A reverse Sidorenko inequality

Let $H$ be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph $G$ without isolated vertices, the weighted number of graph homomorphisms $\hom(G, H)$ satisfies the inequality $$\hom(G, H ) \le \prod_{uv \in E(G)} \hom(K_{d_u,d_v}, H )^{1/(d_ud_v)},$$ where $d_u$ denotes the degree of vertex $u$ in $G$. In particular, one has $$\hom(G, H )^{1/|E(G)|} \le \hom(K_{d,d}, H )^{1/d^2}$$ for every $d$-regular triangle-free $G$. The triangle-free hypothesis on $G$ is best possible. More generally, we prove a graphical Brascamp-Lieb type inequality, where every edge of $G$ is assigned some two-variable function. These inequalities imply tight upper bounds on the partition function of various statistical models such as the Ising and Potts models, which includes independent sets and graph colorings. For graph colorings, corresponding to $H = K_q$, we show that the triangle-free hypothesis on $G$ may be dropped; this is also valid if some of the vertices of $K_q$ are looped. A corollary is that among $d$-regular graphs, $G = K_{d,d}$ maximizes the quantity $c_q(G)^{1/|V(G)|}$ for every $q$ and $d$, where $c_q(G)$ counts proper $q$-colorings of $G$. Finally, we show that if the edge-weight matrix of $H$ is positive semidefinite, then $$\hom(G, H) \le \prod_{v \in V(G)} \hom(K_{d_v+1}, H )^{1/(d_v+1)}.$$ This implies that among $d$-regular graphs, $G = K_{d+1}$ maximizes $\hom(G, H)^{1/|V(G)|}$. For 2-spin Ising models, our results give a complete characterization of extremal graphs: complete bipartite graphs maximize the partition function of 2-spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models. These results settle a number of conjectures by Galvin-Tetali, Galvin, and Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture by Kahn.Comment: 30 page

Exponential improvements for superball packing upper bounds

We prove that for all fixed $p > 2$, the translative packing density of unit $\ell_p$-balls in $\mathbb{R}^n$ is at most $2^{(\gamma_p + o(1))n}$ with $\gamma_p < - 1/p$. This is the first exponential improvement in high dimensions since van der Corput and Schaake (1936)

New Clock Comparison Searches for Lorentz and CPT Violation

We present two new measurements constraining Lorentz and CPT violation using the Xe-129 / He-3 Zeeman maser and atomic hydrogen masers. Experimental investigations of Lorentz and CPT symmetry provide important tests of the framework of the standard model of particle physics and theories of gravity. The two-species Xe-129 / He-3 Zeeman maser bounds violations of CPT and Lorentz symmetry of the neutron at the 10^-31 GeV level. Measurements with atomic hydrogen masers provide a clean limit of CPT and Lorentz symmetry violation of the proton at the 10^-27 GeV level.Comment: 11 pages, 5 figures. To appear in the Proceedings of the 3rd International Symposium on Symmetries in Subatomic Physic

You Can Lead a Horse to Water: Mapping Seasonal Water Resources to Predict Wild Horse Movements on Utah Rangelands

All wild horse herd management areas in Utah overlap BLM grazing allotments. Although horses and cattle have similar dietary habits, both species rely heavily on predictable water sources during dry periods. The concentration of wildlife and livestock in mesic areas during droughts can become problems for farmers and livestock producers. We aimed to map the annual distribution of temporary surface water across Utah that cattle, horses, and wildlife could use. Herein we analyzed an 18-year record of satellite imagery to create a statewide map of seasonal surface-water availability for agricultural and wildlife management purposes

Competition between pairing and ferromagnetic instabilities in ultracold Fermi gases near Feshbach resonances

We study the quench dynamics of a two-component ultracold Fermi gas from the weak into the strong interaction regime, where the short time dynamics are governed by the exponential growth rate of unstable collective modes. We obtain an effective interaction that takes into account both Pauli blocking and the energy dependence of the scattering amplitude near a Feshbach resonance. Using this interaction we analyze the competing instabilities towards Stoner ferromagnetism and pairing.Comment: 4+epsilon pages, 4 figure