Orbit Length Research Articles

Spectral theorem for convex monotone homogeneous maps, and ergodic control

We consider convex maps f : R n→ R n that are monotone (i.e., that preserve the product ordering of R n ), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex inf-subsemilattice of R n , whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.

Parry measure and the topological entropy of chaotic repellers embedded within chaotic attractors

We study the topological entropy of chaotic repellers formed by those points in a given chaotic attractor that never visit some small forbidden hole-region in the phase space. The hole is a set of points in the phase space that have a sequence α=( α 0 α 1⋯ α l−1 ) as the first l letters in their itineraries. We point out that the difference between the topological entropies of the attractor and the embedded repeller is for most choices of α approximately equal to the Parry measure corresponding to α, μ P( α). When the hole encompasses a point of a short periodic orbit, the entropy difference is significantly smaller than μ P( α). This discrepancy is described by the formula which relates the length of the short periodic orbit, the Parry measure μ P( α), and the topological entropies of the two chaotic sets.

QED effects in lithiumlike krypton

The ${(2s}_{1/2}{\ensuremath{-}2p}_{1/2})$ Lamb shift in lithiumlike krypton was obtained through a recombination measurement. The Lamb shift was extracted from the recombination spectrum by comparison with a calculation of the dielectronic-recombination resonance positions. The energy calibration used in this work is based on an accurate energy determination of one arbitrary point in the spectrum. Our value for the ${2s}_{1/2}{\ensuremath{-}2p}_{1/2}$ energy difference is 71.243(8) eV. The relativistic many-body contributions to the Lamb shift can be calculated very precisely, which enabled an extraction of an ``experimental'' quantum-electrodynamical contribution of 1.539(8) eV. This precision is beyond what can be achieved with conventional estimates of the screened quantum-electrodynamical effects. The main error originates from the uncertainty in the length of the ion-beam orbit. Improved knowledge of this value can improve our result significantly.

A new species of the genus Acropoma (Perciformes: Acropomatidae) from the Philippines

Acropoma boholensis sp. nov. is described on the basis of two specimens collected from Dumaguete city market, Negros Island, Philippines. It is distinguished from other species of Acropoma by the following combination of characters: anus situated closer to the origin of pelvic fin than to that of anal fin; luminous gland extending from isthmus to anal fin base; scales cycloid except weakly ctenoid scales around anterior part of lateral line and on ventral side of body; proximal radial of first anal fin pterygiophore lacking trough or hollow; body depth 31%–33% SL; head length 40%–42% SL; length of orbit 13%–14% SL.

On extreme values of orbit lengths in $M/G/1$ queues with constant retrial rate

In the design of waiting facilities for the units in a retrial queue, it is of interest to know probability distributions of extreme values of the orbit length. The purpose of this paper is to investigate the asymptotic behavior of the maximum orbit length in the $M/G/1$ queue with constant retrial rate, as the time interval increases. From the classical extreme value theory, we observe that, under standard linear normalizations, the maximum orbit length up to the nth time the positive recurrent queue becomes empty does not have a limit distribution. However, by allowing the parameters to vary with n, we prove the convergence of maximum orbit lengths to three possible limit distributions when the traffic intensity $\rho_n$ approaches 1 from below and n approaches infinity.

A Spectral Theorem for Convex Monotone Homogeneous Maps

We consider convex maps f : Rn → Rn that are monotone (i.e., that preserve the product ordering of Rn), and non-expansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is non-empty, is isomorphic to a convex inf-semilattice, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This bound on the dimension is attained when f is piecewise affine. This yields in particular an uniqueness result for the bias vector of ergodic control problems. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.

Reversible Boolean networks: II. Phase transitions, oscillations, and local structures

We continue our consideration of a class of models describing the reversible dynamics of N Boolean variables, each with K inputs. We investigate in detail the behavior of the Hamming distance as well as of the distribution of orbit lengths as N and K are varied. We present numerical evidence for a phase transition in the behavior of the Hamming distance at a critical value K c≈1.62 and also an analytic theory that yields the exact bounds 1.5< K c<2. We also discuss the large oscillations that we observe in the Hamming distance for K< K c as a function of time as well as in the distribution of cycle lengths as a function of cycle length for moderate K both greater than and less than K c. We propose that local structures, or subsets of spins whose dynamics are not fully coupled to the other spins in the system, play a crucial role in generating these oscillations. The simplest of these structures are linear chains, called linkages, and rings, called circuits. We discuss the properties of the linkages in some detail, and sketch the properties of circuits. We argue that the observed oscillation phenomena can be largely understood in terms of these local structures.

