Search Results (764)

The development of efficient strategies for optimizing and controlling nonlinear bioprocesses remains a significant challenge due to their complex dynamics and sensitivity to operating conditions. This work addresses the problem by proposing a two-step methodology applied to a laboratory-scale fed-batch bioethanol process. The first step employs a dynamic optimization approach based on Fourier parameterization and orthonormal polynomials, which generates smooth and continuous substrate-feed profiles using only three parameters instead of the ten required by piecewise approaches. The second step introduces a controller formulated through basic linear algebra operations, which ensures accurate trajectory tracking of the optimized state variables. Simulation results demonstrate a $3.65\%$ increase in ethanol concentration at the end of the process, together with an accumulated tracking error of only $0.0189$ under nominal conditions. In addition, the closed-loop strategy outperforms open-loop implementation when the initial conditions deviate from their nominal values. These findings highlight that the proposed methodology reduces mathematical complexity and computational effort while producing continuous control profiles suitable for practical application. The combination of optimization and algebraic control thus provides a promising alternative for improving the efficiency of bioethanol-production processes. Full article

There are two ways of linearizing the Klein–Gordon equation: Dirac’s choice, which introduces a matter–antimatter pair, and a second approach using a bivector, which Dirac did not consider. In this paper, we show that a bivector provides the classical origin of quantum spin. At high precessional frequencies, a symmetry transformation occurs in which classical reflection becomes quantum parity. We identify a classical spin-1 boson and demonstrate how bosons deliver energy, matter, and torque to a surface. The correspondence between classical and quantum domains allows spin to be identified as a quantum bivector, $i\sigma$. Using geometric algebra, we show that a classical boson has two blades, corresponding to magnetic quantum number states $m=\pm 1$. We conclude that fermions are the blades of bosons, thereby unifying both into a single particle theory. We compare and contrast the Standard Model, which uses chiral vectors as fundamental, with the Bivector Standard Model, which uses bivectors, with two hands, as fundamental. Full article

In several stream cipher designs, Boolean functions (BFs) play a crucial role as non-linear components, either serving as filtering functions or being used within the combining process. The overall strength of stream ciphers mainly depends on certain cryptographic properties of BFs, including their balancedness, non-linearity, resistance to correlation, and algebraic degrees. In this paper, we present novel findings related to the algebraic degrees of BFs, which play an important role in the design of symmetric cryptographic systems, and propose a novel algorithm to directly deduce the algebraic degree of a Boolean function (BF) from its truth table. We also explore new results concerning balanced Boolean functions, specifically characterizing them by establishing new results regarding their support. Additionally, we propose a new approach for a subclass of affine equivalent Boolean functions and discuss well-known cryptographic properties in a very simple and lucid manner using this newly introduced approach. Moreover, we propose the first algorithm in the literature to construct non-quadratic balanced Boolean functions (NQBBFs) that possess no linear structure where their derivative equals 1. Finally, we discuss the complexity of this algorithm and present a table that shows the time taken by this algorithm, after its implementation in SageMath, for the generation of Boolean functions corresponding to different values of n (i.e., number of variables). Full article

Maximum Rank-Distance (MRD) codes are a class of optimal error-correcting codes that achieve the Singleton-like bound for rank metric, making them invaluable in applications such as network coding, cryptography, and distributed storage. While algebraic constructions of MRD codes (e.g., Gabidulin codes) are well-studied for specific parameters, a comprehensive theory for their existence and structure over arbitrary finite fields remains an open challenge. Recent advances have expanded MRD research to include twisted, scattered, convolutional, and machine-learning-aided approaches, yet many parameter regimes remain unexplored. This paper introduces a computational optimization framework for constructing MRD codes using Particle Swarm Optimization (PSO), a bio-inspired metaheuristic algorithm adept at navigating high-dimensional, non-linear, and discrete search spaces. Unlike traditional algebraic methods, our approach does not rely on prescribed algebraic structures; instead, it systematically explores the space of possible generator matrices to identify MRD configurations, particularly in cases where theoretical constructions are unknown. Key contributions include: (1) a tailored finite-field PSO formulation that encodes rank-metric constraints into the optimization process, with explicit parameter control to address convergence speed and global optimality; (2) a theoretical analysis of the adaptability of PSO to MRD construction in complex search landscapes, supported by experiments demonstrating its ability to find codes beyond classical families; and (3) an open-source Python toolkit for MRD code discovery, enabling full reproducibility and extension to other rank-metric scenarios. The proposed method complements established theory while opening new avenues for hybrid metaheuristic–algebraic and machine learning–aided MRD code construction. Full article

