Terms Of Algebra Research Articles

Same same but Different - Definition and Explicit Notation of an Orientation in MTEX vs. Quasi-Standard

Even though the descriptive definition of orientation is the same in both settings, the explicitnotation of a crystallographic orientation as (3 3) matrix in terms of Euler angles featuredby the popular MATLAB toolbox MTEX differs by an inversion from the quasi-standard notation datedback to the early days of quantitative texture analysis championed by H.-J. Bunge. The origin of thisdiscrepancy is revealed by an enlightening view provided in algebraic terms of a change of basis.Understanding the effect of inversion is instrumental to do proper computations with crystallographicorientations and rotations, e.g. when multiplying with elements of a crystallographic symmetry group,and to compare results of texture analyses accomplished in different settings.

Dynamic event-triggered neuroadaptive fault-tolerant control of quadrotor UAV with a novel cosine kernel

The fault-tolerant control (FTC) for trajectory tracking of a quadrotor unmanned aerial vehicle (UAV) has attracted researchers. The non-linear model of the UAV, coupled with model uncertainties, external disturbances, and actuator failures, requires the function approximation of the lumped non-linearity for controller design. One of the most efficient ways to approximate non-linearity is using radial basis function neural networks (RBFNNs). To date, RBFNNs have been formulated, trained, and used directly for function approximation, which requires considerable computation to derive the control laws. The proposed control and parameter estimation laws approximate the non-linearity by using RBFNNs indirectly. The proposed laws with virtual parameter estimation do not require the actual formulation of RBFNNs and their weights, thus saving on computational resources. For kernel optimization, the Gaussian kernels with exponential terms in RBFNNs are replaced by cosine kernels with algebraic terms, which shows faster convergence as per simulation results. To save on communication bandwidth, static event-triggering communication mechanisms (SECM) and dynamic event-triggering mechanisms (DECM) have been proposed. As DECM works on dynamically changing variables, it saves more communication bandwidth, as tested in simulation. Lyapunov stability analysis proves that errors are uniformly ultimately bounded (UUB). The performance of the proposed algorithm has been tested through numerical simulations, which show superior performance when compared with similar studies. The proposed algorithm has been validated in a real-time Gazebo simulator.

Hilbert’s tenth problem for term algebras with a substitution operator

Hilbert’s tenth problem for term algebras with a substitution operator

Loading conditions for self‐organization in the BML model with stochastic direction choice

A dynamical system is considered such that, in this system, particles move on a toroidal lattice of the dimension according to a version of the rule of particle movement in Biham–Middleton–Levine traffic model. We introduce a stochastic case with direction choice for particles. Particles of the first type move along rows, and the particles of the second type move along columns. The goal is to find conditions of self‐organization system for any lattice dimension. We have proved that the BML model as a dynamical system is a special case of Buslaev nets. This equivalence allows us to use of Buslaev net analysis techniques to investigate the BML model. In Buslaev nets conception, the self‐organization property of the system corresponds to the existence of velocity single point spectrum equal to 1. In the paper, we consider the model version when one notable aspect is that a particle may change its type. Exactly, we assume a constant probability that a particle changes type at each step. In the case where , the system corresponds to the classical version of the BML model. We define a state of the system where all particles continue to move indefinitely, in both the present and the future, as a state of free movement. A sufficient condition for the system to result in a state of free movement from any initial state (condition for self‐organization) has been found. This condition is that the number of particles be not greater than half the greatest common divisor of the numbers . It has been proved that, if , and whether or and there are both at least one particle of the first type and at least particle of the second type, then a necessary condition for a state of free movement to exist is the greatest common divisor of and be not less than 3. The theorems are formulated in terms of the algebraic structures and terms of the dynamical system. The spectrum of particle velocities has been found for the net . This approach allows us to hope that the spectrum of dynamical system can be studied for an arbitrary dimension of the net.

