The present study addresses the extraction of nanosilica (NPs) from sugarcane bagasse ash through a two-step process involving alkali extraction and acid precipitation. The following parameters affecting silica yield and purity were evaluated: ash:NaOH ratio, fusion temperature and time, reflux time, acid used in precipitation (type and concentration) and gelation pH. The synthesized NPs samples were characterized by conventional physical and chemical analysis techniques. The statistical analysis showed that ash:NaOH ratio was the factor with the greatest influence on the silica synthesis yield. The optimum conditions was obtained by 1:2 ash:NaOH ratio (w:w), fusion temperature of 300^∘C for 30 min, 1-h reflux, precipitation using 8 mol L⁻¹ sulfuric acid, gelation pH 4, resulting in 88% silica recovery and 96% purity. Silica prepared under optimized conditions was examined for removal of dyes from aqueous solution. The material showed efficient methylene blue and crystal violet adsorption with Langmuir adsorption capacity of 108.4 mg⋅g⁻¹ and 102.0 mg⋅g⁻¹, respectively. These results confirm that silica derived from sugarcane bagasse ash represents an effective and low-cost adsorbent for removing dyes from wastewater contributing to circular economy strategies and promoting sustainable waste management.

The prebiotic formation and persistence of carbohydrates remain central challenges in origin-of-life chemistry owing to their instability and structural diversity in aqueous environments. Among potential stabilizing agents, boron species—primarily boric acid and borate—exhibit a unique capacity to form reversible, stereoselective complexes with cis-diol-containing molecules, including sugars and low-molecular-weight polyols. Here we examine the coordination chemistry of boron–diol interactions and explore their implications for prebiotic chemical evolution. By preferentially stabilizing specific sugar configurations, particularly furanose forms, boron may bias the composition of prebiotic mixtures, acting as a primitive chemical “editor.” These dynamic interactions—encompassing mono- and diester formation, as well as higher-order assemblies—are modulated by environmental factors such as pH and evaporative concentration. We propose that boron-mediated complexation constitutes a form of thermodynamic selection that enriches biologically relevant carbohydrates, including ribose, while disfavoring less stable isomers.

This paper introduces the \(Z_6\) transformation for the first time as a constructive framework for generating and solving differential equations. Its core idea is to construct a new solvable equation from two known solvable equations through a predefined transformation between the dependent and independent variables. This framework unifies two classic techniques-coordinate transformation and substitution of the dependent variable-into a single, systematic procedure. Furthermore, by iterating this construction process and combining it with the principle of common solutions proposed in this paper, an infinite number of solvable equations can be generated. Using this framework, we have obtained, for the first time, general solutions for a variety of differential equations, including general solutions for the Laplace equation in cylindrical and spherical coordinates, as well as general solutions for several classes of linear and nonlinear second-order partial differential equations with varying coefficients, among others. We also solved two boundary value problems exactly, demonstrating the practicality of this framework. The \(Z_6\) transformation, together with the \(Z_A\) method we previously proposed, provides two complementary approaches for systematically expanding the range of solvable differential equations, offering powerful tools for both theoretical analysis and practical applications.

Abstract: This paper develops a mathematics-first modular framework for completion quotients, Real-commutant parameter spaces, positive-frequency polar selection, temporal spectral triples, and conditional cone readouts for orthogonal Hilbert representations. The opening part states the ontological and operational principles used to organize the construction: generative histories are separated from completed records, observable dynamics is obtained by quotienting through completed-future tests, phase is selected only after a real orthogonal carrier has positive time orientation, and Lorentzian readout is treated as a downstream cone geometry obtained from length-delay inequalities. The theorem-bearing core concerns orthogonal Hilbert representations. For a real representation \(U:G\to O(\mathcal H_{\mathbb R})\), with complexification \(V\) on \(\mathcal H_{\mathbb C}\) and canonical conjugation \(\mathcal C\), the Real commutant

