Elementary Example Research Articles

Ellipsoidal nested sampling, expression of the model uncertainty and measurement

The measurand value, the conclusions, and the decisions inferred from measurements may depend on the models used to explain and to analyze the results. In this paper, the problems of identifying the most appropriate model and of assessing the model contribution to the uncertainty are formulated and solved in terms of Bayesian model selection and model averaging. As computational cost of this approach increases with the dimensionality of the problem, a numerical strategy, based on multimodal ellipsoidal nested sampling, to integrate over the nuisance parameters and to compute the measurand post-data distribution is outlined. In order to illustrate the numerical strategy, by use of MATHEMATICA an elementary example concerning a bimodal, two-dimensional distribution has also been studied.

Electron-photon coupling in mesoscopic quantum electrodynamics

Understanding the interaction between cavity photons and electronic nanocircuits is crucial for the development of Mesoscopic Quantum Electrodynamics (QED). One has to combine ingredients from atomic Cavity QED, like orbital degrees of freedom, with tunneling physics and strong cavity field inhomogeneities, specific to superconducting circuit QED. It is therefore necessary to introduce a formalism which bridges between these two domains. We develop a general method based on a photonic pseudo-potential to describe the electric coupling between electrons in a nanocircuit and cavity photons. In this picture, photons can induce simultaneously orbital energy shifts, tunneling, and local orbital transitions. We study in details the elementary example of a single quantum dot with a single normal metal reservoir, coupled to a cavity. Photon-induced tunneling terms lead to a non-universal relation between the cavity frequency pull and the damping pull. Our formalism can also be applied to multi quantum dot circuits, molecular circuits, quantum point contacts, metallic tunnel junctions, and superconducting nanostructures enclosing Andreev bound states or Majorana bound states, for instance.

The contribution of phenotypic plasticity to complementary light capture in plant mixtures.

Interspecific differences in functional traits are a key factor for explaining the positive diversity-productivity relationship in plant communities. However, the role of intraspecific variation attributable to phenotypic plasticity in diversity-productivity relationships has largely been overlooked. By taking a wheat (Triticum aestivum)-maize (Zea mays) intercrop as an elementary example of mixed vegetation, we show that plasticity in plant traits is an important factor contributing to complementary light capture in species mixtures. We conceptually separated net biodiversity effect into the effect attributable to interspecific trait differences and species distribution (community structure effect), and the effect attributable to phenotypic plasticity. Using a novel plant architectural modelling approach, whole-vegetation light capture was simulated for scenarios with and without plasticity based on empirical plant trait data. Light capture was 23% higher in the intercrop with plasticity than the expected value from monocultures, of which 36% was attributable to community structure and 64% was attributable to plasticity. For wheat, plasticity in tillering was the main reason for increased light capture, whereas for intercropped maize, plasticity induced a major reduction in light capture. The results illustrate the potential of plasticity for enhancing resource acquisition in mixed stands, and indicate the importance of plasticity in the performance of species-diverse plant communities.

On weakly [formula omitted]-differentiable operators

Let D be a self-adjoint operator on a Hilbert space H and a a bounded operator on H. We say that a is weakly D-differentiable, if for any pair of vectors ξ,η from H the function 〈eitDae−itDξ,η〉 is differentiable. We give an elementary example of a bounded operator a, such that a is weakly D-differentiable, but the function eitDae−itD is not uniformly differentiable. We show that weakD-differentiability may be characterized by several other properties, some of which are related to the commutator (Da−aD).

Length-expanding Lipschitz maps on totally regular continua

The tent map is an elementary example of an interval map possessing many interesting properties, such as dense periodicity, exactness, Lipschitzness and a kind of length-expansiveness. It is often used in constructions of dynamical systems on the interval/trees/graphs. The purpose of the present paper is to construct, on totally regular continua (i.e. on topologically rectifiable curves), maps sharing some typical properties with the tent map. These maps will be called length-expanding Lipschitz maps, briefly LEL maps. We show that every totally regular continuum endowed with a suitable metric admits a LEL map. As an application we obtain that every non-degenerate totally regular continuum admits an exactly Devaney chaotic map with finite entropy and the specification property.

Partially Gaussian stationary stochastic processes in discrete time

We present here an elementary example, for every fixed positive integer $k,$ of a strictly stationary nongaussian stochastic process in discrete time, all of whose $k$-marginals are gaussian.

A Formalism for Expressing the Probability Density Functions of Interrelated Quantities

In this paper we address measurement problems involving several quantities that are interrelated by model equations. Available knowledge about some of these quantities is represented by probability density functions (PDFs), which are then propagated through the model in order to obtain the PDFs attributed to the quantities for which nothing is initially known. A formalism for analyzing such models is presented. It comprises the concept of a „base parameterization“, which is used in conjunction with the change-of-variables theorem. The calculation procedure that results from this formalism is described in very general terms. Guidance is given on how to employ it in practice by presenting both an elementary example and a much more involved one.

