## This equation arises from the following variational characterization of the first nonzero eigenvalue given by.

First eigenvalue of the -Laplacian on Kähler manifolds

## 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.

The first nonzero eigenvalue of the $p$-Laplacian on Differential forms

## Then for a bounded domain $\Omega \subset\mathbb{M}$ with smooth boundary, we prove that the first nonzero Neumann eigenvalue $\mu_{1}(\Omega) \leq \mathcal{C} \mu_{1}(B_{k}(R))$.

An upper bound for the first nonzero Neumann eigenvalue.

## 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.

Generalization of Philippin’s results for the first Robin eigenvalue and estimates for eigenvalues of the bi-drifting Laplacian

## 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.

Infinite Orbit depth and length of Melnikov functions.

## 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).

The first nonzero Melnikov function for a family of good divides

10.1016/j.geomphys.2020.103838

## Then for a bounded domain $\Omega \subset\mathbb{M}$ with smooth boundary, we prove that the first nonzero Neumann eigenvalue $\mu_{1}(\Omega) \leq \mathcal{C} \mu_{1}(B_{k}(R))$.

An upper bound for the first nonzero Neumann eigenvalue.

10.1007/S10455-019-09652-1

## 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.

Generalization of Philippin’s results for the first Robin eigenvalue and estimates for eigenvalues of the bi-drifting Laplacian

10.1016/J.BULSCI.2019.02.001

## In 1996 Francoise gave an algorithm for calculating the first nonzero term of the displacement function Δ along γ of such deformations.

Godbillon–Vey sequence and Françoise algorithm

10.1007/S00229-019-01119-8

## 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.

Evolution of the Steklov eigenvalue under geodesic curvature flow

## This equation arises from the following variational characterization of the first nonzero eigenvalue given by.

First eigenvalue of the -Laplacian on Kähler manifolds

10.1088/1742-5468/ab3aea

## 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.

Correlations in the shear flow of athermal amorphous solids: A principal component analysis

10.1007/S00233-019-10028-X

## The first nonzero element of S is called the multiplicity of S and is denoted by m(S).

Numerical semigroups and Kunz polytopes

10.2140/pjm.2020.309.213

## 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.

The first nonzero eigenvalue of the $p$-Laplacian on Differential forms

10.1016/j.physletb.2019.134994

## We compute the first nonzero power correction.

Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment

10.1016/J.ANIHPC.2019.07.003

## 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.

Infinite Orbit depth and length of Melnikov functions.

10.1007/S12220-019-00255-7

## We then estimate the first nonzero eigenvalue of Hodge–de Rham Laplacian of $$(X,\omega _0)$$ ( X , ω 0 ).

Results Related to the Chern–Yamabe Flow

10.1007/S13398-019-00636-1

## 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).

The first nonzero Melnikov function for a family of good divides

10.7153/jmi-2019-13-72

## 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.

Sharp L^p Hardy type and uncertainty principle inequalities on the sphere