A Novel Image Encryption Technique Based on Mobius Transformation

The nonlinear transformation concedes as S-box which is responsible for the certainty of contemporary block ciphers. Many kinds of S-boxes are planned by various authors in the literature. Construction of S-box with a powerful cryptographic analysis is the vital step in scheming block cipher. Through this paper, we give more powerful and worthy S-boxes and compare their characteristics with some previous S-boxes employed in cryptography. The algorithm program planned in this paper applies the action of projective general linear group P G L 2 , G F 2 8 on Galois field G F 2 8 . The proposed S-boxes are constructed by using Mobius transformation and elements of Galois field. By using this approach, we will encrypt an image which is the preeminent application of S-boxes. These S-boxes offer a strong algebraic quality and powerful confusion capability. We have tested the strength of the proposed S-boxes by using different tests, BIC, SAC, DP, LP, and nonlinearity. Furthermore, we have applied these S-boxes in image encryption scheme. To check the strength of image encryption scheme, we have calculated contrast, entropy, correlation, energy, and homogeneity. The results assured that the proposed scheme is better. The advantage of this scheme is that we can secure our confidential image data during transmission.

Möbius Transformation-Induced Distributions Provide Better Modelling for Protein Architecture

Proteins are found in all living organisms and constitute a large group of macromolecules with many functions. Proteins achieve their operations by adopting distinct three-dimensional structures encoded within the sequence of the constituent amino acids in one or more polypeptides. New, more flexible distributions are proposed for the MCMC sampling method for predicting protein 3D structures by applying a Möbius transformation to the bivariate von Mises distribution. In addition to this, sine-skewed versions of the proposed models are introduced to meet the increasing demand for modelling asymmetric toroidal data. Interestingly, the marginals of the new models lead to new multimodal circular distributions. We analysed three big datasets consisting of bivariate information about protein domains to illustrate the efficiency and behaviour of the proposed models. These newly proposed models outperformed mixtures of well-known models for modelling toroidal data. A simulation study was carried out to find the best method for generating samples from the proposed models. Our results shed new light on proposal distributions in the MCMC sampling method for predicting the protein structure environment.

The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.

Visualization of Mandelbrot and Julia Sets of Möbius Transformations

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

On a Rigidity Problem of Beardon and Minda

In this paper, we give a positive answer to a rigidity problem of maps on the Riemann sphere related to cross-ratios, posed by Beardon and Minda (Proc Am Math Soc 130(4):987–998, 2001). Our main results are: (I) Let $$E\not \subset {\hat{\mathbb {R}}}$$ E ⊄ R ^ be an arc or a circle. If a map $$f:{\hat{\mathbb {C}}}\mapsto {\hat{\mathbb {C}}}$$ f : C ^ ↦ C ^ preserves cross-ratios in E, then f is a Möbius transformation when $${\bar{E}}\not =E$$ E ¯ ≠ E and f is a Möbius or conjugate Möbius transformation when $${\bar{E}}=E$$ E ¯ = E , where $${\bar{E}}=\{{\bar{z}}|z\in E\}$$ E ¯ = { z ¯ | z ∈ E } . (II) Let $$E\subset {\hat{\mathbb {R}}}$$ E ⊂ R ^ be an arc satisfying the condition that the cardinal number $$\#(E\cap \{0,1,\infty \})<2$$ # ( E ∩ { 0 , 1 , ∞ } ) < 2 . If f preserves cross-ratios in E, then f is a Möbius or conjugate Möbius transformation. Examples are provided to show that the assumption $$\#(E\cap \{0,1,\infty \})<2$$ # ( E ∩ { 0 , 1 , ∞ } ) < 2 is necessary.

Liouville quantum gravity and the Brownian map III: the conformal structure is determined

Previous works in this series have shown that an instance of a $$\sqrt{8/3}$$ 8 / 3 -Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the $$\sqrt{8/3}$$ 8 / 3 -LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and $$\sqrt{8/3}$$ 8 / 3 -LQG surfaces with other topologies.

Two meromorphic functions on annuli sharing some pairs of small functions or values

<abstract><p>In this paper, we prove that two admissible meromorphic functions on an annulus must be linked by a quasi-Möbius transformation if they share some pairs of small function with multiplicities truncated by $ 4 $. We also give the representation of Möbius transformation between two admissible meromorphic functions on an annulus if they share four pairs of values with multiplicities truncated by $ 4 $. In our results, the zeros with multiplicities more than a certain number are not needed to be counted if their multiplicities are bigger than a certain number.</p></abstract>

Antipodal Identification in the Schwarzschild Spacetime

"Through a Möbius transformation, we study aspects like topology, ligth cones, horizons, curvature singularity, lines of constant Schwarzschild coordinates r and t, null geodesics, and transformed metric, of the spacetime (SKS/2)^' that results from: i) the antipode identification in the Schwarzschild-Kruskal-Szekeres (SKS) spacetime, and ii) the suppression of the consequent conical singularity. In particular, one obtains a non simply-connected topology: (SKS/2)^' = R^2* ×S^2 and, as expected, bending light cones."