A Note on Observability Canonical Forms for Nonlinear Systems

For nonlinear systems affine in the input with state x ɛ ℛn, input u ɛ ℛ and output y ɛ ℛ, it is a well-known fact that, if the function mapping (x, u,…, u(n–1)) into (u,…, u(n–1) y,…, y(n–1)) is an injective immersion, then the system can be locally transformed into an observability normal form with a triangular structure appropriate for a high-gain observer. In this technical note we extend this result to the case of systems not necessarily affine in the input and such that the injectivity condition holds for the function mapping (x, u,…, u(p–1)) into (u,…, u(p–1) y,…, y(p–1)) with p ⩾ n. The forced uncertain harmonic oscillator is taken as elementary example to illustrate the theory.

Systematic Implementation of a Tier 2 Behavior Intervention

Schools are increasingly adopting tiered models of prevention to meet the needs of diverse populations of students. This article outlines the steps involved in designing and implementing a systematic Tier 2 behavior intervention within a tiered service delivery model. An elementary school example is provided to outline the identification, implementation, and data-based decision-making process. Recommendations are provided for teachers related to supporting the effective and systematic implementation of Tier 2 behavior supports.

A Causality Approach to Particle Dynamics for Systems

Causality in physics is an old idea which emphasizes the notion that there are ‘causes’ and ‘responses’ to those causes; this relationship is expressed through a fundamental equation which is supposed to describe the evolution of a dynamical system of particles. In a most elementary example, one may identify the power input to a system as the ‘cause’ and the rate of change of the internal plus kinetic energy of the system as representing the ‘response’. In this case, the balance of one against the other is a statement of balance of energy for the dynamical system; such balance is often postulated as fundamental in mechanics. It is the purpose of this paper to investigate a more fundamental causality based approach for describing the dynamics of a system of particles. Forces intrinsic and extrinsic to the system are distinguished and an evolutionary causality law, which is form-invariant under a change of frame, is proposed. We assume, as is common, that intrinsic forces are objective under a general change of frame. Extrinsic forces are assumed to be objective only under a Galilean (that is, inertial) change of frame. This limited objectivity will play a major role in reducing the fundamental proposed ‘law of causality’ to what is recognized as the classical balance of energy for the system. The concept of mass is not introduced as primitive in this work; the existence of ‘inertial constants’, each of which associates with a specific particle of the system, is a result of the theory. Double and triple binding forces between pairs and triples of particles are admitted as fundamental to the intrinsic force structure of this theory. The balance of linear momentum for the system is derived and a generalized intrinsic force action-reaction principle is obtained. The emergence from the causality theory of pair- and triple-particle potential functions is discussed and these potential functions are shown to be intimately related to the internal binding force structure. Moreover, in Theorem 6.1 the underlying invariance structure of the theory shows that the double and triple binding forces must be separately ‘moment-balanced’ and that this gives rise to the balance of moment-of-momentum for the system. Finally, a constitutive theory for the binding forces is posited and it is shown that the double binding forces are determined by the pair-particle potential function and that the triple binding forces are determined by the triple-particle potential function.

The Binomial k‐Clique

AbstractA finite collection C of k‐sets, where is called a k‐clique if every two k‐sets (called lines) in C have a nonempty intersection and a k‐clique is a called a maximal k‐clique if and C is maximal with respect to this property. That is, every two lines in C have a nonempty intersection and there does not exist A such that , and for all . An elementary example of a maximal k‐clique is furnished by the family of all the k‐subsets of a ‐set. This k‐clique will be called the binomial k‐clique. This paper is intended to give some combinatorial characterizations of the binomial k‐clique as a maximal k‐clique. The techniques developed are then used to provide a large number of examples of mutually nonisomorphic maximal k‐cliques for a fixed value of k.

Robust Stability Analysis of a Linear Time-PeriodicActive Helicopter Rotor

Robust stability of a helicopter hingeless rotormodel used for air resonance alleviation is investigated in this paper. This aeroelastic phenomenon can be described naturally considering the rotor as a whole through a time-periodic change of coordinates called multiblade coordinates transformation. To assess robust stability of the resulting continuous linear time-periodic system, this paper defines a set of uncertainties specific to the helicopter rotor problem. The uncertain system is written in the form of a linear fractional representation and transformed using a frequency-lifting technique. This leads to a time-invariant representation of the uncertain system, the so-called harmonic transfer function. The extension of the standard -analysis to such lifted systems is proposed in the paper. Compared with a similar approach from literature based on an elementary example, the technique requires a moderate increase of the number of inputs and outputs, thus reducing the numerical effort involved in the computation of the -bounds. Themethodology, applied to the uncertain closed-loop helicopter rotor, shows robust stability for the defined set of uncertainty. The results are compared to an analysis performed on the base of a linear time-invariant system to quantify how strongly robust stability bounds are underestimated when the periodicity is not accounted for.

