Hasil untuk "Mathematics"

I. Kra, S. R. Simanca

G. Berkolaiko, P. Kuchment

G. Aubert, Pierre Kornprobst

F. Cucker, S. Smale

(1) A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference). We try to emphasize relations of the theory of learning to the mainstream of mathematics. In particular, there are large roles for probability theory, for algorithms such as least squares, and for tools and ideas from linear algebra and linear analysis. An advantage of doing this is that communication is facilitated and the power of core mathematics is more easily brought to bear. We illustrate what we mean by learning theory by giving some instances. (a) The understanding of language acquisition by children or the emergence of languages in early human cultures. (b) In Manufacturing Engineering, the design of a new wave of machines is anticipated which uses sensors to sample properties of objects before, during, and after treatment. The information gathered from these samples is to be analyzed by the machine to decide how to better deal with new input objects (see [43]). (c) Pattern recognition of objects ranging from handwritten letters of the alphabet to pictures of animals, to the human voice. Understanding the laws of learning plays a large role in disciplines such as (Cognitive) Psychology, Animal Behavior, Economic Decision Making, all branches of Engineering, Computer Science, and especially the study of human thought processes (how the brain works). Mathematics has already played a big role towards the goal of giving a universal foundation of studies in these disciplines. We mention as examples the theory of Neural Networks going back to McCulloch and Pitts [25] and Minsky and Papert [27], the PAC learning of Valiant [40], Statistical Learning Theory as developed by Vapnik [42], and the use of reproducing kernels as in [17] among many other mathematical developments. We are heavily indebted to these developments. Recent discussions with a number of mathematicians have also been helpful. In

Michael O. Martin, A. Beaton, E. González et al.

P. K. Maji, A. R. Roy, R. Biswas

H. Stichtenoth

J. Vázquez

I. Blake, G. Seroussi, N. Smart

W. Thurston

G. Pólya

J. Moon, G. Dantzig, C. L. Liu

M. Kline

A. Schoenfeld

Chuangxia Huang, Xiaodan Ding, Qian Wang

This study is devoted to the stability issue of a time-varying delayed Nicholson's blowflies equation of neutral type. By the aid of the Lyapunov stability theory and novel analysis techniques, two sharp conditions ensuring separately the global asymptotic stability of the zero equilibrium and positive equilibrium are derived, which completely cover the corresponding ones in the associated non-neutral equation. In addition, a numerical example is taken to support the availability of the theoretical results.

K. Hart, Csms Mathematics Team

Fatrima Santri Syafri, Agus Susanta, Irwan Koto

Background: This research develops a foundational mathematics textbook for PGMI students based on the Rigorous Mathematical Thinking (RMT) framework, contextualized within houses of worship to foster deeper understanding of mathematical concepts. Aim: The study aims to design and evaluate the feasibility and effectiveness of the textbook while gauging students’ responses and performance improvements. Method: Using a research and development (R&D) model, this study validated the textbook through expert reviews and tested its practicality and effectiveness in small-scale and broader applications. Data were obtained through a combination of observations, interviews, surveys, and tests measuring students' conceptual comprehension. Statistical evaluations, including descriptive analysis and t-tests, were employed. Results: The textbook received validation from subject matter, presentation, and language experts, who confirmed its content and structural accuracy. Trial phases categorized the textbook as highly practical, while broader field testing indicated that students using the textbook demonstrated significantly greater improvement in conceptual understanding compared to those without it. Conclusion: The RMT-based textbook aligns with its intended objectives and provides a contextually appropriate tool for improving mathematical comprehension. These findings highlight its potential as an effective resource for mathematics learning in PGMI contexts.

Thenmozhi D, M. Eswara Rao, Ch. Nagalakshmi et al.

The current study presents a novel examination of heat transfer properties in a magnetohydrodynamic (MHD) flow of Casson fluid across a porous stretching sheet, uniquely incorporating the effects of heat source and variable viscosity. Unlike previous studies, this research employs the Lie similarity transformation to convert the governing equations into a dimensionless form. These transformed equations are then solved using advanced numerical techniques, specifically the fourth-order Runge-Kutta (RK) along with shooting method. The findings reveal that the velocity decreases with the adjustment of significant parameters such as the Casson fluid properties, variable viscosity, heat source, magnetic field, and porosity, leading to an inverse increase in temperature within the convection system. As the Prandtl number increases, the temperature gradient and thermal boundary layer thickness decrease, resulting in reduced heat transfer rates within the convection system. Likewise, an increase in the Schmidt number decreases the concentration gradient and mass transfer rate within the fluid. This novel approach provides new insights into the behavior of Casson fluids, with significant applications in industrial processes, energy systems, environmental engineering, material science, and aerospace and automotive industries, where understanding heat transfer mechanisms in complex systems can enhance efficiency, performance, and safety.

Li-Hong Jiang, Hong-Yu Wu, Peng Dong et al.

Many two-place physical problems can be explicitly presented as related events model named Alice–Bob systems. In this paper, an integrable Alice–Bob Boussinesq system is introduced via the Boussinesq equation with parameters, which may meet the symmetry transformation of Psˆx (parity with a shift) and Tdˆt (time reversal with a delay). After constructing an Bäcklund transformation, the system has rich symmetry solutions with the aid of auxiliary functions. The structures of obtained soliton solutions, such as the breathers, lumps and their hybrids, are all satisfied the Pˆsx or Tˆdt symmetry. To illustrate the symmetric characteristic, some lower-order solutions and the related dynamic structures are explicitly presented. The residual symmetry and its finite transformation for this system are also verified.

Vladimir Barannik, Serhii Sidchenko, Dmitriy Barannik et al.

The subject of research in the article are the video images compression and encryption processes during the critically important objects managing process. The goal is to develop a method for compressing video images based on floating positional coding with an uneven codegrams length to simultaneously ensure information reliability and confidentiality during its transmission with a given time delay. Objectives: analyzing existing approaches to ensuring the video images confidentiality; development a method for compressing video images based on floating positional coding with an uneven codegrams length; evaluate the developed method effectiveness. The methods used are: digital image processing methods, digital image compression methods, image encryption and scrambling methods, structural-combinatorial coding methods, statistical analysis methods. The following results are obtained. The technology of floating encoding of an uneven sequence of blocks is proposed. Code values are formed from elements of different video image blocks. For this, a scheme for linearizing an image point coordinates from its four-dimensional representation on a plane into a one-dimensional element coordinate in a vector has been developed. The four-dimensional element coordinate on the plane describes the image block coordinates and the coordinates of the element in this block. Code values are formed under conditions of control their binary representation's length. At the same time, coding is implemented for an indeterminate number of video image elements. The number of elements depends on the length of the code word. Accordingly, codegrams with an indeterminate length are formed. Their length depends on the service data values, generated during the encoding process. Service data acts as a key element. Conclusions. The one-stage polyadic image encoding method in a differentiated basis has been further improved. The developed encoding method provides image compression without information quality loss. The original images volume compression provides by 3–20 % better compared to the TIFF data presentation format and by 4–15 % compared to the PNG format. The overhead amount is less than 2.5 % of the entire codestream size.