Our main result generalizes two different kinds of theorems: the **fixed**-**point**- **like** **theorem** by Hirsh, Magill, Mas-Colell [12] or Husseiny, Lasry, Magill [13] and the **fixed**-**point** **theorem** by Gale, Mas-Colell [8] (which generalizes Kakutani’s **Theorem** [14]). As in [12] and [13], we shall mainly use techniques from degree theory. As a consequence of our main result we first deduce the standard **fixed**- **point** theorems when the variable is in a convex domain (such as Brouwer’s and Kakutani’s **theorem**) and second Borsuk-Ulam’s **theorem**.

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Then there exists (p, E) ∈ S ++ L−1 × G J
(R S ) such that z(p, E) = 0 and SpanV (p) ⊂ E.
The (first) proof of Duffie and Shafer (32) uses modulo 2 degree theory. Geanakoplos and Shafer (43) first prove that the set of solutions of the equation span V (p) ⊂ E is a manifold, then they prove that there exists a solution of z(p, E) = 0 on this manifold, using the homotopy invariance of topological degree; Husseiny et al. (50) use characteristic classes of vector bundles, and Hirsch, Magill and Mas-Colell (48) use intersection theory of vector bundles. A byproduct is the following **fixed**-**point**-**like** **theorem** on subspaces that, as remarked by all, admits as corollaries the Brouwer and Borsuk–Ulam theorems.

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Section 2 gives the necessary preliminaries. In Section 3, we recall what ultralimits are and prove **Theorem** 1.5. Finally, Section 3 is devoted to applications to Properties (FH) and (FRA).
Acknowledgements. I would **like** to thank Benjamin Delay for having pointed out a mistake in the proof of **Theorem** 3.12 and for his suggestions about a first version of this paper. Thanks are also due to Yves de Cornulier for references [Cor, Gro03, Laf06], to Vincent Guirardel for a useful discussion about the results in this text, and to the referee for her/his valuable suggestions. Finally, I am particularly grateful to Nicolas Monod and Alain Valette for their very valuable advices and hints, and for their remarks about previous versions of the text.

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to operate directly over an algorithmic specification is considered as being a very im- portant **point**, yet current WLO tools suffer from severe limitations. In particular, they require the flattening of the program by unrolling the loops, which makes them very sen- sitive to the number of iterations and they face severe scaling issues. Moreover, they are not applicable if some program parameters, **like** iteration bounds, are unknown at design time. Therefore, a lot of open problems remain to be addressed before such approaches can be fully integrated within a compiler.

Keywords: Banach space - **fixed** **point** - expansive mappings - contraction
mappings.
Résumé
Dans ce travail, nous avons étudié un ensemble de théories liées au **point** fixe et nous avons distingué trois cas : théorèmes de points fixes pour des applications expansive, et des théorèmes de points fixes pour un somme de deux applications, dont l'une est expansive. En plus des théories communes des points fixes pour les applications d'expansion. Dans ce dernier cas, nous avons appliqué une de ces théories sur les équations différentielles retardées et les équations intégrales retardées pour prouver l'existence et l'unité de la solution.

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Figure 4 shows selected melting plateaus measured for both CoC-2 and CoC-3 using Pt/Pd-4. These four plateaus represent the maximal variations in the inflection **point** observed during the measurements. The plateaus with the highest and lowest inflection points differed by 160 mK. The inflection **point** of all the melting curves was determined in three ways: a cubic polynomial was fit to the plateau and the minimum of the derivative was used; a numerical derivative was taken with the minimum determining the inflection **point**; the plateau data was sorted into bins of 0.17 V, a histogram was plotted, and the peak of the histogram taken as the inflection **point**. Figure 5 shows the inflection **point** determined for each melting plateau measured. Each **point** represents the average of the three methods used to determine the inflection **point** for each melting plateau. The average of all the measurements of the Co-C **fixed** **point** was found to be 1323.99 C. The standard deviation of the all the inflection points was 0.12 C. It appears from Figure 5 that there may be some drift to higher temperature of either the cell melting temperature or the Pt/Pd thermocouple, although the magnitude of this drift is small compared to the

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3. inversion of some simple operations of the original algorithm
The specific AD tool that we develop basically relies on the first strategy (1). There- fore the structure of an adjoint code consists of a forward (FWD) sweep that runs the original code and stores the needed values, followed by a backward (BWD) sweep that retrieves the stored values and computes the derivatives. The above strategies are general, and as such are unable to take advantage of algorithmic knowledge of the specific simulation. On the other hand, exploiting knowledge of the algorithm and the structure of the given simulation code can yield a huge per- formance improvement in the Adjoint AD code. For instance, special strategies are available for parallel loops, long unsteady iterative loops, linear solvers... We fo- cus on the particular case of **Fixed**-**Point** (FP) loops, i.e. loops that iteratively refine a value until it becomes stationary. We call state the variable that progressively evolves as the FP loop runs till it reaches a stationary value and parameters the variables used by the FP iteration that influence the result but are never modified during the FP loop.

