Multi-criteria decision-making (MCDM) over heterogeneous spatial data presents two fundamental challenges that remain insufficiently addressed in the literature: the optimal fusion of subjective and objective weighting schemes, and the explicit incorporation of spatial dependence into distance-based ranking models. This paper proposes a hybrid spatial MCDM framework that addresses both issues through rigorous mathematical construction. First, we formulate combined weighting as a minimum Kullback-Leibler (KL) divergence optimization problem and derive a closed-form solution via the Lagrange multiplier method, yielding a weighted geometric mean combination that optimally balances expert judgment against data-driven information. Second, we introduce a spatial-weighted TOPSIS model that explicitly integrates a spatial weight matrix into the Euclidean distance computation to the positive and negative ideal solutions, thereby encoding Tobler's first law of geography into the ranking mechanism. Third, we establish several theoretical properties of the framework: a sensitivity theorem bounding the error propagation under data perturbation, a robustness theorem for the KL-divergence combined weights, and a rank-reversal analysis proving that a globally bounded normalization scheme eliminates the classical TOPSIS rank-reversal pathology. The framework's practical utility is demonstrated through a large-scale numerical experiment involving 2,502 spatial alternatives with 10 heterogeneous attributes derived from multi-source geographic data. Comparative analysis against standard AHP-TOPSIS and entropy-TOPSIS baselines confirms that the proposed framework produces spatially coherent, numerically stable, and theoretically grounded priority rankings. The framework is general and transferable to any spatial MCDM problem characterized by heterogeneous attribute data and spatial dependence.
Let Γ \Gamma be a finitely generated group and G G a complex Lie group with Lie algebra g \mathfrak {g} . The complex variety of group homomorphisms H o m ( Γ , G ) Hom(\Gamma ,\,G) is equipped with an action of G G by inner automorphism and thus it defines a complex-analytic stack. We associate to each Φ ∈ ( S y m n g ) G \Phi \,\in \, (Sym^{n} \mathfrak {g})^{G} a closed holomorphic n n -form on this stack with values in H n ( Γ , C ) H^{n}(\Gamma ,\,\mathbb {C}) . This construction is multiplicative so that we get a graded algebra homomorphism. We obtain this as a universal refinement of a Chern-Weil homomorphism. If we take n = 2 n \,=\, 2 and let Γ \Gamma be the fundamental group of a closed orientable surface, then we recover a classical construction due to Goldman.
We study topological properties of complex hyperplane arrangements that are decomposable. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G G decomposes (in degrees greater than 1 1 ) as a direct product of free Lie algebras. It follows that the I I -adic completion of the Alexander invariant B ( G ) B(G) also decomposes as a direct sum of “local” invariants and the Chen ranks of G G are the sums of the local contributions. Moreover, if B ( G ) B(G) is separated, then the degree 1 1 cohomology jump loci of the arrangement complement have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1 1 .
The synergistic interplay of ciliary motion and Magnetohydrodynamics (MHD) in facilitating the peristaltic transport of nanofluid presented a transformative approach for bio-applications. Our research enhanced medical diagnostics and increased the precision of drug delivery. Our analysis also examined the interactions between cilia-induced fluid motion and the influence of magnetic fields on nanofluid flow in a vertical endoscopic tube. A Mathematical model was created and solved through analytical as well as numerical procedures. We examined the impacts of involving factors on flow efficiency, heat transfer and particle dispersion. The results indicated that coordinated action of ciliary beating and magnetic field gradient significantly enhanced fluid propulsion and control. Our research indicated that combining ciliary mechanisms with an MHD-driven nanofluid system improved the movement of biofluid in a clinical situation. Magnetic force is observed to reduce velocity while buoyancy force tend to tends to increase it. A wider section of the tube exhibits a lower pressure gradient and facilitates smoother fluid flow with reduced resistance.
