Black-Box Complexity: Breaking the O(nlog⁡n)O(n \log n) Barrier of LeadingOnes

We show that the unrestricted black-box complexity of the $n$-dimensional XOR- and permutation-invariant LeadingOnes function class is $O(n \log (n) / \log \log n)$. This shows that the recent natural looking $O(n\log n)$ bound is not tight. The black-box optimization algorithm leading to this bound can be implemented in a way that only 3-ary unbiased variation operators are used. Hence our bound is also valid for the unbiased black-box complexity recently introduced by Lehre and Witt (GECCO 2010). The bound also remains valid if we impose the additional restriction that the black-box algorithm does not have access to the objective values but only to their relative order (ranking-based black-box complexity).Comment: 12 pages, to appear in the Proc. of Artificial Evolution 2011, LNCS 7401, Springer, 2012. For the unrestricted black-box complexity of LeadingOnes there is now a tight $\Theta(n \log\log n)$ bound, cf. http://eccc.hpi-web.de/report/2012/087

Weighted Automata and Logics for Infinite Nested Words

Nested words introduced by Alur and Madhusudan are used to capture structures with both linear and hierarchical order, e.g. XML documents, without losing valuable closure properties. Furthermore, Alur and Madhusudan introduced automata and equivalent logics for both finite and infinite nested words, thus extending B\"uchi's theorem to nested words. Recently, average and discounted computations of weights in quantitative systems found much interest. Here, we will introduce and investigate weighted automata models and weighted MSO logics for infinite nested words. As weight structures we consider valuation monoids which incorporate average and discounted computations of weights as well as the classical semirings. We show that under suitable assumptions, two resp. three fragments of our weighted logics can be transformed into each other. Moreover, we show that the logic fragments have the same expressive power as weighted nested word automata.Comment: LATA 2014, 12 page

Fourth order indirect integration method for black hole perturbations: even modes

On the basis of a recently proposed strategy of finite element integration in time domain for partial differential equations with a singular source term, we present a fourth order algorithm for non-rotating black hole perturbations in the Regge-Wheeler gauge. Herein, we address even perturbations induced by a particle plunging in. The forward time value at the upper node of the $(r^*,t)$ grid cell is obtained by an algebraic sum of i) the preceding node values of the same cell, ii) analytic expressions, related to the jump conditions on the wave function and its derivatives, iii) the values of the wave function at adjacent cells. In this approach, the numerical integration does not deal with the source and potential terms directly, for cells crossed by the particle world line. This scheme has also been applied to circular and eccentric orbits and it will be object of a forthcoming publication.Comment: This series of papers deals with EMRI for LISA. With the respect to the v1 version, the algorithm has been improved; convergence tests and references have been added; v2 is composed by 23 pages, and 6 figures. Paper accepted by Class. Quantum Gravity for the special issue on Theory Meets Data Analysis at Comparable and Extreme Mass Ratios (Capra and NRDA) at Perimeier Institute in June 201

The theory of the exponential differential equations of semiabelian varieties

The complete first order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arises from an amalgamation-with-predimension construction in the style of Hrushovski. The theory includes necessary and sufficient conditions for a system of equations to have a solution. The necessary condition generalizes Ax's differential fields version of Schanuel's conjecture to semiabelian varieties. There is a purely algebraic corollary, the "Weak CIT" for semiabelian varieties, which concerns the intersections of algebraic subgroups with algebraic varieties.Comment: 53 pages; v3: Substantial changes, including a completely new introductio

Evolutionary design of a full-envelope full-authority flight control system for an unstable high-performance aircraft

The use of an evolutionary algorithm in the framework of H1 control theory is being considered as a means for synthesizing controller gains that minimize a weighted combination of the infinite norm of the sensitivity function (for disturbance attenuation requirements) and complementary sensitivity function (for robust stability requirements) at the same time. The case study deals with a complete full-authority longitudinal control system for an unstable high-performance jet aircraft featuring (i) a stability and control augmentation system and (ii) autopilot functions (speed and altitude hold). Constraints on closed-loop response are enforced, that representing typical requirements on airplane handling qualities, that makes the control law synthesis process more demanding. Gain scheduling is required, in order to obtain satisfactory performance over the whole flight envelope, so that the synthesis is performed at different reference trim conditions, for several values of the dynamic pressure, used as the scheduling parameter. Nonetheless, the dynamic behaviour of the aircraft may exhibit significant variations when flying at different altitudes, even for the same value of the dynamic pressure, so that a trade-off is required between different feasible controllers synthesized at different altitudes for a given equivalent airspeed. A multiobjective search is thus considered for the determination of the best suited solution to be introduced in the scheduling of the control law. The obtained results are then tested on a longitudinal non-linear model of the aircraft

A dynamic logic for every season

This paper introduces a method to build dynamic logics with a graded semantics. The construction is parametrized by a structure to support both the spaces of truth and of the domain of computations. Possible instantiations of the method range from classical assertional) dynamic logic to less common graded logics suitable to deal with programs whose transitional semantics exhibits fuzzy or weighted behaviour.This leads to the systematic derivation of program logics tailored to specific program classes

Relativistic Celestial Mechanics with PPN Parameters

Starting from the global parametrized post-Newtonian (PPN) reference system with two PPN parameters $\gamma$ and $\beta$ we consider a space-bounded subsystem of matter and construct a local reference system for that subsystem in which the influence of external masses reduces to tidal effects. Both the metric tensor of the local PPN reference system in the first post-Newtonian approximation as well as the coordinate transformations between the global PPN reference system and the local one are constructed in explicit form. The terms proportional to $\eta=4\beta-\gamma-3$ reflecting a violation of the equivalence principle are discussed in detail. We suggest an empirical definition of multipole moments which are intended to play the same role in PPN celestial mechanics as the Blanchet-Damour moments in General Relativity. Starting with the metric tensor in the local PPN reference system we derive translational equations of motion of a test particle in that system. The translational and rotational equations of motion for center of mass and spin of each of $N$ extended massive bodies possessing arbitrary multipole structure are derived. As an application of the general equations of motion a monopole-spin dipole model is considered and the known PPN equations of motion of mass monopoles with spins are rederived.Comment: 71 page