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The function f has a least **fixed** **point** obtained by an iterative process on {x i } starting from x 0 . Moreover, if f is continuous then
the least **fixed** **point** is obtained in at most ω iterations.
To use this **theorem**, one should be able to express that the domain of interest has the required completeness property and that the function being considered is continuous. If the goal is to define a partial recursive function then this requires using axioms of classical logic, and for this reason the step is seldom made in the user community of type-theory based **theorem** proving. However, adding classical logic axioms to the constructive logic of type theory can often be done safely to retain the consistency of the whole system.

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Moreover, in FxP arithmetic, the developer has to deal with the binary-**point** alignment, **fixed**-**point** conversion, etc., while in floating-**point** arithmetic this is done by the hardware.
The two main bottlenecks for the **Fixed**-**Point** implementa- tion are the setting of the FxP formats (they depend on the magnitude of each variable) and the roundoff error analysis. We are interested in determining the impact of roundoff errors on the output(s) of the implemented system, therefore we need to rigorously analyse the difference between the exact system and the implemented one. These two issues may be not so easy to deal reliably with, especially in the cases where the system has an internal feedback, **like** for the Infinite Impulse Response (IIR) filters.

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The low-level representation quickly becomes infeasible for image processing applications as they have far more operations. For instance, Gaussian blur filters for N × M images have O(NM) number of operations. However, it can be represented as a single two-dimensional convolution when modeled with convolutions. While many computations may not be “convolutions” from a strict, mathematical, **point** of view, any weighted sum can be viewed as convolutions. Perceiving them as convolutions enables us to analyze many signal/image processing applications in a unified manner. This is the key insight behind this work.

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Matrix multiplication in **fixed**-**point** A compact algorithm
Analysis of CompactProduct
For square matrices of size n, only one call to the cgpeGenDotProduct is issued
Ï The dot product uses around 2n instructions (n multiplications + n additions)

ization for k > 1 seems suitable to deal with bifurcation problems (see [35] for example).
160
Another interesting generalization of multi-dimensional secant approaches has been originally proposed by Anderson [4]. This method has been directly built for vector sequences and is based, **like** the nonlinear hybrid approach [9], on a minimization process. This approach is a one-step method as it is applied every **fixed** **point** iteration. As it has been pointed out in some recent arti-

This thesis analyzes, compares, and contrasts five duality models for variable dimension fixed point algorithms, namely, primal-dual subdivided manifolds, primal-dual pseudomanifo[r]

determinantal **point** processes (DPPs for short), that are an important example of negatively associated **point** processes. DPPs are a type of repulsive **point** processes that were first introduced by Macchi [ 32 ] in 1975 to model systems of fermions in the context of quantum mechanics. They have been extensively studied in Probability theory with applications ranging from random matrix theory to non-intersecting random walks, random spanning trees and more (see [ 26 ]). From a statistical perspective, DPPs have applications in machine learning [ 29 ], telecommunication [ 15 , 33 , 22 ], biology, forestry [ 30 ] and computational statistics [ 2 ]. As a first result, we relate the association property of a **point** process to its α-mixing properties. First introduced in [ 36 ], α-mixing is a measure of dependence between random variables, which is actually more popular than PA or NA. It has been used extensively to prove central limit theorems for dependent random variables [ 7 , 16 , 23 , 27 , 36 ]. More details about mixing can be found in [ 8 , 16 ]. We derive in Section 2 an important covariance inequality for associated **point** processes (**Theorem** 2.5 ), that turns out to be very similar to inequalities established in [ 18 ] for weakly dependent continuous random processes. We show that this inequality implies α-mixing and precisely allows to control the α-mixing coefficients by the first two intensity functions of the **point** process. This result for **point** processes is in contrast with the case of random fields where it is known that association does not imply α-mixing in general (see Examples 5.10-5.11 in [ 10 ]). However, this implication holds true for integer-valued random fields (see [ 17 ] or [ 10 ]). As explained in [ 17 ], this is because the σ-algebras generated by countable sets are much poorer than σ-algebras generated by continuous sets. In fact, by this aspect and some others (for instance our proofs boil down to the control of the number of points in bounded sets), **point** processes are very similar to discrete processes.

