Algebraic Conditions Research Articles

Deciding finiteness of bosonic dynamics with tunable interactions

Abstract In this work we are motivated by factorization of bosonic quantum dynamics and we study the corresponding Lie algebras, which can potentially be infinite dimensional. To characterize such factorization, we identify conditions for these Lie algebras to be finite dimensional. We consider cases where each free Hamiltonian term is itself an element of the generated Lie algebra. In our approach, we develop new tools to systematically divide skew-hermitian bosonic operators into appropriate subspaces, and construct specific sequences of skew-hermitian operators that are used to gauge the dimensionality of the Lie algebras themselves. The significance of our result relies on conditions that constrain only the independently controlled generators in a particular Hamiltonian, thereby providing an effective algorithm for verifying the finiteness of the generated Lie algebra. In addition, our results are tightly connected to mathematical work where the polynomials of creation and annihilation operators are known as the Weyl algebra. Our work paves the way for better understanding factorization of bosonic dynamics relevant to quantum control and quantum technology.

Laumon parahoric local models via quiver Grassmannians

Local models of Shimura varieties in type A can be realized inside products of Grassmannians via certain linear algebraic conditions. Laumon suggested a generalization which can be identified with a family over a line whose general fibers are quiver Grassmannians for the loop quiver and the special fiber is a certain quiver Grassmannian for the cyclic quiver. The whole family sits inside the Gaitsgory central degeneration of the affine Grassmannians. We study the properties of the special fibers of the (complex) Laumon local models for arbitrary parahoric subgroups in type A using the machinery of quiver representations. We describe the irreducible components and the natural strata with respect to the group action for the quiver Grassmannians in question. We also construct a cellular decomposition and provide an explicit description for the corresponding poset of cells. Finally, we study the properties of the desingularizations of the irreducible components and show that the desingularization construction is compatible with the natural projections between the parahoric subgroups.

Impulse Controllability for Singular Hybrid Coupled Systems

This study examines the concept of impulse controllability within singular hybrid coupled systems through the utilisation of decentralised proportional plus derivative (P-D) output feedback. By employing the Differential Mean Value Theorem, the nonlinear model can be converted into a linear parameter-varying large-scale system. Our analysis leads to the establishment of algebraic conditions that are both necessary and sufficient for the existence of a decentralised P-D output feedback controller that can guarantee impulse controllability in these complex systems. Moreover, we address the issue of admissibility within these systems by employing matrix trace inequalities. We present a novel sufficient condition for impulse controllability, which offers a new perspective on addressing this challenging problem. To validate our findings, we present numerical examples that demonstrate the effectiveness of the proposed methodologies in practice.

Polyvector deformations of Type IIB backgrounds

We develop a formalism of poly-vector deformations for Type IIB backgrounds with a block diagonal metric and non-vanishing self-dual 5-form RR field strength. Making use of the embedding of the Type IIB theory into the E6(6)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{E}_{6(6)}$$\\end{document} exceptional field theory we derive explicit transformation rules for four-vector deformations. We prove that the algebraic condition following from the Type IIB realization of exceptional Drinfeld algebras is sufficient for the transformation to generate a solution.

Min–max group consensus of discrete-time multi-agent systems under directed random networks

This paper studies the min–max group consensus of discrete-time multi-agent systems under a directed random graph, where the presence of each directed edge is randomly determined by a probability and independent of the presence of other edges. Firstly, we propose a min–max consensus protocol without memory, and give the necessary and sufficient conditions to ensure that the multi-agent system can achieve the min–max group consensus in the sense of almost sure and mean square, respectively. Secondly, we design a novel consensus protocol with memory and a behavior mechanism. Using the stochastic analysis theory and the extremal algebra, some necessary and sufficient conditions are obtained for achieving the min–max group consensus in the sense of almost sure and mean square, respectively. It is shown that the protocol with memory can solve the loss problem of the maximum and minimum initial states. Finally, the effectiveness of the two group consensus protocols and the behavior mechanism is verified by four numerical simulations.

Non-affine n-valued maps on tori

In this paper we construct n-valued maps on k-dimensional tori, where n,k≥2, that are not homotopic to affine n-valued maps. This is in high contrast with the single valued case, where any such map is homotopic to an affine (even linear) map. We do this by investigating necessary and sufficient algebraic conditions on certain induced morphisms.

Static analysis of violin bow behavior under playing loads.

After discussing the relationship between taper and camber typically used by bowmakers, energy methods are then employed to develop nonlinear boundary value problems describing both the hair tightening problem and static deflection analysis of the violin bow under playing loads, including an analysis of possible hair tension asymmetry by employing a linear springs in series model of the hair. A simple algebraic condition for the distance the frog travels in reaching maximum tension is also presented. Boundary value problems are then applied using tools in scilab, comparing the idealized Tourte taper exhibiting a linear stiffness profile and an alternative design with a significantly different profile, revealing small differences in their behavior under static playing loads and validating previously observed significant increase in hair tension when approaching the tip. While the difference in vertical hair deflection as a function of bow position is extremely small, the Tourte design exhibits somewhat more compliance in the stick when approaching the tip, possibly desirable by players given increasing hair tension. An interesting small loss of hair tension in the lower section of the bow stroke is also discussed, representing additional slight differences in the initial compliance felt by the player when engaging the strings.

