Constant mean curvature hypersurfaces with constant δ-invariant

We completely classify constant mean curvature hypersurfaces (CMC) with constant δ-invariant in the unit 4-sphere S4 and in the Euclidean 4-space E4.

Invariant-free deduction systems for temporal logic

In this thesis we propose a new approach to deduction methods for temporal logic. Our proposal is based on an inductive definition of eventualities that is different from the usual one. On the basis of this non-customary inductive definition for eventualities, we first provide dual systems of tableaux and sequents for Propositional Linear-time Temporal Logic (PLTL). Then, we adapt the deductive approach introduced by means of these dual tableau and sequent systems to the resolution framework and we present a clausal temporal resolution method for PLTL. Finally, we make use of this new clausal temporal resolution method for establishing logical foundations for declarative temporal logic programming languages. The key element in the deduction systems for temporal logic is to deal with eventualities and hidden invariants that may prevent the fulfillment of eventualities. Different ways of addressing this issue can be found in the works on deduction systems for temporal logic. Traditional tableau systems for temporal logic generate an auxiliary graph in a first pass.Then, in a second pass, unsatisfiable nodes are pruned. In particular, the second pass must check whether the eventualities are fulfilled. The one-pass tableau calculus introduced by S. Schwendimann requires an additional handling of information in order to detect cyclic branches that contain unfulfilled eventualities. Regarding traditional sequent calculi for temporal logic, the issue of eventualities and hidden invariants is tackled by making use of a kind of inference rules (mainly, invariant-based rules or infinitary rules) that complicates their automation. A remarkable consequence of using either a two-pass approach based on auxiliary graphs or aone-pass approach that requires an additional handling of information in the tableau framework, and either invariant-based rules or infinitary rules in the sequent framework, is that temporal logic fails to carry out the classical correspondence between tableaux and sequents. In this thesis, we first provide a one-pass tableau method TTM that instead of a graph obtains a cyclic tree to decide whether a set of PLTL-formulas is satisfiable. In TTM tableaux are classical-like. For unsatisfiable sets of formulas, TTM produces tableaux whose leaves contain a formula and its negation. In the case of satisfiable sets of formulas, TTM builds tableaux where each fully expanded open branch characterizes a collection of models for the set of formulas in the root. The tableau method TTM is complete and yields a decision procedure for PLTL. This tableau method is directly associated to a one-sided sequent calculus called TTC. Since TTM is free from all the structural rules that hinder the mechanization of deduction, e.g. weakening and contraction, then the resulting sequent calculus TTC is also free from this kind of structural rules. In particular, TTC is free of any kind of cut, including invariant-based cut. From the deduction system TTC, we obtain a two-sided sequent calculus GTC that preserves all these good freeness properties and is finitary, sound and complete for PLTL. Therefore, we show that the classical correspondence between tableaux and sequent calculi can be extended to temporal logic. The most fruitful approach in the literature on resolution methods for temporal logic, which was started with the seminal paper of M. Fisher, deals with PLTL and requires to generate invariants for performing resolution on eventualities. In this thesis, we present a new approach to resolution for PLTL. The main novelty of our approach is that we do not generate invariants for performing resolution on eventualities. Our method is based on the dual methods of tableaux and sequents for PLTL mentioned above. Our resolution method involves translation into a clausal normal form that is a direct extension of classical CNF. We first show that any PLTL-formula can be transformed into this clausal normal form. Then, we present our temporal resolution method, called TRS-resolution, that extends classical propositional resolution. Finally, we prove that TRS-resolution is sound and complete. In fact, it finishes for any input formula deciding its satisfiability, hence it gives rise to a new decision procedure for PLTL. In the field of temporal logic programming, the declarative proposals that provide a completeness result do not allow eventualities, whereas the proposals that follow the imperative future approach either restrict the use of eventualities or deal with them by calculating an upper bound based on the small model property for PLTL. In the latter, when the length of a derivation reaches the upper bound, the derivation is given up and backtracking is used to try another possible derivation. In this thesis we present a declarative propositional temporal logic programming language, called TeDiLog, that is a combination of the temporal and disjunctive paradigms in Logic Programming. We establish the logical foundations of our proposal by formally defining operational and logical semantics for TeDiLog and by proving their equivalence. Since TeDiLog is, syntactically, a sublanguage of PLTL, the logical semantics of TeDiLog is supported by PLTL logical consequence. The operational semantics of TeDiLog is based on TRS-resolution. TeDiLog allows both eventualities and always-formulas to occur in clause heads and also in clause bodies. To the best of our knowledge, TeDiLog is the first declarative temporal logic programming language that achieves this high degree of expressiveness. Since the tableau method presented in this thesis is able to detect that the fulfillment of an eventuality is prevented by a hidden invariant without checking for it by means of an extra process, since our finitary sequent calculi do not include invariant-based rules and since our resolution method dispenses with invariant generation, we say that our deduction methods are invariant-free.

Permutationally invariant state reconstruction

Feasible tomography schemes for large particle numbers must possess, besides an appropriate data acquisition protocol, an efficient way to reconstruct the density operator from the observed finite data set. Since state reconstruction typically requires the solution of a nonlinear large-scale optimization problem, this is a major challenge in the design of scalable tomography schemes. Here we present an efficient state reconstruction scheme for permutationally invariant quantum state tomography. It works for all common state-of-the-art reconstruction principles, including, in particular, maximum likelihood and least squares methods, which are the preferred choices in today's experiments. This high efficiency is achieved by greatly reducing the dimensionality of the problem employing a particular representation of permutationally invariant states known from spin coupling combined with convex optimization, which has clear advantages regarding speed, control and accuracy in comparison to commonly employed numerical routines. First prototype implementations easily allow reconstruction of a state of 20 qubits in a few minutes on a standard computer

Optimally robust shortcuts to population inversion in two-level quantum systems

19 p.

Geometric multiplicity margin for a submatrix

AMS Classification: 15A18, 15A21, 15A60.

Properties of convergence of a class of iterative processes generated by sequences of self-mappings with applications to switched dynamic systems

This article investigates the convergence properties of iterative processes involving sequences of self-mappings of metric or Banach spaces. Such sequences are built from a set of primary self-mappings which are either expansive or non-expansive self-mappings and some of the non-expansive ones can be contractive including the case of strict contractions. The sequences are built subject to switching laws which select each active self-mapping on a certain activation interval in such a way that essential properties of boundedness and convergence of distances and iterated sequences are guaranteed. Applications to the important problem of stability of dynamic switched systems are also given.

Hyers-Ulam-Rassias Stability of Functional Differential Systems with Point and Distributed Delays

This paper investigates stability and asymptotic properties of the error with respect to its nominal version of a nonlinear time-varying perturbed functional differential system subject to point, finite-distributed, and Volterra-type distributed delays associated with linear dynamics together with a class of nonlinear delayed dynamics. The boundedness of the error and its asymptotic convergence to zero are investigated with the results being obtained based on the Hyers-Ulam-Rassias analysis.

Possible antigravity regions in F (R) theory?

We construct an F(R) gravity theory corresponding to the Weyl invariant two scalar field theory. We investigate whether such F (R) gravity can have the antigravity regions where the Weyl curvature invariant does not diverge at the Big Bang and Big Crunch singularities. It is revealed that the divergence cannot be evaded completely but can be much milder than that in the original Weyl invariant two scalar field theory. (C) 2014 The Authors. Published by Elsevier B.V.

Convergence Properties and Fixed Points of Two General Iterative Schemes with Composed Maps in Banach Spaces with Applications to Guaranteed Global Stability

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems.