Many concepts in mathematics are not fully defined, and their properties are implicit, which leads to paradoxes. New foundations of mathematics were formulated based on the concept of innate programs of behavior and thinking. The basic axiom of mathematics is proposed, according to which any mathematical object has a physical carrier. This carrier can store and process only a finite amount of information. As a result of the D-procedure (encoding of any mathematical objects and operations on them in the form of qubits), a mathematical object is digitized. As a consequence, the basis of mathematics is the interaction of brain qubits, which can only implement arithmetic operations on numbers. A proof in mathematics is an algorithm for finding the correct statement from a list of already-existing statements. Some mathematical paradoxes (e.g., Banach–Tarski and Russell) and Smale’s 18th problem are solved by means of the D-procedure. The axiom of choice is a consequence of the equivalence of physical states, the choice among which can be made randomly. The proposed mathematics is constructive in the sense that any mathematical object exists if it is physically realized. The consistency of mathematics is due to directed evolution, which results in effective structures. Computing with qubits is based on the nontrivial quantum effects of biologically important molecules in neurons and the brain.