The geometry of artistic space in the works by Valentin Louis Georges Eugène Marcel Proust

The article considers the peculiarities of the artistic system in the works by the French writer of the turn of the 19th–20th centuries, VLGE Marcel Proust. The foundations of his aesthetic views, which are manifested primarily at the level of the structural organisation in the novel cycle “In Search of Lost Timeˮ are studied. The specificity of the narrative, where the main role is played by involuntary memory, allows us to speak about the special geometry of the artistic space in this work. It happens due to Proust's rejection of “plane psychologyˮ in favour of “psychology in time”. It is shown how on such a basis, thanks to the mnemonic mechanism, a complex connection of spontaneously arising spatial fragments with the temporal moments of existence arises and the chronotopic structure of the novel, built on the principle of relativity, is constructed where time becomes, in fact, the fourth dimension of space.

Manin Involutions for Elliptic Pencils and Discrete Integrable Systems

We contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.

The Theory and Demonstration of the Solid-Fluid Transformation of Ice Water

In this paper, a mathematical model for describing the solid-fluid transformation of ice water is put forward based on the special geometry cases. The correctness of the obtained model is verified through comparison with numerical analysis and experiments. The good agreement indicates that the obtained model is available for the study of the solid-fluid transformation of ice water. The theory derived in this paper lays a foundation for the research of solid-fluid transformation phenomena of other materials and may have important applications in engineering areas such as rheology, creep, and instability of materials.

Stationary States in Infinite Volume with Non-zero Current

We study the Ginzburg–Landau stochastic models in infinite domains with some special geometry and prove that without the help of external forces there are stationary measures with non-zero current in three or more dimensions.