Reverse Theorem Research Articles

A reverse Denjoy theorem II

For α satisfying 0 < α < π, suppose that C 1 and C 2 are rays from the origin, C 1: z = re i(π−α) and C 2: z = re i(π+α), r ≥ 0, and that D = {z: | arg z − π| < α}. Let u be a nonconstant subharmonic function in the plane and define B(r, u) = sup|z|=r u(z) and A D (r, u) = $$ \inf _{z \in \bar D_r } $$ u(z), where D r = {z: z ∈ D and |z| = r}. If u(z) = (1 + o(1))B(|z|, u) as z → ∞ on C 1 ∪ C 2 and A D (r, u) = o(B(r, u)) as r → ∞, then the lower order of u is at least π/(2α).

A reverse Denjoy theorem

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, has measure at most 2α, where 0 0 (or lim r→∞ log B(r, u)/log r ≥ π/2α).

Effective method for constrained minimum — reverse bridge theorem

We propose reverse bridge theorem (RBTH) and give the demonstration in this work. RBTH is an effective method for constrained minimum. It is necessary and sufficient condition. RBTH bases on Extended saddle-point condition (ESPC), which is proposed by Yixin Chen. SGPlan according to ESPC dose the best in satisfying the International Planning Competition 2006 and in Suboptimal Metric Track of International Planning Competition 2004. RBTH comes into all advantage of ESPC and is different from ESPC, which can be concluded from two points. First, the core inequality of ESPC is formed by two sub-inequalities, and the core inequality of RBTH contains only one sub-inequality and one sub-equality. Generally, equality is easier to handle than inequality. RBTH should be better for planner than ESPC. The core inequality changes and is no longer saddle shape. We analyze why the core inequality changes. Second, continuous RBTH (CRBTH) does not need constraint-qualification condition which is necessary for continuous ESPC. This can be authenticated in two respects. One is the proof of CRBTH; the other is KKT which has relationship with ESPC. RBTH is real necessary and sufficient condition and ESPC is not. So RBTH can be used more widely. Besides, we show some errors in ESPC. We define point domain and Shrink-point domain to explain error. We also analyze the cause of errors, and provide an example.

INDEPENDENCE AND MARKOVIANITY FOR STATES ON VON NEUMANN ALGEBRAS

Several possible generalizations of the classical notion of markovianity are given for states defined on a von Neumann algebras generated on a triple of subalgebras. Their mutual relation is discussed in the particular case in which they mutually commute, and the generalization of the classical; time reversal theorem is proved. A structure theorem for a class of Markov chains is also proved.

Recovering the Acoustic Green’s Function from Ambient Noise Cross Correlation in an Inhomogeneous Moving Medium

We study long-range correlation of diffuse acoustic noise fields in an arbitrary inhomogeneous, moving fluid. The flow reversal theorem is used to show that the cross-correlation function of ambient noise provides an estimate of a combination of the Green's functions corresponding to sound propagation in opposite directions between the two receivers. Measurements of the noise cross correlation allow one to quantify flow-induced acoustic nonreciprocity and evaluate both spatially averaged flow velocity and sound speed between the two points.

Emergence of the Green’s function from ambient noise in moving media

A rather general proof of a connection between two-point correlation function of ambient noise and the elastodynamic Green’s function relies on the reciprocity principle [K. Wapenaar, Phys. Rev. Lett. 93, 254301 (2004)]. In this paper, the flow reversal theorem [O. A. Godin, Wave Motion 25, 143–167 (1997)] is applied to extend this result to moving, inhomogeneous fluids and problems involving flow-structure interactions. In the motionless case, the time derivative of the cross-correlation function of ambient noise gives an estimate of the sum of the causal and acausal time-dependent Green’s functions between the two receivers. In the moving medium, the result is shown to be similar, but the acausal Green’s function corresponds to sound propagation in a medium with reversed flow. Therefore, instead of redundant information, combination of the causal and acausal Green’s functions contains additional useful data on flow velocity field in the medium. The data contained in the two-point cross-correlation function of ambient noise are analogous to the data that would be obtained in a reciprocal transmission experiment employing acoustic transceivers located at the two points. Possible applications of the theoretical results obtained to passive remote sensing of ocean and atmosphere are discussed. [Work supported in part by ONR.]

