The Research of Sequence Shadowing Property and Regularly Recurrent Point on the Double Inverse Limit Space

According to the definition of sequence shadowing property and regularly recurrent point in the inverse limit space, we introduce the concept of sequence shadowing property and regularly recurrent point in the double inverse limit space and study their dynamical properties. The following results are obtained: (1) Regularly recurrent point sets of the double shift map σ f ∘ σ g are equal to the double inverse limit space of the double self-map f ∘ g in the regularly recurrent point sets. (2) The double self-map f ∘ g has sequence shadowing property if and only if the double shift map σ f ∘ σ g has sequence shadowing property. Thus, the conclusions of sequence shadowing property and regularly recurrent point are generalized to the double inverse limit space.

G-Expansibility and G-Almost Periodic Point under Topological Group Action

Firstly, the new concepts of G − expansibility, G − almost periodic point, and G − limit shadowing property were introduced according to the concepts of expansibility, almost periodic point, and limit shadowing property in this paper. Secondly, we studied their dynamical relationship between the self-map f and the shift map σ in the inverse limit space under topological group action. The following new results are obtained. Let X , d be a metric G − space and X f , G ¯ , d ¯ , σ be the inverse limit space of X , G , d , f . (1) If the map f : X ⟶ X is an equivalent map, then we have A P G ¯ σ = Lim ← A p G f , f . (2) If the map f : X ⟶ X is an equivalent surjection, then the self-map f is G − expansive if and only if the shift map σ is G ¯ − expansive. (3) If the map f : X ⟶ X is an equivalent surjection, then the self-map f has G − limit shadowing property if and only if the shift map σ has G ¯ − limit shadowing property. The conclusions of this paper generalize the corresponding results given in the study by Li, Niu, and Liang and Li . Most importantly, it provided the theoretical basis and scientific foundation for the application of tracking property in computational mathematics and biological mathematics.

Classification of backward filtrations and factor filtrations: examples from cellular automata

We consider backward filtrations generated by processes coming from deterministic and probabilistic cellular automata. We prove that these filtrations are standard in the classical sense of Vershik’s theory, but we also study them from another point of view that takes into account the measure-preserving action of the shift map, for which each sigma-algebra in the filtrations is invariant. This initiates what we call the dynamical classification of factor filtrations, and the examples we study show that this classification leads to different results.

Efficient JPEG Encoding Using Bernoulli Shift Map for Secure Communication

Digital data must be compressed and encrypted to maintain confidentiality and efficient bandwidth usage. These two parameters are essential for information processing in most communication systems. Image compression and encryption may result in reduced restoration quality and degraded performance. This study attempts to provide a compression and encryption scheme for digital data named as Secure-JPEG. This scheme is built on the JPEG compression format, the most widely used lossy compression scheme. It extends the standard JPEG compression algorithm to encrypt data during compression. Secure-JPEG scheme provides encryption along with the process of compression, and it could be altered easily to provide lossless compression. On the other hand, the lossless compression provides less compression ratio and is suitable only in specific scenarios. The paper address the problem of security lacks due to the use of a simple random number generator which can not be cryptographically secure. The improved security characteristics are provided through Generalized Bernoulli Shift Map, which has a chaotic system with demonstrated security. The algorithm's security is tested by several cryptographic tests and the chaotic system’s behavior is also analyzed.

Probing bacterial cell wall growth by tracing wall-anchored protein complexes

The dynamic assembly of the cell wall is key to the maintenance of cell shape during bacterial growth. Here, we present a method for the analysis of Escherichia coli cell wall growth at high spatial and temporal resolution, which is achieved by tracing the movement of fluorescently labeled cell wall-anchored flagellar motors. Using this method, we clearly identify the active and inert zones of cell wall growth during bacterial elongation. Within the active zone, the insertion of newly synthesized peptidoglycan occurs homogeneously in the axial direction without twisting of the cell body. Based on the measured parameters, we formulate a Bernoulli shift map model to predict the partitioning of cell wall-anchored proteins following cell division.