Linear systems with finite-group symmetry, from geometric invariances to repetitive structural patterns, are ubiquitous in science and engineering, and standard computational software rarely exploits that structure. This paper introduces PySymmetry, an open-source Sage/Python framework that applies the constructive algorithms of classical representation theory to the symmetry reduction of G-equivariant linear systems. Projection operators built from the characters of the irreducible representations of a finite group generate a symmetry-adapted basis, which carries an equivariant operator into block-diagonal form and decouples one large problem into several small independent ones. A single character-theoretic construction serves two paths inside one open environment: an exact symbolic path over an exact ring, and a numerical path interoperable with NumPy, SciPy, and sparse matrix formats. Three case studies exercise both. The GF method of molecular vibrations illustrates the workflow. A non-Hermitian Schr & ouml;dinger eigenvalue benchmark shows the reduced path to be nearly 17 times faster than a general-purpose dense eigensolver on the largest grid tested. A symbolic investigation produced the first complete analytical classification of the nested cube central configuration in celestial mechanics. Source code, documentation, and the notebooks that reproduce the results are publicly available. PROGRAM SUMMARY Program Title: PySymmetry CPC Library link to program files: https://doi.org/10.17632/xm3skv4v68.1 Developer's repository link: https://github.com/ldsufrpe/pysymmetry Licensing provisions: MIT Programming language: SageMath 10.4, Python 3.11 Supplementary material: The GitHub repository includes Jupyter notebooks that reproduce the figures and tables generated by the framework; the Software Availability section maps each one to the notebook that produces it. Nature of problem: Linear systems governed by operators with discrete symmetries (G-equivariant systems) are common in physics and engineering, from quantum mechanics to structural analysis. Solving these systems, such as large-scale eigenvalue problems (Mx = Ax) or linear equations (Mx = b), can be computationally prohibitive. In numerical contexts, the cost of standard dense eigensolvers scales poorly with system size. In symbolic contexts, the problem is often intractable due to intermediate expression swell, where symbolic calculations exhaust memory and processing resources. This framework addresses the need for a tool that can systematically exploit these inherent symmetries to reduce computational complexity in both numerical and symbolic domains. Solution method: The PySymmetry framework implements the constructive algorithms of group representation theory to simplify G-equivariant linear operators. Given a finite group G and a representation, the method first uses characters and the conjugacy classes of the group to determine the structure of the representation's isotypic decomposition. It then algorithmically constructs projection operators to generate a symmetry-adapted basis for each isotypic component. By applying a similarity transformation with this basis, the original equivariant operator is transformed into a block-diagonal form. This procedure effectively decouples a single, large-scale problem into a set of smaller, independent subproblems that can be solved more efficiently, either numerically or symbolically. Additional comments including restrictions and unusual features: A key feature of PySymmetry is its hybrid symbolic-numeric architecture, built on the SageMath environment to provide seamless interoperability with the scientific Python ecosystem (NumPy, SciPy). It is designed with an object-oriented API and supports sparse matrix formats for efficiency. A significant advantage is the reusability of the computed symmetry-adapted basis: for parametric studies where the operator changes but the symmetry remains, the expensive preprocessing step is performed only once and its cost is amortized over numerous subsequent rapid calculations. The resulting block-diagonal structure is also inherently suited for parallel computation ("embarrassingly parallel"), further enhancing its performance potential on modern hardware.
The microscopic mechanisms by which Ti substitution modifies the structural, electronic and magnetic properties of M-type barium hexaferrite remain incompletely understood. In this work, first-principles density functional theory (DFT) calculations combined with structural and magnetic measurements were employed to establish the relation between Ti site occupancy, electronic structure, exchange interactions and magnetic behavior in BaFe12-xTixO19. The formation-energy calculations reveal that Ti4+ preferentially occupies the 12k sites at low concentrations and progressively extends to the 2b sites with increasing Ti content. Occupation matrix analysis indicates that charge compensation occurs through localized Fe3+-> Fe2+ reduction exclusively at the 12k sublattice. This electronic redistribution modifies the Fe-O hybridization, producing a half-metallic electronic structure for 0.5 <= x <= 1.5, selectively weakens the dominant Fe-O-Fe superexchange interactions, and progressively suppresses the magnetocrystalline anisotropy. The X-ray diffraction and magnetic measurements performed on co-precipitated BaFe12-xTixO19 nanopowders confirm the predicted increase in the cell parameters, the systematic reduction of the saturation magnetization, magnetocrystalline anisotropy constant, anisotropy field, remanent magnetization and coercive field with increasing Ti content. The excellent agreement between theory and experiment establishes a unified microscopic mechanism linking Ti site occupancy, charge compensation, electronic structure, exchange interactions and magnetic softening in M-type barium hexaferrite.