A REVERSE FLOW THEOREM AND ACOUSTIC RECIPROCITY IN COMPRESSIBLE POTENTIAL FLOWS IN DUCTS

A reverse flow theorem for acoustic propagation in compressible potential flow has been obtained directly from the field equations without recourse to energy conservation arguments. A reciprocity theorem for the scattering matrix for the propagation of acoustic modes in a duct with either acoustically rigid walls or acoustically absorbing walls follows. It is found that for a source at a specific end of the duct, suitably scaled reflection matrices in direct and reverse flow have a reciprocal relationship. Scaled transmission matrices obtained for direct flow and reversed flow with simultaneous switching of source location from one end to the other also have a reciprocal relationship. A related reverse flow theorem specialized to one-dimensional acoustic propagation has also been obtained. Reciprocity relationships for the scattering coefficients for propagation are derived, and are found to be similar though much simpler than in the case of multi-mode propagation. In one-dimensional flow, reciprocal relations and power conservation arguments are used to show that scaled power reflection and transmission coefficients are invariant to flow reversal and switching of source location from one end of the duct to the other. Numerical verification of the reciprocal relationships is given in a companion paper.

NUMERICAL EXPERIMENTS ON ACOUSTIC RECIPROCITY IN COMPRESSIBLE POTENTIAL FLOWS IN DUCTS

A reciprocity theorem for the scattering matrix for the propagation of acoustic modes in a duct with acoustically hard walls or with acoustically absorbing walls has been given in a companion publication. It was found that for a source at a specified end of the duct, suitably scaled reflection matrices in direct and reverse flow have a reciprocal relationship. Scaled transmission matrices obtained for direct flow and reversed flow with simultaneous switching of source location from one end to the other also have a reciprocal relationship. A reverse flow theorem for the equivalent one-dimensional propagation model, which is a good approximation to the three-dimensional model at low frequencies, was also obtained. In this case, using reciprocity and acoustic power conservation arguments it is additionally found that the acoustic power transmission coefficient is the same for a source at either end of the duct for a given flow direction. This result leads to an invariance theorem which relates acoustic power propagated due to sources of equal pressure amplitude at the two ends of the duct. A numerical verification of these reciprocal relationships is given here for propagation in axially symmetric (circular and annular) ducts with multi-modal propagation and at low frequencies when a one-dimensional model is appropriate.

Distortion analysis of periodically switched nonlinear circuits using time-varying Volterra series

This paper presents a new frequency-domain method for distortion analysis of general periodically switched nonlinear circuits. It generalizes Zadeh's time-varying network functions and bifrequency transfer functions from linear time-varying systems to nonlinear time-varying systems. The periodicity of time-varying network functions of linear and nonlinear periodically time-varying systems is investigated using time-varying Volterra series. We show that a periodically switched nonlinear circuit can be characterized by a set of coupled periodically switched linear circuits. Distortion of the periodically switched nonlinear circuit is obtained by solving these linear circuits. This result is a generalization of the multi-linear theory known for nonlinear time-invariant circuits. We also show that the aliasing effect encountered in noise analysis of switched analog circuits exists in distortion analysis of periodically switched nonlinear circuits. Computation associated with the folding effect can be minimized by using the adjoint network of periodically switched linear circuits, in particular, the frequency reversal theorem. The method presented in this paper has been implemented in a computer program. Distortion of practical switched circuits is analyzed and the results are compared with SPICE simulation.

Energy-Conserving and Reciprocal Solutions for Higher-Order Parabolic Equations

The energy conservation law and the flow reversal theorem are valid for underwater acoustic fields. In media at rest the theorem transforms into well-known reciprocity principle. The presented parabolic equation (PE) model strictly preserves these important physical properties in the numerical solution. The new PE is obtained from the one-way wave equation by Godin12 via Pade approximation of the square root operator and generalized to the case of moving media. The PE is range-dependent and explicitly includes range derivatives of the medium parameters. Implicit finite difference scheme solves the PE written in terms of energy flux. Such formalism inherently provides simple and exact energy-conserving boundary condition at vertical interfaces. The finite-difference operators, the discreet boundary conditions, and the self-starter are derived by discretization of the differential PE. Discreet energy conservation and flow reversal theorem are rigorously proved as mathematical properties of the finite-difference scheme and confirmed by numerical modeling. Numerical solution is shown to be reciprocal with accuracy of 10–12 decimal digits, which is the accuracy of round-off errors. Energy conservation and wide-angle capabilities of the model are illustrated by comparison with two-way normal mode solutions including the ASA benchmark wedge.

