In this paper, the authors employ the splitting method to address support vector machine within a reproducing kernel Banach space framework, where a lower semi-continuous loss function is utilized. They translate support vector machine in reproducing kernel Banach space with such a loss function to a finite-dimensional tensor optimization problem and propose a splitting method based on the alternating direction method of multipliers. Leveraging Kurdyka-Lojasiewicz property of the augmented Lagrangian function, the authors demonstrate that the sequence derived from this splitting method is globally convergent to a stationary point if the loss function is lower semi-continuous and subanalytic. Through several numerical examples, they illustrate the effectiveness of the proposed splitting algorithm.

The present paper is devoted to the well-posedness of a type of multi-dimensional backward stochastic differential equations (BSDE for short) with a diagonally quadratic generator. The author gives a new priori estimate, and prove that the BSDE admits a unique solution on a given interval when the generator has a sufficiently small growth of the off-diagonal elements (i.e., for each i, the i-th component of the generator has a small growth of the j-th row z ^j of the variable z for each j ≠ i). Finally, a solvability result is given when the diagonally quadratic generator is triangular.

In this paper, the authors study the persistence approximation property for quantitative K-theory of filtered L ^p operator algebras. Moreover, they define quantitative assembly maps for L ^p operator algebras when p ∈ [1, ∞). Finally, in the case of L ^p crossed products and L ^p Roe algebras, sufficient conditions for the persistence approximation property are found. This allows to give some applications involving the L ^p (coarse) Baum-Connes conjecture.

In this paper, the authors consider the spectra of second-order left-definite difference operator with linear spectral parameters in two boundary conditions. First, they obtain the exact number of this kind of eigenvalue problem, and prove these eigenvalues are all real and simple. In details, they get that the number of the positive (negative) eigenvalues is related to not only the number of positive (negative) elements in the weight function, but also the parameters in the boundary conditions. Second, they obtain the interlacing properties of these eigenvalues and the sign-changing properties of the corresponding eigenfunctions according to the relations of the parameters in the boundary conditions.

Costa first constructed a family of complete minimal surfaces which have genus 1 and 4 planar ends by use of Weierstrass-℘ functions. They are Willmore tori of Willmore energy 16π. In this paper, the authors consider the geometry of conjugate surfaces of these surfaces. It turns out that these conjugate surfaces are doubly periodic minimal surfaces with flat ends in ℝ³. Moreover, the authors can also perform a Lorentzian deformation on these Costa’s minimal tori, which produce a family of complete space-like stationary surfaces (i.e., of zero mean curvature) with genus 1 and 4 planar ends in 4-dimensional Lorentz-Minkowski space ℝ₁ ⁴.

In this paper, the author verifies the ℓ ^p coarse Baum-Connes conjecture for open cones and shows that the K-theory for ℓ ^p Roe algebras of open cones is independent of p ∈ [1, ∞). Combined with the result of Fukaya and Oguni, he gives an application to the class of coarsely convex spaces that includes geodesic Gromov hyperbolic spaces, CAT(0)-spaces, certain Artin groups and Helly groups equipped with the word length metric.

Note that some classic fluid dynamical systems such as the Navier-Stokes equations, Magnetohydrodynamics (MHD for short), Boussinesq equations and etc., are observably different from each other but obey some energy inequalities of the similar type. In this paper, the authors attempt to axiomatize the extending mechanism of solutions to these systems, merely starting from several basic axiomatized conditions such as the local existence, joint property of solutions and some energy inequalities. The results established have nothing to do with the concrete forms of the systems and, thus, give the extending mechanisms in a unified way to all systems obeying the axiomatized conditions. The key tools are several new multiplicative interpolation inequalities of Besov type, which have their own interests.

The authors introduce the Sobolev space $H^{{1\over 2}}(\Gamma)$ on a quasi-circle Γ and give a fast approach to the jump formula which gives a decomposition of an element in $H^{{1\over 2}}(\Gamma)$ as the boundary values of two Dirichlet functions in the complementary domains of Γ.