Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge

For a supersonic Euler flow past a straight-sided wedge whose vertex angle is less than the extreme angle, there exists a shock-front emanating from the wedge vertex, and the shock-front is usually strong especially when the vertex angle of the wedge is large. In this paper, we establish the L¹ well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose boundary slope function has small total variation, when the total variation of the incoming flow is small. In this case, the Lipschitz wedge perturbs the flow, and the waves reflect after interacting with the strong shock-front and the wedge boundary. We first obtain the existence of solutions in BV when the incoming flow has small total variation by the wave front tracking method and then establish the L¹ stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated from the wedge boundary to develop a Lyapunov functional between two solutions containing strong shock-fronts, which is equivalent to the L¹ norm, and prove that the functional decreases in the flow direction. Then the L¹ stability is established, so is the uniqueness of the solutions by the wave front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained.     

The pebbling threshold of the square of cliques

Given an initial configuration of pebbles on a graph, one can move pebbles in pairs along edges, at the cost of one of the pebbles moved, with the objective of reaching a specified target vertex. The pebbling number of a graph is the minimum number of pebbles so that every configuration of that many pebbles can reach any chosen target. The pebbling threshold of a sequence of graphs is roughly the number of pebbles so that almost every (resp. almost no) configuration of asymptotically more (resp. fewer) pebbles can reach any chosen target. In this paper we find the pebbling threshold of the sequence of squares of cliques, improving upon an earlier result of Boyle and verifying an important instance of a probabilistic version of Graham's product conjecture.     

Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps

A strong solutions approximation approach for mild solutions of stochastic functional differential equations with Markovian switching driven by Lévy martingales in Hilbert spaces is considered. The Razumikhin–Lyapunov type function methods and comparison principles are studied in pursuit of sufficient conditions for the moment exponential stability and almost sure exponential stability of equations in which we are interested. The results of [A.V. Svishchuk, Yu.I. Kazmerchuk, Stability of stochastic delay equations of Itô form with jumps and Markovian switchings, and their applications in finance, Theor. Probab. Math. Statist. 64 (2002) 167–178] are generalized and improved as a special case of our theory.     

Curve forecasting by functional autoregression

This paper deals with the prediction of curve-valued autoregression processes. It develops a novel technique, predictive factor decomposition, for the estimation of the autoregression operator. The technique is based on finding a reduced-rank approximation to the autoregression operator that minimizes the expected squared norm of the prediction error.Implementing this idea, we relate the operator approximation problem to the singular value decomposition of a combination of cross-covariance and covariance operators. We develop an estimation method based on regularization of the empirical counterpart of this singular value decomposition, prove its consistency and evaluate convergence rates.The method is illustrated by an example of the term structure of the Eurodollar futures rates. In the sample corresponding to the period of normal growth, the predictive factor technique outperforms the principal components method and performs on a par with custom-designed prediction methods.     

Rational points of bounded height on projective surfaces

We give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces.     

Global s-solvability and global s-hypoellipticity for certain perturbations of zero order of systems of constant real vector fields

The main purpose of this paper is to show, in the two-dimensional torus, a necessary and sufficient condition in order to certain perturbations of zero order of a system of constant real vector fields to be globally s-solvable. We are also interested in studying its global s-hypoellipticity. We present connections between these global concepts and a priori estimates. We also present two applications of our results for systems of operators with variable coefficients.     

On the distribution tail of an integrated risk model: A numerical approach

We consider an insurance risk process with the possibility to invest the capital reserve into a portfolio consisting of a risky asset and a riskless asset. The stock price is modelled by an exponential Lévy process and the riskless interest rate is assumed to be constant. We aim at the risk assessment of the integrated risk process in terms of a high quantile or the far out distribution tail. We indicate an application to an optimal investment strategy of an insurer.     

Secondary Metabolites from the Fungus Chaetomium brasiliense

Two new depsidones, mollicellins I and J ( 1 and 2 , resp.), and a new chromone, 2‐(hydroxymethyl)‐6‐methylmethyleugenin ( 3 ), along with six known compounds, 4 – 9 , were isolated from the ethyl acetate extract of a solid‐state fermented culture of Chaetomium brasiliense. Their structures were elucidated based on spectroscopic analysis. Mollicellins I and H ( 5 ) exhibited significant growth inhibitory activity against human breast cancer (Bre04), human lung (Lu04), and human neuroma (N04) cell lines with GI₅₀ values between 2.5–8.6 μg/ml.