is a real \(C^*\)-algebra canonically isomorphic to the real commutant of \(U\). Symmetry-compatible orthogonal complex structures are precisely the skew-adjoint square roots of \(-\operatorname{Id}\) in this algebra, naturally under real unitary intertwiners. Abelian representations yield partial spectral phase operators, orthogonal flows yield the positive-frequency polar factorization \(A=J_A|A|=|A|J_A\) on the common domain \(\operatorname{Dom}(A)=\operatorname{Dom}(|A|)\) after removing the kernel, and compact groups yield a Peter–Weyl/Frobenius–Schur bounded-product formula in complex, real, and quaternionic multiplicity algebras. Downstream of this phase layer, the phase-balanced real tensor product is identified with the real form of the usual complex Hilbert tensor product, and a temporal shift spectral triple is built from a concrete dense \(*\)-algebra with compact resolvent, bounded generator commutators, exact diagonal Connes distance \(|m-n|/\kappa\) on the diagonal record algebra, and spectral dimension governed by an explicit counting-function hypothesis. The final module proves a conditional cone lemma: once a length budget, completion delay, and Euclidean readout are supplied, completed displacements lie in the finite cone \(\|\Delta x\|\le c_*\Theta\); the standard automorphism statement gives Lorentz similitudes, and a smooth Lorentzian representative requires additional smooth geometric input. The broader physical interpretation motivates the axioms but is not used as a hidden premise.

We study window-weighted Wijsman statistical convergence for sequences of non-empty closed sets in hyperspaces associated with idempotent bicomplex metric spaces. The ambient distance is assumed to have the form \(\rho=d_1e_1+d_2e_2,\) where \(d_1\) and \(d_2\) are ordinary metrics on the same underlying set. Thus the present paper concerns this idempotent, componentwise class of bicomplex-valued metrics, rather than arbitrary bicomplex-valued metric structures. For a closed set \(A\), the point-to-set distance is represented by the two scalar profiles \(d_1(x,A)\) and \(d_2(x,A)\). We define window-weighted Wijsman statistical convergence and the corresponding pre-Cauchy condition by requiring simultaneous control of these two profiles along weighted moving windows. The main results give exact componentwise characterizations, establish invariance under weighted-equivalent changes of the window scheme, and show that a window-weighted Wijsman pre-Cauchy sequence becomes statistically convergent whenever it has an ordinarily Wijsman convergent trace of positive lower window weight. We also prove stability under negligible perturbations and show that ordinary Wijsman convergence on a trace of full window density determines the statistical convergence of the whole sequence. Examples illustrate the dependence on the window scheme, the strict weakness of the pre-Cauchy condition, and the effect of imposing simultaneous closedness with respect to two non-equivalent component metrics.

The aim of this paper is to explore some aspects of the time-frequency analysis associated with the multidimensional Fourier–Bessel wavelet transform including the spectral analysis associated with the time-frequency operators and the scalogram analysis. The results of this paper are illustrated by some examples and figures.

In this paper, we introduce a summability-based functional, called the sampling-induced statistical Riemann sum functional, generated by uniform partitions with a fixed sampling rule. For a function \(f:[a,b]\to\mathbb{R}\), we consider the sequence of uniformly sampled right-endpoint Riemann sums and define the functional value as the statistical limit of this sequence, whenever it exists. We show that every classically Riemann integrable function yields the same value as the classical integral, while the converse fails. Unlike the classical Riemann integral, which is tag-independent and intrinsic, the functional studied here is sampling-dependent and may change under modifications of the function on countable sets. We illustrate these phenomena with examples, discuss the relationship with the statistical derivative, and highlight the structural compatibility issues that arise when attempting a statistical Fundamental Theorem of Calculus. The paper concludes with open problems concerning sampling-scheme independence and the characterization of functions for which the functional is tag-invariant.

In this paper, we study complete minimal submanifolds of Riemannian manifolds by employing a generalized Bochner technique. First, we provide a generalization of the classical theorem of Chern, Kobayashi, and do Carmo on compact minimal submanifolds to the case of complete parabolic minimal submanifolds. Second, we investigate the rigidity of complete stable minimal parabolic hypersurfaces in Riemannian manifolds, showing that under certain curvature conditions such hypersurfaces must be totally geodesic.