On tt-extensions of the Hankel determinants of certain automatic sequences

In 1998, Allouche, Peyri\`ere, Wen and Wen considered the Thue--Morse sequence, and proved that all the Hankel determinants of the period-doubling sequence are odd integral numbers. We speak of $t$-extension when the entries along the diagonal in the Hankel determinant are all multiplied by~$t$. Then we prove that the $t$-extension of each Hankel determinant of the period-doubling sequence is a polynomial in $t$, whose leading coefficient is the {\it only one} to be an odd integral number. Our proof makes use of the combinatorial set-up developed by Bugeaud and Han, which appears to be very suitable for this study, as the parameter $t$ counts the number of fixed points of a permutation. Finally, we prove that all the $t$-extensions of the Hankel determinants of the regular paperfolding sequence are polynomials in $t$ of degree less than or equal to $3$

Existence and uniqueness of global weak solutions to a Cahn-Hilliard-Stokes-Darcy system for two phase incompressible flows in karstic geometry

We study the well-posedness of a coupled Cahn-Hilliard-Stokes-Darcy system which is a diffuse-interface model for essentially immiscible two phase incompressible flows with matched density in a karstic geometry. Existence of finite energy weak solution that is global in time is established in both 2D and 3D. Weak-strong uniqueness property of the weak solutions is provided as well

When bigger isn’t better : bailouts and bank behaviour

Lending retail deposits to SMEs and household borrowers may be the traditional role of commercial banks: but banking in Britain has been transformed by increasing consolidation and by the lure of high returns available from wholesale Investment activities. With appropriate changes to the baseline model of commercial banking in Allen and Gale (2007), we show how market power enables banks to collect „seigniorage‟; and how „tail risk‟ investment allows losses to be shifted onto the taxpayer. In principle, the high franchise values associated with market power assist regulatory capital requirements to check risk-taking. But when big banks act strategically, bailout expectations can undermine these disciplining devices: and the taxpayer ends up „on the hook‟- as in the recent crisis. That structural change is needed to prevent a repeat seems clear from the Vickers report, which proposes to protect the taxpayer by a „ring fence‟separating commercial and investment banking