Keywords: Quantitative veriﬁcation, system distance, distance hierarchy, **linear** **time**, branching **time**
1. Introduction
For rigorous design and veriﬁcation of embedded systems, both qualitative and quantitative information and constraints have to be taken into account [23, 26, 30]. This applies to the models considered, to the properties one wishes to be satisﬁed, and to the verification itself. Hence the question asked in quantitative veriﬁcation is not “Does the system satisfy the requirements?”, but rather “To which extent does the system satisfy the requirements?” Standard qualitative veriﬁcation techniques are inherently fragile: either the requirements are satisﬁed, or they are not, regardless of how close the actual system might come to the speciﬁcation. To overcome this lack of robustness, notions of distance between systems are essential.

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5 Conclusions
We have presented a technique called geometric sampling to construct **linear**-**time** CUR algorithms for admissible blocks of a hierarchical matrix coming from the discretization of a BEM problem. We have presented a relative error bound for geometric column sampling, which we then extended to a bound for a CUR approximation. Also, this bound can directly be used for truncated QR factorizations and inter- polative decompositions. Numerical experiments showed good performance for different integral kernels evaluated on challenging domains. We compared two CUR algorithms created with geometric sampling against ACA with partial pivoting technique. The results showed that our main algorithm CUR_GCS is very efficient and even can handle convergence issues of ACA with partial pivoting, having accuracy comparable with quadratic cost algorithms QRCP and ACA with full pivoting.

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IV. C ONCLUSION
A full characterization of the equivalence of **linear** **time**-delay sys- tems has been given. We claim that the necessary and sufficient condi- tions proposed in this paper are easy to test for the case of equivalence to delay-free systems. The quantity and complexity of equations to solve is increased for the case of delay-reduction. However, the approach is useful and gives a constructive way to find the transformation. Theorem 2 is a full answer to a longstanding open problem. Further research is required to extend this result to a broad class of nonlinear systems.

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Within this **time** model, a number of extensions of the logic and the automata model have been studied. But one can also consider more general models of **time**: general **linear** **time** could be useful in different settings, including concurrency, asynchronous communi- cation, and others, where using the set of integers can be too simplistic. Possible choices include ordinals, the reals, or even arbitrary **linear** orderings. In terms of expressivity, while LTL with Until and Since is expressively complete (i.e. equivalent to FO) on Dedekind- complete orderings (which includes the ordering of the reals as well as all ordinals), this does not hold in the general case. Two more connectives, the future and past Stavi opera- tors, are necessary to handle gaps [9] when considering arbitrary **linear** orderings.

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Splitting a Delaunay Triangulation in Linear Time Bernard Chazelle, Olivier Devillers, Ferran Hurtado, Mercè Mora, Vera Sacristán, Monique Teillaud.. To cite this version: Bernard Chazel[r]

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W [left (i), 0] . . . W [left (i), t − 1] and
W [right (i), 0] . . . W [right (i), t − 1]
Our lower bound proceeds by showing that this correspondence is in some sense inherent, as we reduce (max, +) convolution to tree sparsity over a binary tree of depth Θ(k) with three long paths from the root to the leaves. In turn, our approximation algorithms are obtained by solving approximate (max, +) convolutions or (min, +)-convolutions. Such approximations are known to be computable in nearly-**linear** **time** [Zwi98, PS16]. For completeness, we include simpler (albeit slightly slower) algorithms performing those tasks in the appendix.

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(Received 00 Month 200x; revised 00 Month 200x; in final form 00 Month 200x)
Monotonicity is a simple yet significant qualitative characteristic. We consider the problem of segmenting a sequence in up to K segments. We want segments to be as monotonic as possible and to alternate signs. We propose a quality metric for this problem using the l ∞ norm,
and we present an optimal **linear** **time** algorithm based on novel formalism. Moreover, given a precomputation in **time** O(n log n) consisting of a labeling of all extrema, we compute any optimal segmentation in constant **time**. We compare experimentally its performance to two piecewise **linear** segmentation heuristics (top-down and bottom-up). We show that our algorithm is faster and more accurate. Applications include pattern recognition and qualitative modeling.

