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CHAPTER XI - BASIC TOPOLOGICAL CONCEPTS

In this paper, we give a necessary and sufficient condition under which a variational inequality defined on unbounded sets in a Banach space has a solution. Furthermore, we establish a necessary and sufficient condition under which the minimax equality on unbounded sets is true.

We propose a hierarchy of semidefinite programming (SDP) relaxations for polynomial optimization with sparse patterns over unbounded feasible sets. The convergence of the proposed SDP hierarchy is established for a class of polynomial optimization problems. This is done by employing known sums-of-squares sparsity techniques of Kojima and Muramatsu Comput Optim Appl 42(1):31---41, (2009) and Lasserre SIAM J Optim 17:822---843, (2006) together with a representation theorem for polynomials over unbounded sets obtained recently in Jeyakumar et al. J Optim Theory Appl 163(3):707---718, (2014). We demonstrate that the proposed sparse SDP hierarchy can solve some classes of large scale polynomial optimization problems with unbounded feasible sets using the polynomial optimization solver SparsePOP developed by Waki et al. ACM Trans Math Softw 35:15 (2008).

This paper studies generalized truncated moment problems with unbounded sets. First, we study geometric properties of the truncated moment cone and its dual cone of nonnegative polynomials. By the technique of homogenization, we give a convergent hierarchy of Moment-SOS relaxations for approximating these cones. With them, we give a Moment-SOS method for solving generalized truncated moment problems with unbounded sets. Finitely atomic representing measures, or certificates for their nonexistence, can be obtained by the proposed method. Numerical experiments and applications are also given.

Finite control set model predictive control (FCS-MPC), especially the reference-voltage-vector-based model predictive control (MPC) (RVV-MPC), considers a limited set of input vectors. However, the control performance is not satisfactory enough due to a severe lack of voltage input variants. In particular, additional restraints, such as limiting the currents within setting boundaries, could not be supported with RVV-MPC. In this article, to truly comply with the boundary settings, a reference variant FCS-MPC approach is introduced. In the proposed strategy, reference currents are considered under the maximum torque per current (MTPC) control. As boundary settings could not be supported in RVV-MPC, it is better to search for the optimal current reference that is feasible within the imposed boundaries. Therefore, rather than fixing the current reference, a well-considered neighborhood of the MTPC reference is introduced, and these variants are processed, resulting in a set of optimal candidate inputs. As multiple, but still finite, optimal solutions become possible, additional efforts can be made within the objective function to restrain the current and reduce the torque ripple. The size of the current reference space can be automatically adapted to generate current references that correspond to truly bounded current vectors. A trade-off is then made between control performance and meeting the boundary settings.

Results on the blow-up of nontrivial nonnegative solutions for several classes of nonlinear partial differential inequalities with singularities on unbounded sets are obtained.

. We study the condenser capacity cap p ⁡(E, Ω) on unbounded open sets Ω in a proper connected metric space X equipped with a locally doubling measure supporting a local p-Poincaré inequality, where 1 < p < ∞ . Using a new definition of capacitary potentials, we show that cap p is countably subadditive and that it is a Choquet capacity. We next obtain formulas for the capacity of superlevel sets for the capacitary potential. These are then used to show that p-harmonic Green functions exist in an unbounded domain Ω if and only if either X is p-hyperbolic or the Sobolev capacity C p ( X ∖ Ω ) > 0 . As an application, we deduce new results for Perron solutions and boundary regularity for the Dirichlet boundary value problem for p-harmonic functions in unbounded open sets.

We consider the following distributed parameter linear control system $$ z_{xy} + A\left( {x,y} \right)z_x + B\left( {x,y} \right)z_y + C\left( {x,y} \right)z = F\left( {x,y} \right)U\left( {x,y} \right). $$ Here (x, y) ranges over the unbounded set $$ L_{I,J} = \bigcup\limits_{\left( {u,v} \right) \in I \times J} {l\left( {u,v} \right),} $$ where $$ l\left( {u,v} \right) = \left( {\left[ {u, + \infty } \right[x\left\{ v \right\}} \right) \cup \left( {\left\{ u \right\} \times \left[ {v, + \infty } \right[} \right), \left( {u,v} \right) \in \mathbb{R}^2 , $$ and I, J are two non-degenerate intervals of ℝ. The state vector function z belongs to the Sobolev type functional space $$ W_{p,loc}^* \left( {L_{I,J} ,\mathbb{R}^n } \right) = \left\{ {z \in L_{loc}^p \left( {L_{I,J} ,\mathbb{R}^n } \right):z_x ,z_y ,z_{xy} \in L_{loc}^p \left( {L_{I,J} ,\mathbb{R}^n } \right)} \right\} $$ and the control vector function U is in L loc p (L I,J, ℝm). Moreover, for every (u, v) ∈ I × J, the trace of z on l(u, v) is taken as the system state corresponding to the values x = u, y = v of the parameters. All these traces belong to a functional space of Sobolev type, which does not depend on (u,v).

