Variable System Research Articles

Many-body quantum chaos and space-time translational invariance

We study the consequences of having translational invariance in space and time in many-body quantum chaotic systems. We consider ensembles of random quantum circuits as minimal models of translational invariant many-body quantum chaotic systems. We evaluate the spectral form factor as a sum over many-body Feynman diagrams in the limit of large local Hilbert space dimension q. At sufficiently large t, diagrams corresponding to rigid translations dominate, reproducing the random matrix theory (RMT) behaviour. At finite t, we show that translational invariance introduces additional mechanisms via two novel Feynman diagrams which delay the emergence of RMT. Our analytics suggests the existence of exact scaling forms which describe the approach to RMT behavior in the scaling limit where both t and L are large while the ratio between L and LTh(t), the many-body Thouless length, is fixed. We numerically demonstrate, with simulations of two distinct circuit models, that the resulting scaling functions are universal in the scaling limit.

On State-Space Representations of General Discrete-Time Dynamical Systems

In this article, we establish that every (deterministic) nonautonomous, discrete-time, causal, time invariant system has a state-space representation, and discuss its minimality.

Configuration-Dependent Substructuring as a Tool to Predict the Vibrational Response of Mechanisms

Dynamic substructuring allows us to predict the dynamic behavior of mechanical systems built by linking together several subsystems, whose dynamic behavior is known. The classical formulation, originally conceived for invariant systems, was extended by the authors to include mechanical systems made by invariant subsystems that may be coupled in different configurations. A mechanism is a typical example of a mechanical system built by coupling together invariant subsystems; during its motion, it can take several configurations that significantly affect its vibrational behavior. Therefore, the configuration-dependent substructuring approach can provide meaningful insights into the dynamic behavior of the mechanism. In this paper, the proposed approach is exploited to evaluate the vibrational behavior of a three-point linkage, a widely used mechanism to connect agricultural tractors to operating machines, considering a significant range of operative configurations. The proposed substructuring approach is able to predict the frequency response functions, the natural frequencies and the mode shapes of the mechanism in a wide range of configurations.

On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.

Primary–auxiliary model scheduling based estimation of the vertical wheel force in a full vehicle system

In this work we study the estimation problem of the vertical wheel force in a vehicle system subject to non-stationary and unknown excitation, which is a common and critical problem in vehicle durability design. The vehicle system can be approximated by a linear time invariant system under each working condition. However, the working conditions are unknown during the estimation process. A primary–auxiliary model scheduling procedure based on time-domain transmissibilities is proposed and performed to identify the unknown working conditions and estimate the unknown vertical wheel force: In addition to constructing a primary transmissibility family from the pseudo-inputs to the output during the offline stage, an auxiliary transmissibility family is constructed by further decomposing the pseudo-input vector into two parts. The auxiliary family enables identifying the unknown working condition under which the system is currently running, and then an appropriate transmissibility from the primary transmissibility family for estimating the unknown output can be selected during the online estimation stage. Moreover, Finite Impulse Response (FIR) models, ridge regression, and Bayes classifiers are applied to realize the model scheduling procedure. A numerical example and a real-world application to the estimation of the vertical wheel force in a full vehicle system are, respectively, conducted to demonstrate the effectiveness of the proposed method. The results show that the estimates are in good agreement with reference measurements.

Tight computationally efficient approximation of matrix norms with applications

We address the problems of computing operator norms of matrices induced by given norms on the argument and the image space. It is known that aside of a fistful of ``solvable cases,'' most notably, the case when both given norms are Euclidean, computing operator norm of a matrix is NP-hard. We specify rather general families of norms on the argument and the images space (``ellitopic'' and ``co-ellitopic,'' respectively) allowing for reasonably tight computationally efficient upper-bounding of the associated operator norms. We extend these results to bounding ``robust operator norm of uncertain matrix with box uncertainty,'' that is, the maximum of operator norms of matrices representable as a linear combination, with coefficients of magnitude $\leq1$, of a collection of given matrices. Finally, we consider some applications of norm bounding, in particular, (1) computationally efficient synthesis of affine non-anticipative finite-horizon control of discrete time linear dynamical systems under bounds on the peak-to-peak gains, (2) signal recovery with uncertainties in sensing matrix, and (3) identification of parameters of time invariant discrete time linear dynamical systems via noisy observations of states and inputs on a given time horizon, in the case of ``uncertain-but-bounded'' noise varying in a box.

Iterative learning control applied to distributed‐order linear time invariant MIMO systems to achieve learnability

AbstractSufficient and necessary conditions for a distributed‐order linear time invariant system to be positive real are derived in terms of linear matrix inequalities. The positive realness condition is derived for three of the most usual cases presented in literature, in the realm of distributed‐order linear time invariant systems. As an additional product of this paper, the strictly positive realness condition can be derived. In addition, the concept of learnability of fractional‐order multi‐input multi‐output systems is extended to the case of distributed‐order systems, which is approached from the concept of output‐dissipativity by using an iterative learning scheme.

