In Thurston’s notes, he gives two different definitions of the Gromov norm (also called simplicial volume) of a manifold and states that they are equal but does not prove it. Gromov proves it in the special case of hyperbolic manifolds as a consequence of his proof that simplicial volume is proportional to volume. We give a proof for all differentiable manifolds. MSC: 57N65

In this paper, we study (uniformly) mean ergodic composition operators on $$H^\infty (\mathbb {B}_n)$$ . Under some additional assumptions, it is shown that have norm convergent iterates in , and they are always uniformly ergodic.

Recently the L∞-norm optimization has been introduced to multi-view geometry to achieve global optimality. It is solved through solving a sequence of SOCP (second order cone programming) feasibility problem which needs sophisticated solvers and time consuming. This paper presents an efficient smooth approximation for L∞-norm optimization in multi-view geometry using log-sum-exp functions. We ha...

We propose a practical method for L0 norm regularization for neural networks: pruning the network during training by encouraging weights to become exactly zero. Such regularization is interesting since (1) it can greatly speed up training and inference, and (2) it can improve generalization. AIC and BIC, well-known model selection criteria, are special cases of L0 regularization. However, since...

In this paper, a new affine projection sign algorithm with a variable step-size, robust in impulsive noise environments, is proposed. For that purpose, l1-norm of the a posteriori error vector is minimized under a box constraint on the step-size. Since the proposed l1-norm minimization problem is non-differentiable convex one, an efficient numerical procedure is developed for the affine project...

Interval-gradient cuts are (nonlinear) valid inequalities for nonconvex NLPs defined for constraints g(x) ≤ 0 with g being continuously differentiable in a box [x, x̄]. In this paper we define intervalsubgradient cuts, a generalization to the case of nondifferentiable g, and show that no-good cuts (which have the form ‖x−x̂‖ ≥ ε for some norm and positive constant ε) are a special case of interva...

and Applied Analysis 3 2. Main Results Firstly, we give some notation. The space of all infinitely differentiable functions φ t, x with compact support in 0, ∞ ×R is denoted byC∞ 0 . L L R 1 ≤ p < ∞ is the space of all measurable functions h such that ‖h‖pLp ∫ R |h t, x |pdx < ∞. We define L∞ L∞ R with the standard norm ‖h‖L∞ infm e 0supx∈R\e|h t, x |. For any real number s, H H R denotes the S...

This paper introduces an approach for the minimization of the discrepancy norm. The general idea is to replace the infinity norms appearing in the definition by L norms which are differentiable and to make use of this approximation for local optimization. We will show that the discrepancy norm can be approximated up to any ε and the robustness of this approximation is shown. Moreover, analytica...

This paper proposes a new approach to solve the problem of designing optimal proportional-integral-derivative (PID) controllers that minimize an H2-norm associated with the set-point response while subjecting to an H∞-norm on the load disturbance rejection. The proposed design approach consists of constructing the feasible domain in the controller gain space and searching over the domain the op...

Let S be a discrete semigroup, m (S) the space of all bounded real functions on S with the usualsupremum norm. Let Acm (S) be a uniformly closed left invariant subalgebra of m (S) with 1 c A. We say that A is extremely left amenable if there isamultiplicative left invariant meanon A. Let P = {h ?A: h =|g-1,g | forsome g ?A, s ?S}. It isshown that . A is extremely left amenable if and only ...