This paper presents an innovative numerical method for solving two-dimensional weakly singular Volterra integral equations, including fractional Volterra integral equations with weak singularities. Solving these equations in higher dimensions and in the presence of fractional and weak singularities is highly challenging. The proposed approach uses Euler wavelets (EWs) within an operational matrix (OM) framework combined with advanced numerical techniques, initially transforming these equations into a linear algebraic system and then solving it efficiently. This method offers very high accuracy, strong computational efficiency, and simplicity of implementation, making it suitable for a wide range of such complex problems, especially those requiring high speed and precision in the presence of intricate features. Full article

Inner products of spherical tensor operators have been used since the early eighties to define orthogonal operators. However, the basic theory and properties are largely missing in the literature. An inner product in any configuration is directly proportional to the inner product taken in the most basic configuration in which it can occur. The formula for the proportionality factor in question is presented for the first time. This allows the inner products in and between arbitrary configurations to be calculated in advance. In addition, inner products are shown to be independent of the coupling scheme used to construct the state functions. Applications such as the orthogonal operator method and projections of ab initio calculations check for the completeness of the used basis of operators and, importantly, check the matrix elements in any arbitrary configuration, as discussed and illustrated with examples. Closed formulae for the inner products of the well-known Slater and spin–orbit operators are given. Full article

We consider mixed Hamming packings, addressing the maximal cardinality of codes with a minimum codeword Hamming distance. We do not rely on any algebraic structure of the alphabets. We extend known-integer linear programming models of the problem to be efficiently tractable using standard ILP solvers. This is achieved by adopting the concept of contact graphs from classical continuous sphere packing problems to the present discrete context, resulting in a reduction technique for the models which enables their efficient solution as well as their decomposition to smaller subproblems. Based on our calculations, we provide a systematic summary of all lower and upper bounds for packings in the smallest Hamming spaces. The known results are reproduced, with some bounds found to be sharp, and the upper bounds improved in some cases. Full article

Mutually unbiased bases (MUBs) are essential tools in quantum information science, with applications in state tomography, quantum cryptography, and quantum error correction. In this work, we introduce a constructive framework for generating MUBs using linear bipermutive cellular automata (LBCAs). By leveraging the algebraic structure of generalized Pauli operators over finite fields, we show that disjoint families of LBCAs correspond to commuting sets of such operators (CSPOs), which, in turn, generate MUBs. This correspondence enables the systematic construction of complete or incomplete sets of MUBs, depending on the number of disjoint LBCAs available in a given dimension. We also provide algebraic conditions to verify disjointness and discuss how the finite dimensionality constrains MUB completeness. Our approach reinterprets classical combinatorial structures in a quantum setting, offering new computational pathways for exploring MUBs through discrete dynamical systems. Full article

Assume that L is a simple Lie algebra of Cartan-type over an algebraically closed field with a characteristic $p>3$. We demonstrate that all symmetric biderivations vanish by using weight space decompositions relative to a suitable torus and the standard $\mathbb{Z}$-grading structures of L. We then conclude that every biderivation of L is inner, based on a general result concerning skew-symmetric biderivations. As the direct applications, we determine the linear commuting maps and commutative post-Lie algebra structures on L completely. Full article

Analog computing has re-emerged as a powerful tool for solving complex problems in various domains due to its energy efficiency and inherent parallelism. This paper summarizes recent advancements in analog computing, exploring discrete time and continuous time methods for solving combinatorial optimization problems, solving partial differential equations and systems of linear equations, accelerating machine learning (ML) inference, multi-beam beamforming, signal processing, quantum simulation, and statistical inference. We highlight CMOS implementations that leverage switched-capacitor, switched-current, and radio-frequency circuits, as well as non-CMOS implementations that leverage non-volatile memory, wave physics, and stochastic processes. These advancements demonstrate high-speed, energy-efficient computations for computational electromagnetics, finite-difference time-domain (FDTD) solvers, artificial intelligence (AI) inference engines, wireless systems, and related applications. Theoretical foundations, experimental validations, and potential future applications in high-performance computing and signal processing are also discussed. Full article

This paper presents a combined theoretical, numerical, and experimental analysis to investigate the lateral plastic crushing behavior and energy absorption of circular and non-circular thin-walled rings between two rigid plates. Theoretical solutions incorporating both linear material hardening and power-law material hardening models are solved via numerical shooting methods. The theoretically predicted force-denting displacement relations agree excellently with both FEA and experimental results. The FEA simulation clearly reveals the coexistence of an upper moving plastic region and a fixed bottom plastic region. A robust automatic extraction method of the fully plastic region at the bottom from FEA is proposed. A modified criterion considering the unloading effect based on the resultant moment of cross-section is proposed to allow accurate theoretical estimation of the fully plastic region length. The detailed study implies an abrupt and almost linear drop of the fully plastic region length after the maximum value by the proposed modified criterion, while the conventional fully plastic criterion leads to significant over-estimation of the length. Evolution patterns of the upper and lower plastic regions in FEA are clearly illustrated. Furthermore, the distribution of plastic energy dissipation is compared in the bottom and upper regions through FEA and theoretical results. Purely analytical solutions are formulated for linear hardening material case by elliptical integrals. A simple algebraic function solution is derived without necessity of solving differential equations for general power-law hardening material case by adopting a constant curvature assumption. Parametric analyses indicate the significant effect of ovality and hardening on plastic region evolution and crushing force. This paper should enhance the understanding of the crushing behavior of circular and non-circular rings applicable to the structural engineering and impact of the absorption domain. Full article

