Related papers: A sharp square function estimate for the moment cu…
We study the curvature of a smooth algebraic surface $X\subset \mathbb R^3$ of degree $d$ from the point of view of algebraic geometry. More precisely, we consider umbilical points and points of critical curvature. We prove that the number…
The $\epsilon$-subdifferential of convex univariate piecewise linear-quadratic functions can be computed in linear worst-case time complexity as the level-set of a convex function. Using dichotomic search, we show how the computation can be…
We find the sharp constants $C_p$ and the sharp functions $C_p=C_p(x)$ in the inequality $$|u(x)|\leq \frac{C_p}{(1-|x|^2)^{(n-1)/p}}\|u\|_{h^p(B^n)}, u\in h^p(B^n), x\in B^n,$$ in terms of Gauss hypergeometric and Euler functions. This…
Assuming the Riemann Hypothesis we obtain an upper bound for the moments of the Riemann zeta-function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments…
In this paper we discuss approximation of partially smooth functions. The problem arises naturally in the study of laminated currents.
A polygonal curve $P$ with $n$ vertices is $c$-packed, if the sum of the lengths of the parts of the edges of the curve that are inside any disk of radius $r$ is at most $cr$, for any $r>0$. Similarly, the concept of $c$-packedness can be…
We obtain sharp embeddings from the Sobolev space $W^{k,2}_0(-1,1)$ into the space $L^1(-1,1)$ and determine the extremal functions. This improves on a previous estimate of the sharp constants of these embeddings due to Kalyabin.
A telegraphic survey of some of the standard results and conjectures about the set $C({\bf Q})$ of rational points on a smooth projective absolutely connected curve $C$ over ${\bf Q}$.
We determine the maximal number of smooth rational degree d curves on a complex K3-surface of degree 2n provided n is sufficiently large as compared to d>1. We obtain precise characterization of configurations of rational degree d curves…
Jorge-Koutrofiotis and Pigola-Rigoli-Setti proved sharp sectional curvature estimates for extrinsically bounded submanifolds. Alias, Bessa and Montenegro showed that these estimates hold on properly immersed cylindrically bounded…
We prove new $L^p(\mathbb{R}^3)$ bounds on Stein's square function for $p\geq3.25$. As an application, it improves the maximal Bochner-Riesz conjecture to the same range of $p$.
The analysis of curves has been routinely dealt with using tools from functional data analysis. However its extension to multi-dimensional curves poses a new challenge due to its inherent geometric features that are difficult to capture…
Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of…
In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…
Li-Vogelius and Li-Nirenberg gave a gradient estimate for solutions of strongly elliptic equations and systems of divergence forms with piecewise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be…
A new algorithm for computing a point on a polynomial or rational curve in B\'{e}zier form is proposed. The method has a geometric interpretation and uses only convex combinations of control points. The new algorithm's computational…
We obtain an asymptotic formula with an error term $O(X^{\frac{1}{2} + \varepsilon})$ for the smoothed first moment of quadratic twists of modular $L$-functions. We also give a similar result for the smoothed first moment of the first…
We verify the conjecture of [CFKRS] for the mean square near the critical point of Dirichlet L-functions for a composite modulus q. We also prove a kind of reciprocity formula when the second moment for a prime modulus is twisted by a…
Let $(\Omega, \mathcal{F}, \mathbf{P})$ be a probability space, $\xi$ be a random variable on $(\Omega, \mathcal{F}, \mathbf{P})$, $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$, and let $\mathbf{E}^\mathcal{G} = \mathbf{ E}(\cdot…
We prove that the fourth moment of holomorphic Hecke cusp forms is bounded provided that the Riemann Hypothesis holds for an appropriate degree 8 L-function. We accomplish this using Watson's formula, which translates the question in hand…