Neutrosophic estimation of unknown population parameters has become one of the most recent areas of focus in sampling theory over the past few years. The growing interest in this approach stems from the limitations of classical statistics when dealing with unreliable, indeterminate and uncertain data. Classical statistics only handles with accurate, crisp and unambiguous information. This study introduces an innovative generalised estimator for estimating the population mean under simple random sampling. The proposed generalised estimator can generate a variety of ratio and product exponential-type families of estimators, which are based on both conventional and non-conventional auxiliary information. Expressions for the bias, mean squared error and minimum mean square error have been derived. The proposed estimators provide an interval within which the population parameter is likely to lie with maximum probability. The superiority of the suggested estimators is demonstrated through both natural and simulated data.
The unique continuation property for solutions of second-order elliptic equations has been widely studied owing to its fundamental role in partial differential equations and related fields. While extensive results are available for strictly elliptic operators, comparatively little is known in the degenerate setting. In this paper we establish a strong unique continuation property for a class of degenerate elliptic operators involving potentials in the Kato class. The analysis is carried out under minimal regularity assumptions and the results extend several known contributions in the literature. In particular, we show that solutions that vanish in a neighbourhood, or satisfy appropriate vanishing conditions, must be identically zero. The theoretical findings are further supported by an illustrative example, highlighting the behaviour of solutions and confirming the robustness of the unique continuation property even in the presence of degeneracy and singular potentials.
This study develops a duality theory for paranormed sequence spaces whose terms take values in the non-commutative algebra of quaternions. Starting from Maddox-type positive exponent sequences, we introduce quaternion-valued analogues of the bounded, convergent, null and p-absolutely summable sequence classes. We then determine the corresponding a-, beta- and y-duals of these spaces. The main novelty of the study is that the duality problem is not treated as a direct scalar-field replacement of the classical real or complex case. Instead, the non-commutativity of quaternion multiplication is explicitly incorporated into the structure of the dual spaces. In particular, the convergence and boundedness conditions appearing in the definitions of the beta- and y-duals require a distinction between left and right multiplication, a phenomenon absent from classical sequence-space theory over commutative scalar fields. The obtained results extend Maddox-type paranormed sequence-space duality to the quaternionic setting and clarify the joint role of the paranorm topology and the order of quaternionic multiplication.
Caragana is an important genus in the Fabaceae family with significant ecological, ornamental and economic values. It includes over 100 species, many of which exhibit highly similar morphological characteristics, making species identification using traditional taxonomic methods quite challenging. Therefore, employing scientific, accurate and efficient methods for the identification of Caragana plants is of great significance. In this study the complete chloroplast genome of Caragana bicolor was sequenced using the Illumina HiSeq 4000 platform, and a systematic analysis was conducted by combining the data with the published chloroplast genomes of 56 closely related species. The results indicated that the complete chloroplast genome of C. bicolor is 131,396 bp in length, with an overall guanine and cytosine content of 35.14%. The chloroplast genome of C. bicolor does not exhibit the typical quadripartite structure, suggesting possible degeneration of its inverted repeat regions. A total of 110 genes were identified in the C. bicolor chloroplast genome. Additionally, 33 simple sequence repeat loci were detected in the C. bicolor chloroplast genome, with lengths ranging from 10 to 113 bp. Phylogenetic analysis revealed that C. bicolor is most closely related to C. tibetica, and this sister clade, together with species such as C. jubata and C. sinica, forms the core branch of the genus Caragana, confirming the taxonomic placement of C. bicolor within the genus. This study elucidates the key characteristics of the C. bicolor chloroplast genome, providing a reference for phylogenetic analyses of Caragana using chloroplast genomes and laying the foundation for further investigation into the origin, conservation and utilisation of C. bicolor.
We introduce a class of semi-r-clean rings, defined by the property that every element can be expressed as the sum of a periodic element and a von Neumann regular element. We establish fundamental properties of these rings, including their behaviour under homomorphic images, direct products and various ring-theoretic constructions. We prove that polynomial rings over commutative rings are never semi-r-clean and investigate the semi-rclean property in triangular matrix rings and ideal extensions. Additionally, we explore the relationship between semi-r-clean rings and the previously studied class of semi-clean rings.