Algebraic conditions for stability in Runge-Kutta methods and their certification via semidefinite programming

In this work, we present approaches to rigorously certify A- and A(α)-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adopt two approaches. The first is based on sum-of-squares programming applied to the Runge-Kutta E-polynomial and is applicable to both A- and A(α)-stability. In the second, we sharpen the algebraic conditions for A-stability of Cooper, Scherer, Türke, and Wendler to incorporate the Runge-Kutta order conditions. We demonstrate how the theoretical improvement enables the practical use of these conditions for certification of A-stability within a computational framework. We then use both approaches to obtain rigorous certificates of stability for several diagonally implicit schemes devised in the literature.

On algebraic conditions for the non-vanishing of linear forms in Jacobi theta-constants

On algebraic conditions for the non-vanishing of linear forms in Jacobi theta-constants

From area metric backgrounds to the cosmological constant and corrections to the Polyakov action

Area metrics and area metric backgrounds provide a unified framework for quantum gravity. They encode physical degrees of freedom beyond those of a metric. These nonmetric degrees of freedom must be suppressed by a potential at sufficiently high energy scales to ensure that in the infrared regime classical gravity is recovered. On this basis, we first study necessary and sufficient algebraic conditions for an area metric to be induced by a metric. Second, we consider candidate potentials for the area metric and point out a possible connection between the reduction of area metric geometry to metric geometry on the one hand, and the smallness of the cosmological constant on the other. Finally, we consider modifications of the Nambu-Goto action for a string from a metric background to an area metric background. We demonstrate that area metric perturbations introduce an interaction corresponding to a singular vertex operator in the classically equivalent Polyakov action. The implications of these types of vertex operators for the quantum theory remain to be understood. Published by the American Physical Society 2024

Performance of GRKM-method for solving classes of ordinary and partial differential equations of sixth-orders

Abstract A general quasilinear sixth-order ordinary differential equation (ODE) is an important class of ODEs. The primary objective of this study is to establish a numerical method for solving a general class of quasilinear sixth-order partial differential equations (PDEs) and ODEs. However, the Runge–Kutta method (RKM) approach for solving special classes of ODEs has been generalized as an effort to solve the general class of ODEs. Nonlinear algebraic order condition (OCs) equations have been obtained up to the tenth order using the Taylor-series expansion methodology which is used to derive the novel generalized Runge–Kutta method (GRKM). In this study, a GRKM integrator has been derived for solving a general class of quasilinear sixth-order ODEs and then this method is modified subsequently to solve a class of PDEs. Accordingly, the proposed GRKM is modified to solve a quasilinear sixth-order PDE by converting it to a system of sixth-ODEs using the method of lines. Nine problems have been implemented to prove the efficiency and accuracy of the proposed method. Simulation results of these problems showed that the proposed numerical GRKM is an accurate and efficient method. In contrast, by comparing the proposed GRKM numerical approach with the classical RK method, the numerical results demonstrate that the direct integrator outperforms the indirect classical RK method in terms of algorithm complexity and function evaluations, proving that the numerical GRKM is efficient.

Synthesis of robust memory modes for linear quantum systems with unknown inputs

In this paper, the synthesis of robust memory modes for linear quantum passive systems in the presence of unknown inputs has been studied, aimed at facilitating secure storage and communication of quantum information. In particular, we can switch on decoherence-free (DF) modes in the storage stage by placing the poles on the imaginary axis via a coherent feedback control scheme, and these memory modes can further be simultaneously made robust against perturbations to the system parameters by minimizing the condition number associated with imaginary poles. The DF modes can also be switched off by tuning the controller parameters to place the poles in the left half of the complex plane in the writing/reading stage. We develop explicit algebraic conditions guiding the design of such a coherent quantum controller, which involves employing an augmented system model to counter the influence of unknown inputs. Examples are provided to illustrate the procedure of synthesizing robust memory modes for linear optical quantum systems.

Dupin Cyclides Passing through a Fixed Circle

Dupin cyclides are classical algebraic surfaces of low degree. Recently, they have gained popularity in computer-aided geometric design (CAGD) and architecture owing to the fact that they contain many circles. We derive algebraic conditions that fully characterize the Dupin cyclides passing through a fixed circle. The results are applied to the basic problem in CAGD of the blending of Dupin cyclides along circles.