A Semi-implicit Finite Volume Scheme for Incompressible Two-Phase Flows

This paper presents a mass and momentum conservative semi-implicit finite volume (FV) scheme for complex non-hydrostatic free surface flows, interacting with moving solid obstacles. A simplified incompressible Baer-Nunziato type model is considered for two-phase flows containing a liquid phase, a solid phase, and the surrounding void. According to the so-called diffuse interface approach, the different phases and consequently the void are described by means of a scalar volume fraction function for each phase. In our numerical scheme, the dynamics of the liquid phase and the motion of the solid are decoupled. The solid is assumed to be a moving rigid body, whose motion is prescribed. Only after the advection of the solid volume fraction, the dynamics of the liquid phase is considered. As usual in semi-implicit schemes, we employ staggered Cartesian control volumes and treat the nonlinear convective terms explicitly, while the pressure terms are treated implicitly. The non-conservative products arising in the transport equation for the solid volume fraction are treated by a path-conservative approach. The resulting semi-implicit FV discretization of the mass and momentum equations leads to a mildly nonlinear system for the pressure which can be efficiently solved with a nested Newton-type technique. The time step size is only limited by the velocities of the two phases contained in the domain, and not by the gravity wave speed nor by the stiff algebraic relaxation source term, which requires an implicit discretization. The resulting semi-implicit algorithm is first validated on a set of classical incompressible Navier-Stokes test problems and later also adds a fixed and moving solid phase.

An Improved Blended Numerical Root-Solver for Nonlinear Equations

This study presents a novel three-step iterative approach for solving nonlinear equations inthe domains of science and engineering. It represents a notable change from traditional methodslike Halley by eliminating the need for second derivatives. The suggested method exhibits asixth order of convergence and only requires five function evaluations, showcasing its efficiencywith an index of roughly 1.430969. The suggested method effectively solves nonlinear problemsinvolving equations with algebraic and transcendental terms. Comparative analysis againstexisting root-solving algorithms demonstrates their superior performance. The results not onlyconfirm the strength and effectiveness of the three-step iterative approach but also highlight itspotential for wide-ranging use in many scientific and technical situations.

A discrete basis for celestial holography

Celestial holography provides a reformulation of scattering amplitudes in four dimensional asymptotically flat spacetimes in terms of conformal correlators of operators on the two dimensional celestial sphere in a basis of boost eigenstates. A basis of massless particle states has been previously identified in terms of conformal primary wavefunctions labeled by a boost weight ∆ = 1+iλ with λ ∈ ℝ. Here we show that a discrete orthogonal and complete basis exists for ∆ ∈ ℤ. This new basis consists of a tower of discrete memory and Goldstone observables, which are conjugate to each other and allow to reconstruct gravitational signals belonging to the Schwartz space. We show how generalized dressed states involving the whole tower of Goldstone operators can be constructed and evaluate the higher spin Goldstone 2-point functions. Finally, we recast the tower of higher spin charges providing a representation of the w1+∞ loop algebra (in the same helicity sector) in terms of the new discrete basis.

Tropical Modeling of Battery Swapping and Charging Station

We propose and investigate a queueing model of a battery swapping and charging station (BSCS) for electric vehicles (EVs). A new approach to the analysis of the queueing model is developed, which combines the representation of the model as a stochastic dynamic system with the use of the methods and results of tropical algebra, which deals with the theory and applications of algebraic systems with idempotent operations. We describe the dynamics of the queueing model by a system of recurrence equations that involve random variables (RVs) to represent the interarrival time of incoming EVs. A performance measure for the model is defined as the mean operation cycle time of the station. Furthermore, the system of equations is represented in terms of the tropical algebra in vector form as an implicit linear state dynamic equation. The performance measure takes on the meaning of the mean growth rate of the state vector (the Lyapunov exponent) of the dynamic system. By applying a solution technique of vector equations in tropical algebra, the implicit equation is transformed into an explicit one with a state transition matrix with random entries. The evaluation of the Lyapunov exponent reduces to finding the limit of the expected value of norms of tropical matrix products. This limit is then obtained using results from the tropical spectral theory of deterministic and random matrices. With this approach, we derive a new exact formula for the mean cycle time of the BSCS, which is given in terms of the expected value of the RVs involved. We present the results of the Monte Carlo simulation of the BSCS’s operation, which show a good agreement with the exact solution. The application of the obtained solution to evaluate the performance of one BSCS and to find the optimal distribution of battery packs between stations in a network of BSCSs is discussed. The solution may be of interest in the case when the details of the underlying probability distributions are difficult to determine and, thus, serves to complement and supplement other modeling techniques with the need to fix a distribution.

