arXiv subjects

Mostafa Salarinoghabi

Publications and source records attributed to Mostafa Salarinoghabi.

8 recordsLinked to original sources

arXiv2026

On the contact of surfaces in $4$-space with 2-planes and their apparent contours

We investigate the local geometry of generic smooth surfaces in $\mathbb R^4$ via the contact with $2$-planes and the associated apparent contour. We study the $\mathcal A$-singularities of parallel projections of such surfaces along planes to transverse planes. When the projection plane does not contain an asymptotic direction, we show that, at hyperbolic or elliptic (respectively, parabolic) points, there exist up to ten (respectively, seven) tangent directions determining planes along which the projection exhibits $\mathcal A$-singularities of type butterfly or worse. Moreover, we prove that the locus of points where the discriminant of the equation defining these directions vanishes generically forms regular curves in the hyperbolic and elliptic regions, and isolated points in the parabolic set of $M$. When the projection plane contains an asymptotic direction, we establish connections between the singularities of parallel projections, orthogonal projections to hyperplanes, and height functions. We further study the apparent contour associated with the parallel projection. When the projection is a fold, we prove a Koenderink-type theorem relating the Lipschitz--Killing curvature of the surface to the curvature of the apparent contour. Moreover, we show that elliptic or inflections points of the surface give rise to vertices of the apparent contour, whereas hyperbolic and parabolic points give rise to inflections. These phenomena are then characterized in terms of the singularities of the projection. In the non-fold case, we show that the apparent contour admits singularities of type $(t^k,t^\ell)$-cusp associated with the singularities of the projection.

math.DG↗

arXiv2026

A chaotic flux cipher based on the random cubic family $f_{c_n}(z)=z^3+c_n z$

This paper presents a symmetric stream cipher that utilizes the dynamic properties of random cubic mappings in the complex plane to generate pseudo-random key streams. The system is based on the iterations of the random cubic polynomial $f_n(z)=z^3+c_n z$, where the parameters $c_n$ are chosen randomly from a disc of radius $δ$ and with center at the origin, aiming to improve the chaotic behaviour and, consequently, the randomness of the generated sequence. The stability of the Julia set under small parameter perturbations, when $δ< δ_0\simeq 0.89$, is considered to ensure key consistency in noisy environments, such as 5G networks. On the other hand, for $δ> 3$, the system exhibits instability and chaos, ideal for generating ultra-secure keys. The Python implementation integrates secure key derivation, robust key stream generation via warmed-up iteration, and an authenticated encryption scheme using the modern cryptographic primitives (\texttt{HKDF} and\texttt{HMAC-SHA-256}), to ensure message integrity and authenticity. Statistical analyses, including chi-square test and entropy calculation, are performed on the output of the key stream generator to evaluate its randomness and distribution. In addition, a complete statistical validation, compliant with \texttt{NIST SP 800-22} standards in modern cryptography, was performed to enhance the proposed system's credibility.

cs.CR↗

arXiv2026

Random Dynamics of a Family of Cubic Polynomials

In this work, we study the non-autonomous dynamics generated by random iterations of the cubic family of the form $z^3 + cz$. The parameter sequence is chosen randomly from a bounded Borel subset of $\mathbb{C}$. We investigate topological properties of the corresponding Julia sets, with particular emphasis on conditions leading to total disconnectedness. We prove that the set of parameter sequences for which the Julia set is totally disconnected is dense in the parameter space. We also construct examples where the Julia set is totally disconnected but the associated non-autonomous system is not hyperbolic. Finally, under suitable probabilistic assumptions on the parameter distribution, we show that almost every sequence produces a totally disconnected Julia set.

math.DS↗

arXiv2025

Some results on evolutoids of convex curves in $2$-dimensional space forms

Let $M_c$ be a $2$-dimensional space form of constant curvature $c=-1,0,1$ and $γ$ a smooth, closed, convex curve in $M_c$. We explicitly parametrize the \textit{$α$-evolutoid} of $γ$, i.e.\ the closed curve $γ_α$ describing the envelope of all geodesics $σ_s=σ_s(t)$ such that $σ_s(0)=γ(s)$ and $\sphericalangle(σ_s'(0),γ'(s))=α$, with $α\in[0,π/2]$ fixed and determine its lenght. Also, we deduce that for each $s$ the points $γ(s),γ_α(s),γ_{π/2}(s)$ belong to a distinct geodesic circle. A constraint for the smoothness of $γ_α$ is calculated and, using tools from singularity theory, we prove that its singularities present cuspidal features, which mimics the classical evolute ($α=π/2$) in the plane case. Also, we define the \textit{$α$-involutoids} of a given curve $η$ in $M_c$ to be any curve $γ$ in $M_c$ such that $γ_α=η$ and study some of its properties. In particular, we prove that any convex, closed curve in $M_{-1,0}$ has associated to itself exactly one closed $α$-involutoid. Finally, we show that the evolutoids can be seen as singular sets of wavefronts.

math.DG↗

arXiv2025

Extended Cellular Automata

In this work, the one-dimensional Cellular Automaton is extended to one that involves two sets of symbols and two global rules. As a main result, the Extended Curtis-Hedlund-Lyndon Theorem is demonstrated. Such constructions can be useful in studying complex systems involving two related phenomena and provide a way to their co-study.

nlin.CG↗

arXiv2020

On evolutoids of regular surfaces in Euclidean 3-space

Inspired by the concept of evolutoids of planar curves, we present the concept of evolutoids for regular surfaces as an envelope of a two-parameter family of lines in Euclidean 3-space. We give an explicit parametrization for such evolutoids. Besides, we used the theory of singularities to study the local behavior of regular points of this object and presented some relations between the geometry of the regular surface and its evolutoid.

math.DG↗

arXiv2019

Envelope of intermediate lines of a plane curve

For a pair of points in a smooth closed convex planar curve $γ$, its mid-line is the line containing its mid-point and the intersection point of the corresponding pair of tangent lines. It is well known that the envelope of the mid-lines ($EML$) is formed by the union of three affine invariants sets: Affine Envelope Symmetry Sets ($AESS$); Mid-Parallel Tangent Locus ($MPTL$) and Affine Evolute of $γ$. In this paper, we generalized these concepts by considering the envelope of the intermediate lines. For a pair of points of $γ$, its intermediate line is the line containing an intermediate point and the intersection point of the corresponding pair of tangent lines. Here, we present the envelope of intermediate lines ($EIL$) of the curve $γ$ and prove that this set is formed by three disconnected sets when the intermediate point is different from the mid-point: Affine Envelope of Intermediate Lines ($AEIL$); the curve $γ$ itself and the Intermediate-Parallel Tangent Locus ($IPTL$). When the intermediate point coincides with the mid-point, the $EIL$ coincides with the $EML$, and thus these sets are connected. Moreover, we introduce some standard techniques of singularity theory and use them to explain the local behavior of this set.

math.DG↗

arXiv2017

On the $FRS$-generic family of space cusps

We consider in this paper the $FRS$-deformations of a family of space curves with codimension $\leq 3$. Some geometric aspects of a space curve such as flattenings, vertices and twistings points has been studied.

math.DG↗