This work establishes a multivariable Wold-type decomposition for left-inverse commuting n-tuples of bounded linear operators, built on the hypothesis that each component admits a Wold-type decomposition. For pairs of operators, we obtain a complete analytic model: every left-inverse commuting analytic toral 2-isometric pair is unitarily equivalent to the pair of multiplication operators by co-ordinate functions (Mz1,Mz2) acting on some E-valued Dirichlet-type space DE(μ1,μ2) associated with two finite positive operator-valued Borel measures μ1 and μ2 on the unit circle. An explicit functional model is further derived for the non-analytic case.
Biochar-poly(lactic acid) (PLA) composites are emerging as waste-derived biocomposites that integrate biomass valorization, biodegradable polymer development, and circular bioeconomy strategies. This review critically synthesizes how biochar feedstock, pyrolysis temperature, ash content, inorganic composition, surface chemistry, particle size, filler loading, and processing route influence the thermal, mechanical, degradability, and functional performance of PLA-based composites. Current evidence shows that optimized biochar incorporation can improve stiffness, tensile or flexural modulus, crystallization behaviour, impact resistance, dimensional stability, and composting-driven degradation. These benefits are mainly linked to biochar's carbon-rich structure, porous morphology, nucleating ability, surface functionality, and interfacial interactions with PLA. However, performance gains are not universal. Excessive loading or poor dispersion can reduce tensile strength, elongation at break, thermal stability, melt flow, and processability because of particle agglomeration, weak filler-matrix adhesion, moisture sensitivity, pore blockage, and processing-induced PLA chain scission. Particular attention is given to ash and inorganic residues, including alkali and alkaline-earth metals, carbonates, phosphates, silicates, and metal oxides, which may either promote crystallization and char formation or catalyze PLA degradation depending on their speciation, concentration, and dispersion. The review compares solvent casting, melt mixing, extrusion, compression and injection molding, filament production, and additive manufacturing, highlighting their advantages and processing constraints. Application opportunities in packaging, agriculture, water treatment, construction-related materials, biomedical systems, and 3D printing are discussed alongside food-contact safety, migration, durability, biocompatibility, regulatory, and end-of-life considerations. Wider adoption requires feedstock standardization, ash chemistry control, improved interfacial design, application-specific validation, and life-cycle assessment.
Calcium (Ca2⁺) signaling plays a pivotal role in plant defense responses against pests and pathogens, serving as an early and highly coordinated intracellular messenger. Upon biotic stress perception, Ca2⁺ channels mediate rapid Ca2⁺ influx, triggering downstream signaling pathways that activate defense-associated genes, secondary metabolite production, and hormonal pathways. This review explores the molecular mechanisms underlying Ca2⁺ channel activation, the spatiotemporal dynamics of intracellular Ca2⁺ fluxes, and their crosstalk with phytohormones and other signaling pathways. We further highlight the role of key Ca2⁺ sensors, such as calmodulins (CaMs), calcineurin B-like proteins (CBLs) and CBL-interacting protein kinase (CBL-CIPK) complexes, and calcium-dependent protein kinases (CDPKs) in decoding Ca2⁺ signals. In addition to this, emerging biotechnological approaches to enhance Ca2⁺-mediated resistance in crops like Nicotiana tabacum, Zea mays, and Arabidopsis thaliana has also been discussed. Understanding these mechanisms will provide valuable insights for developing new strategies to enhance plant resilience against evolving pest and pathogen threats under changing environmental conditions.
Pedicels, the short specialized stems that support reproductive structures in flowering plants, play diverse roles throughout plant development, yet many of their functions remain insufficiently understood. Morphologically and anatomically, they share several features with the main stem, including similar epidermal, cortical, and vascular traits. Variation in pedicel length, branching angle, and orientation influences floral and inflorescence architecture, thereby shaping floral display patterns and ultimately affecting reproductive success. Functionally, pedicels provide mechanical support, maintain vascular continuity between flowers and inflorescence axes, and facilitate the transport of nutrients, water, hormones, and signaling molecules to flowers and developing fruits. Recent studies highlight their additional roles in pollen presentation and seed dispersal, in which pedicel orientation aids effective interactions with pollinators and dispersal agents. Pedicels also serve as important sites of abscission, initiating enzymatic processes that lead to the detachment of flower or fruit. Advances in developmental genetics have identified numerous genes and transcription factors involved in pedicel formation, orientation, and functional modulation. Despite these insights, significant gaps remain, particularly regarding how pedicels adapt structurally and functionally to different pollination strategies. Further integrative research is required to clarify the developmental, ecological, and evolutionary significance of this understudied floral organ.
Insects, as diverse and ecologically dominant organisms, rely exclusively on innate immunity to defend against a wide array of microbial threats. This paper presents an integrative review of insect immune mechanisms, highlighting the molecular, cellular, and systemic components that underpin host defense. The immune response is orchestrated through physical barriers, cellular processes and humoral factors. Evolutionarily conserved pattern recognition receptors (PRRs) are essential to these processes. Emphasis is laid on pivotal functions of hemocytes, the significance of microbiome interactions in immune regulation, and the emerging influence of non-coding RNAs. Furthermore, the paper explores defensive symbiosis, environmental and evolutionary influences on immune dynamics, and applications in biotechnology and pest management. Model organisms, such as Drosophila melanogaster and Bombyx mori, serve as critical systems for unravelling innate immunity, with translational relevance to vertebrate immunology and vector control strategies. Understanding these mechanisms offers valuable insights into conserved immune pathways and holds promise for advancing strategies in human disease prevention, therapeutic innovation, and global health.
