arXiv subjects

Bernhard R. Parodi

Publications and source records attributed to Bernhard R. Parodi.

3 recordsLinked to original sources

arXiv2026

From Ponzi Schemes to Benign Investment Dynamics: modelling Collapse, Stability, and a Path to Sustainability

The population and capital dynamics of three stylized investment systems are mathematically described using discrete-time difference equations with closed-form solutions. The models share a common capital budget equation but differ in their demographic laws, which are geometric, quasi-logistic, or epidemiologic (SIR-based). The quasi-logistic model is designed as an analytically tractable non-Ponzi investment system: it generalizes the geometric model (and, in the limit of a constant growth rate, reproduces classical Ponzi dynamics) while closely mirroring the behaviour of an SIR-based model with decreasing effective growth. In all cases, promised returns are modeled as fixed per-period payouts on initial investment with principal repaid upon exit, so that aggregate liabilities depend only on the current number of active investors. Within this unified framework, classical Ponzi schemes arise as special cases that inevitably collapse, while suitable parameter choices in the quasi-logistic and SIR-based versions generate finite-horizon, legally benign "no-Ponzi game" investment schemes with analytically transparent conditions for collapse, stability, and sustained operation.

math.HO↗

arXiv2021

Structure-dynamics relations for late-type spiral and dwarf irregular galaxies revisited

Scaling relations among structural and kinematical features of 79 late-type spiral and dwarf irregular galaxies of the SPARC sample are revisited or newly established. The mean central surface brightness <$μ_{0,[3.6]}$> = 19.63 $\pm$ 0.11 mag arcsec$^{-2}$ allows for a clear-cut distinction between low and high surface brightness galaxies irrespective of luminosity. The geometry of rotation curves is characterized by the relation $dv(R_d)/dr \approx v_{max}/R_{max}$. For the rotation curve decompositions we apply dark matter halos of Burkert and of pseudo-isothermal type. The disk mass-to-light ratios exhibit an asymmetric bimodal distribution with the dominant peak located at 0.2. The baryonic mass fraction at intermediate radii is included to address both an adjusted baryonic Tully-Fisher relation and the significance of deviations from the mean radial acceleration relation. The mean radial decrease of the baryonic mass fraction within galaxies is quantified. The Burkert halo parameters obey $ρ_0 \propto r_0^{-1.5}$ with considerable scatter, but allowing for $v_{max}$ as a third variable we find $ρ_0 \propto r_0^{-1.84}v_{max}^{2.00}$ with small scatter. The halo central surface density $ρ_0r_0$ strongly correlates with the observed radial acceleration at different galactocentric radii. We introduce an alternative universal rotation curve that is based on the non-singular total matter density profile $ρ_{total}(r) \propto (v_{max}^2/r^2)[1-(1-r/r_c)\exp(-r/r_c)]^2$, with the scaling parameter $r_c$ correlating with the halo core size $r_0$ and with $R_{max}$. Fitting the synthetic URC to a selection of galaxies, the co-added doubly-normalized rotation curves exhibit a high degree of similarity.

astro-ph.GA↗

arXiv2020

A generalized Fibonacci spiral

As a generalization of planar Fibonacci spirals that are based on the recurrence relation $F_n=F_{n-1}+F_{n-2}$, we draw assembled spirals stemming from analytic solutions of the recurrence relation $G_n=a\, G_{n-1}+b\, G_{n-2}+c\, d\,^n$, with positive real initial values $G_0$ and $G_1$ and coefficients $a$, $b$, $c$, and $d$. The principal coordinates given in closed-form correspond to finite sums of alternating even- or alternating odd-indexed terms $G_{n}$. For rectangular spirals made of straight line segments (a.k.a. spirangles), the even-indexed and the odd-indexed directional corner points asymptotically lie on mutually orthogonal oblique lines. We calculate the points of intersection and show them in the case of inwinding spirals to coincide with the point of convergence. In the case of outwinding spirals, an $n$-dependent quadruple of points of intersection may form. For arched spirals, interpolation between principal coordinates is performed by means of arcs of quarter-ellipses. A three-dimensional representation is exhibited, too. The continuation of the discrete sequence $\{G_n\}$ to the complex-valued function $G(t)$ with real argument $t$$\in$$R$, exhibiting spiral graphs and oscillating curves in the Gaussian plane, subsumes the values $G_n$ for $t$$\in$$N_0$ as the zeros. Besides, we provide a matrix representation of $G_n$ in terms of transformed Horadam numbers, retrieve the Shannon product difference identity as applied to $G_n$, and suggest a substitution method for finding a variety of other identities and summations related to $G_n$.

math.HO↗