B -值双随机Dirichlet级数的收敛性

主要研究了B -值双随机Dirichlet级数在不同条件(i) {X_n}服从强大数定律,且0<\mathop{\underline{\lim}}\limits_{n-->\infty}\Big\|\frac{\sum\limits_{i=1}^n EX_i}{n}\Big\|\leq \mathop{\overline{\lim}}\limits_{n\to\infty}\Big\|\frac{\sum\limits_{i=1}^n EX_i}{n}\Big\|<+\infty.(ii) {X_{n}}独立不同分布，且\mathop{\underline{\lim}}\limits_{n-->\infty}E||X_n||>0,\quad \sup\limits_{n\geq 1}E||X_n||^p <+\infty \quad (p >1)等条件下的收敛性,得出了收敛横坐标的简洁公式.     

有界完全Reinhardt域上推广的Roper-Suffridge算子

在$\C^n$中的有界完全Reinhardt域$\Omega$上推广的Roper-Suffridge算子$\Phi(f)$定义为 \begin{eqnarray*} \Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)(z)\!=\!\Big(rf\Big(\frac{z_1}{r}\Big), \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_2}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_2}z_2,\ldots, \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_n}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_n}z_n \Big), \end{eqnarray*} 其中 $n\geq2$, $(z_1, z_2,\ldots, z_n)\in \Omega$, $r=r(\Omega)=\sup\{|z_1|: (z_1, z_2,\ldots, z_n)\in \Omega\}, 0\leq \gamma_j\leq 1-\beta_j, 0\leq \beta_j\leq 1$, 这里选取幂函数的单值解析分支, 使得 $(\frac{f(z_1)}{z_1})^{\beta_j}|_{z_1=0}= 1$ 和 $(f’(z_1))^{\gamma_j}|_{z_1=0}=1, j=2,\ldots, n$. 证明了 $\Omega$上的算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 是将 $S^*_\alpha(U)$ 的子集映入$S^*_\alpha\,(\Omega)\,(0\leq \alpha<1)$, 且对于一些合适的常数 $\beta_j, \gamma_j, p_j$, $D_p$上的这个算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 保持$\alpha$阶星形性或保持$\beta$ 型螺形性, 其中 $ D_p=\bigg\{(z_1, z_2,\ldots, z_n)\in \C^n: \he{j=1}{n}|z_j|^{p_j}<1\bigg\},\quad p_j>0, j=1, 2,\ldots, n, $ $U$是复平面$\C$上的单位圆, $S^*_\alpha(\Omega)$ 是 $\Omega$ 上所有正规化$\alpha$阶星形映射所成的类. 也得到: 对于某些合适的常数 $\beta_j, \gamma_j, p_j$ 和 在$\C^n$中的有界完全Reinhardt域$\Omega$上推广的Roper-Suffridge算子$\Phi(f)$定义为 \begin{eqnarray*} \Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)(z)\!=\!\Big(rf\Big(\frac{z_1}{r}\Big), \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_2}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_2}z_2,\ldots, \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_n}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_n}z_n \Big), \end{eqnarray*} 其中 $n\geq2$, $(z_1, z_2,\ldots, z_n)\in \Omega$, $r=r(\Omega)=\sup\{|z_1|: (z_1, z_2,\ldots, z_n)\in \Omega\}, 0\leq \gamma_j\leq 1-\beta_j, 0\leq \beta_j\leq 1$, 这里选取幂函数的单值解析分支, 使得 $(\frac{f(z_1)}{z_1})^{\beta_j}|_{z_1=0}= 1$ 和 $(f’(z_1))^{\gamma_j}|_{z_1=0}=1, j=2,\ldots, n$. 证明了 $\Omega$上的算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 是将 $S^*_\alpha(U)$ 的子集映入$S^*_\alpha\,(\Omega)\,(0\leq \alpha<1)$, 且对于一些合适的常数 $\beta_j, \gamma_j, p_j$, $D_p$上的这个算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 保持$\alpha$阶星形性或保持$\beta$ 型螺形性, 其中 $ D_p=\bigg\{(z_1, z_2,\ldots, z_n)\in \C^n: \he{j=1}{n}|z_j|^{p_j}<1\bigg\},\quad p_j>0, j=1, 2,\ldots, n, $ $U$是复平面$\C$上的单位圆, $S^*_\alpha(\Omega)$ 是 $\Omega$ 上所有正规化$\alpha$阶星形映射所成的类. 也得到: 对于某些合适的常数 $\beta_j, \gamma_j, p_j$ 和 在C~n中的有界完全Reinhardt域Ω上推广的Roper-Suffridge算子Φ(f)定义为Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)(z)=(rf(z_1/r),((rf(z_1/r))/z_1)~(β_2)(f′(z_1/r))~γ_2_(z_2,…,)((rf(z_1/r))/z_1)~(β_n)(f′(z_1/r))~(γ_n)_(z_n),其中n≥2,(z_1,z_2,…,z_n)∈Ω,r=r(Ω)=sup{|z_1|:(z_1,z_2,…,z_n)∈Ω},0≤γ_j≤1-β_j,0≤β_j≤1,这里选取幂函数的单值解析分支,使得((f(z_1))/z_1)~(β_j)|_(z_1=0)=1和(f′(z_1))~(γ_j)|_(z_1=0)=1,j= 2,…,n.证明了Ω上的算子Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)是将S_α~*(U)的子集映入S_α~*(Ω)(0≤α<1),且对于一些合适的常数β_j,γ_j,p_j,D_p上的这个算子Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)保持α阶星形性或保持β型螺形性,其中(?) U是复平面C上的单位圆,S_α~*(Ω)是Ω上所有正规化α阶星形映射所成的类.也得到:对于某些合适的常数β_j,γ_j,p_j和0≤α<1,Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)∈S_α~*(D_p)当且仅当f∈S_α~*(U).     

