Information percolation and cutoff for the random‐cluster model

We consider the random‐cluster model (RCM) on with parameters p∈(0,1) and q ≥ 1. This is a generalization of the standard bond percolation (with edges open independently with probability p) which is biased by a factor q raised to the number of connected components. We study the well‐known Fortuin‐Kasteleyn (FK)‐dynamics on this model where the update at an edge depends on the global geometry of the system unlike the Glauber heat‐bath dynamics for spin systems, and prove that for all small enough p (depending on the dimension) and any q>1, the FK‐dynamics exhibits the cutoff phenomenon at with a window size , where λ_∞ is the large n limit of the spectral gap of the process. Our proof extends the information percolation framework of Lubetzky and Sly to the RCM and also relies on the arguments of Blanca and Sinclair who proved a sharp mixing time bound for the planar version. A key aspect of our proof is the analysis of the effect of a sequence of dependent (across time) Bernoulli percolations extracted from the graphical construction of the dynamics, on how information propagates.     

Perturbative manifolds and the Noether generators of nth-order Poisson equations

A manifold that contains small perturbations will induce a perturbed partial differential equation. The partial differential equation that we select is the Poisson equation – in order to explore the interplay between the geometry of the manifold and the perturbations. Specifically, we show how the problem of symmetry determination, for higher-order perturbations, can be elegantly expressed via geometric conditions.     

Two-dimensional vortex sheets for the nonisentropic Euler equations: Nonlinear stability

We show the short-time existence and nonlinear stability of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on the weakly linear stability result of Morando and Trebeschi (2008) [20]. The missing normal derivatives are compensated through the equations of the linearized vorticity and entropy when deriving higher-order energy estimates. The proof of the resolution for this nonlinear problem follows from certain a priori tame estimates on the effective linear problem in the usual Sobolev spaces and a suitable Nash–Moser iteration scheme.     

General non local boundary value problem for second order elliptic equation

In this paper we study a class of second order coefficient operators differential equation with general (possibly non local) boundary conditions. We obtain new results extending those given in a previous paper 1 . Existence, uniqueness and optimal regularity of the strict solution are proved in UMD spaces, using the well‐known Dore–Venni theorem.     

Asymmetric Cross [10+2] Cycloadditions of 2-Alkylidene-1-indanones and Activated Alkenes under Phase-Transfer Catalysis

Isobenzofulvene species are versatile synthons in organic chemistry, which have been employed in diverse challenging higher-order cycloaddition reactions. Here, the first chemoselective and asymmetric cross [10+2] cycloaddition reaction between activated 2-alkylidene-1-indanones and a variety of electron-deficient alkenes has been developed, relying on the in situ generation of dearomative 1-hydroxyl isobenzofulvene anion intermediates under the catalysis of a newly designed bulky cinchona-derived phase-transfer compound. An array of fused frameworks with multifunctionalities were generally furnished in excellent diastereo- and enantioselectivity, even at 1 mol % catalyst loadings.     

Curing behavior and properties of high biosourced epoxy resin blends based on a triepoxy monomer and a tricarboxylic acid hardener from 10-undecenoic acid

Renewable propane-1,2,3-triyl tris(9-(oxiran-2-yl) nonanoate) (EGU, 100 wt% biogenic) and a tricarboxylic acid triglyceride (CGTU) hardener (85.7 wt% biogenic) were synthesized from 10-undecenoic acid (10-UDA) and used to produce epoxy resins with 52–92 wt% biobased carbon. CGTU was prepared by thermally activated thiol-ene coupling of thioglycolic acid onto propane-1,2,3-triyl tris(undec-10-enoate), (GUD) in the absence of solvent. The characterized CGTU was used as a green hardener of blends based on EGU and a conventional bisphenol A-based epoxy pre-polymer (DGEBA) at various mass percentages (0–100 wt%) with an stoichiometric epoxy/acid equivalent ratio. Calorimetric studies revealed higher peak temperature, lower reaction heats, and longer gelation times in resins with high EGU proportion, evidencing the lower reactivity of aliphatic EGU compared with aromatic DGEBA. Cured resins were yellowish transparent rubber-like materials with glass transition temperatures (Tg) varying from −14 °C to −42 °C and tensile strength in the range of 1750 kPa–790 kPa, for 0 and 100 wt % EGU, respectively. The soluble fraction of all resins was less than 4.3%, reflecting a high level of crosslinking. Thermosets with high biobased content showed both UV-light protection and visible light transparency.     

Gaps in the number of generators of monomial Togliatti systems

Let $I_{d,n}?k[x_0,?,x_n]$ be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I_{d,n})$ of $I_{d,n}$ lies in the interval $[2n+1,\left(\begin{array}{c}n+d?1\\ n?1\end{array}\right)]$. In this paper, we prove that for $n\ge 4$ and $d\ge 3$, the integer values in $[2n+3,3n?1]$ cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{d,n})=2n+2$ or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for $n=4$, $d\ge 3$ and $\mu \in [9,\left(\begin{array}{c}d+3\\ 3\end{array}\right)]?\{11\}$ there exists a minimal monomial Togliatti system $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{n,d})=\mu$.