Invariant Functions Research Articles

Entwining Yang–Baxter maps over Grassmann algebras

In this work we construct novel solutions to the set-theoretical entwining Yang–Baxter equation. These solutions are birational maps involving non-commutative dynamical variables which are elements of the Grassmann algebra of order n. The maps arise from refactorisation problems of Lax supermatrices associated to a nonlinear Schrödinger equation. In this non-commutative setting, we construct a spectral curve associated to each of the obtained maps using the characteristic function of its monodromy supermatrix. We find generating functions of invariants for the entwining Yang–Baxter maps from the moduli of the spectral curves. Moreover, we show that a hierarchy of birational entwining Yang–Baxter maps with commutative variables can be obtained by fixing the order n of the Grassmann algebra, and we present the cases n=1 (dual numbers) and n=2. Then we discuss the integrability properties, such as Lax matrices, invariants, and measure preservation, for the obtained discrete dynamical systems.

An Adaptive Spatial-Terminal Iterative Learning Strategy in Roll-to-Roll Control Problems

Abstract Roll-to-Roll (R2R) systems, featuring motorized or idle rollers, are crucial for high-volume, continuous production of flexible substrates. A significant challenge in R2R printing processes is maintaining tight alignment tolerances for multi-layer printed electronics. This alignment, known as registration, is complicated by the deformability of flexible substrates and complex roller dynamics, leading to registration errors (RE) caused by variations in substrate tensions and speeds. Despite using real-time feedback controllers like PID or MPC for tension control, these systems struggle with transient and angle-periodic disturbances common in R2R systems. We introduce a Spatial-Terminal Iterative Learning Control (STILC) method with an adaptive basis function to eliminate RE in R2R gravure printing. This function, along with the registration error, updates the STILC compensation profile via a P-type Iterative Learning Control law. Our numerical experiments demonstrate that STILC with the adaptive basis function effectively eliminates RE caused by roller-motor axis mismatches and provides better convergence than STILC with an invariant basis function. This novel approach shows promise for various industrial applications involving spatially periodic disturbances, particularly those with unknown dynamics and without given state trajectories to track rigorously, which is common in engineering practice.

Hopf link invariants and integrable hierarchies

The goal of this note is to study integrable properties of a generating function of the HOMFLY-PT invariants of the Hopf link colored with different representations. We demonstrate that such a generating function is a τ-function of the KP hierarchy. Furthermore, this Hopf generating function in the case of composite representations, which is a generating function of the 4-point functions in topological string (corresponding to the resolved conifold with branes on the four external legs), is a τ-function of the universal character(UC) hierarchy put on the topological locus. We also briefly discuss a simple matrix model associated with the UC hierarchy.

Consequences of Invariant Functions for the Riemann Hypothesis

Consequences of Invariant Functions for the Riemann Hypothesis

Wignerian symplectic covariance approach to the interaction-time problem

The concept of the symplectic covariance property of the Wigner distribution function and the symplectic invariance of the Wigner–Rényi entropies has been leveraged to estimate the interaction time of the moving quantum state in the presence of an absolutely integrable time-dependent potential. For this study, the considered scattering centre is represented initially by the Gaussian barrier. Two modifications of this potential energy are considered: a sudden change from barrier to barrier and from barrier to well. The scattering state is prepared in the form of a Schrödinger cat, and the Moyal equation governs its further time evolution. The whole analysis of the considered scattering problem is conducted in the above-barrier regime using the phase-space representation of quantum theory. The presented concept shifts the focus from the dynamics of the quantum state to a symplectically invariant state functions in the form of the Wigner–Rényi entropies of order one and one-half. These quantities serve as indicators of the beginning and end of the interaction of the non-Gaussian state with the sufficiently fast decaying potential representing the scattering centre. The presented approach is significant because it provides a new way to estimate the interaction time of moving quantum states with the dynamical scattering centre. Moreover, it allows for studying scattering processes in various physical systems, including atoms, molecules, and condensed matter systems.

Correlative Dynamics of Complex Systems: A Multifractal Perspective of Motion Based on SL(2R) Symmetry

By assimilating any complex system into a multifractal, a new approach for describing the dynamics of such systems is proposed by means of the Multifractal Theory of Motion. In such context, the description of these dynamics is accomplished through continuous and non-differentiable curves (multifractal curves), giving rise to two scenarios. The first scenario is a Schrödinger-type multifractal scenario, a situation in which the motion laws can be related to the SL(2R) algebra invariant functions. The second scenario is a Madelung-type multifractal scenario, a situation in which if the differentiable and non-differentiable components of the velocity field satisfy a particular restriction, an SL(2R) symmetry can also be highlighted. Moreover, correlative dynamics in either of the two scenarios, based on the same SL(2R) symmetry, can be obtained by Riccati-type gauges, which imply Stoler coherent states. Several cases induced by the SL(2R) symmetry are also analyzed.

