High energy femtosecond supercontinuum light generation in large mode area photonic crystal fiber

A large mode area photonic crystal fiber (LMA PCF) with an effective area of 180 μm² is used to generate a high energy, micro-joule range, flat, octave spanning supercontinuum (SC) extending from ~ 600 nm to ~ 1720 nm. A train of femtosecond pulses from a widely-tunable parametric amplifier pumped by a Ti:Sapphire regenerative amplifier system are coupled into a 20 cm length of LMA PCF generating a SC of 1.4 μJ energy. We present an experimental study of the high energy SC as a function of the input power and the pumping wavelength. The spectrum obtained at a pump wavelength of 1260 nm presents spectral flatness variation less than 12 dB over more than 1.1 octave bandwidth. The physical processes behind the SC formation are described in the normal and the anomalous dispersion regions. Our experimental results are successfully compared with the numerical solution of the nonlinear Schrödinger equation.     

Simulated annealing for Lévy-driven jump-diffusions

We consider a one-dimensional dynamical system driven by a vector field −U^′$-U^{\prime }$, where U$U$ is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a stable symmetric non-Gaussian Lévy process whose scale decreases as a power function of time. It turns out that the limiting behaviour of the perturbed dynamical system is different for slow and fast decrease rates of the noise intensity. As opposed to the well-studied Gaussian case, the support of the limiting law is not located in the set of global minima of U$U$.     

Differentiability of Spectral Functions for Relativistic <Emphasis Type="Italic">α</Emphasis>-Stable Processes with Application to Large Deviations

We consider the relativistic α-stable process, a pure jump Markov process generated by \(\mathcal{H}^{\alpha} = (-\Delta + m^{2/\alpha})^{\alpha /2}-m\). Let ?C(λ) be the bottom of spectrum of Schrödinger type operator \(\mathcal{H}^{\lambda \mu} = \mathcal{H}^{\alpha} - \lambda \mu\), where μ is a signed Kato measure. We prove the differentiability of C(λ). As an application of it, we establish a large deviation principle for the additive functional \(A_t^{\mu}\) corresponding to the measure μ.     

Another construction for large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection , where every is a KTS(v) and all form a partition of all triples on X. In this article, we give a new construction for LKTS(6v + 3) via OLKTS(2v + 1) with a special property and obtain new results for LKTS, that is there exists an LKTS(3v) for , where p, q ≥ 0, r _i , s _j ≥ 1, q _i is a prime power and mod 12.      

Adaptive wavelet algorithms for elliptic PDE's on product domains

With standard isotropic approximation by (piecewise) polynomials of fixed order in a domain , the convergence rate in terms of the number of degrees of freedom is inversely proportional to the space dimension . This so-called curse of dimensionality can be circumvented by applying sparse tensor product approximation, when certain high order mixed derivatives of the approximated function happen to be bounded in . It was shown by Nitsche (2006) that this regularity constraint can be dramatically reduced by considering best -term approximation from tensor product wavelet bases. When the function is the solution of some well-posed operator equation, dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms, assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible. Applying piecewise smooth wavelets, we verify this compressibility for general, non-separable elliptic PDEs in tensor domains. Applications of the general theory developed include adaptive Galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust, i.e., independent of the scale parameters, resp. of the size of the anisotropy.

     

On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game

In the framework of the game-theoretic probability of Shafer and Vovk it is of basic importance to construct an explicit strategy weakly forcing the strong law of large numbers in the bounded forecasting game. We present a simple finite-memory strategy based on the past average of Reality’s moves, which weakly forces the strong law of large numbers with the convergence rate of ${O(\sqrt{\log n/n})}In the framework of the game-theoretic probability of Shafer and Vovk it is of basic importance to construct an explicit strategy weakly forcing the strong law of large numbers in the bounded forecasting game. We present a simple finite-memory strategy based on the past average of Reality’s moves, which weakly forces the strong law of large numbers with the convergence rate of . Our proof is very simple compared to a corresponding measure-theoretic result of Azuma (T?hoku Mathematical Journal, 19, 357–367, 1967) on bounded martingale differences and this illustrates effectiveness of game-theoretic approach. We also discuss one-sided protocols and extension of results to linear protocols in general dimension.     

Large solutions to non-monotone semilinear elliptic systems

We present the existence of entire large positive radial solutions for the non-monotonic system Δu=p(|x|)g(v), Δv=q(|x|)f(u) on Rn where n?3. The functions f and g satisfy a Keller-Osserman type condition while nonnegative functions p and q are required to satisfy the decay conditions and . Further, p and q are such that min(p,q) does not have compact support.     

The economics of forestry and a set-valued turnpike of the classical type

In recent work, the authors set classical turnpike theory in the context of the economics of forestry, as developed by Mitra and Wan, and presented two far-reaching results. In this paper, we present a conceptual generalization that takes this theory and configures it around a set in the space of forest configurations rather than around the golden-rule forest configuration. Our set-valued analysis hinges on periodicity and yields the earlier results as corollaries under a non-interiority condition on the felicity function that shrinks the set to the point. The question that we pose, and answer, has obvious relevance to more general contexts and, in particular, to turnpike theory as developed by Samuelson, Gale, McKenzie, and their followers.     

Large eddy simulation of coherent structures in rectangular methane non-premixed flame

Large eddy simulation of a three-dimensional spatially developing transitional free methane non-premixed flame is performed. The solver of the governing equations is based upon a projection method. The Smagorinsky model is utilized for the turbulent subgrid scale terms. A global reaction mechanism is applied for the simulation of methane/air combustion. Simulation results clearly illustrate the coherent structure of the rectangular non-premixed flame, consisting of three distinct zones in the near field. Periodic characteristics of the coherent structures in the rectangular non-premixed flame are discussed. The predicted structure of the flame is in good agreement with the experimental results. Distributions of species concentrations across the flame surfaces are illustrated and typical flame structures in the far field are analyzed. Local mass fraction analysis and flow visualization indicate that the black spots of the flames are due to strong entrainment of oxygen into the central jet by streamwise vortices, and breaking up of the flame is caused by an enormous amount of entrainment of streamwise vortices as well as stretching of spanwise vortices at the bottom of the flame.     

Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer n and a prime p, let . In this paper, we present a construction for vertex-transitive self-complementary k-uniform hypergraphs of order n for each integer n such that for every prime p, where ?=max{k₍₂₎,(k−1)₍₂₎}, and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k=2^? or k=2^?+1 due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary k-hypergraphs which have prime order p in the case where k=2^? or k=2^?+1 and , and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order , up to isomorphism.