Local scour depth at piles group exposed to regular waves: On the assessment of expressions based on classification concepts and evolutionary algorithms

An accurate estimation of local scour depth around piles group is inevitably essential to provide stability of marine structures. Over the past decades, many investigations have been made to understand the scouring process at piles group exposed to waves for field and experimental scales. This study aims to predict the wave-induced local scour depth by various robust Data Driven Models (DDMs) and Machine Learning Models (MLMs) developed by classification and evolutionary concepts: Model Tree (MT), Evolutionary Polynomial Regression (EPR), Multivariate Adaptive Regression Spline (MARS), and Gene-Expression Programming (GEP). From relevant literature, 125 data series were employed to provide empirical relationships for scour depth predictions. The raw variables were related to bed sediment, pile configuration, pile geometry, approaching flow, and wave characteristics. Non-dimensional parameters have been obtained through the Buckingham theorem in order to control local scour depth. In this way, the ratio of spacing between piles to pile diameter (G/D), sediment number (Ns), pile arrangement number (ratio of the number of piles parallel to the flow; m; to the number of piles normal to the flow; n), Shields parameter (θ), and Keulegan-Carpenter (KC) were considered. From the training and testing stages of Artificial Intelligence Models (AIMs), it was found that regression equation given by MARS model provided better performance (e.g., Correlation Coefficient [R] = 0.9297, Root Mean Square Error [RMSE] = 0.3489, and Scatter Index [SI] = 0.2765) than other AI models’ efficiency. Additionally, the performance of AI models was assessed for various ranges of dimensionless parameters (i.e., G/D, Ns, and θ) versus KC variation. MARS model had the best performance for G/D = 0–1 and KC < 10 (R = 0.9917 and RMSE = 0.2198) and 0.4≤θ < 0.5 and 15 < KC < 25 whereas EPR model had promising efficiency in the its highest level for Ns = 1–3 and 10 < KC ≤ 15 (R = 0.9941 and RMSE = 0.0771) than other AI models. For the sensitivity analysis, the mutation rate and number of chromosomes are examined for the GEP model, the K-fold number and number of parameters for the MARS model, and the number of algebraic terms and number of chromosomes for the EPR model. Furthermore, ranking analysis of AI models indicated that MARS model had the best performance (Ranking Performance Index [RPI] = 0.7348) and followed by GEP (0.3412), M5MT (0.2999), and EPR (0.2848).

Products, Polynomials and Differential Equations in the Stream Calculus

We study connections among polynomials, differential equations, and streams over a field 𝕂, in terms of algebra and coalgebra. We first introduce the class of (F,G) - products on streams, those where the stream derivative of a product can be expressed as a polynomial function of the streams and their derivatives. Our first result is that, for every (F,G) -product, there is a canonical way to construct a transition function on polynomials such that the resulting unique final coalgebra morphism from polynomials into streams is the (unique) commutative 𝕂-algebra homomorphism—and vice versa. This implies that one can algebraically reason on streams via their polynomial representation. We apply this result to obtain an algebraic-geometric decision algorithm for polynomial stream equivalence, for an underlying generic (F,G) -product. Finally, we extend this algorithm to solve a more general problem: finding all valid polynomial equalities that fit in a user specified polynomial template.

Compositional quantum field theory: An axiomatic presentation

We introduce Compositional Quantum Field Theory (CQFT) as an axiomatic model of quantum field theory, based on the principles of locality and compositionality. Our model is a refinement of the axioms of general boundary quantum field theory, and is phrased in terms of correspondences between certain commuting diagrams of gluing identifications between manifolds and corresponding commuting diagrams of state-spaces and linear maps, thus making it amenable to formalization in terms of involutive symmetric monoidal functors and operad algebras. The underlying novel framework for gluing leads to equivariance of CQFT. We study CQFTs in dimension 2 and the algebraic structure they define on open and closed strings. We also consider, as a particular case, the compositional structure of 2-dimensional pure quantum Yang–Mills theory.

Algebras of full terms constructed from transformations with fixed set

Algebras of full terms constructed from transformations with fixed set

Weighted composition operators on harmonic Fock spaces

ABSTRACT In this paper, we study weighted composition operators on Fock spaces of harmonic functions (briefly as harmonic Fock spaces). The basic properties of such operators (boundedness, compactness, invertibility and symmetries) are characterized completely in simple algebraic terms, involving their symbols.

