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The best bounds of the form $B(\alpha,\beta,\gamma,x)=(\alpha+\sqrt{\beta^2+\gamma^2 x^2})/x$ for ratios of modified Bessel functions are characterized: if $\alpha$, $\beta$ and $\gamma$ are chosen in such a way that $B(\alpha,\beta,\gamma,x)$ is a sharp approximation for $\Phi_{\nu}(x)=I_{\nu-1} (x)/I_{\nu}(x)$ as $x\rightarrow 0^+$ (respectively $x\rightarrow +\infty$) and the graphs of the functions $B(\alpha,\beta,\gamma,x)$ and $\Phi_{\nu}(x)$ are tangent at some $x=x_*>0$, then $B(\alpha,\beta,\gamma,x)$ is an upper (respectively lower) bound for $\Phi_{\nu}(x)$ for any positive $x$, and it is the best possible at $x_*$. The same is true for the ratio $\Phi_{\nu}(x)=K_{\nu+1} (x)/K_{\nu}(x)$ but interchanging lower and upper bounds (and with a slightly more restricted range for $\nu$). Bounds with maximal accuracy at $0^+$ and $+\infty$ are recovered in the limits $x_*\rightarrow 0^+$ and $x_*\rightarrow +\infty$, and for these cases the coefficients have simple expressions. For the case of finite and positive $x_*$ we provide uniparametric families of bounds which are close to the optimal bounds and retain their confluence properties.

We construct a Lipschitz-free space that is locally almost square but not weakly almost square; this is the first example of such a Banach space. We also prove a result, which indicates that geodesic metric spaces are a potential metric characterization for weakly almost square Lipschitz-free spaces. Lastly, we prove that a Lipschitz-free space can not have the symmetric strong diameter $2$ property.

International audience; The paper is concerned with the homogenization of the equations describing the flow of a ferrofluid through a heterogeneous porous medium $\Omega$ in the presence of an applied magnetic field. We discuss two models where the magnetization $M$ is parallel to the magnetic field $H$. In the first one $M$ and $H$ satisfy the relation$M=\lambda_0 \, {1}_{\Omega_f} H\, \mbox{ in } \Omega,$where $\lambda_0$ is a positive constant and ${1}_{\Omega_f}$ is the characteristic function of $\Omega_f$ (the pore space). In the second model, $M$ and $H$ satisfy the Langevin magnetization law$M= M_s \frac{L(b_1\, |H|)}{|H|} {1}_{\Omega_f} \, H,$where ${L}$ is the Langevin function given by $L(x)= \frac{1}{\tanh x} - \frac{1}{ x}$,$M_s$ is the saturation magnetization and $b_1$ is a positive physical constant.The velocity and the pressure satisfy the Stokes equation with a Kelvin magnetic force. We perform the homogenization of the equations of each of the two models. Using the two-scale convergence method, we rigorously derive the homogenized equation for the magnetic potential and determine the asymptotic limit of the magnetization. Then we rigorously derive a Darcy law.

The semi-group of weighted composition operators $(W_n)_{n\geq 1}$ where \[ W_nf(z)=(1+z+\ldots+z^{n-1})f(z^n) \] on the classical Hardy-Hilbert space $H^2$ of the open unit disk is related to the Riemann Hypothesis (RH) (see \cite{Waleed}). The semigroup $(W_n)_{n\geq 1}$ is also closely related to the Invariant Subspace Problem (ISP) and the Periodic Dilation Completeness Problem (PDCP). We obtain results on cyclic vectors, spectra, invariant and reducing subspaces. In particular, we show that several basic questions related to the semigroup $(W_n)_{n\geq 1}$ are equivalent to the RH and provide generalizations of the B\'aez-Duarte criterion for the RH (see \cite{Baez-Duarte})., Comment: 10 pages

The ergodic properties of the shift on both full and $m$-truncated $t$-free $C^*$-algebras are analyzed. In particular, the shift is shown to be uniquely ergodic with respect to the fixed-point algebra. In addition, for every $m\geq 1$, the invariant states of the shift acting on the $m$-truncated $t$-free $C^*$-algebra are shown to yield a $m+1$-dimensional Choquet simplex, which collapses to a segment in the full case. Finally, the spectrum of the position operators on the $m$-truncated $t$-free Fock space is also determined., Comment: 15 pages

We consider Schrödinger equations with real quadratic Hamiltonians, for which the Wigner distribution of the solution at a given time equals, up to a linear coordinate transformation, the Wigner distribution of the initial condition. Based on Hardy's uncertainty principle for the joint time-frequency representation, we prove a uniqueness result for such Schrödinger equations, where the solution cannot have strong decay at two distinct times. This approach reproduces known, sharp results for the free Schrödinger equation and the harmonic oscillator, and we also present an explicit scheme for quadratic systems based on positive definite matrices.