The blessing of dimensionality for the analysis of climate data

We give a simple description of the blessing of dimensionality with the main focus on the concentration phenomena. These phenomena imply that in high dimensions the lengths of independent random vectors from the same distribution have almost the same length and that independent vectors are almost orthogonal. In the climate and atmospheric sciences we rely increasingly on ensemble modelling and face the challenge of analysing large samples of long time series and spatially extended fields. We show how the properties of high dimensions allow us to obtain analytical results for e.g. correlations between sample members and the behaviour of the sample mean when the size of the sample grows. We find that the properties of high dimensionality with reasonable success can be applied to climate data. This is the case although most climate data show strong anisotropy and both spatial and temporal dependence, resulting in effective dimensions around 25–100.

Positive solutions for a degenerate Kirchhoff problem

By assuming that the Kirchhoff term has $K$ degeneracy points and other appropriated conditions, we have proved the existence of at least $K$ positive solutions other than those obtained in Santos Júnior and Siciliano [Positive solutions for a Kirchhoff problem with vanishing nonlocal term, J. Differ. Equ. 265 (2018), 2034–2043], which also have ordered $H_{0}^{1}(\Omega )$ -norms. A concentration phenomena of these solutions is verified as a parameter related to the area of a region under the graph of the reaction term goes to zero.

Concentration phenomena for fractional magnetic NLS equations

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

Mass Concentration and Asymptotic Uniqueness of Ground State for 3-Component BEC with External Potential in ℝ2

We investigate the ground states of 3-component Bose–Einstein condensates with harmonic-like trapping potentials in ℝ 2 {\mathbb{R}^{2}} , where the intra-component interactions μ i {\mu_{i}} and the inter-component interactions β i ⁢ j = β j ⁢ i {\beta_{ij}=\beta_{ji}} ( i , j = 1 , 2 , 3 {i,j=1,2,3} , i ≠ j {i\neq j} ) are all attractive. We display the regions of μ i {\mu_{i}} and β i ⁢ j {\beta_{ij}} for the existence and nonexistence of the ground states, and give an elaborate analysis for the asymptotic behavior of the ground states as β i ⁢ j ↗ β i ⁢ j * := a ∗ + 1 2 ⁢ ( a ∗ - μ i ) ⁢ ( a ∗ - μ j ) {\beta_{ij}\nearrow\beta_{ij}^{*}:=a^{\ast}+\frac{1}{2}\sqrt{{(a^{\ast}-\mu_{i% })(a^{\ast}-\mu_{j})}}} , where 0 < μ i < a ∗ := ∥ w ∥ 2 2 {0<\mu_{i}<a^{\ast}:=\|w\|_{2}^{2}} are fixed and w is the unique positive solution of Δ ⁢ w - w + w 3 = 0 {\Delta w-w+w^{3}=0} in H 1 ⁢ ( ℝ 2 ) {H^{1}(\mathbb{R}^{2})} . The energy estimation as well as the mass concentration phenomena are studied, and when two of the intra-component interactions are equal, the nondegeneracy and the uniqueness of the ground states are proved.

Multiplicity and Concentration of Solutions for Kirchhoff Equations with Magnetic Field

In this paper, we study the following nonlinear magnetic Kirchhoff equation: { - ( a ⁢ ϵ 2 + b ⁢ ϵ ⁢ [ u ] A / ϵ 2 ) ⁢ Δ A / ϵ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( | u | 2 ) ⁢ u in ⁢ ℝ 3 , u ∈ H 1 ⁢ ( ℝ 3 , ℂ ) , \left\{\begin{aligned} &\displaystyle{-}(a\epsilon^{2}+b\epsilon[u]_{A/% \epsilon}^{2})\Delta_{A/\epsilon}u+V(x)u=f(\lvert u\rvert^{2})u&&\displaystyle% \phantom{}\text{in }\mathbb{R}^{3},\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3},\mathbb{C}),\end{aligned}\right. where ϵ > 0 {\epsilon>0} , a , b > 0 {a,b>0} are constants, V : ℝ 3 → ℝ {V:\mathbb{R}^{3}\rightarrow\mathbb{R}} and A : ℝ 3 → ℝ 3 {A:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}} are continuous potentials, and Δ A ⁢ u {\Delta_{A}u} is the magnetic Laplace operator. Under a local assumption on the potential V, by combining variational methods, a penalization technique and the Ljusternik–Schnirelmann theory, we prove multiplicity properties of solutions and concentration phenomena for ϵ small. In this problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

Strengthening Mechanism of Studs for Embedded-Ring Foundation of Wind Turbine Tower

The embedded-ring wind turbine foundations were widely applied in the early development stage of wind power industries because of its properties such as easy installation and adjustment. However, different damages occurred on some embedded-ring wind turbine foundations in recent years. Based on the common damage phenomena of embedded-ring wind turbine foundations, the structural defects and damage mechanisms of embedded-ring wind turbine foundations are analyzed in a gradual way. Cheese head studs are proposed to be welded on the lateral wall of the steel ring to strengthen the connection between the steel ring and the foundation concrete. The foundation pier is elevated 1 m to increase the embedded depth of the steel ring. The circumferential confining pressure is applied on the lateral side of the foundation pier to lead it into a state of pressure. One simplified method is proposed to calculate the contribution of welding studs in this strengthening method. Taking an embedded-ring wind turbine foundation as an example, the numerical analyses for the original foundation and the reinforced one are carried out to compare the stress and strain distribution changes. Based on the numerical results corresponding to the peak and valley value loads, the fatigue life of the concrete and studs are evaluated according to the relevant design codes. The numerical results show that this strengthening method can coordinate the deformation of the embedded steel ring and the foundation concrete by circumferential prestressing and welding studs. The maximum principal stresses of the foundation pier and the fatigue stress range of the concrete around the bottom of the steel ring have been greatly reduced after strengthening. The gaps between the embedded steel ring and the foundation pier are also obviously decreased because of these strengthening measures. The stress concentration phenomena of the concrete around the T-shaped flange have been remarkably improved. The fatigue life can meet the requirements of relevant design codes after strengthening. Therefore, it can be concluded that the safety performance and service life of the embedded-ring foundation can be guaranteed.