It is well known that the second-order cone and the circular cone have many analogous properties. In particular, there exists an important distance inequality associated with the second-order cone and the circular cone. The inequality indicates that the distances of arbitrary points to the second-order cone and the circular cone are equivalent, which is crucial in analyzing the tangent cone and normal cone for the circular cone. In this paper, we provide an alternative approach to achieve the aforementioned inequality. Although the proof is a bit longer than the existing one, the new approach offers a way to clarify when the equality holds. Such a clarification is helpful for further study of the relationship between the second-order cone programming problems and the circular cone programming problems.

First-order methods have been popularly used for solving large-scale problems. However, many existing works only consider unconstrained problems or those with simple constraint. In this paper, we develop two first-order methods for constrained convex programs, for which the constraint set is represented by affine equations and smooth nonlinear inequalities. Both methods are based on the classic augmented Lagrangian function. They update the multipliers in the same way as the augmented Lagrangian method (ALM) but employ different primal variable updates. The first method, at each iteration, performs a single proximal gradient step to the primal variable, and the second method is a block update version of the first one.

Motivated by the nuclear magnetic resonance (NMR) spectroscopy of biofluids (urine and blood serum), we present a recursive blind source separation (rBSS) method for nonnegative and correlated data. BSS problem arises when one attempts to recover a set of source signals from a set of mixture signals without knowing the mixing process. Various approaches have been developed to solve BSS problems relying on the assumption of statistical independence of the source signals. However, signal independence is not guaranteed in many real-world data like the NMR spectra of chemical compounds. The rBSS method introduced in this paper deals with the nonnegative and correlated signals arising in NMR spectroscopy of biofluids. The statistical independence requirement is replaced by a constraint which requires dominant interval(s) from each source signal over some of the other source signals in a

In this paper, we will discuss some upper bound formulas for Ramsey numbers.At first, we will derive a more unified form from two parameter formulas which was obtained by Yiru Huang etc. Then, similar to their methods, we introduce two new bound formulas for Ramsey numbers in this paper. At last, as the applications of our new bound formulas, some upper bounds for small Ramsey numbers will be obtained.

We prove that Mark Gross’ [8] topological Calabi-Yau compactifications can be made into symplectic compactifications. To prove this we develop a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. This produces pairs of compact symplectic 6-manifolds homeomorphic to mirror pairs of Calabi-Yau 3-folds together with Lagrangian fibrations whose underlying integral affine structures are dual.