Invariant Relations Research Articles

The Relationship between Strength and Toughness in Tempered Steel: Trade‐Off or Invariable?

Two different types of strength‐toughness relations are investigated in tempered steel (tempered at 25–420 °C). It is found that there is a trade‐off relation between strength‐fracture toughness and becomes invariable relation between strength‐impact toughness. The effect of microstructure on the materials toughness is related to the manner of energy consumption, which should be attributed to the notch root radius of the specimen. The coarse carbides and dislocation density inside the martensite lath have distinct effects on the energy for crack initiation (impact toughness) and crack growth (fracture toughness). Additionally, the mechanism of coarse carbides on the tempered martensite embrittlement is elaborated deeply.

Stochastic local operations and classical communication invariants via square matrix

In this paper, we raise a few corrections from the Li et al paper (2015 Phys. Rev. A 91 012302). We obtain some stochastic local operations and classical communication (SLOCC) invariants from second degree to sixth degree by defining square matrices. The relation of SLOCC invariants and the character polynomial of the square matrix are defined for three and four qubit states. This method comes as a useful resource in determining SLOCC invariants for n-qubit states.

Construction of Linear Invariant Relations of Kirchhoff Equations

For the Kirchhoff equations in the general case, when the cross-term matrix of the Hamiltonian can be asymmetric, the existence conditions for a linear invariant relation of a general form that connect the impulsive moment and the impulsive force are obtained. It is shown that equations with such an invariant relation can be transformed into the equations with an invariant relation for an impulsive moment, the existence conditions of which with a symmetric cross-term matrix coincide with the Chaplygin case. A description of a set of Hamiltonians that admit the existence of a linear invariant relation of a general form is given. The coordinate form of the invariant relation and its existence conditions is obtained. The total number of conditions is six (in contrast to the eight conditions in the Chaplygin case). Reduction to the Riccati equation is preformed. It is shown that if there is a linear integral, then the Kirchhoff equations are reduced to the Kirchhoff case.

Research on the Motion of a Body in a Potential Force Field in the Case of Three Invariant Relations

Research on the Motion of a Body in a Potential Force Field in the Case of Three Invariant Relations

Möbius--Lie Geometry and Its Extension

These lectures review the classical Moebius-Lie geometry and recent work on its extension. The latter considers ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. "to be orthogonal", "to be tangent", etc.), as new objects in an extended Moebius--Lie geometry. It is shown on examples, that such ensembles of cycles naturally parameterise many other conformally-invariant families of objects, two examples---the Poincare extension and continued fractions are considered in detail. Further examples, e.g. loxodromes, wave fronts and integrable systems, are published elsewhere. The extended Moebius--Lie geometry is efficient due to a method, which reduces a collection of conformally invariant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation. The algorithmic nature of the method allows to implement it as a C++ library, which operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures. Numeric calculations can be done in exact or approximate arithmetic. In the two- and three-dimensional cases illustrations and animations can be produced. An interactive Python wrapper of the library is provided as well.

ИДЕНТИФИКАЦИЯ ХАРАКТЕРИСТИК ДИССИПАЦИИ ОСЦИЛЛЯТОРОВ ВАН ДЕР ПОЛЯ

In many applications of physics, biology, and other sciences, an approach based on the concept of model equations is used as an approximate model of complex nonlinear processes. The basis of this concept is the provision that a small number of characteristic types movements of simple mathematical models inherent in systems gives the key to understanding and exploring a huge number of different phenomena. With this approach it is a priori assumed that the entire physical manifold can be represented in the form of fairly simple model equations. It is contributes to a qualitative study of complex systems for various physical nature since basic models individually are well studied, and their parameters have a physical interpretation. In particular, it is well known that oscillatory motion of various systems with a stable limit cycle can be modeled by a system consisting of one or more coupled van der Pol oscillators. This systems are widely represented, for example, in the study and modeling of some biological functions of the body, such as cardiac activity, respiration, locomotor activity, etc. Therefore, the tasks of determining in real time the state and parameters of such systems based on the results of measuring the output signals are relevant. One of these problems, namely, the problem of identification some parameters of an oscillatory system, is considered in this article. In the paper it is proposed Aa method is proposed for obtaining asymptotic estimates of parameters that determine the nonlinear nature of oscillations for a system of interconnected van der Pol oscillators by information about their motion. On the first step the corresponding identification problem was solved for one van der Pol oscillator; further, the results obtained results are extended to a system of interconnected oscillators. The unknown parameter characterizes the nonlinear term in the van der Pol equation and determines the threshold value for deviations at which the damping value in the system changes sign. The method of invariant relations is used for identification scheme design. Such approach allows us to synthesize additional relations arising between known and unknown quantities during the observed motion of the system considered. The constructed identifier provides an asymptotic estimation of unknown parameters for oscillatory networks of arbitrary structure. The numerical simulation confirms the operability of the proposed scheme for solving the identification problem.