Polystability in positive characteristic and degree lower bounds for invariant rings

We develop a representation theoretic technique for detecting closed orbits that is applicable in all characteristics. Our technique is based on Kempf’s theory of optimal subgroups and we make some improvements and simplify the theory from a computational perspective. We exhibit our technique in many examples and in particular, give an algorithm to decide if a symmetric polynomial in n -variables has a closed \operatorname{SL}_n -orbit. As an important application, we prove exponential lower bounds on the maximal degree of a system of generators of invariant rings for two actions that are important from the perspective of Geometric Complexity Theory (GCT). The first is the action of \operatorname{SL}(V) on S^3(V)^{\oplus 3} , the space of 3 -tuples of cubic forms, and the second is the action of \operatorname{SL}(V) \times \operatorname{SL}(W) \times \operatorname{SL}(Z) on the tensor space (V \otimes W \otimes Z)^{\oplus 5} . In both these cases, we prove an exponential lower degree bound for a system of invariants that generate the invariant ring or that define the null cone.

Cultural Code in Engineering Design

The subject of the study is a cultural code that gives the key to deciphering the cultural text of technical design. The object of the study is the language of technical design. The research uses methods of analyzing visual symbols used in technical design, the method of analyzing previous studies, the method of synthesis in conclusions regarding the concept of cultural code. The authors consider the following aspects of the topic: semiotic approach to the theory of culture and the definition of cultural code in visual semiosis; morphology and semantics of technical symbols in engineering design; syntax and pragmatics of cultural text in engineering design. Special attention is paid to the theory of complex analysis of a technical scheme as an example of a holistic cultural text. The main conclusions of the study are: 1. The semiotic approach to the definition of culture involves considering the sign system from the point of view of the diversity of meanings of its elements (symbols), and identifying three categories of symbols: semantics, pragmatics and syntax. The cultural text of technical design (device diagrams) is (in comparison with the text of art) a stricter, invariant system; however, as in any language, the genesis associated with changes in syntactics can be traced in it. 2. The syntax of the cultural text of technical design can be analyzed only if there is a cultural code of this technical language. This code includes the "alphabet" of symbol elements, their semantics (variety of meanings), pragmatics (scope of application) and syntax (relationships that form a unique schema text). A special contribution of the author to the study of the topic is the systematization of the morphology of pictorial symbols in engineering design from the point of view of cultural studies. The scientific novelty of the research lies in the culturological reading of the technical text.

Constrained Dynamics and Directed Percolation

In a recent work [A. Deger et al., Phys. Rev. Lett. 129, 160601 (2022).PRLTAO0031-900710.1103/PhysRevLett.129.160601] we have shown that kinetic constraints can completely arrest many-body chaos in the dynamics of a classical, deterministic, translationally invariant spin system with the strength of the constraint driving a dynamical phase transition. Using extensive numerical simulations and scaling analyses we demonstrate here that this constraint-induced phase transition lies in the directed percolation universality class in both one and two spatial dimensions.

Complete Systems of Galileo Invariants of a Motion of Parametric Figure in the Three Dimensional Euclidean Space

Let $E^{3}$ be the 3-dimensional Euclidean space and $S$ be a set with at least two elements. The notions of an $S$-parametric figure and the motion of an $S$-parametric figure in $E^{3}$ are defined. Complete systems of invariants of an $S$-parametric figure in $E^{3}$ for the orthogonal group $O(3,R)$ , the special orthogonal group $SO(3,R)$, Euclidean group $MO(3,R)$, the special Euclidean group $MSO(3,R)$ and Galileo groups $Gal_{1}(3,R)$ , $Gal^{+}_{1}(3,R)$ are obtained.

Semi-Explicit Model Predictive Control of Quasi Linear Parameter Varying Systems

Explicit linear model predictive control (MPC) is a proven method for constrained optimal control of fast, linear time invariant (LTI) dynamic systems. However, an extension to quasi-linear parameter varying (Q-LPV) systems requires overcoming significant challenges, since neither the explicit MPC law is piecewise linear in state nor are the critical regions polytopic. In this work, we propose a semi-explicit solution to a sub-optimal MPC solution for a class of Q-LPV systems, which are obtained as multiple linear model approximations to nonlinear systems. The sub-optimal MPC for Q-LPV systems, which has been shown to be stabilizing in an earlier work, requires an online solution of a quadratic program (QP). Casting the QP into a multiparametric form leads to a new class of multiparametric QP (mpQP), termed herein as generalized mpQP (GmpQP). Information about the critical regions and the MPC law are obtained offline using a semi-explicit solution of the GmpQP based on an implicit enumeration of the active sets. Three algorithms are compared for the online evaluation of the control law. Offline and online complexity of the proposed method are presented for an example, including comparison of computational speeds with well-known implicit QP solvers.

