Monotone Case Research Articles

Complexity of DNF minimization and isomorphism testing for monotone formulas

We investigate the complexity of finding prime implicants and minimum equivalent DNFs for Boolean formulas, and of testing equivalence and isomorphism of monotone formulas. For DNF related problems, the complexity of the monotone case differs strongly from the arbitrary case. We show that it is DP-complete to check whether a monomial is a prime implicant for an arbitrary formula, but the equivalent problem for monotone formulas is in L. We show PP-completeness of checking if the minimum size of a DNF for a monotone formula is at most k, and for k in unary, we show that the complexity of the problem drops to coNP. In Christopher Umans [Christopher Umans, The minimum equivalent DNF problem and shortest implicants, Journal of Computer and System Sciences 63 (4) (2001) 597–611] a similar problem for arbitrary formulas was shown to be ∑ 2 p -complete. We show that calculating the minimum equivalent DNF for a monotone formula is possible in output-polynomial time if and only if P = NP. Finally, we disprove a conjecture from Steffen Reith [Steffen Reith, On the complexity of some equivalence problems for propositional calculi, in: Proceedings of the 28th International Symposium on Mathematical Foundations of Computer Science (MFCS), vol. 2747, Lecture Notes in Computer Science, Springer, 2003, pp. 632–641] by showing that checking whether two formulas are isomorphic has the same complexity for arbitrary formulas as for monotone formulas.

Global Existence Results and Uniqueness for Dislocation Equations

We are interested in nonlocal Eikonal Equations arising in the study of the dynamics of dislocations lines in crystals. For these nonlocal but also non monotone equations, only the existence and uniqueness of Lipschitz and local-in-time solutions were available in some particular cases. In this paper, we propose a definition of weak solutions for which we are able to prove the existence for all time. Then we discuss the uniqueness of such solutions in several situations, both in the monotone and non monotone case.

MP1

This tutorial assumes basic knowledge or experience with the Ada programming language. Basic understanding of tasking and real-time constraints -- at the level of tutorial MA1 -- would be helpful. We assume that attendees have a minimal background on real time operating systems, Ada tasking and critical real time applications/systems; however, a (very) short presentation of such topics will be included in the tutorial. This tutorial deals with real time scheduling theory and its use with Ada. Real time scheduling theory was firstly proposed in 1974, but stays mostly unused by many software designers. Too many practitioners have no background on real time scheduling. This tutorial should help them to understand the basis of such a theory and to apply it with Ada technologies. This tutorial is expected to help Ada engineers to predict the performances of their critical Ada real time applications. This tutorial will be organized in three parts: 1. First, we define what a real time system is and then, the theoretical results of the real time scheduling theory are presented. The basic task models and schedulers are explained (their behavior and their properties). The tutorial focusses on fixed priority scheduler and its feasibility tests (eg. Rate monotonic and worst case task response time) which is the most useful scheduler with current real-life applications. However, a dynamic priority scheduler is also presented (eg. EDF). At the end of this first part, attendees should understand what a feasibility test is and how it can help them to predict performances of their real time applications. This part is ended with a short introduction on shared resource support (eg. what a priority inversion is, how PIP/PCP work). 2. Second, we show how real time scheduling features are taken into account in Ada international standards such as ISO/IEC Ada 1995 and 2005, the POSIX 1003 Ada binding and ARINC 653 (which is an avionic operating system standard providing an Ada binding). In this part, we try to show the relationships between the real time scheduling theory and the implementation of real time schedulers into practical systems. 3. Finally, some Ada program examples are presented and explained.

Extension of Arrow's theorem to symmetric sets of tournaments

Arrow's impossibility theorem [K.J. Arrow, Social Choice and Individual Values, Wiley, New York, NY, 1951] shows that the set of acyclic tournaments is not closed to non-dictatorial Boolean aggregation. In this paper we extend the notion of aggregation to general tournaments and we show that for tournaments with four vertices or more any proper symmetric (closed to vertex permutations) subset cannot be closed to non-dictatorial monotone aggregation and to non-neutral aggregation. We also demonstrate a proper subset of tournaments that is closed to parity aggregation for an arbitrarily large number of vertices. This proves a conjecture of Kalai [Social choice without rationality, Reviewed NAJ Economics 3(4)] for the non-neutral and the non-dictatorial and monotone cases and gives a counter example for the general case.

Finite element response sensitivity analysis of continuous steel-concrete composite girders

The behavior of steel-concrete composite beams is strongly influenced by the type of shear connection between the steel beam and the concrete slab. For accurate analytical predictions, the structural model must account for the interlayer slip between these two components. This paper focuses on a procedure for response sensitivity analysis using state-of-the-art finite elements for composite beams with deformable shear connection. Monotonic and cyclic loading cases are considered. Realistic cyclic uniaxial constitutive laws are adopted for the steel and concrete materials as well as for the shear connection. The finite element response sensitivity analysis is performed according to the Direct Differentiation Method (DDM); its analytical derivation and computer implementation are validated through Forward Finite Difference (FFD) analysis. Sensitivity analysis results are used to gain insight into the effect and relative importance of the various material parameters in regards to the nonlinear monotonic and cyclic response of continuous composite beams, which are commonly used in bridge construction.

