Search Results (18)

This study presents a hybrid sensor placement methodology that combines criterion-based candidate selection with advanced optimization algorithms. Four established selection criteria—modal kinetic energy (MKE), modal strain energy (MSE), modal assurance criterion (MAC) sensitivity, and mutual information (MI)—are used to evaluate DOF sensitivity and generate candidate pools. These are followed by one of four optimization algorithms—greedy, genetic algorithm (GA), particle swarm optimization (PSO), or simulated annealing (SA)—to identify the optimal subset of sensor locations. A key feature of the proposed approach is the incorporation of constraint dynamics using the Udwadia–Kalaba (U–K) generalized inverse formulation, which enables the accurate expansion of structural responses from sparse sensor data. The framework assumes a noise-free environment during the initial sensor design phase, but robustness is verified through extensive Monte Carlo simulations under multiple noise levels in a numerical experiment. This combined methodology offers an effective and flexible solution for data-driven sensor deployment in structural health monitoring. To clarify the rationale for using the Udwadia–Kalaba (U–K) generalized inverse, we note that unlike conventional pseudo-inverses, the U–K method incorporates physical constraints derived from partial mode shapes. This allows a more accurate and physically consistent reconstruction of unmeasured responses, particularly under sparse sensing. To clarify the benefit of using the U–K generalized inverse over conventional pseudo-inverses, we emphasize that the U–K method allows the incorporation of physical constraints derived from partial mode shapes directly into the reconstruction process. This leads to a constrained dynamic solution that not only reflects the known structural behavior but also improves numerical conditioning, particularly in underdetermined or ill-posed cases. Unlike conventional Moore–Penrose pseudo-inverses, which yield purely algebraic solutions without physical insight, the U–K formulation ensures that reconstructed responses adhere to dynamic compatibility, thereby reducing artifacts caused by sparse measurements or noise. Compared to unconstrained least-squares solutions, the U–K approach improves stability and interpretability in practical SHM scenarios. Full article

In this paper, we explore the application of fuzzy set theory in the context of triangular norms, with a focus on strong Sheffer stroke NMV-algebras. We introduce the concepts of $\mathfrak{T}$-fuzzy subalgebras and $\mathfrak{T}$-fuzzy filters, analyze their properties, and provide several illustrative examples. Our study demonstrates that $\mathfrak{T}$-fuzzy subalgebras and filters generalize classical subalgebras and filters, with level subsets preserving algebraic structures under t-norms. Notably, $\mathfrak{T}$-fuzzy sets exhibit strong closure properties, and homomorphisms between SSNMV-algebras extend naturally to fuzzy settings. Furthermore, we examine the relationships between $\mathfrak{T}$-fuzzy subalgebras (or filters) and their classical counterparts, as well as their corresponding level subsets and homomorphisms. These results contribute to refined uncertainty modeling in logical systems, with potential applications in fuzzy control and AI. Full article

To systematically address the intricate multiple criteria decision-making (MCDM) challenges to practical situations where uncertain and hesitant information plays a critical role in guiding optimal choices. In this article, we introduce the concept of $\mathfrak{m}$-polar Q-hesitant fuzzy ($\mathcal{MPQHF}$) $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras, combining $\mathfrak{m}$-PFS theory with Q-hesitant fuzzy set theory in the framework of $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras. This innovative approach enhances the attitudes of uncertainty, vagueness, and hesitance of data in decision-making processes. We investigate the features and actions of this proposed hybrid approach to fuzzy sets and hesitant fuzzy sets, focusing on $\mathcal{MPQHF}$ subalgebras, and explore the characteristics of several kinds of ideals under $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras. It also showed that it can better represent complex levels of uncertainty than regular sets. The proposed method’s theoretical framework offers a better way to show uncertain data in areas like engineering, computer science, and computational mathematics. By linking theoretical advancements of $\mathcal{MPQHF}$ sets with practical applications, we highlight the benefits and challenges of this approach. Demonstrating the practical uses of the $\mathcal{MPQHF}$ sets aims to encourage broader adoption. Symmetry plays a vital role in algebraic structure and is used in various fields like decision-making, encryption, pattern recognition problems, and automata theory. Furthermore, this work enhances the understanding of algebraic structures and offers a robust tool for career exploration and development through improved decision-making methodologies. Full article

The HX-groups represent a generalization of the group notion. The Chinese mathematicians Mi Honghai and Li Honxing analyzed this theory. Starting with a group $(G,·)$, they constructed another group $(\mathcal{G},\ast )\subset {\mathcal{P}}^{\ast }\left(G\right)$, where ${\mathcal{P}}^{\ast }\left(G\right)$ is the set of non-empty subsets of G. The hypercomposition “$\ast$” is thus defined for any A, B from G, $A\ast B=\{a·b|a\in A,b\in B\}.$ In this article, we consider a particular group, G, to be the dihedral group $D_n,n$ is a natural number, greater than 3, and we analyze the HX-groups with the dihedral group $D_n$ as a support. The HX-groups were studied algebraically, but the novelty of this article is that it is a computer analysis of the HX-groups by creating a program in $C++$. This code aims to improve the calculation time regarding the composition of the HX-groups. In the first part of the paper, we present some results from the hypergroup theory and HX-groups. We create another hyperstructure formed by reuniting all the HX-groups associated with a dihedral group $D_n$ as a support for a natural fixed number n. In the second part, we present the $C++$ code created in the Microsoft Visual Studio program, and we provide concrete examples of the program’s application. We created this program because the code aims to improve the calculation time regarding the composition of HX-groups. Full article