This research presents a nonlinear mathematical approach that integrates smoking and vaping dynamics in a one population. The model is defined as a system of nonlinear ordinary differential equations and rigorously analyzed to provied the existence, uniqueness and boundedness of solutions. The disease-free and endemic equilibrium states are obtained to investigate the major dynamical characteristics of the disease as well as the basic reproduction number is obtained using the next-generation matrix method. Stability analyses encompassing both local and global perspectives are conducted to detail the threshold conditions and the future behavior of the system. To examine the structure and the interaction of the model diagonally, a signal flow graph is plotted and analyzed using spectral graph theory. The measures include graph energy and Estrada index, which are used to establish the connectivity of the network, the feedback and flow of information among the variables. Moreover, sensitivity analysis is conducted to indicate the key parameters which have the greatest influence on the transmission of smoking and vaping, thereby causing the control strategies to be developed. The numerical experiments confirm the analytical findings besides showing the strength of the suggested framework to represent the complex correlation between the behaviors of smoking and vaping accurately. The simulations were further supported by a MATLAB Simulink model that demonstrated the system equations as an animated block diagram which besides allowing the visual exploration of state transitions and parameter sensitivity in real-time also provides a user-friendly tool for scenario testing and educational demonstration.
As the demand for renewable energy increases, it becomes more vital to efficiently integrate different resources such as wind and solar energy. This study illustrates a novel integrated mechanism for analyzing a green energy generation dynamical system utilizing the Fractal-Fractional (FF) sense of operators. We give a robust deep learning framework designed to successfully model the complex interactions and nonlinear characteristics of these energy sources. This mechanism leverages FF-calculus to improve the model's efficiency, incorporating comprehensive error analysis and advanced pattern recognition techniques. Our mathematical analysis confirms the feasibility of both the initial and parametric values of the model, demonstrating its robustness. The existence criteria, uniqueness results, and stability were also confirmed in the theoretical aspect of the work. Computational results show that the proposed hybrid system of energy significantly outperforms traditional solar-only systems, achieving an efficiency improvement of approximately 25% in energy generation. This work not only supports the existing knowledge in renewable energy but also provides a practical approach for optimizing green power generation systems by the use of artificial intelligence as a process innovation
Over the past decades, the study of Hardy-type inequalities has led to significant developments in both theory and applications, giving rise to a wide range of refinements and generalizations in weighted and unweighted settings. Building upon the weighted Hardy-type inequalities established by Sulaiman and the generalized forms proposed by Sroysang, this paper develops new sharp inequalities within the framework of time scale calculus. The approach unifies continuous and discrete settings by employing delta integrals and accommodating general weight functions under the assumption monotonicity. Our main theorems extend the continuous inequalities of Sulaiman and Sroysang to arbitrary time scales, and yield new discrete analogues as direct consequences. Proofs rely on H & ouml;lder's inequality, the chain rule, and integration by parts adapted to the time scale context. Several corollaries and illustrative examples are included, highlighting that various known continuous and discrete inequalities appear as exceptional cases of the presented framework.
We develop an exact transient analysis for a special class of continuous time random walks on the d d -dimensional integer lattice, driven by an independent family of Poisson processes. We derive closed form solutions to their transition probabilities by applying complex analytic techniques tied directly to their Markovian random walk structure. These results are achieved without the use of generating functions, Laplace transforms, or special functions. Our analysis goes directly from the sample path structure to a spectral decomposition of the eigenvalues for the underlying Markov matrix-generator. These random walks have a canonical structure that leads to a natural set of group actions on the lattice state space. The resulting group symmetries then allow us to solve for the transition probabilities of various stopped random walks. These stopping times correspond to the boundary absorption or interior exit times within distinguished lattice subsets. As a result, these random walk solutions give us an exact transient absorption analysis for a large family of queueing network models. The analysis of this paper extends and simplifies the original work for these transient absorbing probabilities found in papers [J. Appl. Probab. 24 (1987), no. 1, 226–234] and [Theoret. Comput. Sci. 125 (1994), no. 1, 149–165]. Moreover, papers [Queueing Syst. 103 (2023), no. 1-2, 1–43] and [ A transient symmetry analysis for absorbed random walks on two-dimensional integer lattices , 2026] show that one and two dimensional examples of these random walks and queueing networks have transportation service applications.