Symplectic Cuts and Open/Closed Strings I

This paper introduces a concrete relation between genus zero closed Gromov–Witten invariants of Calabi–Yau threefolds and genus zero open Gromov–Witten invariants of a Lagrangian A-brane in the same threefold. Symplectic cutting is a natural operation that decomposes a symplectic manifold (X,ω) with a Hamiltonian U(1) action into two pieces glued along an invariant divisor. In this paper we study a quantum uplift of the cut construction defined in terms of equivariant gauged linear sigma models. The nexus between closed and open Gromov–Witten invariants is a quantum Lebesgue measure associated to a choice of cut, that we introduce and study. Integration of this measure recovers the equivariant quantum volume of the whole CY3, thereby encoding closed Gromov–Witten invariants. Conversely, the monodromies of the quantum measure around cycles in Kähler moduli space encode open Gromov–Witten invariants of a Lagrangian A-brane associated to the cut. Both in the closed and the open string sector we find a remarkable interplay between worldsheet instantons and semiclassical volumes regularized by equivariance. This leads to equivariant generating functions of GW invariants that extend smoothly across the entire moduli space, and which provide a unifying description of standard GW potentials. The latter are recovered in the non-equivariant limit in each of the different phases of the geometry.

Observer Design for State and Parameter Estimation for Two-Time-Scale Nonlinear Systems

The design and calculation of nonlinear observers for parameter estimation in multi-time-scale nonlinear systems present significant challenges due to the inherent complexity and stiffness of such systems. This study proposes a framework for designing observers for two-time-scale nonlinear systems, with the objective of overcoming the aforementioned challenges. The design procedure involves reducing the original two-time-scale nonlinear system to a lower-dimensional model that captures only the slow dynamics while approximating the fast states through the use of an algebraic slow motion invariant manifold function. Subsequently, an exponential observer can be devised for this reduced system, which is valid for both state and parameter estimation. By employing the output from the original system, this observer can be adapted for online state and parameter estimation for the detailed two-time-scale system. The challenges associated with estimating parameters in two-time-scale nonlinear systems, the complexities of designing observers for such systems, and the computational burden associated with computing observers for ill-conditioned systems can be effectively addressed through the application of this design framework. A rigorous error analysis validates the convergence of the proposed observer towards the states and parameters of the original system. The viability and necessity of this observer design framework are demonstrated through a numerical example and an anaerobic digestion process. This study presents a practical approach for state and parameter estimation with observers for two-time-scale nonlinear systems.

Lie symmetry method to discuss the optimal system of Lie subalgebras and similarity solutions for shock propagation in a perfectly conducting dusty gas with magnetic field in rotating medium

In this paper, the similarity solutions for the shock wave propagation in a perfectly conducting dusty gas under the effect of axial or azimuthal magnetic field in the case of rotating medium by using Lie symmetry method have been discussed. The optimal system of one-dimensional Lie subalgebras related to the hyperbolic system of nonlinear partial differential equations had been constructed by using general invariant function and general adjoint transformation matrix. The optimal system is the collection of inequivalent optimal classes, which provides crucial information regarding different types of the similarity solutions. On the basis of the optimal system, we have discussed all possible similarity solutions for inequivalent optimal classes individually. Also, we have found that the similarity solutions exist in two cases for power law shock path and in one case with exponential law shock path. The shock wave decay is due to an increment in the physical parameter such as shock Cowling number or non-idealness parameter or adiabatic exponent or rotational parameter. For smaller value of the ratio of density of solid particles to initial species density of the gas i.e. [Formula: see text], the shock strength decreases with an increment in the mass fraction of solid particles in the mixture [Formula: see text], but the shock strength decreases as [Formula: see text] increases for [Formula: see text]. Also, the shock strength increases with an increment in [Formula: see text] at constant [Formula: see text]. It is observed that the shock wave is stronger in the presence of axial magnetic field as compared to azimuthal magnetic field.

