Ergodic,primal convergence in dual subgradient schemes for convex programming,II: the case of inconsistent primal problems

Consider the utilization of a Lagrangian dual method which is convergent for consistent convex optimization problems. When it is used to solve an infeasible optimization problem, its inconsistency will then manifest itself through the divergence of the sequence of dual iterates. Will then the sequence of primal subproblem solutions still yield relevant information regarding the primal program? We answer this question in the affirmative for a convex program and an associated subgradient algorithm for its Lagrange dual. We show that the primal–dual pair of programs corresponding to an associated homogeneous dual function is in turn associated with a saddle-point problem, in which—in the inconsistent case—the primal part amounts to finding a solution in the primal space such that the Euclidean norm of the infeasibility in the relaxed constraints is minimized; the dual part amounts to identifying a feasible steepest ascent direction for the Lagrangian dual function. We present convergence results for a conditional \(\varepsilon \)-subgradient optimization algorithm applied to the Lagrangian dual problem, and the construction of an ergodic sequence of primal subproblem solutions; this composite algorithm yields convergence of the primal–dual sequence to the set of saddle-points of the associated homogeneous Lagrangian function; for linear programs, convergence to the subset in which the primal objective is at minimum is also achieved.     

Inviscid symmetry breaking with non-increasing energy

In a recent article, C. Bardos et al. constructed weak solutions of the three-dimensional incompressible Euler equations which emerge from two-dimensional initial data yet become fully three-dimensional at positive times. They asked whether such symmetry-breaking solutions could also be constructed under the additional condition that they should have non-increasing energy. In this note, we give a positive answer to this question and show that such a construction is possible for a large class of initial data. We use convex integration techniques as developed by De Lellis and Székelyhidi.     

A causal continuous-time stochastic model for the turbulent energy cascade in a helium jet flow

We discuss continuous cascade models and their potential for modelling the energy dissipation in a turbulent flow. Continuous cascade processes, expressed in terms of stochastic integrals with respect to Lévy bases, are examples of ambit processes. These models are known to reproduce experimentally observed properties of turbulence: the scaling and self-scaling of the correlators of the energy dissipation and of the moments of the coarse-grained energy dissipation. We compare three models: a normal model, a normal inverse Gaussian model, and a stable model. We show that the normal inverse Gaussian model is superior to both, the normal and the stable models, in terms of reproducing the distribution of the energy dissipation; and that the normal inverse Gaussian model is superior to the normal model and competitive with the stable model in terms of reproducing the self-scaling exponents. Furthermore, we show that the presented analysis is parsimonious in the sense that the self-scaling exponents are predicted from the one-point distribution of the energy dissipation, and that the shape of these distributions is independent of the Reynolds number.     

Sobolev estimates for the complex green operator on a class of weakly pseudoconvex boundaries

On the boundary of a smooth bounded convex domain in C_n (more generally, a domain that admits a defining funtion that is plurisubharmonic on the boundary), the canonicl (narm-minimizing) solution operator of the -equation, the inverse of □_b (the complex Gereen Operator) and the Szegö projection are continuous is Sobolev norms.     

Unique Identification of Supramolecular Structures in Amyloid Fibrils by Solid‐State NMR Spectroscopy

The fibril structure formed by the amyloidogenic fragment SNNFGAILSS of the human islet amyloid polypeptide (hIAPP) is determined with 0.52 Å resolution. Symmetry information contained in the easily obtainable resonance assignments from solid‐state NMR spectra (see picture), along with long‐range constraints, can be applied to uniquely identify the supramolecular organization of fibrils.

     

Oxo-biodegradable polyolefins show continued and increased thermal oxidative degradation after exposure to light

Samples of polyethylene (PE) and polypropylene (PP) with 2% Renatura™ pro-oxidant additive were subjected to varying times of weathering (UVCON; ISO 4892-3), transferred to dark thermal exposure (at 70 °C) and the subsequent degradation monitored by measuring elongation at break and tensile strength during intervals up to 45 days. The results demonstrate that the oxidative degradation, after initial light exposure, continues more rapidly in the dark thermal conditions and that increased exposure to light increases thermal oxidative degradation.     

Heteroaromatic Tosylates as Electrophiles in Regioselective Mizoroki–Heck‐Coupling Reactions with Electron‐Rich Olefins

Easy and direct : Regioselective Mizoroki–Heck‐coupling reactions using heteroaromatic tosylates as electrophiles were achieved, thus providing direct and easy access to highly functionalized α‐heteroarylvinyl amides and ethers.