Reversible Boolean networks I: distribution of cycle lengths

We consider a class of models describing the dynamics of N Boolean variables, where the time evolution of each depends on the values of K of the other variables. Previous work has considered models with dissipative dynamics. Here, we consider time-reversible models, which necessarily have the property that every possible point in the state space is an element of one and only one cycle. The orbits can be classified by their behavior under time reversal. The orbits that transform into themselves under time reversal have properties quite different from those that do not; in particular, a significant fraction of latter-type orbits have lengths enormously longer than orbits that are time-reversal symmetric. For large K and moderate N, the vast majority of points in the state space are on one of the time-reversal singlet orbits, and a random hopping model gives an accurate description of orbit lengths. However, for any finite K, the random hopping approximation fails qualitatively when N is large enough ( N≫2 2 K ). As in the dissipative case, when K is large, typical orbit lengths grow exponentially with N, whereas for small enough K, typical orbit lengths grow much more slowly with N. The numerical data are consistent with the existence of a phase transition at which the average orbit length grows as a power of N at a value of K between 1.4 and 1.7. However, in the reversible models, the interplay between the discrete symmetry and quenched randomness can lead to enormous fluctuations of orbit lengths and other interesting features that are unique to the reversible case.

Finite-precision stationary states at and away from equilibrium

We study the precision dependence of equilibrium and nonequilibrium phase-space distribution functions for time-reversible dynamical systems simulated with finite, computational precision. The conservative and dissipative cases show different behavior, with substantially reduced period lengths in the dissipative case. The main effect of finite precision is to change the phase-space fraction occupied by the distributions. The convergence of thermodynamic averages is only slightly affected. We introduce and discuss a simple stochastic model which is nicely consistent with all of our numerical results. This model links the length of periodic orbits to the strange attractor's correlation dimension.

Characterization of translation planes by orbit lengths. II. Even order

Characterization of translation planes by orbit lengths. II. Even order

Stochastic analysis of a single server retrial queue with general retrial times

Retrial queueing systems are widely used in teletraffic theory and computer and communication networks. Although there has been a rapid growth in the literature on retrial queueing systems, the research on retrial queues with nonexponential retrial times is very limited. This paper is concerned with the analytical treatment of an M/G/1 retrial queue with general retrial times. Our queueing model is different from most single server retrial queueing models in several respectives. First, customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served) discipline and only the customer at the head of the queue is allowed for access to the server. Besides, a retrial time begins (if applicable) only when the server completes a service rather upon a service attempt failure. We carry out an extensive analysis of the queue, including a necessary and sufficient condition for the system to be stable, the steady state distribution of the server state and the orbit length, the waiting time distribution, the busy period, and other related quantities. Finally, we study the joint distribution of the server state and the orbit length in non-stationary regime. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 561–581, 1999

Stochastic analysis of a single server retrial queue with general retrial times

Retrial queueing systems are widely used in teletraffic theory and computer and communication networks. Although there has been a rapid growth in the literature on retrial queueing systems, the research on retrial queues with nonexponential retrial times is very limited. This paper is concerned with the analytical treatment of an M/G/1 retrial queue with general retrial times. Our queueing model is different from most single server retrial queueing models in several respectives. First, customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served) discipline and only the customer at the head of the queue is allowed for access to the server. Besides, a retrial time begins (if applicable) only when the server completes a service rather upon a service attempt failure. We carry out an extensive analysis of the queue, including a necessary and sufficient condition for the system to be stable, the steady state distribution of the server state and the orbit length, the waiting time distribution, the busy period, and other related quantities. Finally, we study the joint distribution of the server state and the orbit length in non-stationary regime. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 561–581, 1999

Application of RF electric field to a charged-particle trap in toroidal geometry

In a toroidal magnetic configuration, charged particles have long orbit lengths Kondoh and Yoshida (Nucl. Instr. and Meth. in Phys. Res. A 382 (1996) 561), and hence, a slow method of modulating the kinetic energy can be applied to confine the particles in the trapping region. Numerical simulation shows that a spatially inhomogeneous radio-frequency electric field induces effective diffusion of the particles toward the trapping region.