Assessment of material damage in a nickel-base superalloy using nonlinear Rayleigh surface waves

A reliable laser-based ultrasonic technique is developed to measure the second order harmonic amplitude of a Rayleigh surface wave propagating in metallic specimens. Rayleigh waves are experimentally generated with a wedge transducer and detected with a heterodyne laser interferometer. The capability of this system to measure the nonlinear contribution present in Rayleigh surface waves is demonstrated, and these results are interpreted in terms of a parameter developed for Rayleigh surface waves which corresponds to the nonlinear parameter of a longitudinal wave, β. The proposed measurement technique is used to assess damage in nickel-base high temperature alloy specimens, and the evolution of material nonlinearity under various loading conditions is quantitatively measured in terms of the increasing amplitude of the second order harmonic. These results show that there is a significant increase in the second order harmonic amplitude at monotonic tensile loads above the material’s yield stress, and that during low cycle fatigue tests, the increase in the second order harmonic amplitude is considerable, although less significant than that in the monotonic loading case. The results from this study show the major role that accumulated plasticity plays in the increase in material nonlinearity, and demonstrate the effectiveness of the proposed experimental procedure to track damage in high temperature alloys.

Calabi quasi-morphisms for some non-monotone symplectic manifolds

In this work we construct Calabi quasi-morphisms on the universal cover of the group Ham(M) of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasi-morphisms descend to non-trivial homomorphisms on the fundamental group of Ham(M).

A projection algorithm for general variational inequalities with perturbed constraint sets

We consider the problem of general variational inequalities, GVI, with nonmonotone operator, in a finite dimensional space. We propose a method to solve GVI that at each iteration considers only one projection on an easy approximation of the constraint set, which is important from a practical point of view. We analyse the convergence of the algorithm under a weak cocoercivity condition, using variational metric analysis. Computational experience is reported and comparative analysis with other two algorithms is also given for the monotone case.

Computable convergence rates for sub-geometric ergodic Markov chains

In this paper, we give quantitative bounds on the f-total variation distance from convergence of a Harris recurrent Markov chain on a given state space under drift and minorization conditions implying ergodicity at a subgeometric rate. These bounds are then specialized to the stochastically monotone case, covering the case where there is no minimal reachable element. The results are illustrated with two examples, from queueing theory and Markov Chain Monte Carlo theory.

A global stability criterion for a family of delayed population models

For a family of single-species delayed population models, a new global stability condition is found. This condition is sharp and can be applied in both monotone and nonmonotone cases. Moreover, the consideration of variable or distributed delays is allowed. We illustrate our approach on the Mackey-Glass equations and the Lasota-Wazewska model.

Gromov--Witten invariants of symplectic quotients and adiabatic limits

We study pseudoholomorphic curves in symplectic quotients as adiabatic limits of solutions to the symplectic vortex equations. Our main theorem asserts that the genus zero invariants of Hamiltonian group actions defined by these equations are related to the genus zero Gromov--Witten invariants of the symplectic quotient (in the monotone case) via a natural ring homomorphism from the equivariant cohomology of the ambient space to the quantum cohomology of the quotient.

Enlarging the Hamiltonian group

This paper investigates ways to enlarge the Hamiltonian subgroup Ham of the symplectomorphism group Symp(M) of the symplectic manifold (M,ω) to a group that both intersects every connected component of Symp(M) and characterizes symplectic bundles with fiber M and closed connection form. As a consequence, it is shown that bundles with closed connection form are stable under appropriate small perturbations of the symplectic form. Further, the manifold (M,ω) has the property that every symplectic M -bundle has a closed connection form if and only if the flux group vanishes and the flux homomorphism extends to a crossed homomorphism defined on the whole group Symp(M). The latter condition is equivalent to saying that a connected component of the commutator subgroup [Symp,Symp] intersects the identity component of Symp only if it also intersects Ham. It is not yet clear when this condition is satisfied. We show that if the symplectic form vanishes on 2-tori the flux homomorphism extends to the subgroup of Symp acting trivially on π1(M). We also give an explicit formula for the Kotschick–Morita extension of Flux in the monotone case. The results in this paper belong to the realm of soft symplectic topology, but raise some questions that may need hard methods to answer.

Forcing relation on interval patterns

We consider-without restriction to the piecewise monotone case—a forc- ing relation on interval (transitive, roof, bottom) patterns. We prove some basic properties of this type of forcing and explain when it is a partial ordering. Finally, we show how our approach relates to the results known from the literature.

