Reverse Theorem Research Articles

Noise predictions of a Mach 0.9 round jet using tailored adjoint Green’s functions

The turbulent mixing noise radiated by a Mach 0.9 jet is investigated. The focus is put on the proper calculation of acoustic propagation effects by means of adjoint Green’s function that are tailored to the jet mean flow. Tam and Auriault’s statistical mixing noise model is recast for Pierce’s wave equation that is energy preserving. An unconditionally stable formulation to compute propagation effects is thus obtained. Adjoint fields are computed from the direct problem with help of the flow reversal theorem. A finite element solver is used to solve tailored adjoint Green’s functions, and corresponding adjoint fields are displayed. Acoustic predictions are carried out for a wide range of polar angles, and compared to measurements. A particular attention is given to predictions achieved at upstream observer angles. At these angles, the present model describes the physics of upstream travelling guided jet waves. The adjoint method provides a suitable framework to split the generation of sound from its propagation. It is illustrated how tailored adjoint Green’s functions filter the radiating part of Tam and Auriault’s sound source model, by weighting with propagation effects.

Reverse comparison theorems with upper integral Ricci curvature condition

Reverse comparison theorems with upper integral Ricci curvature condition

Sound propagation using an adjoint-based method

Abstract

Positive equilibria of weakly reversible power law kinetic systems with linear independent interactions

In this paper, we extend our study of power law kinetic systems whose kinetic order vectors (which we call “interactions”) are reactant-determined (i.e. reactions with the same reactant complex have identical vectors) and are linear independent per linkage class. In particular, we consider PL-TLK systems, i.e. such whose T-matrix (the matrix with the interactions as columns indexed by the reactant complexes), when augmented with the rows of characteristic vectors of the linkage classes, has maximal column rank. Our main result states that any weakly reversible PL-TLK system has a complex balanced equilibrium. On the one hand, we consider this result as a “Higher Deficiency Theorem” for such systems since in our previous work, we derived analogues of the Deficiency Zero and the Deficiency One Theorems for mass action kinetics (MAK) systems for them, thus covering the “Low Deficiency” case. On the other hand, our result can also be viewed as a “Weak Reversibility Theorem” (WRT) in the sense that the statement “any weakly reversible system with a kinetics from the given set has a positive equilibrium” holds. According to the work of Deng et al. and more recently of Boros, such a WRT holds for MAK systems. However, we show that a WRT does not hold for two proper MAK supersets: the set PL-NIK of non-inhibitory power law kinetics (i.e. all kinetic orders are non-negative) and the set PL-FSK of factor span surjective power law kinetics (i.e. different reactants imply different interactions).

Localization of sources of noise in moving atmospheres using time reversal and geometric acoustics

Due to acoustic refraction and convection in moving, inhomogeneous atmospheres, the backazimuths of recorded signals can be significantly different from the source backazimuth, posing a significant difficulty in source localization. Although an inverse problem, localization of sources of infrasound in such atmospheres has mostly been made using the forward ray-theoretic model of sound propagation. This approach uses search grids for the unknown model parameters that are subsequently estimated by optimizing an objective function. This approach can give rise to a large computational time as well as to the issue of uniqueness, stability, and accuracy of the solutions. The current paper presents a technique which avoids these issues, and which combines time reversal and geometric acoustics for source localization in such atmospheres. This technique, which can also be useful for source localization of other forms of acoustic signals in refractive media, effectively transforms the inverse problem into a forward one by utilizing a flow reversal theorem in the ray tracing equations. Beside the atmospheric conditions, the only parameters needed in this technique are the trace velocities of the signals recorded by at least two arrays. Comparison of numerical simulations with observational data will be presented.