This paper presents a novel four-dimensional autonomous conservative model characterized by an infinite set of equilibrium points and an unusual algebraic structure in which all eigenvalues of the Jacobian matrix are zero. The linearization of the proposed model implies that classical stability analysis is inadequate, as only the center manifolds are obtained. Consequently, the stability of the system is investigated through both analytical and numerical methods using Lyapunov functions and numerical simulations. The proposed model exhibits rich dynamics, including hyperchaotic behavior, which is characterized using the Lyapunov exponents, bifurcation diagrams, sensitivity analysis, attractor projections, and Poincaré map. Moreover, in this paper, we explore the model with fractional-order derivatives, demonstrating that the fractional dynamics fundamentally change the geometrical structure of the attractors and significantly change the system stability. The Grünwald–Letnikov formulation is used for modeling, while numerical integration is performed using the Caputo operator to capture the memory effects inherent in fractional models. Finally, an analog electronic circuit realization is provided to experimentally validate the theoretical and numerical findings. Full article

This study presents a hyperbolic three-dimensional telegraph interface model with regular interfaces, numerically solved using a hybrid scheme that integrates Haar wavelets and the finite difference method. Spatial derivatives are approximated via a truncated Haar wavelet series, while temporal derivatives are discretized using the finite difference method. For linear problems, the resulting algebraic system is solved using Gauss elimination; for nonlinear problems, Newton’s quasi-linearization technique is applied. The method’s accuracy and stability are evaluated through key performance metrics, including the maximum absolute error, root mean square error, and the computational convergence rate $R_c\left(M\right)$, across various collocation point configurations. The numerical results confirm the proposed method’s efficiency, robustness, and capability to resolve sharp gradients and discontinuities with high precision. Full article

Number sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements of the k-Fibonacci and k-Oresme sequences were analyzed using matrix algebra through matrix structures created by connecting the characteristic equations and roots of these sequences. In this context, using the properties of these matrices, the identities $A_n^2-A_{n+1}{A}_{n-1}=k^{-2n}$, $A_n^2-A_n{A}_{n-1}+{\displaystyle \frac{1}{k^2}}{A}_{n-1}^2=k^{-2n}$, and $B_n^2-B_n{B}_{n-1}+{\displaystyle \frac{1}{k^2}}{B}_{n-1}^2=-(k^2-4)k^{-2n}$, and some generalizations such as $B_{n+m}^2-(k^2-4)A_{n-t}{B}_{n+m}{A}_{t+m}-(k^2-4)k^{2t-2n}{A}_{t+m}^2=k^{-2m-2t}{B}_{n-t}^2$, $A_{t+m}^2-B_{t-n}{A}_{n+m}{A}_{t+m}+k^{2n-2t}{A}_{n+m}^2=k^{-2n-2m}{A}_{t-n}^2$, and more were derived, where $m,n,t\in \mathbb{Z}$ and $t\ne n$. In addition to this, the solution pairs of the algebraic equations $x^2-B_{p}xy+k^{-2p}{y}^2=k^{-2q}{A}_p^2$, $x^2-(k^2-4)A_{p}xy-(k^2-4)k^{-2p}{y}^2=k^{-2q}{B}_p^2$, and $x^2-B_{p}xy+k^{-2p}{y}^2=-(k^2-4)k^{-2q}{A}_p^2$ are presented, where $A_p$ and $B_p$ are k-Oresme and k-Oresme–Lucas numbers, respectively. Full article

Let X be a complex projective variety embedded in a complex projective space. The dimensions of the secant varieties of X have an expected value, and it is important to know if they are equal or at least near to this expected value. Blomenhofer and Casarotti proved important results on the embeddings of G-varieties, G being an algebraic group, embedded in the projectivations of an irreducible G-representation, proving that no proper secant variety is a cone. In this paper, we give other conditions which assure that no proper secant varieties of X are a cone, e.g., that X is G-homogeneous. We consider the Segre product of two varieties with the product action and the case of toric varieties. We present conceptual tests for it, and discuss the information we obtained from certain linear projections of X. For the Segre–Veronese embeddings of ${\mathbb{P}}^n\times {\mathbb{P}}^n$ with respect to forms of bidegree $(1,d)$, our results are related to the simultaneous rank of degree d forms in $n+1$ variables. Full article