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In previous work [2,3], we attempted to provide tools that stay closer to the level of expertise of programmers in conventional functional programming. The key **point** is to start from the recursive equation and to generate the recursive function definition from this equation. Users still need to prove that the recursive calls happen on predecessors of the initial input for a chosen well-founded rela- tion, but these requirements appear as proof obligations that are generated as a complement of the recursive equation. The tool produces the recursive function and a proof of the recursive equation. The technique is also based on iterating the functional that occurs in the recursive equation, but no continuity argument is required (instead we use a well-founded relation). With respect to all this body of work, our work is original in that it concentrates on describing potentially non-terminating functions, not by adding extra input arguments to describe the domain, but by adding an element to the target type to denote non-termination. We have only done the minimal amount of domain theory to just make it possible to define potentially non-terminating fuctions and perform basic rea- soning steps on these functions. More complete studies of domain theory have been performed in the LCF system [19]. It was also formalized in Isabelle’s HOL instanciation to provide a package known as HOLCF [20,13]. We believe these other experiments can give us guidelines to make it easier for programmers to prove the continuity requirements.

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Cette thèse est composée de quatre articles qui sont présentés dans quatre chapitres. Dans le chapitre 1, nous établissons des résultats de **point** fixe pour des fonctions multivoques, appelées G-contractions faibles. Celles-ci envoient des points connexes dans des points connexes et contractent la longueur des chemins. Les ensembles de points fixes sont étudiés. La propriété d’invariance homo- topique d’existence d’un **point** fixe est également établie pour une famille de G- contractions multivoques faibles. Dans le chapitre 2, nous établissons l’existence de solutions pour des systèmes d’inclusions intégrales de Hammerstein sous des conditions de type de monotonie mixte. L’existence de solutions pour des sys- tèmes d’inclusions différentielles avec conditions initiales ou conditions aux limites périodiques est également obtenue. Nos résultats s’appuient sur nos théorèmes de **point** fixe pour des G-contractions multivoques faibles établis au chapitre 1. Dans le chapitre 3, nous appliquons ces mêmes résultats de **point** fixe aux sys- tèmes de fonctions itérées assujettis à un graphe orienté. Plus précisément, nous construisons un espace métrique muni d’un graphe G et une G-contraction ap- propriés. En utilisant les points fixes de cette G-contraction, nous obtenons plus d’information sur les attracteurs de ces systèmes de fonctions itérées. Dans le chapitre 4, nous considérons des contractions multivoques définies sur un espace de jauges muni d’un graphe. Nous prouvons un résultat de **point** fixe pour des fonctions multivoques qui envoient des points connexes dans des points connexes et qui satisfont une condition de contraction généralisée. Ensuite, nous étudions des systèmes infinis de fonctions itérées assujettis à un graphe orienté (H-IIFS). Nous donnons des conditions assurant l’existence d’un attracteur unique à un

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placed within the main furnace tube. A stainless-steel, water-cooled cap was attached and sealed to this tube by way of a Teflon ring. The cap’s central hole allowed the thermocouple protection tube to be inserted and sealed via an o-ring. The cap also contained inlet and outlet connections for evacuation and inert gas flow. The sealed protection tube containing the **fixed** **point** was evacuated and flushed with Ar twice prior to each heating cycle. A slow Ar flow was

The **fixed**-**point** simulation can be accelerated by executing it on a more adequate machine **like** a **fixed**-**point** DSP [78, 73, 39, 82, 37] or an FPGA [38] through hardware acceleration. In the case of hardware implementation, the operator word-length, the supplementary elements for overflow and quantization modes are adjusted to comply exactly with the **fixed**-**point** specification which has to be simulated. In the case of software implementation, the operator and register word- lengths are **fixed**. When the word-length of the **fixed**-**point** data is lower than the data word-length supported by the target machine, different degrees of freedom are available to map the **fixed**-**point** data into the target storage elements. In [39], to optimize this mapping, the execution time of the **fixed**-**point** simulation is minimized. The cost integrates the data alignment and the overflow and quantization mechanism. This combinatorial optimization problem is solved by a divide and conquer technique and several heuristics to limit the search space are used. In [82] a technique is proposed to minimize the execution time due to scaling operations according to the shift capa- bilities of the target architecture. In the same way, the aim of the Hybris simulator [76] [36] is to optimize the mapping of the **fixed**-**point** data described with SystemC into the target architecture register. All compile-time information are used to minimize the number of operations required to carry-out the **fixed**-**point** simulation. The overflow and quantization operations are implemented by conditional structures, a set of shift operations or bit mask operations. Nevertheless, to obtain fast simulation, some quantization modes are not supported. In [143], the binary **point** alignment is formulated as a combinatorial optimization problem and an integer linear programming ap- proach is used to solve it. But, this approach is limited to simple applications to obtain reasonable optimization times. These methods reduce the execution time of the **fixed**-**point** simulation but, this optimization needs to be performed every time that the **fixed** **point** configuration changes. Accordingly, it might not compensate for the execution time gain of the **fixed**-**point** simulation when involving complex optimizations.

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Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - [r]