张量衍生不变关系下的磁源单点定位

张量衍生不变关系下的磁源单点定位

Estimation of oscillation velocities of oscillator network

The study of the collective behavior of multiscale dynamic processes is currently one of the most urgent problems of nonlinear dynamics. Such systems arise on modelling of many cyclical biological or physical processes. It is of fundamental importance for understanding the basic laws of synchronous dynamics of distributed active subsystems with oscillations, such as neural ensembles, biomechanical models of cardiac or locomotor activity, models of turbulent media, etc. Since the nonlinear oscillations that are observed in such systems have a stable limit cycle , which does not depend on the initial conditions, then a system of interconnected nonlinear oscillators is usually used as a model of multiscale processes. The equations of Lienar type are often used as the main dynamic model of each of these oscillators. In a number of practical control problems of such interconnected oscillators it is necessary to determine the oscillation velocities by known data. This problem is considered as observation problem for nonlinear dynamical system. A new method – a synthesis of invariant relations is used to design a nonlinear observer. The method allows us to represent unknowns as a function of known quantities. The scheme of the construction of invariant relations consists in the expansion of the original dynamical system by equations of some controlled subsystem (integrator). Control in the additional system is used for the synthesis of some relations that are invariant for the extended system and have the attraction property for all of its trajectories. Such relations are considered in observation problems as additional equations for unknown state vector of initial oscillators ensemble. To design the observer, first we introduce a observer for unique oscillator of Lienar type and prove its exponential convergence. This observer is then extended on several coupled Lienar type oscillators. The performance of the proposed method is investigated by numerical simulations.

Synchronization of oscillations for coupled Van der Pol oscillators by output

A number of automatic control tasks, in particular, the synchronization of trajectories, the tracking task, control by a reference system are associated with the synthesis of control algorithms for dynamic cascade systems, which are a set of interconnected active subsystems. In this paper, the oscillation synchronization problem is considered for two Van der Pol coupled oscillators. It is assumed that the driven subsystem depends on the external control action, in addition, the phase vector is not fully known. On the first step the solution of the problem of synchronization in the form of state feedback is written. The aim of the work is to find the synchronizing control in the form of feedback on the state estimation. Such a formulation is relevant, since for many practical applications of control theory, a typical situation is when the complete state vector of the system is unknown and only some of the functions of the state variables - the outputs of the system are accessible to measurement. One can try to use the control law obtained from feedback by replacing the state with its estimate obtained by observer - a special dynamical system whose state eventually approaches (asymptotically or exponentially) to the state of the original system. In this case a question arises whether such control will be solving the synchronization problem. In mathematical control theory, in particular for the stabilization problem of dynamical systems, similar questions constitute the content of the known principle of separation. For the observation problem solving the apparatus of the method of synthesis of auxiliary invariant relations for constructing a nonlinear observer was used. In accordance with this approach a nonlinear observer is constructed for the system under consideration, which ensures the exponential estimates of the phase vector. It is further shown that the use in the control law instead of the state of the system of its evaluation under simultaneously solving the problems of observation and synchronization leads to the local solution of the problem under consideration.

