Context-aware Distance Measures for Dynamic Networks

Dynamic networks are widely used in the social, physical, and biological sciences as a concise mathematical representation of the evolving interactions in dynamic complex systems. Measuring distances between network snapshots is important for analyzing and understanding evolution processes of dynamic systems. To the best of our knowledge, however, existing network distance measures are designed for static networks. Therefore, when measuring the distance between any two snapshots in dynamic networks, valuable context structure information existing in other snapshots is ignored. To guide the construction of context-aware distance measures, we propose a context-aware distance paradigm, which introduces context information to enrich the connotation of the general definition of network distance measures. A Context-aware Spectral Distance (CSD) is then given as an instance of the paradigm by constructing a context-aware spectral representation to replace the core component of traditional Spectral Distance (SD). In a node-aligned dynamic network, the context effectively helps CSD gain mainly advantages over SD as follows: (1) CSD is not affected by isospectral problems; (2) CSD satisfies all the requirements of a metric, while SD cannot; and (3) CSD is computationally efficient. In order to process large-scale networks, we develop a kCSD that computes top- k eigenvalues to further reduce the computational complexity of CSD. Although kCSD is a pseudo-metric, it retains most of the advantages of CSD. Experimental results in two practical applications, i.e., event detection and network clustering in dynamic networks, show that our context-aware spectral distance performs better than traditional spectral distance in terms of accuracy, stability, and computational efficiency. In addition, context-aware spectral distance outperforms other baseline methods.

The Metarepresentational Role of Mathematics in Scientific Explanations

Several philosophers have argued that to capture the generality of certain scientific explanations, we must count mathematical facts among their explanantia. I argue that we can better understand these explanations by adopting a more nuanced stance toward mathematical representations, recognizing the role of mathematical representation schemata in representing highly abstract features of physical systems. It is by picking out these abstract but non-mathematical features that explanations appealing to mathematics achieve a high degree of generality. The result is a rich conception of the role of mathematics in scientific explanations that does not require us to treat mathematical facts as explanantia.

Friction Factor Comparison Between Mixing Length Theory and Empirical Correlation

Equivalent sand grain roughness is required for estimating friction factor for engineering applications from empirical relation via Haalands equation. The real surfaces are different from the sand grain profile. The correlations for friction factor were derived from use of discrete roughness elements with regular shapes such as cones, bars etc. The purpose of the paper is to derive analytical expression of friction factor for a 2 dimensional semi-cylindrical roughness (not exactly a 3 dimensional sand grain but for the circular profile of cross- section) using Navier Stoke equation and mixing length theory. This is compared with the modified series mathematical representation of Haalands equation for friction factor in terms of equivalent sand grain roughness. The comparison is valid for high Reynolds number where the velocity profile is almost flat beyond boundary layer and approximately linear all throughout the boundary layer. The high Reynolds number approximation for Haalands equation is derived and the series form of the friction factor compares approximately with the series form derived from first principles, where in the exponents of the series expansion are close.

DEVELOPMENT OF STUDENT WORKSHEET WITH SAVI APPROACH TO IMPROVE MATHEMATICS REPRESENTATION ABILITIES IN CYLINDER MATERIALS

This research aimed to develop a valid, practical, and effective worksheets with SAVI approach and can improve students' mathematical representation skills on cylinder material. The method of this research was a Research and Development (R and D) through the development of 4-D (Define, Design, Develop, and Disseminate). This research was conducted at SMP Negeri 1 Pace. The subjects of this research was 32 students of class IX. The results of the validation of the LKS with SAVI approach by the validator get an assessment of 3.8 after experiencing product revision. This shows that the LKS with SAVI approach that was developed is declared valid. The results of the analysis of the practicality of the LKS with SAVI approach were declared practical with the percentage of student response questionnaires of 85.03% and from the teacher's observation sheet of 87.5%. LKS with SAVI approach succeeded in increasing students' representation ability with a percentage increase of 90.625%. The result of the percentage of product completeness is 71.875% so it can be said that the LKS product with SAVI approach is an effective product. These results indicate that worksheets with SAVI approach are considered valid, practical, effective, and can improve mathematical representation skills.

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.

Analisis Kemampuan Pemahaman Konsep Matematis Siswa pada Pembelajaran Matematika Kelas XI IIS SMA PGRI 3 Padang

This research is motivated by the low ability of students to understand mathematical concepts in learning mathematics.The purpose of this study was to determine how thw mathematical ability of class XI IIS students at SMA PGRI 3 Padang. The subjects of this study were students of class XI IIS SMA PGRI 3 Padang, totalling 19 students. The research method used is descriptive method using a qualitative approach. The instrument used in this study was a test of the ability to understand concepts in the form of 4 questions in the form of descriptions and interview. The test results were analysed based on indicators of concept understanding. The results of this study indicate hat the results of data analysis on the ability to understand mathematical concepts of the 19 students tested obtained the total qualification score of 50% in the medium category. The qualifications for the total score of students mathematical concept understanding abilities based on the analysis of eachindicator on concept undestandig are as follow: on the indicator of restating a concept, 54% is in the medium category, classifying objects according to certain properties (according to the concept) is 82% are in the high category, providing examples and non-examples of the concept are 30 in the less category, presenting the concept in the form of a mathematical representation is 35% in the less category, developing necessary or sufficient requirements of a conceptis 73% in the medium category, and applying the concept or problem-solving algorithm is 24% in the lo category.

Polynomial matrix synthesis method of a subordinate control system

The main difference between controlsyn thesis approaches is the various mathematical representation of a plant or system model. The aim of the work is to represent a single channel control plant model by a multichannel one and to obtain an identical design result for a single channel multiloop synthesis method by a multichannel one. For these purposes, direct current motor model is used as an example of a single channel plant. Classical approach to design control system for that kind of plant is to describe it as a serial connected transfer functions and design a multiloop system in accordance with subordinate concept. Polynomial matrix synthesis method with Sylvester matrix is utilized to make identical subordinate regulator. By several transformations, polynomial matrix description was obtained, that describe the plant as one input and three output model and subordinate regulator as a three input and one output model. Arbitrary parameters of regulator were introduced for extended null placement.

Challenges of gated myocardial perfusion SPECT processing

The paper provides an in-depth look at gated myocardial perfusion single photon emission computed tomography (SPECT) data processing. Attention paid to several unmentioned subjects of the quantitative analysis of gated myocardial perfusion SPECT data. The article considers several options in the construction process of the ellipsoid coordinate system of the left ventricle (LV). Mathematical representation of polar maps is given. Formulas of the regional parameters calculation are proposed. Issues of phase analysis are explored.

THE IMPACT OF THE USE OF GEOGEBRA ON STUDENT’S MATHEMATICAL REPRESENTATION SKILLS AND ATTITUDE

<p>Mathematics is important and applies to science, technology, society or the natural sciences. It is applied directly or indirectly. Most students find this to be a very stimulating, complex, and well-understood subject. Maths in high school is extremely important. The study was designed to investigate the impact of students' mathematical representation skills and their attitudes towards GeoGebra. This study was quasi-experimental and carried out on high school students. We have two groups belonging to the same standard class. The control group consisted of 22 students, while the experimental group consisted of 28 participants. The conventional approach was used to teach certain concepts of plane geometry to the students in the control group. On the other hand, the experimental group taught similar teachings using GeoGebra. The results show that students have more skills in mathematical representation using GeoGebra. The semi-empirical test also showed a significant change in students' attitudes between the pre-test and the post-test. Students are more active in mathematical representation skills in GeoGebra.</p><p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/edu_01/0967/a.php" alt="Hit counter" /></p>