Invariant output feedback stabilisability: the scalar case

In this paper we prove that stabilisability is a static output feedback (SOF) invariant, for scalar and multivariable systems. Then we examine scalar stabilisability, from an invariant viewpoint. We prove that the signature of Hermite’s Bezoutian is constant within certain intervals that we call critical, and we give a very simple algebraic stabilisability criterion, consisted of a finite number of stability checks, one for each critical interval. We establish the validity of this criterion with Routh, Hurwitz and Lyapunov methods. We prove that the winding number of the Nyquist plot around points of the real axis, is constant within critical intervals. We correlate the stabilisability within critical intervals with the stability of corresponding critical polynomials defined at their limits. Finally, we discuss why our findings constitute a decisive step towards the analytical solution for the persisting problem of static output feedback stabilizability of linear time invariant multivariable systems.

Computing critical points for invariant algebraic systems

Let K be a field and (f1,…,fs,ϕ) be multivariate polynomials in K[x1,…,xn] (with s<n) each invariant under the action of Sn, the group of permutations of {1,…,n}. We consider the problem of computing the critical points of ϕ restricted to the algebraic set V(f), where f=(f1,…,fs). This is the same as computing the points at which f vanishes and the Jacobian matrix associated to (f1,…,fs,ϕ) is rank deficient, provided that this set is finite.We exploit the invariance properties of the input to split the solution space according to the orbits of Sn. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in ds, (n+dd) and (ns+1) where d is the maximum degree of the input polynomials. When d,s are fixed, this is polynomial in n while when s is fixed and d≃n this yields an exponential speed-up with respect to the usual polynomial system solving algorithms.

Precise determination of pair interactions from pair statistics of many-body systems in and out of equilibrium.

The determination of the pair potential v(r) that accurately yields an equilibrium state at positive temperature T with a prescribed pair correlation function g_{2}(r) or corresponding structure factor S(k) in d-dimensional Euclidean space R^{d} is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato [Phys. Rev. E 101, 032124 (2020)2470-004510.1103/PhysRevE.101.032124] conjectured that any realizable g_{2}(r) or S(k) corresponding to a translationally invariant nonequilibrium system can be attained by a classical equilibrium ensemble involving only (up to) effective pair interactions. Testing this conjecture for nonequilibrium systems as well as for nontrivial equilibrium states requires improved inverse methodologies. We have devised an optimization algorithm to precisely determine effective pair potentials that correspond to pair statistics of general translationally invariant disordered many-body equilibrium or nonequilibrium systems at positive temperatures. This methodology utilizes a parameterized family of pointwise basis functions for the potential function whose initial form is informed by small-, intermediate- and large-distance behaviors dictated by statistical-mechanical theory. Subsequently, a nonlinear optimization technique is utilized to minimize an objective function that incorporates both the target pair correlation function g_{2}(r) and structure factor S(k) so that the small intermediate- and large-distance correlations are very accurately captured. To illustrate the versatility and power of our methodology, we accurately determine the effective pair interactions of the following four diverse target systems: (1) Lennard-Jones system in the vicinity of its critical point, (2) liquid under the Dzugutov potential, (3) nonequilibrium random sequential addition packing, and (4) a nonequilibrium hyperuniform "cloaked" uniformly randomized lattice. We found that the optimized pair potentials generate corresponding pair statistics that accurately match their corresponding targets with total L_{2}-norm errors that are an order of magnitude smaller than that of previous methods. The results of our investigation lend further support to the Zhang-Torquato conjecture. Furthermore, our algorithm will enable one to probe systems with identical pair statistics but different higher-body statistics, which will shed light on the well-known degeneracy problem of statistical mechanics.

Galois quotients of tropical curves and invariant linear systems

For a map $\varphi : \Gamma \rightarrow \Gamma'$ between tropical curves and an isometric action on $\Gamma$ of a finite group $K$, $\varphi$ is a $K$-Galois covering on $\Gamma'$ if $\varphi$ is a harmonic morphism, the degree of $\varphi$ coincides with the order of $K$ and the action of $K$ induces a transitive action on every fibre. We prove that for a tropical curve $\Gamma$ with an isometric action of a finite group $K$, there exists a rational map, from $\Gamma$ to a tropical projective space, which induces a $K$-Galois covering on the image with proper edge-multiplicities. As an application, we also prove that for a hyperelliptic tropical curve without one valent points and of genus at least two, the invariant linear system of the hyperelliptic involution $\iota$ of the canonical linear system, the complete linear system associated with the canonical divisor, induces an $\langle \iota \rangle$-Galois covering on a tree. This is an analogy of the fact that a compact Riemann surface is hyperelliptic if and only if the canonical map, the rational map induced by the canonical linear system, is a double covering on a projective line $\boldsymbol{P}^1$.

