Related papers: A note on the Schur and Phillips lemmas

For decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, we obtain the almost everywhere convergence results for sequences of Schr\"{o}dinger means $e^{it_{n}\Delta}f$, where $f \in H^{s}(\mathbb{R}^{N}), N\geq 2$. The…

In this note, we prove a Schwarz-Pick type lemma for minimal maps between negatively curved Riemannian surfaces. More precisely, we prove that if $f:M \to N$ is a minimal map with bounded Jacobian between two complete negatively curved…

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces $K$ and $L$ the product $K\times L$ admits a sequence…

We show that for a given exact category, there exists a bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed length abelian subcategories). In particular, we show that a…

The long standing Lech's conjecture in commutative algebra states that for a flat local extension $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of Noetherian local rings, we have an inequality on the Hilbert--Samuel multiplicities: $e(R)\leq…

The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem,…

We prove that for arbitrary partitions $\mathbf{\lambda} \subseteq \mathbf{\kappa},$ and integers $0\leq c<r\leq n,$ the sequence of Schur polynomials $S_{(\mathbf{\kappa} + k\cdot \mathbf{1}^c)/(\mathbf{\lambda} + k\cdot…

Different notions for order convergence have been considered by various authors. Associated to every notion of order convergence corresponds a topology, defined by taking as the closed sets those subsets of the poset satisfying that no net…

We study the asymptotic behavior of the $\nu$-symmetric Riemman sums for functionals of a self-similar centered Gaussian process $X$ with increment exponent $0<\alpha<1$. We prove that, under mild assumptions on the covariance of $X$, the…

We prove a gluing lemma for sections of line bundles on a rigid analytic variety. We apply the lemma, in conjunction with a result of Buzzard's, to give a proof of (a generalization) of Coleman's theorem which states that overconvergent…

In the conformal class of the standard metric on the $3$-sphere, we prove a quantitative refinement of the Andrews-De Lellis-Topping inequality in terms of a two-term distance to the set of minimizing conformal factors. This inequality is…

Opial's Lemma is a fundamental result in the convergence analysis of sequences generated by optimization algorithms in real Hilbert spaces. We introduce the concept of Opial sequences - sequences for which the limit of the distance to each…

The Schwarz lemmas are well-known characterizations for holomorphic maps and we exhibit two examples of their applications. For a sequence family of biholomorphisms $f_j$, it is useful to determine the location of $f_j(q)$ for a fixed point…

The law of a finite graph is a probability measure induced by the orbits of the graph under its automorphism group. Every law satisfies the intrinsic mass transport principle, which is also known as unimodularity. We discuss the convergence…

The Euler product formula relates Dirichlet $L(s,\chi)$ functions to an infinite product over primes, and is known to be valid for $\Re (s) >1$, where it converges absolutely. We provide arguments that the formula is actually valid for $\Re…

For any $s \in \mathbb{C}$ with $\Re(s)>0$, denote by $\eta_{n-1}(s)$ the $(n-1)^{th}$ partial sum of the Dirichlet series for the eta function $\eta(s)=1-2^{-s}+3^{-s}-\cdots \;$, and by $R_n(s)$ the corresponding remainder. Denoting by…

Suppose that $\eta$ is a whole-plane space-filling SLE$_\kappa$ for $\kappa \in (4,8)$ from $\infty$ to $\infty$ parameterized by Lebesgue measure and normalized so that $\eta(0) = 0$. For each $T > 0$ and $\kappa \in (4,8)$ we let…

Fatou's lemma states under appropriate conditions that the integral of the lower limit of a sequence of functions is not greater than the lower limit of the integrals. This note describes similar inequalities when, instead of a single…

Let $\mathcal{S} = \{ \tau_n \}_{n=1}^\infty \subset (0,T)$ be an arbitrary countable (dense) set. We show that for any given initial density and momentum, the compressible Euler system admits (infinitely many) admissible weak solutions…

Inspired by the classic apolarity theory of symmetric tensors, the aim of this paper is to introduce the Schur apolarity theory, i.e. an apolarity for any irreducible representation of the special linear group $SL(V)$. This allows to…