Universidad Juárez Autónoma de Tabasco

Let X be a planar vector field with an equilibrium p at which the Jacobian J=DX(p) is nilpotent of rank one, and let q_0 span J. We prove that the two coefficients a and b of the Bogdanov–Takens (BT) normal form are the directional derivatives, along q_0, of the two invariants of the Jacobian: a=-1/2⟨∇ DX(p),q_0⟩, b=⟨∇trDX(p),q_0⟩. The identity is invariant under changes of phase-space coordinates and equivariant under the rescaling of q_0, and the resulting formula requires neither generalized eigenvectors nor the second-order multilinear form. It yields a coordinate-free reading of the BT nondegeneracy conditions in terms of the kernel line field of the projection of the equilibrium manifold onto parameter space, the transformation rule (a,b)↦(h^2a,hb) under orbital equivalence, and the fact that a=0 whenever the vector field factors through a function vanishing at p. We then prove an obstruction of a combinatorial nature. For a Kolmogorov system ẋ=xA/g_1, ẏ=yB/g_2 and an equilibrium p in the torus (ℂ^*)^2 at which the Jacobian is nilpotent and nonzero, the order m=ordf in the Takens normal form is bounded by the mixed volume of the Newton polytopes of A and B. In particular, if MV(NewtA,NewtB)≤2 then a≠0 unless the equilibrium fails to be isolated, and the nilpotent singularities of saddle, focus and elliptic type are unreachable: only cusp-type singularities occur at isolated equilibria, while their exact codimension is not controlled by the mixed-volume bound. The class of systems with cross-product cubic terms, for which the mixed volume equals 2, is treated in detail, and two classical Bazykin models are shown to be instances.

This study presents the design, construction, and thermal evaluation of a solar-powered cocoa roaster based on a Parabolic Cylinder Collector (PCC) with dual-axis solar tracking. The system integrates three functional subsystems: the cylindrical-parabolic reflecting surface, the stainless-steel absorber tube, and a microcontrollerbased tracking mechanism. The prototype enables continuous acquisition of key thermal variables (solar irradiance, ambient temperature, absorber surface temperature, and bean temperature), allowing a detailed characterization of heat transfer processes during roasting. Roasting experiments were conducted at controlled durations of 40, 55, and 70 min between 10:00 and 14:00 h. Maximum roasting temperatures of 125 circle C-137 circle C were reached under average irradiance levels of 685.7-930.5 W m-2. The lowest final moisture content was 2.19%, within the recommended range for high-quality cocoa. Longer roasting durations promoted thermal energy accumulation within the absorber tube, enhancing convective and radiative heat transfer to the bean mass even under fluctuating irradiance. The experimental trends reveal a strong coupling between irradiance variability, absorber temperature, and internal air-beam heat transfer. Comparison with reference parabolic trough collector studies indicate that, although the process-level roasting efficiency (3.83%-7.45%) is lower than conventional collector-level thermal efficiencies, the operating temperatures and moisture-reduction rates align with the thermal requirements of food-processing systems rather than high-enthalpy solar applications. These results also demonstrate the potential of coupling PCC-based solar concentration with low-temperature convective-radiative roasting processes. Overall, the findings confirm the feasibility of implementing PCC-based roasting technologies in rural or off-grid regions, where solar-driven heat transfer offers a sustainable, low-cost alternative to fossil-fuel-based roasting systems, enabling a controlled thermophysical environment for cocoa transformation.

We prove that the critical structure of the prey nullcline constrains where oscillatory instabilities can emerge in planar predator–prey systems of Gause type. Under a predator self–damping hypothesis, the prey coordinate of every Hopf bifurcation point is confined to an ascending branch of the prey nullcline, between consecutive critical points. The mechanism is purely geometric: along the nullcline the diagonal Jacobian entry satisfies J_11=p(x) g'(x) , independently of the bifurcation parameter, while J_22<0 at any coexistence equilibrium; at a critical point ( g'=0 ) the trace is therefore strictly negative, so the critical points act as spectral barriers that preclude oscillatory onset. We illustrate the principle in three canonical families—Bazykin’s model (quadratic nullcline), a Leslie–Holling type IV system with harvesting (cubic nullcline), and the Crowley–Martin model with predator interference (rational nullcline)—for which we obtain closed–form Hopf loci. The mechanism persists in discrete time: for the forward Euler map with step size τ , the exact identity (J^G)-1=τ [tr(J)+τ (J)] shows that the Neimark–Sacker locus is strictly disjoint from the continuous Hopf locus, differs from it by the O(τ ) term τ (J) , and crosses the spectral barrier only in the coarse–step regime. Predator saturation and interference thus not only bound the efficiency of interactions but also imprint a geometric architecture on the bifurcation landscape.

The use of solar energy for the photocatalytic degradation of contaminated water is a sustainable strategy. Water scarcity is becoming increasingly critical in semi-arid regions with high solar radiation. In this study, ZnO particles were synthesized using two techniques, hydrothermal and solvothermal, and the obtained materials were named HZ and SZ, respectively. The influence of the synthesis method on the physicochemical properties and photocatalytic activity is reported in detail. The characterization results revealed that the solvothermal method yielded quasi-spherical nanoparticles (62.31 f 27.95 nm and 88.34 f 46.71 nm) with a crystallite size of 65 f 5 nm, a band gap of 3.18 eV, and showed superior textural properties (13.74 m2/g) compared to those obtained with the hydrothermal method. The photocatalytic activity was evaluated by the degradation of rhodamine B (RhB) under natural Sunlight (UV-A: 25.18 f 4.73 and Vis-NIR: 730.73 f 44.9 W/m2) in Chihuahua city, Mexico. The particles obtained by the solvothermal technique exhibited the highest efficiency under solar radiation conditions, significantly surpassing the HZ sample and conventional UV and visible light sources from lamps. The SZ material was evaluated for reuse testing. This process was carried out for up to 10 consecutive cycles, without treatment after each cycle, and the material demonstrated its potential for industrial applications. Although slight surface contamination slows down the initial reaction kinetics, the material maintained a high final efficiency of 96.5% after 120 min, making it competitive with commercial P25 (TiO2). These experimental results demonstrated the potential of ZnO synthesized by the solvothermal technique as a reusable, cost-effective photocatalyst for water purification under natural solar radiation.

This paper describes the corrosion protection of carbon steel using thin films of TiO2 nanoparticles (NPs). It increases its remaining lifetime and improves its mechanical behaviour. Samples of an oil storage tank were coated. The steel plates have ferrite and pearlite, as well as an agglomeration of cementite and graphite. NPs TiO2 were synthesized using the Sol-Gel technique with a compact tetragonal structure. The samples were treated at 550 degrees C for one hour. The anatase phase of the TiO2 was detected. The surface of the samples was evaluated with SEM and EDS. There are no significant changes to the TiO2 layer. The degradation rate of the coated carbon steel was assessed using linear polarization with a 3.5% NaCl corrosion solution. The corrosion rate was reduced from 0.20 mm/year to 0.14 mm/year at room temperature and from 1.82 mm/year to 1.20 mm/year at a critical operating temperature of 90 degrees C. The above demonstrates the effectiveness of the TiO2 coating in reducing the corrosion rate under normal operating conditions and high temperatures.