Spectral problems for variational inequalities with discontinuous operators

Some spectral problems for variational inequalities with discontinuous nonlinear operators are considered. The variational method is used to prove the assumption that such problems are solvable. The general results are applied to the corresponding elliptic variational inequalities with discontinuous nonlinearities.     

Quantum Hamiltonians with Weak Random Abstract Perturbation. I. Initial Length Scale Estimate

We study random Hamiltonians on finite-size cubes and waveguide segments of increasing diameter. The number of random parameters determining the operator is proportional to the volume of the cube. In the asymptotic regime where the cube size, and consequently the number of parameters as well, tends to infinity, we derive deterministic and probabilistic variational bounds on the lowest eigenvalue, i.e., the spectral minimum, as well as exponential off-diagonal decay of the Green function at energies above, but close to the overall spectral bottom.     

Koplienko Spectral Shift Function on the Unit Circle

The note shows that, for every pair of contractions, a trace formula of Koplienko holds with a summable spectral shift function, see Theorem 1 below. This answers a question by F. Gesztesy, A. Pushnitski, B. Simon (see Zh Mat Fiz Anal Geom 4(1):63–107, 202, 2008, Open Question 11.2).     

A Compact Cauchy-Kovalevskaya Extension Formula in Discrete Clifford Analysis

We prove a compact expression for the Cauchy-Kovalevskaya extension in the setting of discrete Clifford analysis; it seamlessly simplifies to the well-know exponential formula in the continuous setting when the discrete step size tends to zero.     

Piezoelectric property improvement of polyethylene ferroelectrets using postprocessing thermal‐pressure treatment

In this work, biaxially stretched polymer foams with well‐defined cellular structures were prepared from polyethylene via blown‐film extrusion and subjected to corona charging to produce a piezoelectric response. The charging parameters were first optimized in terms of charging voltage and needle distance, as well as the gas type and pressure to investigate their effect on the piezoelectric coefficient (d₃₃). The results show that samples charged under nitrogen (N₂) at 100 kPa had better d₃₃ coefficient than those charged under ambient air or N₂ at 20 kPa. Moreover, 2 different thermal pressure treatments were imposed to obtain an optimized eye‐like cellular structure with different cell aspect ratios (AR). The results showed that when the cells were elongated in both the longitudinal and transverse directions (higher AR), higher d₃₃ coefficients were achieved. From all the samples produced, the best results were obtained for a longitudinal aspect ratio (AR‐L) of 7.1, a transversal aspect ratio (AR‐T) of 4.6, and a relative foam density of 0.52 leading to a d₃₃ coefficient of 935 pC/N. This coefficient was further increased using reverse charging and multilayered films, reaching a maximum of 2550 pC/N. This value is much higher than typical ones reported so far for any polyethylene and polypropylene ferroelectrets. These results could increase the use of polyethylene in piezoelectric applications as these materials are very attractive for the large‐scale production of electret‐based sensors and transducers due to their low cost and easy processing.     

Helicity and other conservation laws in perfect fluid motion

In this review paper, we discuss helicity from a geometrical point of view and see how it applies to the motion of a perfect fluid. We discuss its relation with the Hamiltonian structure, and then its extension to arbitrary space dimensions. We also comment about the existence of additional conservation laws for the Euler equation, and its unlikely integrability in Liouville's sense.