Countably Many Weakenings of Belnap–Dunn Logic

Every Berman’s variety $$\mathbb {K}_p^q$$ which is the subvariety of Ockham algebras defined by the equation $${\sim ^{2p+q}}a = {\sim ^q}a$$ ($$p\ge 1$$ and $$q\ge 0$$) determines a finitary substitution invariant consequence relation $$\vdash _p^q$$. A sequent system $$\mathsf {S}_p^q$$ is introduced as an axiomatization of the consequence relation $$\vdash _p^q$$. The system $$\mathsf {S}_p^q$$ is characterized by a single finite frame $$\mathfrak {F}_p^q$$ under the frame semantics given for the formal language. By the duality between frames and algebras, $$\mathsf {S}_p^q$$ can be viewed as a $$4^{2p+q}$$-valued logic as it is characterized by a distributive lattice of $$4^{2p+q}$$ elements with a unary operator. Moreover, a structural-rule-free, cut-free and terminating sequent system $$\mathsf {G}_p^q$$ is established for $$\vdash _p^q$$. The Craig interpolation property of $$\vdash _p^q$$ is shown proof-theoretically utilizing $$\mathsf {G}_p^q$$.

An inconsistency between light speed invariance and causal relations

An inconsistency between light speed invariance and causal relations

Halo orbits construction based on invariant manifold technique

The current investigation applied the invariant manifold technique to study the halo orbits around the libration points in circular restricted three-body problem. Two dominant (leading) directions were considered as dominant motions for the construction of spatial halo orbits, the third direction motion being the slave (following) motion. The dominant motions correspond to four-dimensional invariant manifolds in the phase space. The invariant nonlinear asymptotic relations (i.e. INARs) between the two dominant motions and the slave motion were established, enabling a transformation from the 3-DOF problem into a 2-DOF problem. Application of the INARs are also discussed. Such invariant nonlinear relations in polynomial expansion form could be used as: (I) approximate analytical solutions; (II) topological constraints to obtain more exact numerical solutions with differential correction. General findings in the current research revealed that the nonlinear asymptotic relations among the directions provided an alternative point of view to explore the overall dynamics of halo orbits around libration points with general rules. The effectiveness of the proposed method was also verified by numerical simulations.

Topological invariants and phase diagrams for one-dimensional two-band non-Hermitian systems without chiral symmetry

We study topological properties of one-dimensional non-Hermitian systems without chiral symmetry and give phase diagrams characterized by topological invariants $\nu_E$ and $\nu_{total}$, associated with complex energy vorticity and summation of Berry phases of complex bands, respectively. In the absence of chiral symmetry, we find that the phase diagram determined by $\nu_E$ is different from $\nu_{tot}$. While the transition between phases with different $\nu_{E}$ is closely related to the band-touching point, the transition between different $\nu_{tot}$ is irrelevant to the band-touching condition. We give an interpretation for the discrepancy from the geometrical view by analyzing the relation of topological invariants with the winding numbers associated with exception points of the system. We then generalize the fidelity approach to study the phase transition in the non-Hermitian system and find that transition between phases with different $\nu_{tot}$ can be well characterized by an abrupt change of fidelity and fidelity susceptibility around the transition point.

A Note on Mean Equicontinuity

In this note, it is shown that several results concerning mean equicontinuity proved before for minimal systems are actually held for general topological dynamical systems. Particularly, it turns out that a dynamical system is mean equicontinuous if and only if it is equicontinuous in the mean if and only if it is Banach (or Weyl) mean equicontinuous if and only if its regionally proximal relation is equal to the Banach proximal relation. Meanwhile, a relation is introduced such that the smallest closed invariant equivalence relation containing this relation induces the maximal mean equicontinuous factor for any system.