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

Slack Hopf monads

Hopf monads generalise Hopf algebras. They clarify several aspects of the theory of Hopf algebras and capture several related structures such as weak Hopf algebras and Hopf algebroids. However, important parts of Hopf algebra theory are not reached by Hopf monads, most noticeably Drinfeld's quasi-Hopf algebras. In this paper we introduce a generalisation of Hopf monads, that we call slack Hopf monads. This generalisation retains a clean theory and is flexible enough to encompass quasi-Hopf algebras as examples. A slack Hopf monad is a colax magma monad T on a magma category C such that the forgetful functor UT:CT→C ‘slackly’ preserves internal Homs. We give a number of different descriptions of slack Hopf monads, and study special cases such as slack Hopf monads on cartesian categories and k-linear exact slack Hopf monads on Vectk, that is comagma algebras such that a modified fusion operator is invertible. In particular, we characterise quasi-Hopf algebras in terms of slackness.

On the non existence of sympathetic Lie algebras with dimension less than 25

In this paper, we investigate the question of the lowest possible dimension that a sympathetic Lie algebra [Formula: see text] can attain, when its Levi subalgebra [Formula: see text] is simple. We establish the structure of the nilradical of a perfect Lie algebra [Formula: see text], as a [Formula: see text]-module, and determine the possible Lie algebra structures that one such [Formula: see text] admits. We prove that, as a [Formula: see text]-module, the nilradical must decompose into at least four simple modules. We explicitly calculate the semisimple derivations of a perfect Lie algebra [Formula: see text] with Levi subalgebra [Formula: see text] and give necessary conditions for [Formula: see text] to be a sympathetic Lie algebra in terms of these semisimple derivations. We show that there is no sympathetic Lie algebra of dimension lower than 15, independently of the nilradical’s decomposition. If the nilradical has four simple modules, we show that a sympathetic Lie algebra has dimension greater or equal than 25.

High-Order ADER Discontinuous Galerkin Schemes for a Symmetric Hyperbolic Model of Compressible Barotropic Two-Fluid Flows

This paper presents a high-order discontinuous Galerkin (DG) finite-element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) model of compressible two-phase flow, introduced by Romenski et al. in [59, 62], in multiple space dimensions. In the absence of algebraic source terms, the model is endowed with a curl constraint on the relative velocity field. In this paper, the hyperbolicity of the system is studied for the first time in the multidimensional case, showing that the original model is only weakly hyperbolic in multiple space dimensions. To restore the strong hyperbolicity, two different methodologies are used: (i) the explicit symmetrization of the system, which can be achieved by adding terms that contain linear combinations of the curl involution, similar to the Godunov-Powell terms in the MHD equations; (ii) the use of the hyperbolic generalized Lagrangian multiplier (GLM) curl-cleaning approach forwarded. The PDE system is solved using a high-order ADER-DG method with a posteriori subcell finite-volume limiter to deal with shock waves and the steep gradients in the volume fraction commonly appearing in the solutions of this type of model. To illustrate the performance of the method, several different test cases and benchmark problems have been run, showing the high order of the scheme and the good agreement when compared to reference solutions computed with other well-known methods.

Automorphisms of the limits for the direct sequences of the Toeplitz-Cuntz algebras

The paper deals with the direct sequences of the Toeplitz-Cuntz algebras whose connecting homomorphisms are defined by tuples consisting of sequences of prime numbers. We study properties of limit ⁎-endomorphisms of the C⁎-algebras that are the direct limits for such direct sequences. The limit ⁎-endomorphisms are induced by morphisms between copies of the same direct sequences of the Toeplitz-Cuntz algebras. We establish the criteria for the limit ⁎-endomorphisms to be automorphisms of the direct limits. These criteria are formulated in number-theoretic, algebraic and functional terms. In particular, it is shown that a limit ⁎-endomorphism is an automorphism if and only if there exist generalized means on the P-adic solenoids.

Gauge-Invariant Lagrangian Formulations for Mixed-Symmetry Higher-Spin Bosonic Fields in AdS Spaces

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux Y(s1,…,sk) with k≥2 rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a k=2 Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic sp(2k) algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the su(1,1) algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for Y(s1,s2) is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin (s1,s2) is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin (1,1) is obtained.

On compatible Leibniz algebras

In this paper, we study compatible Leibniz algebras. We explore the classification of complex Leibniz algebras in dimensions 2 and 3 to provide compatible pairs. We characterize compatible Leibniz algebras in terms of Maurer–Cartan elements of a suitable differential graded Lie algebra. Moreover, we define a cohomology theory of compatible Leibniz algebras, which in particular controls one-parameter formal deformations of this algebraic structure. Motivated by a classical application of cohomology, we moreover study abelian extensions of compatible Leibniz algebras.