Related papers: Quantitative stratification of $F$-subharmonic fun…

Let F(X,n):= X^n-\Delta be the complementary of the union \Delta of the diagonals of X^n and let U be a quotient of F(X,n) (possibly trivial) by a subgroup of the symmetric group S_n. We construct compactifications of U in products of…

We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain…

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{\alpha…

We discuss the structure of shock singularities of the Burgers-Hopf hierarchy. It is shown that the set of singular solutions defines a stratification of the affine space of the flow parameters in the hierarchy. The stratification is…

Let $E$ be a vector bundle of rank $r\geq 2$ on a smooth projective curve $C$ of genus $g \geq 2$ over an algebraically closed field $K$ of arbitrary characteristic. For any integer with $1\le k\le r-1$ we define $${\se}_k(E):=k\deg…

In this paper we investigate how germs of real functions can change under deformation. In particular we look at deformations of germs of isolated singularities from R_n to R_k (n >= k) and the relation with there natural stratification in…

We construct a positive function $u$ supported and solving $(-\Delta)^{s}u=0$ in a Lipschitz cone. Such a function is unique up to a constant multiplication. Moreover, we show that it is homogeneous of some degree $0<\alpha<2s$.

In this paper, we extend the celebrated global regularity theory of Naber-Valtorta [Ann. Math. 2017] to 1/2-harmonic mappings into manifolds. Inspired by their work, we first adapt Lin's defect measure theory [Ann. Math. 1999] to such maps…

The purpose of this paper is twofold: first we study an eigenvalue problem for the fractional $p$-sub-Laplacian over the fractional Folland-Stein-Sobolev spaces on stratified Lie groups. We apply variational methods to investigate the…

Let $B\subset \mathbb{N}$ and let $\eta\in \{0,1\}^\mathbb{Z}$ be the characteristic function of the set $F_B:=\mathbb{Z}\setminus\bigcup_{b}b\mathbb{Z}$ of B-free numbers. Consider $(S,X_\eta)$, where $X_\eta$ is the closure of the orbit…

We study the behavior of the initial coefficients of univalent functions under the Steiner symmetrization, and give some applications to functions of class \Sigma.

In this paper we study class $\mathcal{S}^+$ of univalent functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients. For such functions we give estimates of the Fekete-Szeg\H{o} functional and sharp estimates of their…

Stratifications and iterative differential equations are analogues in positive characteristic of complex linear differential equations. There are few explicit examples of stratifications. The main goal of this paper is to construct…

For separable metrizable spaces $X,Y$ and a metrizable topological group $Z$ by $S(X\times Y,Z)$ we denote the space of all separately continuous functions $f:X\times Y\to Z$ endowed with the topology of layer-wise uniform convergence,…

In this paper, we establish the fractional Morrey-Sobolev type embeddings on stratified Lie groups. This extends and complements the Sobolev type embeddings derived in \cite{GKR}. As an application of the results, we study the following…

We propose to study deformation quantizations of Whitney functions. To this end, we extend the notion of a deformation quantization to algebras of Whitney functions over a singular set, and show the existence of a deformation quantization…

Let U_q(sl_2) be the standard Drinfeld-Jimbo quantized universal enveloping algebra over sl_2, let F_q[SL_2] be the corresponding quantum function algebra, and let R be the ring of Laurent polynomials in q with coefficients in the ring of…

This article is the PhD thesis of the author. It is focused on Type II compactifications because of the potential for the construction of realistic MSSM-like compactifications. In particular we concentrate in Type IIB Calabi-Yau…

We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$ - harmonic functions on the unit polydisc $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$ - Poisson…

In this paper, we give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E) \quad\quad (-\Delta)^s u = |x|^{\theta} u^{p}\quad {\rm in}\ \ B_1\setminus\{0\},\quad u=…