Experimental checking of reversibility theorem for axially symmetric body with empennage

Wind tunnel tests were implemented for experimental verification of the reversibility theoremfor three various shapes of fore body for an axially symmetric thin body in direct and reverse streams. Besides,such tests were implemented for UAV models with three types of wings. Results of investigationsshowed that the error of execution reversibility theorem consequence can achieve 12.5% at the range ofangles of attack from 0 to 20° (accordingly, from 180° to 200°)

Analysis of the practical stability and sensitivity of linear dynamical systems with change of phase space measurability

In this work, the models of linear parametric systems of ordinary differential equations with variable measurability of phase space are investigated. The theorems about the practical stability of linear parametrical systems with variable measurability are proved. It is important that the reverse theorem about practical stability of indicated systems is obtained. The algorithms and criteria of analysis of practical stability of linear parametrical systems with variable measurability of phase space at the presence of constantly occurring perturbations are shown. The matrix equations of sensitivity of linear parametrical systems with variable measurability of the phase space are researched. It was investigated that on the basis of methods of practical stability and conditions which satisfied sensitivity matrices it was possible to effectively find the estimations of parameters for an analysis of the system sensitivity with variable measurability of the phase space. Results of given investigations can be successfully applied in the tasks of digital data processing and pattern recognition.

Conditional reversibility in nonequilibrium stochastic systems.

For discrete-state stochastic systems obeying Markovian dynamics, we establish the counterpart of the conditional reversibility theorem obtained by Gallavotti for deterministic systems [Ann. de l'Institut Henri Poincaré (A) 70, 429 (1999)]. Our result states that stochastic trajectories conditioned on opposite values of entropy production are related by time reversal, in the long-time limit. In other words, the probability of observing a particular sequence of events, given a long trajectory with a specified entropy production rate σ, is the same as the probability of observing the time-reversed sequence of events, given a trajectory conditioned on the opposite entropy production, -σ, where both trajectories are sampled from the same underlying Markov process. To obtain our result, we use an equivalence between conditioned ("microcanonical") and biased ("canonical") ensembles of nonequilibrium trajectories. We provide an example to illustrate our findings.

Local asymptotics for the first intersection of two independent renewals

We study the intersection of two independent renewal processes, ρ = τ ∩σ. Assuming that P(τ 1 = n) = ϕ(n) n −(1+α) and P(σ 1 = n) = ϕ(n) n −(1+ α) for some α, α 0 and some slowly varying ϕ, ϕ, we give the asymptotic behavior first of P(ρ 1 > n) (which is straightforward except in the case of min(α, α) = 1) and then of P(ρ 1 = n). The result may be viewed as a kind of reverse renewal theorem, as we determine probabilities P(ρ 1 = n) while knowing asymptotically the renewal mass function P(n ∈ ρ) = P(n ∈ τ)P(n ∈ σ). Our results can be used to bound coupling-related quantities, specifically the increments |P(n ∈ τ) − P(n − 1 ∈ τ)| of the renewal mass function. 1. Intersection of two independent renewals We consider two independent (discrete) renewal processes τ and σ, whose law are denoted respectively P τ and P σ , and the renewal process of intersections, ρ = τ ∩ σ. We denote P = P τ ⊗ P σ. The process ρ appears in various contexts. In pinning models, for example, it may appear directly in the definition of the model (as in [1], where σ represents sites with nonzero disorder values, and τ corresponds to the polymer being pinned) or it appears in the computation of the variance of the partition function via a replica method (see for example [20]), and is central in deciding whether disorder is relevant or irrelevant in these models, cf. [3]. When τ and σ have the same inter-arrival distribution, ρ 1 is related to the coupling time of τ and σ, if we allow τ and σ to start at different points. In particular, in the case µ := E[τ 1 ] < +∞, the coupling time ρ 1 has been used to study the rate of convergence in the renewal theorem, see [16, 17], using that

Properties of the Simpson discrete Fourier transform

The Simpson discrete Fourier transform (SDFT) and its inverse are transformations relating the time and frequency domains. In this paper we state and prove the important properties of shift, circular convolution, conjugation, time reversal and Plancherel's theorem. In addition, we provide an alternative representation of the duality property.