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Gaujal Bruno, Girault Alain, Plassart Stéphan ∗
Project-Teams Polaris and Spades
Research Report n° 9339 — April 2020 — 23 pages
Abstract: We consider the classical problem of minimizing off-line the total energy consumption required to execute a set of n real-**time** jobs on a single processor with varying speed. Each real-**time** job is defined by its release **time**, size, and deadline (all integers). The goal is to find a processor speed schedule, such that no job misses its deadline and the energy consumption is minimal. We propose a **linear** **time** algorithm that checks the schedulability of the given set of n jobs and computes an optimal speed schedule. The **time** complexity of our algorithm is in O(n), to be compared with O(n log(n)) for the best known solution. Besides the complexity gain, the main interest of our algorithm is that it is based on a completely different idea: instead of computing the critical intervals , it sweeps the set of jobs and uses a dynamic programming approach to compute an optimal speed schedule. Our **linear** **time** algorithm is still valid (with some changes) when switching costs are taken into account.

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The paper is structured as follows: The next section collects the preliminary definitions and gives a precise statement of our main result. In Sec. 2, we establish the left-to-right inclusion of the identity displayed in (1). The rest of the paper is devoted to the converse inclusion, whose proof is far more involved. In Sec. 3 we build a monotonic Horn formula expressing the language of palindromes (a “toy” example) and deduce from it a cellular automaton that recognizes this language in **linear** **time**. Sec. 4 generalizes this construction to any problem defined in mon-ESO-HORN d (∀ d+1 , arity d+1 ), thus completing the proof of (1). In Sec. 5, we conclude by arguing for the optimality of our result.

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Much effort has gone into developing efficient algorithms for minimising the bottleneck distance between two point sets under groups of transformations. However, the algorithms that have thus far been developed suffer from running times that are large polynomials in the size of the input, even for approximate formulations of the problem. In this paper we define a point set similarity measure that includes both the bottleneck distance and the Hausdorff distance as special cases. This measure relaxes the condition that the mapping must be one-to-one, but guarantees that only a few points are mapped to any point. Using a novel application of Hall’s Theorem to reduce the geometric matching problem to a combinatorial matching problem, we present near-**linear** **time** approximation schemes for minimising this distance between two point sets in the plane under isometries; we note here that the best known algorithms for congruence under the bottleneck measure run in **time** O(n 2.5 ).

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Key words. graph theory, graph algorithms, graph decomposition, split decompostion, 1-join decomposition
AMS subject classifications. 68R10, 05C85
1. Introduction. Let us first define two notions that are central in this paper. Two sets overlap if they intersect and neither is included in the other. Given a family F of subsets of a ground set V , its orthogonal is defined to be the family of subsets that do not overlap any element of F. The computation of the orthogonal of a general family F was done in **linear** **time** by R. McConnell in [15] in which it is the core of a **linear** **time** algorithm to test the consecutive ones property of F. The purpose of this article is to explain how this orthogonal tool can be successfully applied to design a simple **linear** **time** split (or 1-join) decomposition of undirected graphs.

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(1 + inf{j | σ j 6= τ j }) −1 . Cantor distance hence measures the (inverse) length of the common preﬁx of the sequences and has been used for veriﬁcation e.g. in [7]. Both Hamming and Cantor distance have applications in information theory and pattern matching.
We will return to our example trace distances in Section 5.2 to show how our framework may be applied to yield concrete formulations of distances in the **linear**-**time**–branching-**time** spectrum relative to these.

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Monotonicity is a simple yet significant quali- tative characteristic. We consider the problem of segmenting an array in up to K segments. We want segments to be as monotonic as possible and to al- ternate signs. We propose a quality metric for this problem, present an optimal **linear** **time** algorithm based on novel formalism, and compare experi- mentally its performance to a **linear** **time** top-down regression algorithm. We show that our algorithm is faster and more accurate. Applications include pattern recognition and qualitative modeling.