Collagenolytic proteases have attracted increasing attention due to their ability to hydrolyze collagen and their relevance in biotechnological processes. In this study, a collagenolytic protease produced by Mucor subtilissimus UCP 1262 was characterized through an integrated evaluation of its biochemical properties, thermal-inactivation thermodynamics, and conformational behavior. The enzyme was produced by solid-state fermentation using wheat bran and purified by DEAE-Sephadex A50 ion-exchange chromatography. Optimal collagenolytic activity was observed at pH 7 and 40 degrees C, and approximately 50% of the initial activity was retained after 24 h at 30-40 degrees C, indicating moderate thermal stability. Fluorescence spectroscopy revealed marked pH- and temperaturedependent changes in fluorescence intensity and emission maxima, with incomplete structural recovery after cooling. Thermodynamic analysis further supported the occurrence of temperature-dependent destabilization. Overall, the results provide new insight into the functional and conformational stability of the collagenolytic protease from M. subtilissimus UCP 1262 and support its further investigation for biotechnological applications under mild processing conditions.
Despite the growing interest in cactus-derived edible coatings and lactic acid bacteria (LAB)-based biocontrol systems, information on the use of cactus mucilage as a functional matrix for LAB cultures for the control of postharvest fungal diseases remains limited. Thus, this study evaluated the efficacy of coatings formulated with the mucilage of Pilosocereus pachycladus (facheiro) and Nopalea cochenillifera (palm) and incorporated with a mixed LAB culture for the control of anthracnose caused by Colletotrichum siamense in 'Margarida' avocado. The mucilages exhibited suitable physicochemical and technological properties for application in coating formulations. In vitro assays showed that cactus mucilage (2-4% w/v) exhibited dose-dependent inhibition of C. siamense, with the highest values observed at 3-4% (w/v), depending on the mucilage matrix, reaching up to 82.0% inhibition. LAB strains also showed strong antifungal activity, with mycelial growth inhibition ranging from 34% to 100%. The three most effective and compatible LAB strains (Lactiplantibacillus plantarum 49, Lactiplantibacillus pentosus 129, and Limosilactobacillus fermentum 263) were combined in equal proportions (1:1:1, v/v) to prepare a mixed culture standardized to 9-10 log CFU mL(-)& sup1; . The interaction between mucilage and the mixed LAB culture was additive (Abbott index, AI = 0.6-1.1), indicating enhanced antifungal effects. The mixed LAB culture maintained high viable cell counts (> 7 log CFU mL(-)& sup1;) for 15 d at 25 degrees C in the coating matrix. In situ assays demonstrated that the coatings significantly reduced anthracnose lesion development (32-100%) on avocados artificially inoculated with C. siamense, with the highest efficacy and greatest stability observed for the palm mucilage + LAB formulation. The coatings also reduced weight loss during storage without negatively affecting other fruit quality attributes. Cactus mucilage coatings, particularly when combined with mixed LAB cultures, show potential for the biological control of postharvest anthracnose in avocado.
Restoring soil structure and organic carbon (C) pools in semiarid environments requires management strategies that balance productivity and water-use efficiency. This study examined how intercropping forage cactus (Opuntia stricta) with Sudan grass (Sorghum sudanense) or buffel grass (Cenchrus ciliaris L.), combined with mixed grass mulch under treated wastewater irrigation, affected soil aggregation and C dynamics. Six treatments were tested (CS0, CS8, CS12, CB0, CB8, and CB12), representing two intercropping systems, CS (cactus-Sudan grass) and CB (cactus-buffel grass), with three application rates of mixed grass mulch (0, 8, and 12 Mg ha(-1), corresponding to 0, 3.5, and 5.3 Mg C ha(-1)). Evaluations included aggregate mass proportions (AM), the aggregate stability index (ASI), carbon preservation capacity (CPC), total and easily extractable glomalin-related soil proteins, and stocks of total organic carbon (SOC), particulate organic carbon (POC), and mineral-associated organic carbon (MAOC) within macroaggregates, along with the biomass productivity of the intercropping systems. Across the 0-40 cm profile, CB12 showed the strongest structural response, increasing the proportion of extra-large macroaggregates by 84% relative to CB0 and reaching an ASI of 41.8% and a CPC of 5.2 g kg(-1). Within extra-large macroaggregates, CB12 also showed the highest SOC (similar to 77 Mg ha(-1)) and MAOC (similar to 72 Mg ha(-1)) stocks, while its total POC stock across macroaggregate classes reached similar to 5.0 Mg ha(-1). CS12 and CB12 increased total glomalin-related soil protein contents, while CB12 also increased easily extractable glomalin-related soil protein relative to the corresponding no-mulch treatment. Soil under CS12 produced 194 Mg ha(-1) of fresh forage cactus biomass and 17 Mg ha(-1) of cactus dry matter. Under treated wastewater irrigation, higher mixed grass mulch inputs enhanced soil aggregation and C retention in cactus-grass intercropping systems, accompanied by greater accumulation of glomalin-related soil proteins and physical protection of SOC within macroaggregates.
