Proof of the Hénon–Lane–Emden conjecture in R3

We study the Hénon–Lane–Emden conjecture, which states that there is no non-trivial non-negative solution for the Hénon–Lane–Emden elliptic system whenever the pair of exponents is subcritical. By scale invariance of the solutions and Sobolev embedding on $S^{N?1}$, we prove this conjecture is true for space dimension $N=3$; which also implies the single elliptic equation has no positive classical solutions in ${\mathbb{R}}^3$ when the exponent lies below the Hardy–Sobolev exponent, this covers the conjecture of Phan–Souplet [22] for ${\mathbb{R}}^3$.     

On Shimura subvarieties generated by families of abelian covers of P1

We investigate the occurrence of Shimura (special) subvarieties in the locus of Jacobians of abelian Galois covers of ${\mathbb{P}}^1$ in $A_g$ and give classifications of families of such covers that give rise to Shimura subvarieties in the Torelli locus $T_g$ inside $A_g$. Our methods are based on Moonen–Oort works as well as characteristic p techniques of Dwork and Ogus and Monodromy computations.     

Global,decaying solutions of a focusing energy-critical heat equation in R4

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\mathrm{\Delta }u?|u{|}^{2}u$ in ${\mathbb{R}}^4$. We show that solutions emanating from initial data with energy and ${\dot{H}}^1$-norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations.     

Hod up to ADR+Θ is measurable

Suppose M is a transitive class size model of $A{D}_{\mathbb{R}}+\text{“Θ is regular”}$. M is a minimal model of $A{D}_{\mathbb{R}}+\text{“Θ is measurable”}$ if (i) $\mathbb{R},Ord?M$ (ii) there is $\mu \in M$ such that $M?\text{“}\mathit{\mu }\text{ is a normal }\mathbb{R}\text{-complete measure on Θ”}$ and (iii) for any transitive class size $N?M$ such that $\mathbb{R}?N$, $N?\text{“there is no }\mathbb{R}\text{-complete measure on Θ”}$. Continuing Trang's work in [8], we compute HOD of a minimal model of $A{D}_{\mathbb{R}}+\text{“Θ is measurable”}$.     

A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in ${\mathbb{Z}}^d$. Denote by $Z_n\left(z\right)$ the number of particles of generation $n$ located at site $z\in {\mathbb{Z}}^d$. We give the second order asymptotic expansion for $Z_n\left(z\right)$. The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on ${\mathbb{Z}}^d$, which is used in the proof of the main theorem and is of independent interest.     

A pairwise almost disjoint subspace of kN of maximal dimension

We show that if k is an infinite field, then there exists a subspace $W?k^{\mathbb{N}}$ of dimension $|k{|}^{{\mathrm{?}}_0}$, such that no nonzero member of W has infinitely many zeros. This generalizes a result from a paper by Bergman and Nahlus, and partly answers another question from the same paper.     

A regularity theory for quasi-linear Stochastic PDEs in weighted Sobolev spaces

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$-domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain $L_p$ and Hölder estimates of both the solution and its gradient.     

On images of locally finite derivations and E-derivations

Some cases of the LFED Conjecture, proposed by the second author [15], for certain integral domains are proved. In particular, the LFED Conjecture is completely established for the field of fractions $k(x)$ of the polynomial algebra $k[x]$, the formal power series algebra $k[[x]]$ and the Laurent formal power series algebra $k[[x]][x^{?1}]$, where $x=(x_1,x_2,\dots ,x_n)$ denotes n commutative free variables and k a field of characteristic zero. Furthermore, the relation between the LFED Conjecture and the Duistermaat–van der Kallen Theorem [3] is also discussed and emphasized.