A model of invariant control system using mean curvature drift from Brownian motion under submersions

Given a Riemannian submersion ϕ : M → N \phi : M \to N , we construct a stochastic process X X on M M such that the image Y ≔ ϕ ( X ) Y≔\phi (X) is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping π : G L ( n ) → G L ( n ) / O ( n ) \pi : GL(n) \to GL(n)/O(n) , whose image is equivalent to the space of n n -by- n n positive definite matrices, S + ( n , n ) \mathcal {S}_+(n,n) , and the said flow has deterministic image. We are able to compute explicitly the mean curvature (and hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization and (ii) in matrix entries, writing mean curvature as the gradient of log volume of orbits. As a consequence, we are able to write down Brownian motions explicitly on several common homogeneous spaces, such as Poincaré’s upper half plane and the Bures-Wasserstein geometry on S + ( n , n ) \mathcal {S}_+(n,n) , on which we can see the eigenvalue processes of Brownian motion reminiscent of Dyson’s Brownian motion. By choosing the background metric via natural G L ( n ) GL(n) action, we arrive at an invariant control system on the G L ( n ) GL(n) -homogenous space G L ( n ) / O ( n ) GL(n)/O(n) . We investigate the feasibility of developing stochastic algorithms using the mean curvature flow.

YOLOv4를 이용한 산양 탐지 시스템

Global industrialization has made human life comfortable, while causing environmental pollution. Environmental pollution has destroyed wildlife habitats, and as a result, several animals are endangered. In particular, long-tailed goral (Naemorhaedus caudatus) is an international endangered species that must be restored. A huge amount of photographic data has been accumulated through camera trap research for the restoration of endangered species. Until now, analysis of such photo data has only relied on experts, so it took a lot of time to analyze camera trap data. The goral detection system developed in this study, based on goral camera trap data, can enable goral data analysis at a faster than conventional methods. YOLOv4 was used as an object detection algorithm for goral detection. YOLOv4 is a network based on CSPDarknet53 and has improved performance through Bos (Bag of Specials) and BoF (Bag of Freebies). As a result, goral detection was possible with high accuracy and high speed. The performance of the goral detection system can be decreased by luminance at night. In this work, we propose a light invariant goral detection system by appropriately distributing day and night datasets for training YOLOv4. Test results on the night dataset showed an mAP of 0.907 and test results on the day dataset showed an mAP of 0.927. As a result, the goral detection system showed robust detection performance at night.

An algorithm for identifying reference signals under the environment of complex fuzzy sets

In this paper, we have discussed some new operations and results of set theory for complex fuzzy sets (CFSs). Moreover, we developed the basic results of CFSs under the basic operations such as complex fuzzy simple difference, bounded sum, bounded difference, dot product, bounded product, union, intersection, and Cartesian product. We explored the CFSs and discussed the related properties with examples such as complex fuzzy bounded sum over the intersection, complex fuzzy dot product over the union, etc. Identifying the reference signals under the environment of CFSs have always been a challenging. Many algorithms based on set theoretic operations and distance measures have been proposed for identifying a reference signal using any common system. But linear time invariant (LTI) system is considered easy to analyze the linear and time-varying signals. We used CFSs in signals and systems. We developed an algorithm based on convolution product and LTI system under the complex fuzzy environment. We identified a high degree of resemblance (reference signal) of the received signals to the reference signal in a linear time-invariant (LTI) system that receives an input signal and produces an output signal.

Four-Channel Vibrating Testbed Design for Full Vehicle Durability Simulation

The four-channel vibrating testbed for full vehicle durability simulation is indispensable for testing vehicle performance. This paper combines theoretical analysis with experimental verification, and the four-channel vibrating simulation testbed is designed. The hardware system consists of three parts: hydraulic mechanical components, sensors, and closed-loop control system, and the software system consists of platform configuration system, platform management system, and basic test system. Based on the design of the hardware system and software system, the emphasis is put on signal processing. Based on the least squares principle, the undetermined coefficients of trend items are determined, the trend items are transformed into the arithmetic mean of signal data, and the models that eliminate constant trend items are established to eliminate the trend items. In order to reflect the fatigue process of parts more truly, the anomaly points were identified and eliminated based on the two-parameter rain-flow counting method. By representing the transfer function in the frequency domain, the measured signal is changed quickly by Fourier, and the uniqueness of the transfer function is ensured by establishing an invariant system. Finally, test verification was performed on the designed testbed with the white powder noise taken as the driving signal. The test results agree well with field testing results. The design of the testbed can meet the requirements of engineering application, and its design method and signal processing method are reasonable, which has certain theoretical research significance and engineering application value.