Solovay proved in 1967 that the axiom of determinateness implies that the filter C generated by closed and unbounded subsets of ω1 is an ultrafilter. It has long been conjectured that a significant part of the theory of the axiom of determinateness should be provable from the hypothesis that C is an ultrafilter, but even the first step of finding inner models with several measurable cardinals has proved elusive. In this paper we show that such models exist. Much of our proof is a modification of Kunen's proof in [3] of the same conclusion from the existence of a measurable cardinal κ such that 2κ &gt; κ+.Since no proof of Solovay's result seems to have been published, we insert a proof here. We want to show that for any set x ⊂ ω1 there is a closed, unbounded set either contained in or disjoint from x. By the lemma of [4] there is a Turing degree d such that either ω1e Є x for all degrees e ≥T d or ω1e ∉ x for all degrees e ≥T d. By a theorem of Sacks [1], [5] every d-admissible is ω1e for some e ≥T d, so it is enough to show that there is a closed, unbounded set of d-admissibles. Let a ⊂ ω have degree d; then is such a set.

CHAPTER XV - COMPLETE SPACES

The differential properties of the minimax solution are investigated in a class of plane Dirichlet problems for the Bellman equation. The class of problems is defined by closed non-convex solid boundary sets whose boundaries contain pseudovertices, which are singular points associated with the singularity of the minimax solution. The differential properties of the solution depend on the order of smoothness of the boundary of the boundary set at the pseudovertices and on the cardinality of the values of the metric projection operator onto this set. The paper distinguishes between situations where the operator has single-point values and when the number of projections is greater than one. Using tools from the theory of alpha sets and Efimov–Stechkin support balls, the features of the characteristic function of a non-convex set are investigated. Formulas for its limit values are found, which in a fairly general case facilitate the construction of a Chebyshev layer of the boundary set, which is a region adjacent to the boundary set in which the minimax solution is differentiable. An example and its meaningful interpretation from the point of view of optimal control are given.

We investigate the descriptional and computational complexity of boundary sets of regular and context-free languages. The right (left, respectively) a-boundary set of a language L are those words that belong to L, where the a-predecessor or the a-successor of these words w.r.t. the prefix (suffix, respectively) relation is not in L. For regular languages described by deterministic finite automata (DFAs) we give tight bounds on the number of states for accepting boundary sets. Moreover, the question whether the boundary sets of a regular language is finite is shown to be NL-complete for DFAs, while it turns out to be PSPACE-complete for nondeterministic devices. Boundary sets for context-free languages are not necessarily context free anymore. Here we find a subtle difference of right and left a-boundary sets. While right a-boundary sets of deterministic context-free languages stay deterministic context free, we give an example of a deterministic context-free language the left a-boundary set of which is already non context free. In fact, the finiteness problem for a-boundary sets of context-free languages becomes undecidable.KeywordsTuring MachineRegular LanguageInput SymbolLanguage TreePushdown AutomatonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Summary Large deformations and discontinuous problems can be calculated using the discontinuous deformation analysis (DDA) method by solving time steps, and this method is suitable for simulating the seismic dynamic response of engineering rock mass structures. However, the boundary setting must be carefully analyzed. In this paper, four boundary settings for the DDA method are investigated. First, the contributions to the DDA equations for nonreflecting boundaries (including the viscous boundary and the viscoelastic boundary) are deduced based on the Newmark method. Second, a free-field boundary is introduced in the DDA method with boundary grid generation and coupling calculation algorithms to accurately simulate external source wave motion, such as earthquakes. Third, seismic input boundary treatments are intensively examined, and the force input method is introduced based on nonreflecting boundaries. Finally, the static-dynamic unified boundary is implemented to ensure consistent boundary transformation. The boundary setting method in the DDA method is discussed, and the suggested treatments are used to analyze the seismic dynamic response of underground caverns. Copyright © 2015 John Wiley & Sons, Ltd.

Three-way decision-based label integration for crowdsourcing

For each positive integer n n , a σ \sigma - Z Z -set B n {B_n} in the Hilbert cube I ∞ {I^\infty } is constructed whose complement s n = I ∞ − B n {s_n} = {I^\infty } - {B_n} is not homeomorphic to the pseudointerior s s of the Hilbert cube though s n {s_n} and B n {B_n} satisfy: (i) every compact subset of s n {s_n} is a Z Z -set in s n {s_n} ; (ii) s n × s n {s_n} \times {s_n} is homeomorphic to s s ; (iii) B n {B_n} admits small maps I ∞ → B n {I^\infty } \to {B_n} ; (iv) s n {s_n} satisfies the discrete n n -cells property; and (v) B n {B_n} is locally ( n − 1 ) (n - 1) -connected in I ∞ {I^\infty } . It is shown that s n {s_n} does not satisfy the discrete ( n + 1 ) (n + 1) -cells property and thus B n {B_n} is not a boundary set, that is, s n {s_n} is not homeomorphic to s s . These examples build upon an example of Anderson, Curtis, and van Mill of a fake boundary set B 0 {B_0} that satisfies (i)-(iv) for n = 0 n = 0 . Their example is not a boundary set since it fails to be locally continuum-connected. The examples constructed herein show that there is a hierarchy of fake boundary sets satisfying (i)-(iv) that satisfy higher and higher orders of a strong form of local connectivity (v).