Trapped antihydrogen: A new frontier in fundamental physics

Antihydrogen, the bound state of an antiproton and a positron, has been produced at low energies at CERN since 2002. Antihydrogen is of interest for use in precision tests of nature's fundamental symmetries. The charge conjugation/parity/time reversal (CPT) theorem, a crucial part of the foundation of the standard model of elementary particles and interactions, demands that hydrogen and antihydrogen have the same spectrum. Given the current experimental precision of measurements on the hydrogen atom, subjecting antihydrogen to rigorous spectroscopic examination would constitute a compelling, model-independent test of CPT. Antihydrogen could also be used to study the gravitational behaviour of antimatter.However, until recently, experiments have produced antihydrogen that was not confined, precluding detailed study of its structure. Experimenters working to trap antihydrogen have faced the challenge of trapping and cooling relativistic antiprotons and using them to make antihydrogen cold enough to be trapped in a magnetic minimum trap with a depth of only 50 μeV. In November 2010 the ALPHA collaboration demonstrated the first trapping of antihydrogen, thus opening the door to precision measurements on anti-atoms which can soon be subjected to many of the same techniques as developed for atoms. The prospect for such measurements improved further with ALPHA's demonstration of 1000 s confinement of the anti-atoms in the summer of 2011 and the recent first detection of resonant quantum interactions in antihydrogen.

Optimality and Duality for Minimax Fractional Semi-Infinite Programming

The purpose of this paper is to consider a class of nonsmooth minimax fractional semi-infinite programming problem. Based on the concept of tangent derivative, a new generalization of convexity, namely generalized uniform invexity, is defined for this problem. For such semi-infinite programming problem, several sufficient optimality conditions are established and proved by utilizing the above defined new classes of functions. The results extend and improve the corresponding results in the literature. Subsequently, these optimality conditions are utilized as a basis for formulating dual problems. Weak, strong and reverse duality theorems are also derived for dual programs, using generalized invexity on the functions involved. Some previous duality results for differentiable minimax fractional programming problems turn out to be special cases for the results described in the paper

Bipolar Quantum Logic Gates and Quantum Cellular Combinatorics—A Logical Extension to Quantum Entanglement

Based on bipolar dynamic logic (BDL) and bipolar quantum linear algebra (BQLA) this work introduces bipolar quantum logic gates and quantum cellular combinatorics with a logical interpretation to quantum entanglement. It is shown that: 1) BDL leads to logically definable causality and generic particle-antiparticle bipolar quantum entanglement; 2) BQLA makes composite atom-atom bipolar quantum entanglement reachable. Certain logical equivalence is identified between the new interpretation and established ones. A logical reversibility theorem is presented for ubiquitous quantum computing. Physical reversibility is briefly discussed. It is shown that a bipolar matrix can be either a modular generalization of a quantum logic gate matrix or a cellular connectivity matrix. Based on this observation, a scalable graph theory of quantum cellular combinatorics is proposed. It is contended that this work constitutes an equilibrium-based logical extension to Bohr’s particle-wave complementarity principle, Bohm’s wave function and Bell’s theorem. In the meantime, it is suggested that the result may also serve as a resolution, rather than a falsification, to the EPR paradox and, therefore, a equilibrium-based logical unification of local realism and quantum non-locality.

A Reverse cos πρ Theorem and a Question of Fryntov

The authors continue their work on reverse Denjoy theorems, proving a reverse cosπρ theorem. The theorem is connected to a question of Fryntov on entire functions with gaps.

Magnification of three-dimensional optical image without distortion in dynamic holographic projection

We propose a simple technique to enlarge the reconstructed three-dimensional (3D) optical image and shorten the reconstructed distance simultaneously in real time holographic projection using a conventional lens or concave reflecting mirror based on the optical reversibility theorem. The main factors causing the longitudinal and transversal distortions of a 3D enlarged optical image are analyzed, and the 3D optical images are enlarged where severe distortions are precompensated by constructing objects with distortions directly instead of computing the precompensated phase iteratively so that it does not increase the computing time. Numerical simulations and optical experiments are performed for magnifying a simple cubic model. The results show that a 3D enlarged optical image is achieved successfully without any distortion and the reconstructed distance is shortened simultaneously. It is believed that this proposed technique is useful for 3D real time holographic projection in the future.