RNA transcripts play a crucial role as witnesses of gene expression health. Identifying disruptive short sequences in RNA transcription and regulation is essential for potentially treating diseases. Let us delve into the mathematical intricacies of these sequences. We have previously devised a mathematical approach for defining a “healthy” sequence. This sequence is characterized by having at most four distinct nucleotides (denoted as $nt\le 4$). It serves as the generator of a group denoted as $f_p$. The desired properties of this sequence are as follows: $f_p$ should be close to a free group of rank $nt-1$, it must be aperiodic, and $f_p$ should not have isolated singularities within its $S{L}_2\left(\mathbb{C}\right)$ character variety (specifically within the corresponding Groebner basis). Now, let us explore the concept of singularities. There are cubic surfaces associated with the character variety of a four-punctured sphere denoted as $S_2^4$. When we encounter these singularities, we find ourselves dealing with some algebraic solutions of a dynamical second-order differential (and transcendental) equation known as the Painlevé VI Equation. In certain cases, $S_2^4$ degenerates, in the sense that two punctures collapse, resulting in a “wild” dynamics governed by the Painlevé equations of an index lower than VI. In our paper, we provide examples of these fascinating mathematical structures within the context of miRNAs. Specifically, we find a clear relationship between decorated character varieties of Painlevé equations and the character variety calculated from the seed of oncomirs. These findings should find many applications including cancer research and the investigation of neurodegenative diseases. Full article

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph $\widehat{G}$ is fully made up of the zero divisors of the modular ring ${\mathbb{Z}}_n$, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs. Full article

The main purpose of the present paper is to introduced the notions of $(\acute{\in },\acute{\in }\vee \acute{q})$-$U{I}_{FS}SAs$ in subtraction BG-algebras. We provide different characterizations and some equivalent conditions of $(\acute{\in },\acute{\in }\vee \acute{q})$-$U{I}_{FS}SAs$ in terms of the level subsets of subtraction BG-algebras. It has been revealed that the $(\acute{q},\acute{q})$-$U{I}_{FS}SA$ are $(\acute{\in },\acute{\in })$-$U{I}_{FS}SA$ but the converse does not hold and an example is provided. We introduced $(\acute{\in },\acute{\in }\vee \acute{q})$-$U{I}_{FS}IDs$ and its some usual properties. In addition, $h^{-1}(\tilde{\mathfrak{N}}[\varsigma ])$ is $(\acute{\in },\acute{\in }\vee \acute{q})$-$U{I}_{FS}ID$. Moreover, if $h^{-1}(\tilde{\mathfrak{N}}[\varsigma ])$ are an $(\acute{\in },\acute{\in }\vee \acute{q})$-$U{I}_{FS}ID$, then $\tilde{\mathfrak{N}}[\varsigma ]$ are an $(\acute{\in },\acute{\in }\vee \acute{q})$-$U{I}_{FS}ID$. Finally, we characterize $(\acute{\in },\acute{\in }\vee {\acute{q}}_{\check{k}})$-$U{I}_{FS}{H}_{ID}$ which is a generalization of $(\acute{\in },\acute{\in }\vee \acute{q})$-$U{I}_{FS}{H}_{ID}$. Full article

In this paper, first derivatives of the Whittaker function ${\mathrm{M}}_{\kappa ,\mu }\left(x\right)$ are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, from the integral representation of ${\mathrm{M}}_{\kappa ,\mu }\left(x\right)$ it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions). These infinite sums and integrals can be expressed in closed form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function are derived, along with finite and infinite integrals containing products of algebraic, exponential, logarithmic, and Bessel functions. Finally, reduction formulas for the Whittaker functions ${\mathrm{M}}_{\kappa ,\mu }\left(x\right)$ and integral Whittaker functions ${\mathrm{Mi}}_{\kappa ,\mu }\left(x\right)$ and ${\mathrm{mi}}_{\kappa ,\mu }\left(x\right)$ are calculated. Full article