Unité de recherche INRIA Lorraine LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex France Unité de recherche[r]

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L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

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Keywords: Helicopter dynamics, **Time**-varying systems, Periodic motion, Resonance, Subspace methods, Statistical inference, Fault detection, Stability limits
1. INTRODUCTION
Instability monitoring is currently the subject of extensive research activities motivated by an increasing requirement on the design of highly reliable control systems, for numer- ous applications such as civil engineering and aeronautics. Under some assumptions, such structures can be modeled by a **linear** **time**-invariant (LTI) model. The goal is to detect any anomalous behavior on these structures, and to alarm the user before any dysfunction or destruction occurs. There is a large amount of literature on the subject of instability monitoring, which has been handled with different approaches (see Angeli and Chatzinikolaou [2004] for details). One particular method, namely the subspace- based method, consists in comparing the characteristics of a system at a reference state with a subspace matrix given by new data corresponding to an unknown, possibly differ- ent state, as explained in Basseville et al. [2000]. To check whether a change has occurred or not, a quasi-distance between the two states - called residual- is defined. Then, a statistical test decides if this residual is significantly dif- ferent from zero. If some threshold is exceeded, the system has changed with respect to the reference. An overview of the different approaches of this general approach can be found in Basseville and Nikiforov [1993]. This method has been largely investigated and successfully tested on LTI systems. In some complex systems, the characterizing model may change due to some internal commands or parameters (Mach number for aircrafts, blade rotational velocity for helicopters...). In Basseville et al. [2007], the ? This work was supported by the European project FP7-NMP CP- IP 213968-2 IRIS. The authors acknowledge Leonardo Sanches for insightful discussions and comments.

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Abstract: Most subspace-based methods enabling instability monitoring are restricted to the **linear** **time**-invariant (LTI) systems. In this paper, a new subspace method of instability monitoring is proposed for the **linear** periodically **time**-varying (LPTV) case. For some LPTV systems, the system transition matrices may depend on some parameter and are also periodic in **time**. A certain range of values for the parameter leads to an unstable transition matrix. Early warning should be given when the system gets close to that region, taking into account the **time** variation of the system. Using the theory of Floquet, some symptom parameter of stability- or residual- is defined. Then, the parameter variation is tracked by performing a set of parallel cumulative sum (CUSUM) tests. Finally, the method is tested on a simulated model of a helicopter with hinged blades, for monitoring the ground resonance phenomenon.

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of the system in the presence of disturbances. Furthermore, self-triggered control schemes have often been coupled with model predictive control, as both use the model to project the behavior of the system up to some future **time** [9], [10].
In this work, we design a self-triggered control algorithm for continuous-**time** **linear** **time**-invariant (LTI) systems. The algorithm predicts the times at which the system’s behavior will infringe some predefined performance measures. We consider that the system is functioning properly when a pseudo-Lyapunov function (PLF) of its states is below a predefined upper bound. The control law is updated when the PLF reaches this upper bound. Predicting the events analytically is a difficult task, and thus, the self-triggered control algorithm computes an approximation of the event times via a minimization algorithm followed by a root-finding algorithm. The root-finding algorithm detects the intersections between the PLF and the upper limit, but needs to be properly initialized to converge to the right value. To do this, we take advantage of the shape of the PLF between two events; after the control is updated, the PLF decreases for some **time**, reaches a minimum and then increases again. This local minimum is easily computed via a minimization algorithm, and provides a good initial iterate for the root-finding algorithm.

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Unfortunately, most of research interest on subspace meth- ods has been given to **linear** **time**-invariant (LTI) systems. In contrast, the literature on **linear** **time**-varying (LTV) case is not abundant. However, most physical phenomena exhibit **time** varying behaviors, mainly due to internal (fa- tigue...) or external (disturbances...) operating conditions. One particular subclass of LTV systems is the **linear** pe- riodically **time** varying (LPTV) systems which are widely common in communications, circuit modeling and rotating machines such as wind turbines and helicopters’ rotors. In **time**-domain, the few works that have been carried out, in order to extend the subspace-based methods to these LPTV cases, can be categorized into two main approaches. The first approach consists in identifying the considered system recursively using adaptive algorithms which is ap- propriate only for slowly-varying dynamics and when a priori information about the variation behavior is avail- ? This work was supported by the European project FP7-NMP CP- IP 213968-2 IRIS. The authors acknowledge Leonardo Sanches for insightful discussions and comments.

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