This is an introduction to Witten’s analytic proof of the Morse inequalities. The text is directed primarily to readers whose main interest is in complex analysis, and the similarities to Hörmander’s $L^2$-estimates for the $\overline{\partial }$-equation is used as motivation. We also use the method to prove $L^2$-estimates for the $d$-equation with a weight $e^{-t\phi }$ where $\phi$ is a nondegenerate Morse function.

Let $P\left(z\right)={\sum }_{\nu =0}^n{a}_{\nu }{z}^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $\left|z\right|<1$, then for $1\le p<\infty$ and $R>1$, Boas and Rahman proved $${∥P\left(Rz\right)∥}_p\le ({∥R^n+z∥}_p/{∥1+z∥}_p){∥P∥}_p.$$ In this paper, we improve the above inequality for $0\le p<\infty$ by involving some of the coefficients of the polynomial $P\left(z\right)$. Analogous result for the class of polynomials $P\left(z\right)$ having no zero in $\left|z\right|>1$ is also given.

En admettant la conjecture de Dickson, nous démontrons que, pour chaque couple d’entiers $q\&gt;0$ et $k\&gt;0$, il existe une partie infinie $L_{q,k}\subset \mathbb{N}$ telle que, pour chacun des entiers $n\in L_{q,k}$ et tout entier $s$ tel que $0\&lt;\left|s\right|\le q$, on ait $n+s=\left|s\right|t_1...t_k$ où $t_1\&lt;...\&lt;t_k$ sont des nombres premiers. De même, pour chaque couple d’entiers $q\&gt;0$ et $k\&gt;0$, il existe une partie infinie $M_{q,k}\subset \mathbb{N}$ telle que, pour chacun des entiers $n\in M_{q,k}$ et tout entier $s$ (nul ou non ) de l’intervalle $\left[-q,q\right]$, on ait $n+s=l{t}_1...t_k$ où $t_1\&lt;...\&lt;t_k$ sont des nombres premiers et l’entier $l$ appartient à l’intervalle $\left[1,2q+1\right]$. La lecture non standard...

In this paper, a new class of Gaussian field is introduced called Lacunary Fractional Brownian Motion. Surprisingly we show that usually their tangent fields are not unique at every point. We also investigate the smoothness of the sample paths of Lacunary Fractional Brownian Motion using wavelet analysis.

In this paper, a new class of Gaussian field is introduced called Lacunary Fractional Brownian Motion. Surprisingly we show that usually their tangent fields are not unique at every point. We also investigate the smoothness of the sample paths of Lacunary Fractional Brownian Motion using wavelet analysis.

We study the large time behaviour of the solutions of a nonlocal regularisation of a scalar conservation law. This regularisation is given by a fractional derivative of order $1+\alpha$, with $\alpha \in (0,1)$, which is a Riesz-Feller operator. The nonlinear flux is given by the locally Lipschitz function ${\left|u\right|}^{q-1}u/q$ for $q>1$. We show that in the sub-critical case, $1<q<1+\alpha$, the large time behaviour is governed by the unique entropy solution of the scalar conservation law. Our proof adapts the proofs of the analogous results for the local...