For a number field F F and an N N -th root of unity q q , put k N = F ( q ) k_N=F(q) and K N = F N K_N=\root {N}\of {F} . We give two homomorphisms mapping an element ∑ i [ x i ] \sum _i[x_i] in the Bloch group of F F to the quotient group K N × / k N × ( K N × ) N K_N^\times /k_N^\times (K_N^\times )^N . The first one is ∏ i f ( x i ; q ) \prod _i f(x_i;q) with f ( a ; q ) = ∏ j = 1 N ( 1 − a q j ) j f(a;q)=\prod _{j=1}^N (1-a q^j)^j , whereas the second assigns Trace ( M N ) N / det ( M N ) \operatorname {Trace}(M_N)^N/\det (M_N) , where the N × N N\times N -matrix M N M_N arises as a product of Morita equivalences relating central simple algebras associated to the terms x i x_i .
This study defines an extended Neutrosophic b-Metric Space (NbMS) and illustrates the characteristics that are essential to its structure. The Fixed Point (FP) theorem has therefore been proved in the context of these Extended Neutrosophic b-Metric Spaces (ENbMS). Our results show symmetrical patterns and features within these mathematical frameworks, both extending and generalizing the results found in the current literature. A nontrivial example is used to highlight the efficacy of the suggested approaches. To emphasize the usefulness of our primary findings, an application to the existence and uniqueness of solutions for a particular class of Fredholm integral equations is investigated.
An extended direct algebraic method is used in this work to examine the soliton solutions of the fractional Hirota-Satsuma coupled Korteweg-de Vries equation. Understanding the dynamic behaviour of solitons in nonlinear systems using analytical solutions is our goal. To obtain precise soliton solutions, we utilise a logistic technique. These solutions are then shown graphically in three dimensions, two dimensions, and contours. Soliton interactions' complex dynamics and stability in fractional nonlinear systems are demonstrated by the results. This study clarifies the underlying dynamics and possible uses of soliton behaviour in complex systems, advancing our understanding of this phenomenon.
This paper is a review of results on the Gauss-Manin connection in noncommutative geometry. The Gauss-Manin connection in periodic cyclic homology was introduced by Ezra Getzler in 1991, then generalized to a superconnection by the author in a joint work with Dolgushev and Tamarkin. The key to these constructions is the Cartan calculus in noncommutative geometry. The original results of this article, namely the comparisons between the noncommutative Cartan calculus and its classical version, are contained in Section \ref{s:Compatibility with HKR}. The rest of the paper is mostly a review of results from \cite{GGM}, \cite{DTTGM}, \cite{NTbook}, \cite{TsyNCcry}, although the approach is somewhat new.
In this research, we propose an inertial-type iterative scheme constructed by an E-type enriched nonexpansive mappings in the framework of uniformly convex Banach spaces. Strong and weak convergence results along with stability result for the proposed algorithm are presented. Numerical examples and comparisons with well known existing iterative schemes with the help of graphs are presented to demonstrate the efficiency of our proposed scheme. It is shown that the algorithm presented herein converges more faster to fixed point and competitive on the examples considered for this class of mappings. As applications, we apply the scheme to solve variational inequality problems, constrained optimization problems, and split feasibility problems. Moreover, we use the proposed approach to solve a functional differential equation.
Fractional-order neural networks are vital for modeling neuronal interactions in information processing, and exploring their existence and stability remains a key research challenge. This paper aims to achieve almost periodic uniform stability for fractional-order stochastic fuzzy neural networks in the quaternion field, which is crucial for ensuring reliable and sustained periodic performance in dynamic environments, essential for applications like simulating biological neural systems. We employ a direct method without decomposing quaternion-valued networks into real-valued counterparts, and by leveraging fixed point theorem, fundamental fractional calculus properties, and various inequality techniques, we establish the existence of a unique almost periodic solution under specific conditions. Additionally, by constructing an error system and applying uniform stability definitions with inequality methods, we derive conditions for almost periodic uniform stability. Finally, MATLAB simulations validate the feasibility and accuracy of our results.
