Mathematics Interdisciplinary Research
Volume:9 Issue: 1, Winter 2024

‎We present an enhanced approach to solving the combined non-linear time-dependent Burgers-Fisher equation‎, ‎which is widely used in mathematical biology and has a broad range of applications‎. ‎Our proposed method employs a modified version of the finite element method‎, ‎specifically the virtual element method‎, ‎which is a robust numerical approach‎. ‎We introduce a virtual process and an Euler-backward scheme for discretization in the spatial and time directions‎, ‎respectively‎. ‎Our numerical scheme achieves optimal error rates based on the degree of our virtual space‎, ‎ensuring high accuracy‎. ‎We evaluate the efficiency and flexibility of our approach by providing numerical results on both convex and non-convex polygonal meshes‎. ‎Our findings indicate that the proposed method is a promising tool for solving non-linear time-dependent equations in mathematical biology‎.

‎The representation theory of groups is one of the most interesting examples of the interaction between physics and pure mathematics‎, ‎where group rings play the main role‎. ‎The group ring $\rga$ is actually an associative ring that inherits the properties of the group $\ga$ and the ring of coefficients $R$‎. ‎In addition to the fact that the theory of group rings is clearly the meeting point of group theory and ring theory‎, ‎it also has applications in algebraic topology‎, ‎homological algebra‎, ‎algebraic K-theory and algebraic coding theory‎.‎In this article‎, ‎we provide a complete description of Gorenstein flat-cotorsion modules over the group ring $\rga$‎,‎where $\ga$ is a group and $R$ is a commutative ring‎. ‎It will be shown that if $\ga'\leqslant \ga$ is a finite-index subgroup‎, ‎then the restriction of scalars along the ring homomorphism $\rga'\rt\rga$ as well as its right adjoint $\rga\otimes_{\rga'}-$‎, ‎preserve the class of Gorenstein flat-cotorsion modules‎. ‎Then‎, ‎as a result‎, ‎Serre's Theorem is proved for the invariant $\Ghcd_{R}\ga$‎, ‎which refines the Gorenstein homological dimension of $\ga$ over $R$‎, ‎$\Ghd_{R}\ga$‎, ‎and is defined using flat-cotorsion modules‎. ‎Moreover‎, ‎we show that the inequality $\GF (\rga)\leqslant \GF (R)+{\cd_{R}\ga}$ holds for the group ring $\rga$‎, ‎where $\GF (R)$ denotes the supremum of Gorenstein flat-cotorsion dimensions of all $R$-modules and $\cd_{R}\ga$ is the cohomological dimension of $\ga$‎ over $R$‎.

‎Ricci bi-conformal vector fields have find their place in geometry as well as in physical applications‎. ‎In this paper‎, ‎we consider the Siklos spacetimes and we determine all the Ricci bi-conformal vector fields on these spaces‎.

‎From the viewpoint of‎ "‎extra dimension detecting,‎" ‎the phenomenon of the transition of the free point particle into 3d space is investigated‎. ‎In this way‎, ‎we formulate the problem using the second-class constrained system‎. ‎To investigate it using a gauge theoretical approach‎, ‎we use two methods to convert its two second-class constraints to first-class ones‎. ‎In symplectic embedding‎, ‎we construct a pair of scaler and vector gauge potentials‎, ‎which can be interpreted as interactions for detecting extra dimensions‎. ‎A Wess-Zumino variable appears as a new coordinate in potentials‎, ‎and the particle's mass plays the role of a globally conserved charge related to the constructed gauge theory for extra dimensions‎.

‎In emergency situations‎, ‎accurate demand forecasting for relief materials such as food‎, ‎water‎, ‎and medicine is crucial for effective disaster response‎. ‎This research is presented a novel algorithm to demand forecasting for relief materials using extended Case-Based Reasoning (CBR) with the best-worst method (BWM) and Hidden Markov Models (HMMs)‎. ‎The proposed algorithm involves training an HMM on historical data to obtain a set of state sequences representing the temporal fluctuations in demand for different relief materials‎. ‎When a new disaster occurs‎, ‎the algorithm first determines the current state sequence using the available data and searches the case library for past disasters with similar state sequences‎. ‎The effectiveness of the proposed algorithm is demonstrated through experiments on real-world disaster data of Iran‎. ‎Based on the results‎, ‎the forecasting error index for four relief materials is less than 10\%; therefore‎, ‎the proposed CBR-BWM-HMM is a strong and robust algorithm‎.

‎The graph $ AG ( R ) $ {of} a commutative ring $R$ with identity has an edge linking two unique vertices when the product of the vertices equals {the} zero ideal and its vertices are the nonzero annihilating ideals of $R$‎.‎The annihilating-ideal graph with {respect to} an ideal $ ( I ) $, ‎which is {denoted} by $ AG_I ( R ) $‎, ‎has distinct vertices $ K $ and $ J $ that are adjacent if and only if $ KJ\subseteq I $‎. ‎Its vertices are $ \{K\mid KJ\subseteq I\ \text{for some ideal}\ J \ \text{and}\ K$‎, ‎$J \nsubseteq I‎, ‎K\ \text{is a ideal of}\ R\} $‎. ‎The study of the two graphs $ AG_I ( R ) $ and $ AG(R/I) $ and {extending certain} prior findings are two main objectives of this research‎. ‎This studys {among other things‎, ‎the} findings {of this study reveal}‎‎that $ AG_I ( R ) $ is bipartite if and only if $ AG_I ( R ) $ is triangle-free‎.