Backstepping Control of a Class of Space-Time-Varying Linear Parabolic PDEs via Time Invariant Kernel Functions

Backstepping Control of a Class of Space-Time-Varying Linear Parabolic PDEs via Time Invariant Kernel Functions

Asymptotic properties of recursive kernel density estimation for long-span high-frequency data

. This article mainly studies the asymptotic properties of recursive kernel density estimation for high-frequency data with a long time span. Under appropriate conditions, the variance, bias, mean squared error, and optimal bandwidth are obtained. The asymptotic normality is proven by using moment inequalities for ρ-mixing high-frequency data. Numerical simulations show that the recursive kernel density estimation provides good fitting results for the invariant density function of the Vasicek diffusion process. Empirical analysis reveals that recursive kernel density estimation not only reflects the distribution characteristics of financial data but also allows us to fit parameter models by minimizing the squared deviation between the recursive kernel density estimation and the parameter model. It provides more applications for the financial field.

A semi-definite optimization method for maximizing the shared band gap of topological photonic crystals

Topological photonic crystals (PCs) can support robust edge modes to transport electromagnetic energy in an efficient manner. Such edge modes are the eigenmodes of the PDE operator for a joint optical structure formed by connecting together two photonic crystals with distinct topological invariants, and the corresponding eigenfrequencies are located in the shared band gap of two individual photonic crystals. This work is concerned with maximizing the shared band gap of two photonic crystals with different topological features in order to increase the bandwidth of the edge modes. We develop a semi-definite optimization framework for the underlying optimal design problem, which enables efficient update of dielectric functions at each time step while respecting symmetry constraints and, when necessary, the constraints on topological invariants. At each iteration, we perform sensitivity analysis of the band gap function and the topological invariant constraint function to linearize the optimization problem and solve a convex semi-definite programming (SDP) problem efficiently. Numerical examples show that the proposed algorithm is superior in generating optimized optical structures with robust edge modes.

Context and Neural Function

Abstract The debate over context and neural function pits contextualists, who maintain that neural functions vary by context, against invariantists, who maintain that they do not. In this article, I defend a moderate position that permits some context sensitive yet invariant functions. I distinguish performance from competence as well as different types of contexts, accounting for many cases of variability by context. The discussion suggests changes in networks can sometimes change the competences of parts. I conclude that some context sensitivity of function is consistent with invariant functions whereas other changes in context imply changes in the competence to perform functions.

Free Reflection Multiarrangements and Quasi-Invariants

Abstract To a complex reflection arrangement with an invariant multiplicity function one can relate the space of logarithmic vector fields and the space of quasi-invariants, which are both modules over invariant polynomials. We establish a close relation between these modules. Berest–Chalykh freeness results for the module of quasi-invariants lead to new free complex reflection multiarrangements. Saito’s primitive derivative gives a linear map between certain spaces of quasi-invariants. We also establish a close relation between non-homogeneous quasi-invariants for root systems and logarithmic vector fields for the extended Catalan arrangements. As an application, we prove the freeness of Catalan arrangements corresponding to the non-reduced root system $BC_{N}$.

Classification of the conjugacy classes of [formula omitted]

In this note, we classify the conjugacy classes of SL˜2(R), the universal covering group of PSL2(R). For any non-central element α∈SL˜2(R), we show that its conjugacy class may be determined by three invariants: (i) Trace: the trace (valued in the set of positive real numbers R+) of its image α¯ in PSL2(R); (ii) Direction type: the sign behavior of the induced self-homeomorphism of R determined by the lifting SL˜2(R)↷R of the action PSL2(R)↷S1; (iii) The functionℓ♯: a conjugacy invariant length function introduced by Mochizuki (2016).

Analysis of a Dry Friction Force Law for the Covariant Optimal Control of Mechanical Systems with Revolute Joints

This contribution shows a geometric optimal control procedure to solve the trajectory generation problem for the navigation (generic motion) of mechanical systems with revolute joints. The mechanical system is analyzed as a nonlinear Lagrangian system affected by dry friction at the joint level. Rayleigh’s dissipation function is used to model this dissipative effect of joint-level friction, and regarded as a potential. Rayleigh’s potential is an invariant scalar quantity from which friction forces derive and are represented by a smooth model that approaches the traditional Coulomb’s law in our proposal. For the optimal control procedure, an invariant cost function is formed with the motion equations and a Riemannian metric. The goal is to minimize the consumed energy per unit time of the system. Covariant control equations are obtained by applying Pontryagin’s principle, and time-integrated using a Finite Elements Method-based solver. The obtained solution is an optimal trajectory that is then applied to a mechanical system using a proportional–derivative plus feedforward controller to guarantee the trajectory tracking control problem. Simulations and experiments confirm that including joint-level friction forces at the modeling stage of the optimal control procedure increases performance, compared with scenarios where the friction is not taken into account, or when friction compensation is performed at the feedback level during motion control.