The stability and uniqueness of the solutions depend on the inequalities. Bounding, simulating non-local behaviors, and resolving systems that integer-order algorithms are unable to handle are among their uses. In this paper, we examine the generalized (k, W)-Conformable Fractional Integrals (CFIs), which are the k-analogues of the recently published generalized conformable fractional integrals. These integrals are reducible to other fractional integrals under specific parameter values. Finally, we present several integral inequalities pertaining to the generalized (k, W)-Conformable Fractional Integrals. The classical integral inequalities can easily be restored by applying certain conditions on parameters k and W.
The goal of this paper is to reveal how we can use data modeling and decision engineering as a unifying language of mathematics for narratives of discovery and insight. We start by collecting data records relevant to problems of interest. Next, we organize the records by discovering patterns in the data. We then infer from these structures domain-specific models that summarize the data. Finally, we analyze these models to determine what might happen for predictions and what should happen for decision-making. This paper introduces simple examples to illustrate how various mathematical tools contribute to the larger fields of statistics and operations research. These ideas can build useful frameworks for interdisciplinary research collaborations. They can also inspire ongoing motivational classroom dialogues about the mathematical sciences.
Starting from a suitable value of Cramèr’s approximation for the probability of ruin of an insurance portfolio, new approximations for a number of ruin probabilities of interest in statistics are obtained by very simple calculations.
In order to systematize statistical analysis in interactive Multi-Attribute Decision Making (MADM) models, the concept of "Monotone Expectation" is considered for the extension of classical statistics in the p, q-Rung Orthopair Fuzzy (p, q-ROF) environment, concretely, the probability mean (expectation), variance, covariance, and correlation coefficient. In the classical view, the operations of subtraction and division are not defined in p, q-ROF values. The operations of pseudo-subtraction and pseudo-division are introduced to define the covariance and correlation coefficient. Based on the Choquet finite integral, extensions of monotone statistics such as F-associated expectation, variance, covariance and correlation coefficient for the p, q-ROF environment are presented. The fuzzy measure Associated Probabilities Class (APC) of a fuzzy measure and the classes of associated statistics: probability averaging class, variances class, covariances class and correlation coefficients class are considered. The relationship between monotone and associated statistics is studied. It is established that monotone statistics can be represented by unique corresponding associated statistics if there is a partial ordering between the arguments of the statistics, as between the p, q-ROF values. That is, the remaining (n! - 1) associated statistics of the corresponding associated class, where n is the number of interacting attributes, become useless in the definitions of monotone statistics. In the interactive MADM environment, in decision-making modeling, the aggregations of monotone statistics cannot fully reflect all possible degrees of interaction of attributes. In more detail: the aggregations with monotone statistical parameters under p, q-ROF environment used in interactive MADM models take into account interactions of the focal elements of only one consonant structure from the n! consonant structures of attributes. The extensions of monotone statistics, called F-associated statistics, which take into account the degrees of interaction of elements of all n! consonant structures of attributes in aggregations are defined. The issue of the correctness of the extensions is discussed. The values of monotone and F-associated statistics coincide if there is a partial ordering between the arguments of the statistics and the Choquet second-order extreme capacities are taken in the role of a fuzzy measure. A calculation scheme for monotone and F-associated statistics is constructed for a simple interactive MADM model under p, q-ROF environment. A simple numerical example is presented to illustrate the obtained results.
It is well known that there is a strong connection between operator theory and complex analysis. The concept of conformable operators is also employed in fractional calculus to define fractional differential and integral operators. In this study, we develop conformable composition operators in a complex domain acting on spaces of analytic functions. The boundedness of weighted composition operators has been extensively studied on various analytic function spaces. The present work investigates the boundedness of two weighted operators with several variable differences on different analytic function spaces, particularly those involving Banach spaces of analytic functions. Our approach is based on the use of M & ouml;bius transformations and Green's functions defined in terms of the pseudo-hyperbolic distance.
I answer a question which Spencer Bloch asked during the Regulators V conference in Pisa.
