This paper examines how logical structures represent sharp and unsharp propositions in classical and quantum systems. Classical sharp logic is modeled by Boolean algebras, where propositions have bivalent truth values, complements are globally defined, and distributivity holds. Quantum sharp logic is modeled by orthomodular lattices of closed Hilbert-space subspaces or projection operators; it retains definite yes–no outcomes for ideal measurements but rejects the Boolean assumptions of global truth valuation and distributivity. Classical unsharp logic uses fuzzy membership or probabilistic truth values to describe vagueness and incomplete information, whereas quantum unsharp logic is expressed through effects and effect algebras, which capture noisy, inefficient, and generalized measurements. The paper’s central question is which algebraic structure is appropriate for each combination of classical or quantum behavior and sharp or unsharp measurement. The analysis shows that the four cases form a coherent hierarchy: Boolean algebras describe deterministic classical events, orthomodular lattices describe ideal quantum propositions, fuzzy or probabilistic models describe classical partial truth, and effect algebras describe realistic quantum effects. This classification clarifies the mathematical and physical role of unsharpness and explains why effect algebras are needed beyond projection-based quantum logic.

Reliable rainfall–runoff estimation is essential for the hydraulic design and planning of canal systems, particularly in ungauged catchments where streamflow records are unavailable. This study aimed to generate design-storm hydrographs and estimate peak runoff for the Narai Canal catchment in Peshawar, Khyber Pakhtunkhwa, Pakistan, using the Log-Pearson Type III (LP-III) distribution and the WinTR-20 model under ungauged-basin conditions. Rainfall frequency analysis was performed using the LP-III distribution to estimate design rainfall for selected return periods. The Soil Conservation Service (SCS) curve number method implemented in WinTR-20 was then used to simulate runoff hydrographs and peak discharges based on watershed characteristics, including land use, curve number, and time of concentration. The analysis produced design-storm hydrographs and corresponding peak discharge estimates that increased with rainfall intensity and return period. The LP-III distribution adequately characterized design rainfall, while the WinTR-20 model provided physically consistent runoff estimates for evaluating watershed response under different design scenarios. The results demonstrate the influence of watershed characteristics on runoff generation and provide preliminary hydrological estimates for the study area, representing a design-scenario analysis for an ungauged basin rather than a validated rainfall–runoff forecasting model. The generated design hydrographs and peak discharge estimates provide useful information for preliminary canal hydraulic design, drainage planning, and flood-risk assessment. Future studies should incorporate observed streamflow data to calibrate and validate the model before it is applied for operational flood forecasting.

The Stochastic Vehicle Routing Problem (VRPSD) has been extensively studied under the assumption that all customer demands are non-negative random variables. In many real-world settings, however, demands may be signed: customers simultaneously act as sources (pickup, increasing vehicle load) and sinks (delivery, decreasing vehicle load), or excess inventory is returned to the depot. This paper introduces the Stochastic Vehicle Routing Problem with Mixed and Signed Demands (SVRP-MD), in which each customer \(i\) has a real-valued random demand \(\xi_i \in \mathbb{R}\), modelled as a net change in an abstract vehicle load-balance process under a reactive, signed detour-to-depot (DTD) recourse policy in which a violating visit triggers a depot round trip before service and the vehicle resumes at the same customer. Our contributions are: (i) a counterexample proving that the classical superadditivity of the recourse function fails when demands are signed; (ii) an incomparability result showing that the positive-part recourse neither upper- nor lower-bounds the signed recourse under the two-sided failure rule, together with a valid lower bound based on the absolute net demand that underlies our cut inequalities; (iii) a family of signed net-demand (SND) incumbent bounds, embedded in valid integer L-shaped optimality cuts in the extended \((x, \theta)\) space; (iv) a Jensen-type lower bound using a convex lower approximation of the recourse cost; and (v) a branch-and-cut sample-average bounding procedure with a Monte Carlo, confidence-based acceptance rule. Preliminary experiments suggest the SND cuts reduce the LP relaxation gap; the reported figures predate the protocol and bound corrections and are being regenerated.

We introduce a degenerate–\(q\) hybrid coefficient series obtained by multiplying a normalised basic-hypergeometric coefficient array by the degenerate rising factorial defined through the Kim–Kim degenerate Gamma function. The normalisation used here deliberately omits the conventional factor \([(-1)^nq^{n(n-1)/2}]^{1+s-r}\) from the fully standard definition of \({}_r\phi_s\); the precise relationship with the standard convention is stated explicitly. Under explicit nonsingularity assumptions on the \(q\)-denominator parameters and the degenerate Gamma factors, we determine the disc of absolute convergence and give a corrected boundary analysis. The coefficient ratio \((\alpha)^*_{\lambda,n+1}/(\alpha)^*_{\lambda,n}\) is encoded as a rational Euler-transport operator, which leads coefficientwise to a hybrid Heine-type \(q\)-difference equation, numerator and denominator contiguity relations, and Jackson \(q\)-derivative identities. In the scalar case we construct two Frobenius-type local series and formulate the connection coefficients only within the corresponding nonresonant Frobenius span. For terminating series we prove genuine degenerate-Gamma moment representations; for nonterminating series the corresponding moment formalism is explicitly interpreted as a meromorphic regularisation rather than as a positive-measure integral. We also state a symmetric Beta-kernel identity with separate complex-analytic and real-positivity hypotheses and record carefully scaled limit transitions.

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.