Enforcing exact permutation and rotational symmetries in the application of quantum neural networks on point cloud datasets

Recent developments in the field of quantum machine learning have promoted the idea of incorporating physical symmetries in the structure of quantum circuits. A crucial milestone in this area is the realization of Sn-permutation equivariant quantum neural networks (QNNs) that are equivariant under permutations of input objects. In this paper, we focus on encoding the rotational symmetry of point cloud datasets into the QNN. The key insight of the approach is that all rotationally invariant functions with vector inputs are equivalent to a function with inputs of vector inner products. We provide a structure of the QNN that is exactly invariant to both rotations and permutations, with its efficacy demonstrated numerically in the problems of two-dimensional image classifications and identifying high-energy particle decays, produced by proton-proton collisions, with the SO(1,3) Lorentz symmetry. Published by the American Physical Society 2024

Banach lattices with upper p-estimates: free and injective objects

AbstractWe study the free Banach lattice $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] with upper p-estimates generated by a Banach space E. Using a classical result of Pisier on factorization through $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) together with a finite dimensional reduction, it is shown that the spaces $$\ell ^{p,\infty }(n)$$ ℓ p , ∞ ( n ) witness the universal property of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] isomorphically. As a consequence, we obtain a functional representation for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] , answering a question from Oikhberg et al. [Free Banach lattices. J Eur Math Soc (JEMS), 2024]. More generally, our proof allows us to identify the norm of any free Banach lattice over E associated with a rearrangement invariant function space. After obtaining the above functional representation, we take the first steps towards analyzing the fine structure of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] . Notably, we prove that the norm for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] cannot be isometrically witnessed by $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) and settle the question of characterizing when an embedding between Banach spaces extends to a lattice embedding between the corresponding free Banach lattices with upper p-estimates. To prove this latter result, we introduce a novel push-out argument, which when combined with the injectivity of $$\ell ^p$$ ℓ p allows us to give an alternative proof of the subspace problem for free p-convex Banach lattices. On the other hand, we prove that $$\ell ^{p,\infty }$$ ℓ p , ∞ is not injective in the class of Banach lattices with upper p-estimates, elucidating one of many difficulties arising in the study of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] .

Constitutive modelling for understanding stress-stretch behaviour of Lennard-Jones non-crystalline molecular Solid

Abstract Non-crystalline molecular solid materials have many scientific and engineering applications. This study develops a constitutive equation for understanding stress-stretch behaviour of non-crystalline molecular solid using Lennard-Jones (LJ) intermolecular interaction. The strain energy derived from Lennard-Jones interactions between molecules. Based on the excluded volume (spherical volume occupied by the molecules maintaining centre to centre distance with a reference molecule) and density of the molecules, strain energy density is developed. In order to relate the molecular approach with continuum approximation, the excluded volume and density are expressed as a function of strain invariants of right Cauchy-Green deformation tensor. Finally, the constitutive equation in the form of Cauchy stress tensor is developed using the present strain energy density function. The present constitutive model is used to study finite deformations of the molecular solid like uniaxial extension. We compare our theoretical results with the experimental data of flexible polyurethane foams and obtain very good agreements. The current constitutive model can predict the deformation of micro/nano engineering system components.

Masked particle modeling on sets: towards self-supervised high energy physics foundation models

Abstract We propose masked particle modeling (MPM) as a self-supervised method for learning generic, transferable, and reusable representations on unordered sets of inputs for use in high energy physics (HEP) scientific data. This work provides a novel scheme to perform masked modeling based pre-training to learn permutation invariant functions on sets. More generally, this work provides a step towards building large foundation models for HEP that can be generically pre-trained with self-supervised learning and later fine-tuned for a variety of down-stream tasks. In MPM, particles in a set are masked and the training objective is to recover their identity, as defined by a discretized token representation of a pre-trained vector quantized variational autoencoder. We study the efficacy of the method in samples of high energy jets at collider physics experiments, including studies on the impact of discretization, permutation invariance, and ordering. We also study the fine-tuning capability of the model, showing that it can be adapted to tasks such as supervised and weakly supervised jet classification, and that the model can transfer efficiently with small fine-tuning data sets to new classes and new data domains.