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## Regularization Methods for Nonexpansive Semigroups in Hilbert Spaces

### Vietnam Journal of Mathematics (2016-09-01) 44: 637-648 , September 01, 2016

The purpose of this paper is to present a regularization method for finding a common fixed point of a nonexpansive semigroup in a real Hilbert space. A combination of the considered regularization scheme with the proximal point algorithm and another one of the regularization method with an iterative scheme is studied in this research. We also discuss an application of the proposed methods for an initial value problem in Hilbert spaces.

## Prescription of Gauss curvature using optimal mass transport

### Geometriae Dedicata (2016-08-01) 183: 81-99 , August 01, 2016

In this paper we give a new proof of a theorem by Alexandrov on the Gauss curvature prescription of Euclidean convex sets. This proof is based on the duality theory of convex sets and on optimal mass transport. A noteworthy property of this proof is that it does not rely neither on the theory of convex polyhedra nor on P.D.E. methods (which appeared in all the previous proofs of this result).

## Book reviews

### Monatshefte für Mathematik (2016-07-01) 180: 661-663 , July 01, 2016

## Sard theorems for Lipschitz functions and applications in optimization

### Israel Journal of Mathematics (2016-05-01) 212: 757-790 , May 01, 2016

We establish a “preparatory Sard theorem” for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R^{d} to R^{p} that can be expressed as finite selections of *C*^{k} functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet–Daniilidis–Dambrine [1] to the case *p* > 1. Applications in semi-infinite and Pareto optimization are given.

## Gradient-constrained discounted Steiner trees I: optimal tree configurations

### Journal of Global Optimization (2016-03-01) 64: 497-513 , March 01, 2016

A *gradient-constrained discounted Steiner tree**T* is a maximum Net Present Value (NPV) tree, spanning a given set N of nodes in space with edges whose gradients are all no more than an upper bound *m* which is the maximum gradient. The nodes in *T* but not in *N* are referred to as *discounted Steiner points*. Such a tree has costs associated with its edges and values associated with its nodes. In order to reach the nodes in the tree, the edges need to be constructed. The edges are constructed in a particular order and the costs of constructing the edges and the values at the nodes are discounted over time. In this paper, we study the optimal tree configurations so as to maximize the sum of all the discounted cash flows, known as the NPV. An application of this problem occurs in underground mining, where we want to optimally locate a junction point in the underground access network to maximize the NPV in the presence of the gradient constraint. This constraint defines the navigability conditions on mining vehicles along the underground tunnels. Labellings are essential for defining a tree configuration and indicate gradients on the edges of the network. An edge in a gradient-constrained discounted Steiner tree is labelled as an *f* edge, an *m* edge or a *b* edge, if the gradient is less, equal or greater than *m* respectively. Each tree configuration is identified by the labellings of its edges. In this paper the non-optimal sets of labellings of edges that are incident with the discounted Steiner point in a gradient-constrained discounted Steiner network are classified. This reduces the number of configurations that need to be considered when optimizing. In addition, the gradient-constrained discounted Steiner point algorithm is outlined.

## Mathematische Grundlagen der Naturwissenschaften

### Mathematische Grundlagen der Naturwissenschaften (2016-01-01) , January 01, 2016

## Control measures of pine wilt disease

### Computational and Applied Mathematics (2016-07-01) 35: 519-531 , July 01, 2016

In this paper, we study a vector–host model of pine wilt disease with vital dynamics to determine the equilibria and their stability by considering standard incidence rates and horizontal transmission. The complete global analysis for the equilibria of the model is analyzed. The explicit formula for the reproductive number is obtained, and it is shown that the “disease-free” equilibrium always exists and is globally asymptotically stable whenever $$R_{0}\le 1$$ . Furthermore, the disease persists at an “ endemic” level when the reproductive number exceeds unity. It will be very helpful in providing a theoretical basis for the prevention and control of the disease.

## A General Vector-Valued Beurling Theorem

### Integral Equations and Operator Theory (2016-11-01) 86: 321-332 , November 01, 2016

Suppose $$\alpha $$ is a rotationally symmetric norm on $$L^{\infty }\left( \mathbb {T}\right) $$ and $$\beta $$ is a “nice” norm on $$L^{\infty }\left( \Omega ,\mu \right) $$ where $$\mu $$ is a $$\sigma $$ -finite measure on $$\Omega $$ . We prove a version of Beurling’s invariant subspace theorem for the space $$L^{\beta }\left( \mu ,H^{\alpha }\right) .$$ Our proof uses the version of Beurling’s theorem on $$H^{\alpha }\left( \mathbb {T}\right) $$ in Chen (Adv Appl Math, 2016) and measurable cross-section techniques. Our result significantly extends a result of Rezaei, Talebzadeh, and Shin (Int J Math Anal 6:701–707, 2012).

## Sparse and Low-Rank Methods

### Generalized Principal Component Analysis (2016-01-01) 40: 291-346 , January 01, 2016

The previous chapter studies a family of subspace clustering methods based on spectral clustering. In particular, we have studied both local and global methods for defining a subspace clustering affinity, and have noticed that we seem to be facing an important dilemma. On the one hand, local methods compute an affinity that depends only on the data points in a local neighborhood of each data point. Local methods can be rather efficient and somewhat robust to outliers, but they cannot deal well with intersecting subspaces. On the other hand, global methods utilize geometric information derived from the entire data set (or a large portion of it) to construct the affinity. Global methods might be immune to local mistakes, but they come with a big price: their computational complexity is often exponential in the dimension and number of subspaces. Moreover, none of the methods comes with a theoretical analysis that guarantees the correctness of clustering. Therefore, a natural question that arises is whether we can construct a subspace clustering affinity that utilizes global geometric relationships among all the data points, is computationally tractable when the dimension and number of subspaces are large, and is guaranteed to provide the correct clustering under certain conditions.