Information-theoretic quantities reveal dependencies among variables in the structure of joint, marginal, and conditional entropies while leaving certain fundamentally different systems indistinguishable. Furthermore, there is no consensus on the correct higher-order generalisation of mutual information (MI). In this manuscript, we show that a recently proposed model-free definition of higher-order interactions among binary variables (MFIs), such as mutual information, is a Möbius inversion on a Boolean algebra, except of surprisal instead of entropy. This provides an information-theoretic interpretation to the MFIs, and by extension to Ising interactions. We study the objects dual to mutual information and the MFIs on the order-reversed lattices. We find that dual MI is related to the previously studied differential mutual information, while dual interactions are interactions with respect to a different background state. Unlike (dual) mutual information, interactions and their duals uniquely identify all six 2-input logic gates, the dy- and triadic distributions, and different causal dynamics that are identical in terms of their Shannon information content. Full article

In many tasks related to an object’s observation or real-time monitoring, the gathering of temporal multimodal data is required. Such data sets are semantically connected as they reflect different aspects of the same object. However, data sets of different modalities are usually stored and processed independently. This paper presents an approach based on the application of the Algebraic System of Aggregates (ASA) operations that enable the creation of an object’s complex representation, referred to as multi-image (MI). The representation of temporal multimodal data sets as the object’s MI yields simple data-processing procedures as it provides a solid semantic connection between data describing different features of the same object, process, or phenomenon. In terms of software development, the MI is a complex data structure used for data processing with ASA operations. This paper provides a detailed presentation of this concept. Full article

Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. To discover the symmetries and invariants of the transcription and regulation at the scale of the genome, in this paper, we introduce tools of infinite group theory and of algebraic geometry to describe both TFs and miRNAs. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group $\pi$, we compute the $SL(2,\mathbb{C})$ character variety, where $SL(2,\mathbb{C})$ is simultaneously a ‘space-time’ (a Lorentz group) and a ‘quantum’ (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when $\pi$ looks similar to a free group $F_r$ ($r=1$ to 3) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non-free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis $\mathcal{G}$ of the variety. A surface with simple singularities (such as the well known Cayley cubic) within $\mathcal{G}$ is a signature of a potential disease even when $\pi$ looks similar to a free group $F_r$ in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in $\{A,T/U,G,C\}$. We produce a few tables of human TFs and miRNAs showing that a disease may occur when either $\pi$ is away from a free group or $\mathcal{G}$ contains surfaces with isolated singularities. Full article

Revealing the time structure of physical or biological objects is usually performed thanks to the tools of signal processing such as the fast Fourier transform, Ramanujan sum signal processing, and many other techniques. For space-time topological objects in physics and biology, we propose a type of algebraic processing based on schemes in which the discrimination of singularities within objects is based on the space-time-spin group $SL(2,\mathbb{C})$. Such topological objects possess an homotopy structure encoded in their fundamental group, and the related $SL(2,\mathbb{C})$ multivariate polynomial character variety contains a plethora of singularities somehow analogous to the frequency spectrum in time structures. Our approach is applied to a model of quantum computing based on an Akbulut cork in exotic $R_4$, to an hyperbolic model of topological quantum computing based on magic states and to microRNAs in genetics. Such diverse topics reveal the manifold of possibilities of using the concept of a scheme spectrum. Full article

In this paper, the Heisenberg doubles and Long dimodules of a BiHom-Hopf algebra are introduced. Then, we discussed the relation between BiHom-Hopf equation and BiHom-pentagon equation, and we obtain the solutions of BiHom-Hopf equation from Heisenberg doubles. We also showed that the parametric generalized Long dimodules can provide the solutions of BiHom-Yang-Baxter equation and generalized $\mathcal{D}$-equation. Full article

The prediction and smoothing fusion problems in multisensor systems with mixed uncertainties and correlated noises are addressed in the tessarine domain, under ${\mathbb{T}}_k$-properness conditions. Bernoulli distributed random tessarine processes are introduced to describe one-step randomly delayed and missing measurements. Centralized and distributed fusion methods are applied in a ${\mathbb{T}}_k$-proper setting, $k=1,2$, which considerably reduce the dimension of the processes involved. As a consequence, efficient centralized and distributed fusion prediction and smoothing algorithms are devised with a lower computational cost than that derived from a real formalism. The performance of these algorithms is analyzed by using numerical simulations where different uncertainty situations are considered: updated/delayed and missing measurements. Full article

The article develops further directions stemming from the arithmetic of extensional fuzzy numbers. It presents the existing knowledge of the relationship between the arithmetic and the proposed orderings of extensional fuzzy numbers—so-called $\mathcal{S}$-orderings—and investigates distinct properties of such orderings. The desirable investigation of the $\mathcal{S}$-orderings of extensional fuzzy numbers is directly used in the concept of $\mathcal{S}$-function—a natural extension of the notion of a function that, in its arguments as well as results, uses extensional fuzzy numbers. One of the immediate subsequent applications is fuzzy interpolation. The article provides readers with the basic fuzzy interpolation method, investigation of its properties and an illustrative experimental example on real data. The goal of the paper is, however, much deeper than presenting a single fuzzy interpolation method. It determines direction to a wide variety of fuzzy interpolation as well as other analytical methods stemming from the concept of $\mathcal{S}$-function and from the arithmetic of extensional fuzzy numbers in general. Full article