ENERGY-CONSERVING AND RECIPROCAL SOLUTIONS FOR HIGHER-ORDER PARABOLIC EQUATIONS

The energy conservation law and the flow reversal theorem are valid for underwater acoustic fields. In media at rest the theorem transforms into well-known reciprocity principle. The presented parabolic equation (PE) model strictly preserves these important physical properties in the numerical solution. The new PE is obtained from the one-way wave equation by Godin12 via Padé approximation of the square root operator and generalized to the case of moving media. The PE is range-dependent and explicitly includes range derivatives of the medium parameters. Implicit finite difference scheme solves the PE written in terms of energy flux. Such formalism inherently provides simple and exact energy-conserving boundary condition at vertical interfaces. The finite-difference operators, the discreet boundary conditions, and the self-starter are derived by discretization of the differential PE. Discreet energy conservation and flow reversal theorem are rigorously proved as mathematical properties of the finite-difference scheme and confirmed by numerical modeling. Numerical solution is shown to be reciprocal with accuracy of 10–12 decimal digits, which is the accuracy of round-off errors. Energy conservation and wide-angle capabilities of the model are illustrated by comparison with two-way normal mode solutions including the ASA benchmark wedge.

Adjoint network of periodically switched linear circuits with applications to noise analysis

In this paper, Tellegen's theorem for multiphase periodically switched linear (PSL) circuits in phasor domain and the adjoint network of PSL circuits are developed. Two new theorems are introduced and theoretical proofs are given. The first theorem, called the transfer function theorem, gives an efficient way to calculate the transfer functions from multiple inputs to a single output of PSL circuits. This theorem is the generalization of a similar result known for linear time-invariant and ideal switched capacitor networks. A major contribution of this paper is the second theorem, called the frequency reversal theorem, which gives an efficient way to compute the aliasing transfer functions of linear periodically time-varying circuits. The use of both of these theorems results in an efficient algorithm for noise analysis of linear periodically time-varying circuits in the frequency domain. The algorithm handles general PSL circuits, including switched capacitor and switched current networks with both white and 1/f noise sources. It has been implemented in a computer program. The output noise power of practical PSL circuits is analyzed and the results are compared with published measurement data.

Noise and sensitivity analysis of periodically switched linear circuits in frequency domain

This paper presents new theories and efficient computational methods for noise and sensitivity analysis of multiphase periodically switched linear (PSL) circuits in frequency domain. Tellegen's theorem for PSL circuits in the phasor domain, frequency reversal theorem, and transfer function theorem, are introduced. The adjoint network of PSL circuits is developed using a frequency-domain approach. An adjoint network-based noise analysis algorithm for PSL circuits is proposed. It is shown that the computational overhead associated with multiple noise sources is eliminated by using the transfer function theorem. It is also shown that the excessive cost of computation due to aliasing effects is significantly reduced when the frequency reversal theorem is employed. In sensitivity analysis, the incremental form of Tellegen's theorem for PSL circuits in the phasor domain is introduced and frequency-domain sensitivity of PSL circuits Is obtained. It is shown that frequency-domain sensitivity of PSL circuits is a series summation of the network variables. Both the baseband and sideband frequency components of the network variables contribute to baseband sensitivity. The method yields sensitivities of one output with respect to all circuit elements in one frequency analysis. Sensitivity networks of PSL circuits are introduced. It is demonstrated that both the adjoint and sensitivity network approaches give the same sensitivity. Numerical results computed using the proposed methods are compared with measurement data and those from other CAD tools.

Laser Control of Atomic Motion inside Diatomic Molecules

Globally optimal solution describing a phase conjugated field of Raman scattering on the resonant B ← X transition of iodine I2 is studied. Maximum optical coherence is found as a top eigenvalue problem. A reversibility theorem has been stated. This provides sufficient conditions for a tightly localized waveform and molecular hologram to exist. A noisy picosecond pulse has been computed to show how femtosecond polarization is regained at target time.

Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid

A flow reversal theorem (FRT) is established in this paper for sound and acoustic-gravity waves in an arbitrary inhomogeneous moving steady ideal fluid. The theorem is an extension for moving fluid of the reciprocity principle valid in quiescent media. The FRT states symmetry of some wave field quantity with respect to interchange of the source and receiver positions and the simultaneous reversal of the ambient flow. A simple but rather general proof of the FRT becomes possible due to a particular choice of a mixed Eulerian-Lagrangian description of fluid motion. Relation between the FRT proved and a number of known FRTs established earlier for various specific cases is analyzed. Wave quasi-energy and wave-action conservation laws, related to the FRT, are proved for linear waves in an inhomogeneous moving steady compressible fluid. Validity domains of the FRT and the conservation laws are discused. Some possible applications of the FRT and the conservation laws in the investigation of sound generation and propagation in a moving medium are considered.