Braess's paradox in oscillator networks, desynchronization and power outage

Robust synchronization is essential to ensure the stable operation of many complex networked systems such as electric power grids. Increasing energy demands and more strongly distributing power sources raise the question of where to add new connection lines to the already existing grid. Here we study how the addition of individual links impacts the emergence of synchrony in oscillator networks that model power grids on coarse scales. We reveal that adding new links may not only promote but also destroy synchrony and link this counter-intuitive phenomenon to Braess's paradox known for traffic networks. We analytically uncover its underlying mechanism in an elementary grid example, trace its origin to geometric frustration in phase oscillators, and show that it generically occurs across a wide range of systems. As an important consequence, upgrading the grid requires particular care when adding new connections because some may destabilize the synchronization of the grid—and thus induce power outages.

Renormalization group procedure for effective particles: Elementary example of an exact solution with finite mass corrections and no involvement of vacuum

Renormalization group procedure for effective particles in the front form of Hamiltonian dynamics is applied to an elementary quantum field theory for two species of particles mixed through a mass-like interaction term. The model interaction generates only finite terms and the procedure yields a whole family of equivalent effective theories. The exact solution for the family is found without involvement of the vacuum state in the dynamics. Physical spectrum is obtained at the end of the procedure in the form of free particles with definite masses. Since the procedure is designed in general terms, it could be used for the purpose of constructing effective dynamics also in other theories than the elementary model.

Theoretical analysis of discrete contact problems with Coulomb friction

A discrete model of the two-dimensional Signorini problem with Coulomb friction and a coefficient of friction F depending on the spatial variable is analysed. It is shown that a solution exists for any F and is globally unique if F is sufficiently small. The Lipschitz continuity of this unique solution as a function of F as well as a function of the load vector f is obtained. Furthermore, local uniqueness of solutions for arbitrary F > 0 is studied. The question of existence of locally Lipschitz-continuous branches of solutions with respect to the coefficient F is converted to the question of existence of locally Lipschitz-continuous branches of solutions with respect to the load vector f. A condition guaranteeing the existence of locally Lipschitz-continuous branches of solutions in the latter case and results for determining their directional derivatives are given. Finally, the general approach is illustrated on an elementary example, whose solutions are calculated exactly.

Non-existence of Shilnikov chaos in continuous-time systems

In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained. Based on this result and an elementary example, it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation (ODE) systems characterized by the nonexistence of homoclinic and heteroclinic orbits.

Quantum dynamics in ultracold atomic physics

We review recent developments in the theory of quantum dynamics in ultra-cold atomic physics, including exact techniques, but focusing on methods based on phase-space mappings that are appli- cable when the complexity becomes exponentially large. These phase-space representations include the truncated Wigner, positive-P and general Gaussian operator representations which can treat both bosons and fermions. These phase-space methods include both traditional approaches using a phase-space of classical dimension, and more recent methods that use a non-classical phase-space of increased dimensionality. Examples used include quantum EPR entanglement of a four-mode BEC, time-reversal tests of dephasing in single-mode traps, BEC quantum collisions with up to 106 modes and 105 interacting particles, quantum interferometry in a multi-mode trap with nonlinear absorp- tion, and the theory of quantum entropy in phase-space. We also treat the approach of variational optimization of the sampling error, giving an elementary example of a nonlinear oscillator.

ROBUSTIFICATION OF CHAOS IN 2D MAPS

Robust chaos is defined as the absence of periodic windows and coexisting attractors in some neighborhood of the parameter space since the existence of such windows in the chaotic region implies fragility of the chaos. In this paper, we introduce a new terminology called robustification of chaos, which means creating robust chaos (in the sense of the above definition) in a dynamical system. As a first step, a new chaotification (robustification) method to generate robust chaos in planar maps is presented using simple piecewise smooth feedback to create a border collision bifurcation in the resulting system under some realizable conditions. The results are applied to an elementary example to illustrate the validity of the proposed method.

Noise and Synchronization of a Single Active Colloid

A two-state oscillator in a viscous liquid is composed of a micron-scale particle whose intrinsic dynamics is defined by linear potentials that undergo configuration-coupled transitions and is externally driven by a piecewise constant periodic force of varying amplitude and frequency. This elementary example of "active matter" has the minimal elements that allow us to study synchronization in the presence of thermal fluctuations. Experiments reveal the presence of synchronized states (and Arnol'd tongues), which we explain using analytical and numerical calculations. The system maintains synchronization by adjusting the phase between the bead and the clock. We discuss the relevance of this model to synchronization in real-world systems, including the role of thermal noise.

Two-center black holes, qubits, and elliptic curves

We relate the $U$-duality invariants characterizing two-center extremal black-hole solutions in the $stu$, $s{t}^{2}$, and ${t}^{3}$ models of $N=2$, $d=4$ supergravity to the basic invariants used to characterize entanglement classes of four-qubit systems. For the elementary example of a D0D4-D2D6 composite in the ${t}^{3}$ model we illustrate how these entanglement invariants are related to some of the physical properties of the two-center solution. Next we show that it is possible to associate elliptic curves to charge configurations of two-center composites. The hyperdeterminant of the hypercube, a four-qubit polynomial invariant of order 24 with 2 894 276 terms, is featuring the $j$ invariant of the elliptic curve. We present some evidence that this quantity and its straightforward generalization should play an important role in the physics of two-center solutions.