由$q$-积分算子定义的某些亚纯函数类的Fekete-Szeg\"{o}问题

本文首先引入满足如下条件$$-\frac{qzD_{q}f(z)}{f(z)}\prec \varphi (z)$$和$$\frac{-(1-\frac{\alpha }{q})qzD_{q}f(z)+\alpha qzD_{q}[zD_{q}f(z)]}{(1-\frac{\alpha}{q})f(z)-\alpha zD_{q}f(z)}\prec \varphi (z)~(\alpha \in\mathbb{C}\backslash (0,1],\ 0相似文献   

ON MILIN-LEBDEV INEQUALITIES

Let \[\varphi (x) = \sum\limits_{k = 1}^\infty {{A_k}} {x^k},\Phi (x) = {e^{\varphi (x)}} = \sum\limits_{k = 1}^\infty {{D_k}} {x^k}\] \[\begin{gathered} \frac{1}{{{{(1 - x)}^\lambda }}} = \sum\limits_{k = 1}^\infty {{d_k}} (\lambda ){x^k} \hfill \ {\overline \Delta _n}(\lambda ) = {\lambda ^{2 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} \mathop {|{A_k}|}\nolimits_{}^p - \sum\limits_{k = 1}^\infty {\frac{1}{k}} \hfill \\ \end{gathered} \] Milin-Lebedey proved that \[\sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^{p - 1}(\lambda )}}} \leqslant \exp \{ {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} |{A_k}{|^p}\} \] where p>l and \[\lambda \]>0. In this paper, we have proved the following theorems； Theorem 1. Let \[p \geqslant 1,\lambda > 0\] and \[F(x) = \sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}} {x^p}\exp \{ - {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}|{A_k}{|^p}{x^k}} \} (2)\] then F(x) is a decreasing function of x on [0, 1]. This theorem is stronger than the result (1). Theorem 2. Let \[p \geqslant 2,\lambda > 0\] and \[{{\bar Q}_n}(\lambda ) = \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}\exp } \{ - \frac{1}{{n + 1}}\sum\limits_{v = 1}^n {\overline {{\Delta _p}} } (\lambda )\} \] then \[{{\bar Q}_n}(\lambda )\] is a decreasing fimctLon of n(n=l, 2，...)In the case p=2 this is contained in the Miiin-Lebedev's result.     

