Spectra of Graphs and the Spectral Criterion for Property (T)

For a finite connected graph $X$, we consider the graph $RX$ obtained from $X$ by associating a new vertex to every edge of $X$ and joining by edges the extremities of each edge of $X$ to the corresponding new vertex. We express the spectrum of the Laplace operator on $RX$ as a function of the corresponding spectrum on $X$. As a corollary, we show that $X$ is a complete graph if and only if \lambda_1(RX)>\frac{1}{2}. We give a re-interpretation of the correspondence $X\mapsto RX$ in terms of the right-angled Coxeter group defined by $X$

Isometric group actions on Banach spaces and representations vanishing at infinity

Our main result is that the simple Lie group $G=Sp(n,1)$ acts properly isometrically on $L^p(G)$ if $p>4n+2$. To prove this, we introduce property ({\BP}_0^V), for $V$ be a Banach space: a locally compact group $G$ has property ({\BP}_0^V) if every affine isometric action of $G$ on $V$, such that the linear part is a $C_0$-representation of $G$, either has a fixed point or is metrically proper. We prove that solvable groups, connected Lie groups, and linear algebraic groups over a local field of characteristic zero, have property ({\BP}_0^V). As a consequence for unitary representations, we characterize those groups in the latter classes for which the first cohomology with respect to the left regular representation on $L^2(G)$ is non-zero; and we characterize uniform lattices in those groups for which the first $L^2$-Betti number is non-zero.Comment: 28 page

Quantification of void networks of as-sprayed and annealed nanostructured yttria-stabilized zirconia (YSZ) deposits manufactured by suspension plasma spraying

International audienceSuspension plasma spraying (SPS) allows processing a stabilized suspension of nanometer-sized feedstock particles to form thick (from 20 to 100 μm, average values) deposits. The void content and porous network of such deposits are difficult to quantify (in terms of void and size distributions, anisotropy, etc.) using conventional techniques due to their low resolution. The combination of ultra-small-angle X-ray scattering (USAXS) and helium pycnometry permits to address some of the characteristics of this void network. Deposits of yttria-partially stabilized zirconia (YSZ) were manufactured by plasma spraying a suspension made of solid sub-micrometer-sized particles (50 and 400 nm) with several sets of spray operating parameters. Results indicate that the average void size exhibits the same scale as the solid structure; i.e., nanometer sizes and multimodal size distribution which varies with spray operating parameters. About 90% of voids (by number) exhibit characteristic dimensions smaller than 40 nm. The cumulative void volume fraction of such as-sprayed deposits varies between about 13 and 20%, depending upon operating parameters. The void network architecture evolves also with annealing conditions: the void size distribution evolves toward higher void characteristic dimensions as a result of sintering of smallest voids but the cumulative void content does not decrease significantly

Quantification of void network architectures of suspension plasma-sprayed (SPS) yttria-stabilized zirconia (YSZ) coatings using ultra-small-angle X-ray scattering (USAXS)

International audienceSuspension plasma spraying (SPS) is able to process a stabilized suspension of nanometer-sized feedstock particles to form thin (from 20 to 100m) coatings with unique microstructures. The void (pore) network structure of these ceramic coatings is challenging to characterize and quantify using commonly used techniques due to small sizes involved. Nevertheless, the discrimination of these pores in terms of their size and shape distribution, anisotropy, specific surface area, etc., is critical for the understanding of processing, microstructure, and properties relationships. We will show that one of suitable combinations of techniques providing sufficient detail is ultra-small-angle X-ray scattering (USAXS) and helium pycnometry, combined with scanning electron microscopy (SEM). Yttria-partially stabilized zirconia (YSZ) coatings were manufactured by plasma processing of suspension of particles with average diameter of ∼50 nm. Several sets of spray parameters (plasma gas mixture, spray distance, electric arc intensity, etc.) were used to generate plasma jets with different mass enthalpies and coefficients of thermal transfer and different heat fluxes transferred to the substrate. Free-standing coatings were studied as-sprayed and annealed at 800 and 1100 ◦C for 10 and 100 h (non-constrained sintering). Results indicate that the SPS coatings exhibit nanosized pore microstructure: average void size was about the same size scale as the feedstock size; i.e., nanometer sizes with multimodal void size distribution. About 80% of the pores (by number) exhibited characteristic dimensions smaller than 30 nm. Total void content of as-sprayed SPS coatings varies between 13% and 20%. Most of the voids were found to be opened with only between one-tenth to one-third of voids volume being inaccessible by intrusion (not connected to either surface). During annealing, even at temperatures as low than 800 ◦C, the microstructure transformed: while the total void content did not change significantly, the void size distribution evolved toward larger sizes. This unique void system, together with the nanometer scale of the particulate matrix itself, gave these coatings very low apparent thermal conductivity (in the order of 0.1Wm−1 K−1), as rarefaction effect and phonon scattering mechanisms are very likely emphasized