A new multi-passage time-of-flight mass spectrometer at JAIST

A new time-of-flight (TOF) mass spectrometer has been constructed at JAIST. The spectrometer consists of six electrostatic analyzers and the shape of the ion orbit resembles a racetrack oval. Produced ions are injected into the ring and fly around the circular orbit . After an appropriate number of turns, ions are ejected from the ring and directed to an ion detector. The size of the instrument is 2.5 m×3.5 m and the length of the circular orbit is 7.4 m. The mass resolution of a TOF mass spectrometer is directly proportional to the path length of the ion trajectory. Thereby, high mass resolution can easily be achieved in multiple passages in the ring. In order to avoid the spread of the ion bunch in the multiple passages, perfect isochronous focusing and spatial focusing have been introduced by a special ion optic design of the ring. In the middle of the instrument, triple isochronous focusing (displacement, angle, energy) and triple spatial focusing (two-dimensional angles, energy) are satisfied. Moreover, second- and higher-order aberrations are effectively small. The new multi-passage TOF mass spectrometer, therefore, realizes high mass resolution and high ion transmission, simultaneously.

Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems

The scattering problems of a scalar point particle from an assembly of 1< n<∞ non-overlapping and disconnected hard disks, fixed in the two-dimensional plane, belong to the simplest realizations of classically hyperbolic scattering systems. Their simplicity allows for a detailed study of the quantum mechanics, semiclassics and classics of the scattering. Here, we investigate the connection between the spectral properties of the quantum-mechanical scattering matrix and its semiclassical equivalent based on the semiclassical zeta-function of Gutzwiller and Voros. We construct the scattering matrix and its determinant for any non-overlapping n-disk system (with n<∞) and rewrite the determinant in such a way that it separates into the product over n determinants of one-disk scattering matrices – representing the incoherent part of the scattering from the n-disk system – and the ratio of two mutually complex conjugate determinants of the genuine multiscattering matrix M which is of Korringa–Kohn–Rostoker-type and which represents the coherent multidisk aspect of the n-disk scattering. Our quantum-mechanical calculation is well-defined at every step, as the on-shell T -matrix and the multiscattering kernel M − 1 are shown to be trace-class. The multiscattering determinant can be organized in terms of the cumulant expansion which is the defining prescription for the determinant over an infinite, but trace-class matrix. The quantum cumulants are then expanded by traces which, in turn, split into quantum itineraries or cycles. These can be organized by a simple symbolic dynamics. The semiclassical reduction of the coherent multiscattering part takes place on the level of the quantum cycles. We show that the semiclassical analog of the mth quantum cumulant is the mth curvature term of the semiclassical zeta function. In this way quantum mechanics naturally imposes the curvature regularization structured by the topological (not the geometrical) length of the pertinent periodic orbits onto the semiclassical zeta function. However, since the cumulant limit m→∞ and the semiclassical limit, ℏ→0 or (wave number) k→∞, do not commute in general, the semiclassical analog of the quantum multiscattering determinant is a curvature expanded semiclassical zeta function which is truncated in the curvature order. We relate the order of this truncation to the topological entropy of the corresponding classical system. We show this explicitly for the three-disk scattering system and discuss the consequences of this truncation for the semiclassical predictions of the scattering resonances. We show that, under the above mentioned truncations in the curvature order, unitarity in n-disk scattering problems is preserved even at the semiclassical level. Finally, with the help of cluster phase shifts, it is shown that the semiclassical zeta function of Gutzwiller and Voros has the correct stability structure and is superior to all the competitor zeta functions studied in the literature.

Characterization of Translation Planes by Orbit Lengths

The Desarguesian, Hall, and Hering translation planes of order q2 are characterized as exactly those translation planes of odd order with spreads in PG (3,q) that admit a linear collineation group with infinite orbits one of length q+1 and i of length (q-q) /i for i=1 or 2.

High orders of Weyl series: resurgence for odd balls

The semiclassical Weyl series for the d-dimension, unit-radius sphere quantum billiard is studied. A conjecture of Berry and Howls (1994 447 527-55) on the late terms of such series for two-dimensional billiards is seen to survive for general (integer) dimension. The conjecture postulates a leading-order, factorial-on-power approximation for the late terms of the Weyl series in terms of the length of a periodic orbit of the classical system. The expansions manifest a difference between odd and even dimensions. The dominating orbit is the diametral, length-4 path in the even-dimension spheres, echoing the known result for the circle billiard. However, when d is odd, it is the next-longest orbit. This surprise can be traced to an `accidental' symmetry in a postulated hyperasymptotic remainder term. Higher-order asymptotic correction terms are found confirming the resurgent link of the Weyl series to the low orders of the oscillatory periodic orbit corrections. From the structure of the latter, it is possible to make further conjectures on the late terms of the periodic orbit corrections themselves. A factorial-on-power behaviour is also found, but now involving the differences between p-bounce orbits and associated whispering-gallery modes.