在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性

设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的     

ON MILIN-LEBDEV INEQUALITIES

Let \[\varphi (x) = \sum\limits_{k = 1}^\infty {{A_k}} {x^k},\Phi (x) = {e^{\varphi (x)}} = \sum\limits_{k = 1}^\infty {{D_k}} {x^k}\] \[\begin{gathered} \frac{1}{{{{(1 - x)}^\lambda }}} = \sum\limits_{k = 1}^\infty {{d_k}} (\lambda ){x^k} \hfill \ {\overline \Delta _n}(\lambda ) = {\lambda ^{2 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} \mathop {|{A_k}|}\nolimits_{}^p - \sum\limits_{k = 1}^\infty {\frac{1}{k}} \hfill \\ \end{gathered} \] Milin-Lebedey proved that \[\sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^{p - 1}(\lambda )}}} \leqslant \exp \{ {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} |{A_k}{|^p}\} \] where p>l and \[\lambda \]>0. In this paper, we have proved the following theorems； Theorem 1. Let \[p \geqslant 1,\lambda > 0\] and \[F(x) = \sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}} {x^p}\exp \{ - {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}|{A_k}{|^p}{x^k}} \} (2)\] then F(x) is a decreasing function of x on [0, 1]. This theorem is stronger than the result (1). Theorem 2. Let \[p \geqslant 2,\lambda > 0\] and \[{{\bar Q}_n}(\lambda ) = \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}\exp } \{ - \frac{1}{{n + 1}}\sum\limits_{v = 1}^n {\overline {{\Delta _p}} } (\lambda )\} \] then \[{{\bar Q}_n}(\lambda )\] is a decreasing fimctLon of n(n=l, 2，...)In the case p=2 this is contained in the Miiin-Lebedev's result.     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

一类深度1的可迁模Lie超代数

建立了满足如下条件的可迁$\mathbb{Z}$-分次模Lie超代数$\frak{g}=\oplus_{-1\leq i\leq r}\frak{g}_{i}$的嵌入定理:(i) $\frak{g}_{0}\simeq \widetilde{\mathrm{p}}(\frak{g}_{-1}) $ 并且$\frak{g}_{0}$-模 $\frak{g}_{-1}$ 同构于$\widetilde{\mathrm{p}}(\frak{g}_{-1})$的自然模;(ii) $\dim \frak{g}_1=\frac 23 n(2n^2+1),$ 其中 $n=\frac{1}{2} \dim \frak{g}_{-1}.$特别地, 证明了满足上述条件的有限维单模Lie超代数同构于奇Hamilton模Lie超代数.对局限Lie超代数也做了相应的讨论.     

$\mathbb{C}^n$中一类星形映射子族的增长定理及推广的Roper-Suffridge算子

在有界星形圆形域上定义了一个新的星形映射子族, 它包含了$\alpha$阶星形映射族和$\alpha$阶强星形映射族作为两个特殊子类. 给出了此类星形映射子族的增长定理和掩盖定理. 另外, 还证明了Reinhardt域$\Omega_{n,p_{2},\cdots,p_{n}}$上此星形映射子族在Roper-Suffridge算子 \begin{align*} F(z)=\Big(f(z_{1}),\Big(\frac{f(z_{1})}{z_{1}}\Big)^{\beta_{2}}(f'(z_{1}))^{\gamma_{2}}z_{2},\cdots, \Big(\frac{f(z_{1})}{z_{1}}\Big)^{\beta_{n}}(f'(z_{1}))^{\gamma_{n}}z_{n}\Big)' \end{align*} 作用下保持不变, 其中 $\Omega_{n,p_{2},\cdots,p_{n}}=\{z\in {\mathbb{C}}^{n}:|z_1|^2+|z_2|^{p_2}+\cdots + |z_n|^{p_n}<1\}$, $p_{j}\geq1$, $\beta_{j}\in$ $[0, 1]$, $\gamma_{j}\in[0, \frac{1}{p_{j}}]$满足$\beta_{j}+\gamma_{j}\leq1$, 所取的单值解析分支使得 $\big({\frac{f(z_{1})}{z_{1}}}\big)^{\beta_{j}}\big|_{z_{1}=0}=1$, $(f'(z_{1}))^{\gamma_{j}}\mid_{{z_{1}=0}}=1$, $j=2,\cdots,n$. 这些结果不仅包含了许多已有的结果, 而且得到了新的结论.     

