The Past and Future of Aesthetics in Mathematics

This paper explores the connotations of mathematical aesthetics and its connections with art,facilitated by collaboration with Ester,an individual with an artistic professional background.It begins by tracing the historical evolution of aesthetics from the classical pursuit of authenticity to the modern shift toward self-expression in art.The discussion then highlights the similarities in the pursuit of truth between mathematics and art,despite their methodological differences.Through an analysis of aesthetic elements in mathematics,such as lines and function graphs,the article illustrates that the beauty of mathematics is not only manifested in cognitive processes but can also be intuitively expressed through visual arts.The paper further examines the influence of mathematics on the development of art,particularly how Leonardo da Vinci applied mathematical principles to his artworks.Additionally,the article addresses art students’perceptions of mathematics,proposes the customization of math courses for art students,and discusses future trends in the integration of mathematics and art,emphasizing the significance of art therapy and the altruistic direction of art.Lastly,the authors use a poster to visually convey the idea that the beauty of mathematics can be experienced through the senses.

Quantitative evaluation of lateral sealing of extensional fault by an integral mathematical-geological model

To evaluate the lateral sealing mechanism of extensional fault based on the pressure difference between fault and reservoir, an integral mathematical-geological model of diagenetic time on diagenetic pressure considering the influence of diagenetic time on the diagenetic pressure and diagenetic degree of fault rock has been established to quantitatively calculate the lateral sealing ability of extensional fault. By calculating the time integral of the vertical stress and horizontal in-situ stress on the fault rock and surrounding rock, the burial depth of the surrounding rock with the same clay content and diagenesis degree as the target fault rock was worked out. In combination with the statistical correlation of clay content, burial depth and displacement pressure of rock in the study area, the displacement pressure of target fault rock was calculated quantitatively. The calculated displacement pressure was compared with that of the target reservoir to quantitatively evaluate lateral sealing state and ability of the extensional fault. The method presented in this work was used to evaluate the sealing of F_(1), F_(2) and F_(3) faults in No.1 structure of Nanpu Sag, and the results were compared with those from fault-reservoir displacement pressure differential methods without considering the diagenetic time and simple considering the diagenetic time. It is found that the results calculated by the integral mathematical-geological model are the closest to the actual underground situation, the errors between the hydrocarbon column height predicted by this method and the actual column height were 0–8 m only, proving that this model is more feasible and credible.

Hidden Properties of Mathematical Physics Equations. Double Solutions. The Realization of Integrable Structures. Emergence of Physical Structures and Observable Formations

With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various structures and formations such as waves, vortices, turbulent pulsations and others. Such properties of the mathematical physics equations, which are hidden (they appear only in the process of solving these equations), depend on the consistency of derivatives in partial differential equations and on the consistency of equations, if the equations of mathematical physics are a set of equations. This is due to the integrability of mathematical physics equations. It is shown that the equations of mathematical physics can have double solutions, namely, the solutions on the original coordinate space and the solutions on integrable structures that are realized discretely (due to any degrees of freedom). The transition from the solutions of the first type to one of the second type describes discrete transitions and the processes of origin of various structures and observable formations. Only mathematical physics equations, on what no additional conditions such as the integrability conditions are imposed, can possess such properties. The results of the present paper were obtained with the help of skew-symmetric differential forms.

Preparing Mathematical Equations in a Document Preparation Environment

WITS-Math is a mathematical equation formatting tool in WITS, a multilingual document preparation environment. WITS-Math includes a library manager and an equation formatter. The main task of WITS-Math is to format diversities of mathematical equations and organize them into an equation library used by other tools in the WITS environment.WITS-Math is a direct manipulation mathematics editor. It uses syntax directed markup language as the internal representation, alld provides an interactive WYSIWYG interface for users to format equations. WITS-Math provides an equation access mechanism. Other tools can access equations in a library by cross-reference from a source file or through data exchange without knowillg the structure of equation libraries. The common data structure and the rendering object in the WITS platform ensure that the formatted equations can be directly used by other tools.