## What is/are First Nonzero?

First Nonzero - Then for a bounded domain $\Omega \subset\mathbb{M}$ with smooth boundary, we prove that the first nonzero Neumann eigenvalue $\mu_{1}(\Omega) \leq \mathcal{C} \mu_{1}(B_{k}(R))$.^{[1]}Another new estimate for $$\lambda _{1,f}(\beta )$$λ1,f(β) with respect to the first nonzero Neumann eigenvalue $$\mu _{2,f}$$μ2,f of the weighted Laplacian $$\Delta _f$$Δf is also obtained.

^{[2]}In 1996 Francoise gave an algorithm for calculating the first nonzero term of the displacement function Δ along γ of such deformations.

^{[3]}On a two-dimensional compact Riemannian manifold with boundary, we prove that the first nonzero Steklov eigenvalue is nondecreasing along the unnormalized geodesic curvature flow if the initial metric has positive geodesic curvature and vanishing Gaussian curvature.

^{[4]}This equation arises from the following variational characterization of the first nonzero eigenvalue given by.

^{[5]}Projections onto these soft modes also correspond to components of the displacement structure factor at the first nonzero wavevectors, in close analogy to PCA results for thermal phase transitions in conserved Ising spin systems.

^{[6]}The first nonzero element of S is called the multiplicity of S and is denoted by m(S).

^{[7]}we introduce a generalization of the $p$-Laplace operator to act on differential forms and generalize an estimate of Gallot-Meyer for the first nonzero eigenvalue on closed Riemannian manifolds.

^{[8]}We compute the first nonzero power correction.

^{[9]}The first nonzero Melnikov function $M_{\mu}=M_{\mu}(F,\gamma,\omega)$ of the Poincare map along a loop $\gamma$ of $dF=0$ is given by an iterated integral.

^{[10]}We then estimate the first nonzero eigenvalue of Hodge–de Rham Laplacian of $$(X,\omega _0)$$ ( X , ω 0 ).

^{[11]}We consider the first nonzero Melnikov function, $$M_{\mu }$$Mμ, of the displacement function $$\Delta (t,\epsilon )=\sum _{j=\mu }^{\infty }\epsilon ^{j}M_{j}(t)$$Δ(t,ϵ)=∑j=μ∞ϵjMj(t), along a cycle $$\gamma (t)$$γ(t) in $$F^{-1}(t)$$F-1(t).

^{[12]}Furthermore, the inequalities obtained are used to derive an uncertainty principle inequality and another inequality involving the first nonzero eigenvalue of the p -Laplacian on the sphere.

^{[13]}

## first nonzero eigenvalue

This equation arises from the following variational characterization of the first nonzero eigenvalue given by.^{[1]}we introduce a generalization of the $p$-Laplace operator to act on differential forms and generalize an estimate of Gallot-Meyer for the first nonzero eigenvalue on closed Riemannian manifolds.

^{[2]}We then estimate the first nonzero eigenvalue of Hodge–de Rham Laplacian of $$(X,\omega _0)$$ ( X , ω 0 ).

^{[3]}Furthermore, the inequalities obtained are used to derive an uncertainty principle inequality and another inequality involving the first nonzero eigenvalue of the p -Laplacian on the sphere.

^{[4]}

## first nonzero neumann

Then for a bounded domain $\Omega \subset\mathbb{M}$ with smooth boundary, we prove that the first nonzero Neumann eigenvalue $\mu_{1}(\Omega) \leq \mathcal{C} \mu_{1}(B_{k}(R))$.^{[1]}Another new estimate for $$\lambda _{1,f}(\beta )$$λ1,f(β) with respect to the first nonzero Neumann eigenvalue $$\mu _{2,f}$$μ2,f of the weighted Laplacian $$\Delta _f$$Δf is also obtained.

^{[2]}

## first nonzero melnikov

The first nonzero Melnikov function $M_{\mu}=M_{\mu}(F,\gamma,\omega)$ of the Poincare map along a loop $\gamma$ of $dF=0$ is given by an iterated integral.^{[1]}We consider the first nonzero Melnikov function, $$M_{\mu }$$Mμ, of the displacement function $$\Delta (t,\epsilon )=\sum _{j=\mu }^{\infty }\epsilon ^{j}M_{j}(t)$$Δ(t,ϵ)=∑j=μ∞ϵjMj(t), along a cycle $$\gamma (t)$$γ(t) in $$F^{-1}(t)$$F-1(t).

^{[2]}