粗糙核分数次积分算子的多线性算子在Hardy空间上的有界性

该文证明带有粗糙核的分数次积分算子的多线性算子\[T_{\Omega,\alpha}^{A}(f)(x)={\rm {\rm p.v.}}\int_{R^{n}}P_{m}(A;x,y)\frac{\Omega(x-y)}{|x-y|^{n-\alpha+m-1}}f(y){\rm d}y\]的$(H^{1}(\rr^{n}),L^{\frac{n}{n-\alpha},\infty}(\rr^{n}))$有界性.     

一个素变量丢番图问题

设k和r是满足k≥3及r≥Ψ(k)+1的正整数,这里当3≤k≤4时,Ψ(k)=2~(k-1);而当k≥5时,Ψ(k)=1/2k(k+1).假定δ和ε是给定的足够小的正数,λ_1,λ_2,…,λ_(r+1)是不全同号且两两之比不全为有理数的非零实数.对于任意实数η与0σ2~(1-2k)/r-1,证明了:存在一个正数序列X→+∞,使得不等式|λ_1p_1~k+λ_2p_2~k+···+λ_rp_r~k+λ_(r+1)p_(r+1)+η|(max(1≤j≤r+1)p_j)~(-σ)有》■X~(■-(2~(1-2k))/(r-1)+ε组素数解(p_1,p_2,…,p_(r+1)),这里(δX)~(1/k)≤p_j≤X~(1/k)(1≤j≤r)及δX≤p_(r+1)≤X.这改进了之前的结果.     

三维液晶关于压力水平梯度的相关正则性准则

This paper is devoted to investigating regularity criteria for the 3-D nematic liquid crystal flows in terms of horizontal derivative components of the pressure and gradient of the orientation field. More precisely, we mainly proved that the strong solution(u, d)can be extended beyond T, provided that the horizontal derivative components of the pressure■ and gradient of the orientation field satisfy■ and■.     

带粗糙初始值向列型液晶流的适定性

考虑了R~n上n(n≥2)维向列型液晶流(u,d)当初值属于Q_α~(-1)(R~n,R~n)×Q_α(R~n,S~2)(其中α∈(0,1))时Cauchy问题的适定性,这里的Q_α(R~n)最早由Essen,Janson,Peng和Xiao(见[Essen M,Janson S,Peng L,Xiao J.Q space of several real variables,Indiana Univ Math J,2000,49:575-615])引入,是指由R~n中满足的所有可测函数f全体所组成的空间.上式左端在取遍Rn中所有以l(I)为边长且边平行于坐标轴的立方体I的全体中取上确界,而Q_α~(-1)(R~n):=▽·Q_α(R~n).最后证明了解(u,d)在类C([0,T);Q_(α,T)~(-1)(R~n,R~n))∩L_(loc)~∞((0,T);L~∞(R~n,R~n))×C([0,T);Q_α,T(R~n,S~2))∩L_(loc)~∞((0,T);W~(1,∞)(R~n,S~2))(其中0T≤∞)中是唯一的.     

Multiple Axially Asymmetric Solutions to a Mean Field Equation on $\mathbb{S}^{2}$

We study the following mean field equation$$\Delta_{g}u+\rho\left(\frac{e^{u}}{\int_{\mathbb{S}^{2}}e^{u}d\mu}-\frac{1}{4\pi}\right)=0\ \ \mbox{in}\ \ \mathbb{S}^{2},$$where $\rho$ is a real parameter. We obtain the existence of multiple axially asymmetric solutions bifurcating from $u=0$ at the values $\rho=4n(n+1)\pi$ for any odd integer $n\geq3$.     