Infinity-free semiclassical evaluation of Casimir effects

Electromagnetic Casimir energies are a quantum effect proportional to $\ensuremath{\Elzxh}.$ We show that in certain cases one can obtain an exact semiclassical expression for them that depends only on periodic orbits of the associated classical problem. A great merit of the approach is that infinities never appear if one considers only periodic orbits that make contact with the boundary surface. This notion is made more precise by classifying the closed orbits in a phase space with boundaries and identifying the classes that contribute to Casimir effects. A semiclassical evaluation of the path integral gives a systematic expansion of the Casimir energy in terms of the lengths of classical periodic orbits. For some simple geometries the semiclassical expansion can be summed and explicitly shown to reproduce known results. This is the case, for example, for the force per unit area between parallel plates at a separation small compared to their linear dimensions. A more interesting example for our purposes is the closely related problem of the force on a conducting sphere arbitrarily close to a conducting wall. We provide a rigorous proof of Derjaguin's result for the leading contribution to the force. The semiclassical approach, which has never been truly exploited in Casimir studies, is relatively simple and transparent, and should have a wide range of applications. The methods presented, however, do not apply to cases where diffraction is important; diffraction can, in principle, also be described within this semiclassical approach, but its implementation presents some technical problems. In cases where diffraction is important, conventional methods of calculating the Casimir energy may often be simpler.

Initial dual-sweep streak camera measurements on the Duke storage ring OK-4 UV/visible FEL

Initial measurements at the Duke storage ring free electron laser (FEL) of both spontaneous emission radiation (SER) from the OK-4 and FEL radiation outcoupled from the oscillator cavity have been done with a dual-sweep streak camera. In the SER case, ANL's Hamamatsu C5680 streak camera was used to characterize stored electron-beam bunch lengths versus rf gap voltage and single-bunch current. Electron-beam bunch lengths were observed in a range from 50 to 300 ps (FWHM). Particle beam energies of 270 and 500 MeV in the storage ring (SR) were used with the OK-4 to generate SER from 500 nm to less than 200 nm. Sensitivity to single-bunch, single-turn SER was shown down to ∼4 μA beam current at λ=450 nm. By operating in the synchroscan mode and at bunch currents less than 1 mA, beam orbit length vs. the resonator round-trip time was investigated by detecting the relative arrival time of the second pass outcoupled from the resonator at λ≈200 nm. These measurements were used in support of first lasing in the UV-visible regime in November 1996. The dual-sweep streak mode was then used to monitor lasing intensity variations on the 1 and 50 ms time scales, to measure a lasing pulse length as short as ∼3 ps ( σ) at λ=388 nm at low beam current, and to track the orbit length detuning curve.

Quantal–Classical Duality and the Semiclassical Trace Formula

We consider Hamiltonian systems which can be described both classically and quantum mechanically. Trace formulas establish links between the energy spectra in the quantum description and the spectrum of actions of periodic orbits in the Newtonian description. This is the duality which we investigate in the present paper. The duality holds for chaotic as well as for integrable systems. Billiard systems are a very convenient paradigm and we use them for most of our discussions. However, we also show how to transcribe the results to general Hamiltonian systems. In billiards, it is natural to think of the quantal spectrum (eigenvalues of the Helmholtz equation) and the classical spectrum (lengths of periodic orbits) as two manifestations of the properties of the billiard boundary. The trace formula express this link since it can be thought of as a Fourier transform relation between the classical and the quantum spectral densities. It follows that the two-point statistics of the quantal spectrum is related to the two-point statistics of the classical spectrum via a double Fourier transform. The universal correlations of the quantal spectrum are well known; consequently one can deduce the classical universal correlations. In particular, an explicit expression for the scale of the classical correlations is derived and interpreted. This allows a further extension of the formalism to the case of complex billiard systems, and in particular to the most interesting case of diffusive system. The effects of symmetry and symmetry-breaking are also discussed. The concept of classical correlations allows a better understanding of the so-called diagonal approximation and its breakdown. It also paves the way towards a semiclassical theory that is capable of global description of spectral statistics beyond the breaktime. An illustrative application is the derivation of the disorder-limited breaktime in the case of a disordered chain, thus obtaining a semiclassical theory for localization. We also discuss other applications such as the two-cell systems, periodic chains, and localization theory in more than one dimension. A numerical study of classical correlations in the case of the 3D Sinai billiards is presented. Here it is possible to test some assumptions and conjectures that underlie our formulation. In particular we gain a direct understanding of specific statistical properties of the classical spectrum, as well as their semiclassical manifestation in the quantal spectrum. We also analyze the spectral duality for integrable systems, and show that the Poissonian statistics of both the classical and the quantum spectra can be traced to the same origin.