Nonlinear Instability in Two Dimensional Ideal Fluids: The Case of a Dominant Eigenvalue

It is proved that any steady 2 dimensional ideal fluid flow is nonlinearly unstable with respect to L ² growth in the velocity, provided there exists an eigenvalue for the linearised Euler equation with Re>. Here is the maximal Lyapunov exponent of the steady flow.     

Attractors of nonautonomous dynamical systems and an estimation of their dimension

Translated from Matematicheskie Zametki, Vol. 51, No. 6, pp. 141–143, June, 1992.     

A tensor-based dictionary learning approach to tomographic image reconstruction

We consider tomographic reconstruction using priors in the form of a dictionary learned from training images. The reconstruction has two stages: first we construct a tensor dictionary prior from our training data, and then we pose the reconstruction problem in terms of recovering the expansion coefficients in that dictionary. Our approach differs from past approaches in that (a) we use a third-order tensor representation for our images and (b) we recast the reconstruction problem using the tensor formulation. The dictionary learning problem is presented as a non-negative tensor factorization problem with sparsity constraints. The reconstruction problem is formulated in a convex optimization framework by looking for a solution with a sparse representation in the tensor dictionary. Numerical results show that our tensor formulation leads to very sparse representations of both the training images and the reconstructions due to the ability of representing repeated features compactly in the dictionary.     

Synthesis and characterization of chromium and molybdenum(0) complexes bearing the (pentafluoroethyl)diphenylphosphine (pfepp) ligand

The synthesis and characterization of two Group VI pentacarbonyl complexes (pfepp)M(CO)₅ (M = Cr 1, Mo 2; pfepp = PPh₂C₂F₅) are reported. Thermolysis of M(CO)₆ and pfepp in refluxing octane afforded 1 and 2 in moderate yields. These complexes were completely characterized by multinuclear NMR, IR and elemental analysis. X-ray structures for these complexes indicated they were isostructural, crystalizing in triclinic unit cells with four molecules per asymmetric unit. A comparison of the bond lengths in 1 and 2 to other (L)M(CO)₅ complexes showed a relationship between the M-C_ax bond length and the electronic influence of the phosphine ligand, and establishes the pfepp ligand as neither electron-rich nor electron-poor. A comparison of IR data with other (L)M(CO)₅ complexes also indicates the pfepp ligand is electronically neutral, with an electronic influence that approximates phosphites.     

On the Mattila Integral Associated with Sign Indefinite Measures

In order to quantitatively illustrate the rôle of positivity in the Falconer distance problem, we construct a family of sign indefinite, compactly supported measures in \({\Bbb R}^d\), such that their Fourier transform and Fourier energy of dimension \(s \in (0, d)\) are uniformly bounded. However, the Mattila integral, associated with the Falconer distance problem for these measures is unbounded in the range \(0 < s < \frac{d^2}{2d-1}\).     

Algebraic cobordisms of a Pfister quadric

In this article we compute the ring of algebraic cobordismsof a Pfister quadric. This is a rare example of a non-cellularvariety where such a computation is known. We consider the algebraiccobordisms * of Levine and Morel, as well as the MGL^{2*, *} ofVoevodsky. The methods of computation in these two cases arequite different. However, the results do agree (which supportsthe expectation that the two theories actually coincide). Weshow that the restriction homomorphism in our case is injectivefor any field extension E/F.     

The functions operating on multiplier algebras

Let 1 < p < ∞ with p ≠ 2. Let G denote one of the groups $\text{T}$ⁿ, $\text{R}$ⁿ, or $\text{Z}$ⁿ. We show that only entire functions operate in certain algebras of multipliers on L_p(G).     

A parameter-choice method that exploits residual information

Most algorithms for choosing the regularization parameter in a discrete ill-posed problem are based on the norm of the residual vector. In this work we propose a different approach that seeks to use all the information available in the residual vector, and we show how to use statistical tools and fast Fourier transforms to extract this information efficiently. This approach leads to a computationally inexpensive parameter-choice rule based on the normalized cumulative periodogram, which is particularly suited for large-scale problems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)