Positive Gegenbauer Polynomial Sums and Applications to Starlike Functions

Let $s_n(f,z):=\sum_{k=0}^{n}a_kz^k$ be the $n$th partial sum of $f(z)=\sum_{k=0}^{\infty{}}a_kz^k$. We show that $\RE s_n(f/z,z)>0$ holds for all $z\in\D,\ n\in\N$, and all starlike functions $f$ of order $\lambda$ iff $\lambda_0\leq\lambda<1$ where $\lambda_0=0.654222...$ is the unique solution $\lambda\in(\frac{1}{2},1)$ of the equation $\int_{0}^{3\pi/2}t^{1-2\lambda}\cos t \,dt=0$. Here $\D$ denotes the unit disk in the complex plane $\C$. This result is the best possible with respect to $\lambda_0$. In particular, it shows that for the Gegenbauer polynomials $C_{n}^{\mu}(x)$ we have $\sum_{k=0}^n C_{k}^{\mu}(x)\cos k \theta>0$ for all $n\in\N,\ x\in[-1,1]$, and $0<\mu\leq\mu_0:=1-\lambda_0=0.345778...$. This result complements an inequality of Brown, Wang, and Wilson (1993) and extends a result of Ruscheweyh and Salinas (2000).     

Regularity of Positive Solutions for an Integral System on Heisenberg Group

In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha 1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2     

Remarks on the regularity criteria of the solutions of the 3D micropolar fluid equations

We provide two regularity criteria for the weak solutions of the 3D micropolar fluid equations, the first one in terms of one directional derivative of the velocity, i.e., $\partial_{3}u$, while the second one is is in terms of the behavior of the direction of the velocity $\frac{u}{|u|}$. More precisely, we prove that if \begin{equation*} \partial_{3}u \in L^{\beta}(0,T;L^{\alpha}(\mathbb{R}^{3}))\quad\text{ with }\frac{2}{\beta}+\frac{3}{\alpha}\leq 1+\frac{1}{\alpha}, 2&lt; \alpha \leq\infty, 2\leq\beta&lt; \infty; \end{equation*} or \begin{equation*} \operatorname{div}\left(\frac{u}{|u|}\right)\in L^{\frac{4}{1-2r}}(0,T;\dot{X}_{r}(\mathbb{R}^{3}))\quad \text{ with } 0\leq r&lt; \frac{1}{2}, \end{equation*} then the weak solution $(u(x,t),\omega(x,t))$ is regular on $\mathbb{R}^{3}\times [0,T]$. Here $\dot{X}_{r}(\mathbb{R}^{3})$ is the multiplier space.     

On the Serrin's Regularity Criterion for the β-Generalized Dissipative Surface Quasi-geostrophic Equation

The authors establish a Serrin's regularity criterion for the β-generalized dissipative surface quasi-geostrophic equation.More precisely,it is shown that if the smooth solution θ satisfies ▽θ∈L~q(0,T;L~P(R~2)) with α/q+2/p≤α+β-1,then the solution θcan be smoothly extended after time T.In particular,when α+β≥2,it is shown that if α_yθ∈L~q(0,T;L~P(R~2)) with α/q+2/p≤α+β-1,then the solution θ can also be smoothly extended after time T.This result extends the regularity result of Yamazaki in 2012.     

ON THE CONTACT COHOMOLOGY OF ISOLATED HYPERSURFACE SINGULARITIES

The author defines, using jets, cohomology $H^p(\Lambda _{f,k-})$ for hypersurfaces, which are invariant under contact transformations. For isolated hypersurface singularities, it is proved that $H^0(\Lambda _{f,k-})=O_{U,0}/f^{k+1}O_{U,0},$ $H^p(\Lambda _{f,k-})=0,1\leq p \leq N-3 or p=N,$ $dimH^{N-2}(\Lambda _{f,k-})-dimH^{N-1}(\Lambda _{f,k-})=\[\left( {\begin{array}{*{20}{c}} k \ N \end{array}} \right)\dim {O_{U,0}}/(f,\frac{{\partial f}}{{\partial {x_1}}}, \cdots ,\frac{{\partial f}}{{\partial {x_N}}}){O_{U,0}}\] $ The algorithm of computation for H^{N-2} and H^{N-1} is given, and it is proved that $H^{N-1}=0$ when f is quasi-homogeneous.