Static and dynamic FE analysis of piezolaminated composite shells considering electric field nonlinearity under thermo-electro-mechanical loads

The present article focuses on the static and dynamic finite element simulations of smart piezolaminated composite shell structures considering strong electric field nonlinearity under thermo-electro-mechanical loads. To model the electromechanical behaviour of piezoelectric patches or layers under large applied electric fields more efficiently, two-way coupled rotationally invariant second-order nonlinear constitutive relations are used in the variational principle approach. Furthermore, the nonlinear piezoelectric element formulations are further extended to capture the response under temperature gradients. Quadratic and cubic polynomial approximations are deemed to represent the electric potential and temperature fields, respectively. Validation of the present element formulation has been done in comparison to experimental and numerical investigations of those available in the literature. Moreover, numerical simulations are performed to study the large electric field nonlinearities of piezolaminated structures in static and dynamic as well as active vibration control problems under both mechanical and thermal loads. The numerical simulations have shown that using the piezoelectric nonlinearity, both the static shape control and vibration suppression either under mechanical or thermal loads can be accomplished at much lower actuation voltages than estimated by the linear model.

Two-Dimensional DOA Estimation for Incoherently Distributed Sources with Uniform Rectangular Arrays.

Aiming at the two-dimensional (2D) incoherently distributed (ID) sources, we explore a direction-of-arrival (DOA) estimation algorithm based on uniform rectangular arrays (URA). By means of Taylor series expansion of steering vector, rotational invariance relations with regard to nominal azimuth and nominal elevation between subarrays are deduced under the assumption of small angular spreads and small sensors distance firstly; then received signal vectors can be described by generalized steering matrices and generalized signal vectors; thus, an estimation of signal parameters via rotational invariance techniques (ESPRIT) like algorithm is proposed to estimate nominal elevation and nominal azimuth respectively using covariance matrices of constructed subarrays. Angle matching method is proposed by virtue of Capon principle lastly. The proposed method can estimate multiple 2D ID sources without spectral searching and without information of angular power distribution function of sources. Investigating different SNR, sources with different angular power density functions, sources in boundary region, distance between sensors and number of sources, simulations are conducted to investigate the effectiveness of the proposed method.

On physical scattering density fluctuations of amorphous samples

Using the rigorous results obtained by Wiener [Acta Math. (1930), 30, 118–242] on the Fourier integral of a bounded function and the condition that small-angle scattering intensities of amorphous samples are almost everywhere continuous, the conditions that must be obeyed by a function η(r) for this to be considered a physical scattering density fluctuation are obtained. These conditions can be recast in the following form: the V → ∞ limit of the modulus of the Fourier transform of η(r), evaluated over a cubic box of volume V and divided by V 1/2, exists and its square obeys the Porod invariant relation. Some examples of one-dimensional scattering density functions obeying the aforesaid condition are numerically illustrated.

Topological dynamics and the complexity of strong types

We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g., we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup. We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over O is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let $$\mathfrak{C}$$ be a monster model of a countable theory, p ∈ S(O), and E be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on p( $$\mathfrak{C}$$ ). Then, exactly one of the following holds: (1) E is relatively definable (on p( $$\mathfrak{C}$$ )), smooth, and has finitely many classes, (2) E is not relatively definable, but it is type-definable, smooth, and has 2ℵ0 classes, (3) E is not type definable and not smooth, and has 2ℵ0 classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups.

Reflections on and of minor-closed classes of multisorted operations

The minor relation of functions is generalized to multisorted functions. Pippenger’s Galois theory for minor-closed sets of functions is extended to multisorted functions and multisorted relation pairs. Reflections of minor-closed sets are again minor-closed, and the effect of reflections on the invariant relation pairs of minor-closed sets of multisorted functions is described.

Definable relations in finite-dimensional subspace lattices with involution

For a large class of finite dimensional inner product spaces V, over division \(*\)-rings F, we consider definable relations on the subspace lattice \(\mathsf{L}(V)\) of V, endowed with the operation of taking orthogonals. In particular, we establish translations between the relevant first order languages, in order to associate these relations with definable and invariant relations on F—focussing on the quantification type of defining formulas. As an intermediate structure we consider the \(*\)-ring \(\mathsf{R}(V)\) of endomorphisms of V, thereby identifying \(\mathsf{L}(V)\) with the lattice of right ideals of \(\mathsf{R}(V)\), with the induced involution. As an application, model completeness of F is shown to imply that of \(\mathsf{R}(V)\) and \(\mathsf{L}(V)\).