Large-scale empirical data, the sample size and the dimension are high, often exhibit various characteristics. For example, the noise term follows unknown distributions or the model is very sparse that the number of critical variables is fixed while dimensionality grows with n. The authors consider the model selection problem of lasso for this kind of data. The authors investigate both theoretical guarantees and simulations, and show that the lasso is robust for various kinds of data.

The authors obtain the estimates of all homogeneous expansions for a subclass of ε quasi-convex mappings on the unit ball in complex Banach spaces. Moreover, the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in ℂ ⁿ are also obtained. Especially, the above estimates are only sharp for a subclass of starlike mappings, quasi-convex mappings and quasi-convex mappings of type $\mathbb{A}$. The results are the generalization of many known results.

Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras. For $A,B \in \mathcal{A}$, define by [A,B]_* = AB − BA* the skew Lie product of A and B. In this article, it is proved that a bijective map $\Phi :\mathcal{A} \to \mathcal{B}$ satisfies Φ([[A,B]_*,C]_*) = [[Φ(A),Φ(B)]_*,Φ(C)]_* for all $A,B,C \in \mathcal{A}$ if and only if Φ is a linear *-isomorphism, or a conjugate linear *- isomorphism, or the negative of a linear *-isomorphism, or the negative of a conjugate linear *-isomorphism.

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L ₂ and L ^p+1 norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.

Let D be a bounded positive (m, p)-circle domain in ℂ². The authors prove that if dim(Iso(D)⁰) = 2, then D is holomorphically equivalent to a Reinhardt domain; if dim(Iso(D)⁰) = 4, then D is holomorphically equivalent to the unit ball in ℂ². Moreover, the authors prove the Thullen’s classification on bounded Reinhardt domains in ℂ² by the Lie group technique.

Let M be an n-dimensional differentiable manifold with an affine connection without torsion and T ₁ ¹ (M) its (1, 1)-tensor bundle. In this paper, the authors define a new affine connection on T ₁ ¹ (M) called the intermediate lift connection, which lies somewhere between the complete lift connection and horizontal lift connection. Properties of this intermediate lift connection are studied. Finally, they consider an affine connection induced from this intermediate lift connection on a cross-section σ _ξ(M) of T ₁ ¹ (M) defined by a (1, 1)-tensor field ξ and present some of its properties.

In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls $\mathbb{B}$ and $\mathbb{B'}$ in separable complex Hilbert spaces $\mathcal{H}$ and $\mathcal{H'}$, respectively. It is found that if the mapping f ∈ C ^1+α at ${z_0} \in \partial \mathbb{B}$ with $f\left( {{z_0}} \right) = {w_0} \in \partial \mathbb{B}'$, then the Fréchet derivative operator Df(z ₀) maps the tangent space ${T_{{z_0}}}(\partial {\mathbb{B}^n})$ to ${T_{{w_0}}}(\partial {\mathbb{B}'})$, the holomorphic tangent space $T_{{z_0}}^{(1,0)}(\partial {\mathbb{B}^n})$ to $T_{{w_0}}^{(1,0)}(\partial {\mathbb{B}'})$, respectively.

This paper deals with Hérmite learning which aims at obtaining the target function from the samples of function values and the gradient values. Error analysis is conducted for these algorithms by means of approaches from convex analysis in the framework of multi-task vector learning and the improved learning rates are derived.

In this paper, some endpoint estimates for the generalized multilinear fractional integrals I _α,m on the non-homogeneous metric spaces are established.

A regime-switching geometric Brownian motion is used to model a geometric Brownian motion with its coefficients changing randomly according to a Markov chain. In this work, the author gives a complete characterization of the recurrent property of this process. The long time behavior of this process such as its p-th moment is also studied. Moreover, the quantitative properties of the regime-switching geometric Brownian motion with two-state switching are investigated to show the difference between geometric Brownian motion with switching and without switching. At last, some estimates of its first passage probability are established.

The authors obtain a complex Hessian comparison for almost Hermitian manifolds, which generalizes the Laplacian comparison for almost Hermitian manifolds by Tossati, and a sharp spectrum lower bound for compact quasi Kähler manifolds and a sharp complex Hessian comparison on nearly Kähler manifolds that generalize previous results of Aubin, Li Wang and Tam-Yu.