Construction of Numerical RKM-Method for Solving A Class of Twelves-Order Ordinary Differential Equations

This paper constructs and generalizes the numerical Runge-Kutta-Mohammed (RKM) method for solving twelve-order ordinary differential equations (ODEs). The novel contribution of this study is the development and generalization of the numerical RKM methods for solving ODEs of the order of less than a tenth. The algebraic order conditions (OCs) for the proposed RKM method are derived up to order thirteen using Taylor expansion. Then, the constructed method has been derived from these order conditions. However, the proposed numerical RKM method has been evaluated at some implementations and compared to an existing Runge-Kutta (RK) method to determine the method's viability. Moreover, this comparison demonstrates the proposed direct method is more efficient than the classical method in terms of efficiency and accuracy. Also, numerical implementations are used to prove the efficiency and time complexity of function evaluations. This direct RKM method is a suggested technique for solving ODEs of twelve orders which has great features like a direct and efficient method. Consequently, the proposed method requires less time complexity of computation than other methods.

Double extensions of left-symmetric structures

The notion of double extension for symplectic Lie algebras was introduced by Medina and Revoy in 1985. In 2021, Valencia gave a condition for symplectic Lie algebras which are obtained through double extension to admit compatible left-symmetric structures. In this paper, we formulate the notion of double extension for left-symmetric structures and give a condition for left-symmetric structures to be obtained through double extension. Moreover, we prove that right nilpotent left-symmetric structures on some solvable Lie algebras are obtained through double extension. We also give a condition for left-symmetric structures which are obtained through double extension to be complete.

New methods for quasi-interpolation approximations: Resolution of odd-degree singularities

In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange functions when the orders of the singularities of the radial function’s Fourier transforms at zero do not match the parity of the dimension of the space, and therefore new expansions and coefficients are needed to overcome this problem. We develop explicit constructions of infinite Fourier expansions that provide these coefficients and make an extensive comparison of the approximation qualities and – with a particular focus – polynomial reproduction and uniform approximation order of the various formulae. One of the interesting observations concerns the link between algebraic conditions of expansion coefficients and analytic properties of localness and convergence.

Behaviors of matrix-weighted networks with antagonistic interactions

This essay considers the bipartite consensus of matrix-weighted second-order multi-agent systems with structurally unbalanced antagonistic interactions networks. Owing to the influence of matrix-weights, bipartite consensus and cluster bipartite consensus can also be reached, which is different from the existing works on signed networks. With the assistance of Lyapunov stability and matrix-valued Gauge transformation, a condition to achieve bipartite consensus is established. Moreover, a relatively straightforward algebraic graph condition for reaching cluster bipartite consensus is attained. Finally, simulation examples are supplied to illustrate the obtained results.

Vivencialidad y Formación Virtual para mejorar la Competencia Matemática

This study proposes, develops and applies the didactic model based on experience and virtual training to improve mathematical competence: solves problems of regularity, equivalence and change in the virtual modality, in the San Juan Alternative Basic Education Center of the city of Trujillo in the year 2020. The study has a mixed approach which type of research was quasi experimental, we chose a sample of 28 second grade students whose average age was 16 years, in the experimental group the proposal of the didactic model was applied, for which a test, a rubric, a research field notebook and evidence registration form were used as measurement instruments, getting as a result a significant improvement in the indicated competence, with respect to the dimensions of data translation and algebraic conditions, mathematical communication, math application and math problem solving.

Dynamics of a Conformable Fractional Order Generalized Richards Growth Model on Star Network with N=20 Nodes

In this study, we analyze dynamical behavior of the conformable fractional order Richards growth model. Before examining the analysis of the dynamical behavior of the fractional continuous time model, the model is reduced to the system of difference equations via utilizing piecewise constant functions. An algebraic condition that ensures the stability of the positive fixed point of the system is obtained. With the center manifold theory, the existence of a Neimark-Sacker bifurcation at the fixed point of the discrete-time system is proven and the direction of this bifurcation is determined. In addition, the discrete dynamical system is also studied on the star network with N=20 nodes. Analysis complex dynamics of Richards growth model into coupled dynamical network shows that the complex star network with N=20 nodes also exhibits Neimark-Sacker bifurcation about the fixed point concerning with parameter c. Numerical simulations are performed to demonstrate the stability, bifurcations and dynamic transition of the coupled network.

On different approaches to integrable lattice models

Interaction-round the face (IRF) models are two-dimensional lattice models of statistical mechanics defined by an affine Lie algebra and admissibility conditions depending on a choice of representation of that affine Lie algebra. Integrable IRF models, i.e. the models the Boltzmann weights of which satisfy the quantum Yang–Baxter equation, are of particular interest. In this paper, we investigate trigonometric Boltzmann weights of integrable IRF models. By using an ansatz proposed by one of the authors in some previous works, the Boltzmann weights of the restricted IRF models based on the affine Lie algebras su(2)k and su(3)k are computed for fundamental and adjoint representations for some fixed levels k. New solutions for the Boltzmann weights are obtained. We also study the vertex-IRF correspondence in the context of an unrestricted IRF model based on su(3)k (for general k) and discuss how it can be used to find Boltzmann weights in terms of the quantum R^ matrix when the adjoint representation defines the admissibility conditions.