A reversibility relationship for two Markovian time series models with stationary geometric tailed distribution

The discrete autoregressive and minification stationary time series models discussed by Little-john (1992a) are generalized to model marginal distributions which have perturbations at the origin. The reversibility theorem relating these processes with geometric marginal distribution is extended to the case where the marginal distribution has geometric tail.

The Kirchhoff–Helmholtz integral theorem and related identities for waves in an inhomogeneous moving fluid.

The Kirchhoff–Helmholtz integral theorem (KHT), which expresses the wave field in a volume inside (or outside) a given surface in terms of the field’s value on the surface, is well known and widely used in acoustics of motionless media. In this paper, an extension of the theorem to acoustic‐gravity waves in an arbitrary inhomogeneous moving fluid is obtained as a corollary of the reciprocity‐type relations constituting the recently established flow reversal theorem (FRT) [O. A. Godin, J. Acoust. Soc. Am. 97, 3396(A) (1995); 98, 2866(A) (1995)]. The KHT takes a concise form when stated in terms of acoustic pressure and oscillatory displacement of fluid particles. The KHT is applied to study uniqueness of solutions to acoustic boundary value problems in moving media and to establish unitarity and other general properties of the scattering matrix for surface and volume scattering as well as in an irregular (range‐dependent) waveguide with flow. Relation of the scattering matrix properties to the FRT and to wave‐action conservation is discussed. [Work supported by NSERC.]

Acoustic reciprocity-type and energy conservation theorems for flow-structure problems

Recently the reciprocity principle was proved for acoustic fields in motionless fluid/solid structures [A. N. Norris and D. A. Rebinsky, J. Acoust. Soc. Am. 94, 1714–1715 (1993)] and a flow reversal theorem (FRT) was established for sound and acoustic-gravity waves in 3-D inhomogeneous, moving fluids [O. A. Godin, J. Acoust. Soc. Am. 97, 3396(A) (1995)]. The FRT is counterpart of the reciprocity principle when ambient flow is present. The theorem states symmetry of some field quantity with respect to interchange of the source and receiver positions and the simultaneous reversal of flow. An FRT will be presented which generalizes results of the above-mentioned works to include waves in flow/solid structures of arbitrary geometry. Parameters of the medium are allowed to be spatially inhomogeneous but time independent. Wave propagation and ambient flow are considered as adiabatic thermodynamic processes. It is assumed the prestresses in the solids due to ambient flow are small. Conservation of a quasienergy for the wave is established and its relation to known acoustic energy corollaries as well as to the FRT is discussed. Some possible applications of the general theorems will be considered. [Work supported by NRC.] a)On leave from P. P. Shirshov Oceanography Institute, Moscow, Russia.

Reciprocity-type relations for waves in a moving inhomogeneous fluid

The reciprocity principle is known to be invalid for acoustic waves in a moving fluid due to differences in up- and down-flow wave propagation velocities. A variation of the reciprocity principle, a flow reversal theorem (FRT), which states symmetry of some field quantity with respect to interchange of the source and receiver positions and the simultaneous reversal of flow, was considered by many authors during the last four decades. It was proven under that or other specific assumptions about fluid parameters and/or flow velocity space dependence or in a framework of some asymptotic representations of the acoustic field. A simple but rather general proof of the FRT will be presented which is valid for sound as well as acoustic-gravity waves in an arbitrary three-dimensional inhomogeneous moving fluid with time-independent parameters provided divergence of the flow velocity is zero. The fluid may be unbounded or have pressure-release, rigid, or impedance boundaries. The key point in the approach is the choice of a set of dependent variables in hydrodynamic equations for describing the wave field. An implication of the FRT on wave action conservation for cw sound in moving fluid is considered. Some possible FRT applications are indicated. [Work supported by NRC.]

Generalized reverse theorems for multipass applications in matrix optics

The reverse theorem that governs backward propagation through optical systems represented by transfer matrices is examined for various matrix theories. We extend several reverse theorems to allow for optical systems represented by matrices that may or may not be unimodular and that may be 2 × 2 or take on an augmented 3 × 3 form. As an example, we use the 3 × 3 form of the reverse theorem to study a laser with intracavity misaligned optics. It is shown that, by tilting one of the laser’s mirrors, we can align the laser output arbitrarily, and the mirror tilt angle is calculated.