Ultrabroadband laser using prism-based “spatially-dispersive” resonator

An ultrabroadband (up to 35 nm FWHM) generation by a pulsed dye laser is reported. The principle of laser operation is based on a recently proposed cavity configuration, which we have termed spatially-dispersive resonator. By means of a prism pair this spreads the various spectral components into different spatial areas in the active medium and thus reduces the competition between them. A generation of a spectrum containing a number of equally spaced lines is also demonstrated.     

Upper-Branch Instability of Flow in Pipes of Large Aspect Ratio with Three-Dimensional Nonlinear Viscous Critical Layers

We consider the upper-branch neutral stability of flow in pipesof large aspect ratio, basically extending the work of F. T.Smith to the nonlinear regime. The inclusion of weak nonlinearityleads to an eigenproblem whose solution depends on the propertiesof three-dimensional nonlinear critical layers. Two specialcases are considered. The first is for very small amplitude perturbations, where R is a Reynolds numberbased on the height of the tube and which is assumed large.Then a fully analytical solution of the three-dimensional criticallayers is possible, from which the linear results of Smith maybe deduced. The second case studied is that of flow in a rectangularpipe, where a solution of the nonlinear critical layer problemcan be obtained. Further analysis of neutral modes in this lattercase suggests the possible existence, inter alia, of neutralmodes for finite aspect ratio tubes. These modes depend on thescaled amplitude and have O(1) wavespeeds.     

On the dynamics of cosmological particle production

The time evolution of the particle number density and spectrum of massive scalar particles, coupled conformally to a classical Friedman-Robertson-Walker gravitational field is followed numerically. Not only for a pure radiation cosmos, but also for one with an inflationary interlude, the density of particles produced remains constant in time. This results in a constant equivalent temperature scale up to the Compton time of the massive particles, and opens the possibility that during a sufficiently long inflation the energy density of the particles produced can exceed that of the radiation background.     

The Asymptotic Solution of a Connection Problem of a Second Order Ordinary Differential Equation

The solutions of the equation are discussed in the limit ρ → 0. The solutions which oscillate about ? |t| as t → ∞ have asymptotic expansions whose leading terms are where Ã₊, , Ã_?, and are constants. The connection problem is to determine the asymptotic expansion at + ∞. In other words, we wish to find (Ã₊, ) as functions of Ã_? and The nonlinear solutions with Ã_± not small are analyzed by using the method of averaging. It is shown that this method breaks down for small amplitudes. In this case a solution can be obtained on [0, ∞) as a small amplitude perturbation about the exact nonoscillating solution W(t) whose asymptotic expansion is This is a solution of (1) which corresponds to Ã₊ ≡ 0 in (2). A quantity which determines the scale of the small amplitude response is ?W'(0). This quantity is found to be exponentially small. The determination of this constant is shown to reduce to a solution of the equation for the first Painlevé transcendent. The asymptotic behavior of the required solution is determined by solving an integral equation.     

Optimal birth control of population dynamics

The authors studied optimal birth control policies for an age-structured population of McKendrick type which is a distributed parameter system involving 1st order partial differential equations with nonlocal bilinear boundary control. The functional analytic approach of Dubovitskii and Milyutin is adopted in the investigation. Maximum principles for problems with a free end condition and fixed final horizon are developed, and the time optimal control problems, the problem with target sets, and infinite planning horizon case are investigated.     

The evaluation of Cauchy principal value integrals involving unknown poles

A method is described for the evaluation of a Cauchy principal value integral of the formf ₀ ^p f(t)dt, wheref is analytic in the interval [0,p] except at a simple pole at an unknown point in (0,p), with an unknown residue. The method is based on the trapezoidal rule.     

Crystal structures of dimethylsuccinate and dimethyloxalate: Carbonyl group orientation for C13 chemical shielding tensor studies

The crystal structures of dimethylsuccinate (DMS) and dimethyloxalate (DMO) have been determined to facilitate the determination of the C-13 chemical shielding tensors of the carbonyl carbon in esters. Crystals of DMS are monoclinic, space groupC2/c,Z=4,a=13.154(4),b=6.156(1),c=9.363(4)Å,=98.53(3)°. The structure was solved by direct methods and refined by leastsquares procedures to giveR=0.071 for 932 observed data. Crystals of DMO are monoclinic space group,P2₁/n,Z=2, witha=3.891(1),b=11.879(2),c=6.213(2) Å,=103.32(2)°. The structure is the same (within experimental error) as that reported by Dougill and Jeffrey (1953) and refined to giveR=0.074 for 395 observed data.     

Some homological properties of commutative semitrivial ring extensions

LetR be a commutative ring with 1 andM anR-module. If:M _R MR is anR-module homomorphism satisfying(mm)=(mm) and(mm)m=m(mm), the additive abelian groupRM becomes a commutative ring, if multiplication is defined by (r,m)(r,m)=(rr+(mm),rm+rm). This ring is called the semitrivial extension ofR byM and and it is denoted byR M. This generalizes the notion of a trivial extension and leads to a more interesting variety of examples. The purpose of this paper is to studyR M; in particular, we are interested in some homological properties ofR M as that of being Cohen-Macaulay, Gorenstein or regular. A sample result: Let (R,m) be a local Noetherian ring,M a finitely generatedR-module and Im() m. ThenR M is Gorenstein if and only if eitherRM is Gorenstein orR is Gorenstein,M is a maximal Cohen-Macaulay module andMM ^*, where the isomorphism is given by the adjoint of.     

Truncation error bounds for modified continued fractions with applications to special functions

Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a _n , 1;x ₁), with andnth approximant

TheL <ω-theory of the class of Archimedian real closed fields

For the classA of uncountable Archimedian real closed fields we show that the statement TheL ^<-theory ofA is complete is independent of ZFC. In particular we have the following results:Assuming the Continuum-Hypothesis (CH) is incomplete. Conversely it is possible to build a model of set theory in which is complete and decidable. The latter can also be deduced from the Proper Forcing Axiom (PFA). In this case turns out to be equivalent to the elementary theory of the real numbers (by a quantifier-elimination procedure).Formally: is incomplete. is complete and decidable.