Let R = C[x1; x2;


; xn]=(f) where f is a weighted homogeneous polynomial dening an isolated singularity at the origin. Then R and Der(R;R) are graded. It is well-known that Der(R;R) does not have a negatively graded component. Wahl conjectured that this is still true for R = C[x1; x2;


; xn]=(f1; f2;


; fm) which denes an isolated, normal and complete intersection singularity and f1; f2;


; fm weighted homogeneous polynomials with the same weight type (w1;w2;


;wn). Here we give a positive answer to the Wahl Conjecture and its generalization (without the condition of complete intersection singularity) for R when the degree of fi; 1  i  m are bounded below by a constant C depending only on the weights w1;w2;


;wn. Moreover this bound C is improved when any two of w1;w2;


;wn are coprime. Since there are counter-examples for the Wahl Conjecture and its generalization when fi are low degree, our theorem is more or less optimal in the sense that only the lower bound constant can be improved.

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We study the asymptotic behaviour of$N$(α)—the number of negative eigenvalues of the operator (-τ)^{$l$}-α$V$in$L$_{2}($R$^{$d$}) for an even$d$and$2l≥d$. This is the only case where the previously known results were far from being complete. In order to describe our results we introduce an auxiliary ordinary differential operator (system) on the semiaxis. Depending on the spectral properties of this operator we can distinguish between three cases where$N$(α) is of the Weyl-type,$N$(α) is of the Weyl-order but not the Weyl-type coefficient and finally where$N$(α)=$O$(α^{q}) with$q>d/2l$.

In this paper, Orlicz valuations compatible with $SL(n)$ transforms are classified. Unlike their $L_p$ analogs, the identity \text{operator} and the reflection operator are the only $SL(n)$ \text{compatible} Orlicz valuations (up to dilations). It turns out that the Orlicz projection body operator, the Orlicz centroid body operator and the Orlicz difference body operator are not Orlicz valuations. The property that the Orlicz difference body operator is not an Orlicz valuation plays an important role in characterizing the identity operator and the reflection operator.