On the occurrence of even harmonics in the shear stress response of viscoelastic fluids in large amplitude oscillatory shear

The non-linear response of polymeric liquids observed experimentally in large amplitude oscillatory shear (LAOS) is generally characterized by the presence of odd harmonics of the excitation frequency in the Fourier spectrum for the shear stress. Even harmonics of relatively smaller amplitude have also been observed, whose appearance is usually attributed to wall slip phenomena. In the present work, we show that wall slip is not a necessary condition for the occurrence of even harmonics. To this end, we perform a non-linear study of planar LAOS flow between two infinite parallel plates using either a monotone or non-monotone viscoelastic constitutive equation (i.e., respectively, the Giesekus and Johnson–Segalman models). The analysis allows for spatially non-homogeneous velocity and stress fields. We assume no-slip boundary conditions, and investigate the combined effects of inertia, elasticity, and shear thinning by means of spectral methods. A regular perturbation analysis is also conducted in the inertialess monotone case. Results for the Giesekus model show that combination of elasticity and shear thinning yields transient even harmonics in shear stress whose life span and intensity are considerably increased by inertia. Furthermore, the one-dimensional flow is unstable to finite two-dimensional perturbations under inertia and at high elasticity. This results in the development of secondary flows and saturation of even harmonics into small but finite values. Simulations for the non-monotone Johnson–Segalman model predict even harmonics of relatively larger amplitude that settle in dynamic equilibrium. Furthermore, the fluid’s response is quasi-periodic with the appearance of incommensurate frequencies.

Observation of Optical Interferometric Band Structure Representing Plastic Deformation Front under Cyclic Loading

An interferometric band structure previously observed under monotonic tensile loading has been observed under cyclic loading as well. Like the previous monotonic loading case, the band structure is found to represent the plastic deformation front, indicating that it can be used for visualization of stress concentration. Observed correlation between the motions of the plastic deformation front and the test machine's crosshead indicates that the plastic deformation front moves in proportion to the strength of the displacement field induced by the test machine. This is consistent with a previous report that the propagation speed of the plastic deformation front varies in proportion to the monotonic loading speed.

The monotone condition for optimal multiple stopping problems

This paper is concerned with the optimal stopping problems for discrete time stochastic processes with the multiple stopping rules. The sufficient condition for the optimality of multiple stopping rules, the so-called monotone condition, is introduced for multiple stopping problems and the optimality of a certain multiple stopping rule is proved in the monotone case.

Spatial modeling and prediction under stationary non-geometric range anisotropy

For modeling spatial processes, we propose a rich parametric class of stationary range anisotropic covariance structures that, when applied in R2, greatly increases the scope of variogram contors. Geometric anisotropy, which provides the most common generalization of isotropy within stationarity, is a special case. Our class is built from monotonic isotropic correlation functions and special cases include the Matern and the general exponential functions. As a result, our range anisotropic correlation specification can be attached to a second order stationary spatial process model, unlike ad hoc approaches to range anisotropy in the literature. We adopt a Bayesian perspective to obtain full inference and demonstrate how to fit the resulting model using sampling-based methods. In the presence of measurement error/microscale effect, we can obtain both the usual predictive as well as the noiseless predictive distribution. We analyze a data set of scallop catches under the general exponential range anisotropic model, withholding ten sites to compare the accuracy and precision of the standard and noiseless predictive distributions.

Solving integer programs over monotone inequalities in three variables: A framework for half integrality and good approximations

We define a class of monotone integer programs with constraints that involve up to three variables each. A generic constraint in such integer program is of the form ax− by⩽ z+ c, where a and b are nonnegative and the variable z appears only in that constraint. We devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables U. The solution is obtained from a minimum cut on a graph with O( nU) nodes and O( mU) arcs where n is the number of variables of the types x and y and m is the number of constraints. Our algorithm is also valid for nonlinear objective functions. Nonmonotone integer programs are optimization problems with constraints of the type ax+ by⩽ z+ c without restriction on the signs of a and b. Such problems are in general NP-hard. We devise here an algorithm, relying on a transformation to the monotone case, that delivers half integral superoptimal solutions in polynomial time. Such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation. When the half integral solution can be rounded to an integer feasible solution, this is a 2-approximate solution. In that the technique is a unified 2-approximation technique for a large class of problems. The results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe. The algorithm described here has a wide array of problem applications. An additional important consequence of our results is that nonmonotone problems in the framework are MAX SNP-hard and at least as hard to approximate as vertex cover. Problems that are amenable to the analysis provided here are easily recognized. The analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality.

On convergence of descent methods for variational inequalities in a Hilbert space

In this paper, properties of differentiable gap functions for variational inequalities and convergence of the corresponding descent methods under a Hilbert space setting are considered. We give various convergence results under different assumptions on the cost mapping, including the monotone case.

Global Convergence of Algorithms with Nonmonotone Line Search Strategy in Unconstrained Optimization

In this paper we state some nonmonotone line search strategies for unconstrained optimization algorithms. Abstracting from the concrete line search strategy we prove two general convergence results. Using this theory we can show the global convergence of the BFGS method with nonmonotone line search strategy. In contrast to some former results about nonmonotone line search strategies, both our convergence results and their proofs are natural generalizations of known results for the monotone case.