QUANTUM f-DIVERGENCES AND ERROR CORRECTION

Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

Trapped antihydrogen

Antimatter was first predicted in 1931, by Dirac. Work with high-energy antiparticles is now commonplace, and anti-electrons are used regularly in the medical technique of positron emission tomography scanning. Antihydrogen, the bound state of an antiproton and a positron, has been produced at low energies at CERN (the European Organization for Nuclear Research) since 2002. Antihydrogen is of interest for use in a precision test of nature's fundamental symmetries. The charge conjugation/parity/time reversal (CPT) theorem, a crucial part of the foundation of the standard model of elementary particles and interactions, demands that hydrogen and antihydrogen have the same spectrum. Given the current experimental precision of measurements on the hydrogen atom (about two parts in 10(14) for the frequency of the 1s-to-2s transition), subjecting antihydrogen to rigorous spectroscopic examination would constitute a compelling, model-independent test of CPT. Antihydrogen could also be used to study the gravitational behaviour of antimatter. However, so far experiments have produced antihydrogen that is not confined, precluding detailed study of its structure. Here we demonstrate trapping of antihydrogen atoms. From the interaction of about 10(7) antiprotons and 7 × 10(8) positrons, we observed 38 annihilation events consistent with the controlled release of trapped antihydrogen from our magnetic trap; the measured background is 1.4 ± 1.4 events. This result opens the door to precision measurements on anti-atoms, which can soon be subjected to the same techniques as developed for hydrogen.

On Quadratic g-Evaluations/Expectations and Related Analysis

In this article we extend the notion of g-evaluation, in particular g-expectation, of Peng [8, 9] to the case where the generator g is allowed to have a quadratic growth (in the variable “z”). We show that some important properties of the g-expectations, including a representation theorem between the generator and the corresponding g-expectation—and consequently the reverse comparison theorem of quadratic BSDEs as well as the Jensen inequality—remain true in the quadratic case. Our main results also include a Doob–Meyer type decomposition, the optional sampling theorem, and the upcrossing inequality. The results of this article are important in the further development of the general quadratic nonlinear expectations (cf. [5]).

A reverse Denjoy theorem III

Suppose that C 1 and C 2 are two simple curves joining 0 to ∞, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, the set {θ: re iθ ∈ $$ \bar D $$ } has measure at most 2α, where 0 < α < π. Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = Φ(|z|) for all large z ∈ C 1 ∪ C 2∪ ∼D, where Φ(|z|) is a convex, non-decreasing function of |z| and ∼D is the complement of D. Let A D (r, u) = inf{u(z): z ∈ $$ \bar D $$ and |z| = r}. It is shown that if A D (r, u) = O(1) then lim infr→∞ B(r, u)/r π/(2α) > 0.

Reverse Bridge Theorem under Constraint Partition

Reverse bridge theorem (RBTH) has been proved to be both a necessary and sufficient condition for solving Nonlinear programming problems. In this paper, we first propose three algorithms for finding constraint minimum points of continuous, discrete, and mixed-integer nonlinear programming problems based on the reverse bridge theorem. Moreover, we prove that RBTH under constraint partition is also a necessary and sufficient condition for solving nonlinear programming problems. This property can help us to develop an algorithm using RBTH under constraints. Specifically, the algorithm first partitions mixed-integer nonlinear programming problems (MINLPs) by their constraints into some subproblems in similar forms, then solves each subproblem by using RBTH directly, and finally resolves those unsatisfied global constraints by choosing appropriate penalties. Finally, we prove the soundness and completeness of our algorithm. Experimental results also show that our algorithm is effective and sound.