Dennis Wegener
email: dennis.wegener@iais.fraunhofer.de

Thierry Sengstag
email: thierry.sengstag@isb-sib.ch

Stelios Sfakianakis

Stefan Rüping
email: stefan.rueping@iais.fraunhofer.de

Anthony Assi
email: anthony.assi@irisa.fr

GridR: An R-based grid-enabled tool for data analysis in ACGT clinicogenomics trials

In this paper, we describe an analysis tool based on the statistical environment R, GridR, which allows using the collection of methodologies available as R packages in a grid environment. The aim of GridR, which was initiated in the context of the EU project Advancing ClinicoGenomics Trials on Cancer (ACGT), is to provide a powerful framework for the analysis of clinicogenomic trials involving large amount of data (e.g. microarraybased clinical trials). As a proof of concept, an example of microarray based analysis taken from the literature was reproduced using GridR. As GridR will ultimately be made available to the ACGT community as a web service, we are sketching the ACGT project and its architecture in the present article as well.

1.Introduction Overview of ACGT

In recent years, the rapid development of high throughput genomics and postgenomics technologies has provided clinicians fighting cancer with new discovery paths and has opened the possibility to develop patientspecific treatment strategies. However, the amount of information now available for each patient (e.g. in microarray context from 10'000s to 100'000s of variables summarizing upto millions of array features) has rendered difficult the isolation of the clinically relevant information from all available data. Considering the current size of clinical trials (hundreds of patients), there is a clear need, both from the viewpoint of the fundamental research and from that of the treatment of individual patients, for a data analysis environment that allows the exploitation of this enormous pool of data 8. This is the aim of the Advancing ClinicoGenomics Trials on Cancer (ACGT) project 8.

ACGT aims at developing an opensource IT infrastructure to provide the biomedical community with the tools needed to integrate complex clinical information and make a concrete step towards the tailorization of treatment to the patient.

On the data side, the ACGT environment is designed to be versatile and will allow the integration of highthroughput databases with data both from existing (e.g. microarrays, imaging) and future technologies (e.g. highthroughput proteomics). The design of the platform considers the integration of private (i.e. trialspecific) databases with public ones, thus making publicly available datasets potentially immediately available for hypothesis validation and metaanalyses.

On the methodology side, the ACGT platform is designed to be modular, allowing to integrate additional data analysis tools (software, both open source and commercial, web services) as plugins, as they become available.

Considering that the amount of data generated is expected to rise to several gigabytes of data per patient in a close future access to highperformance computing resources will be unavoidable. Hence, Grid computing 8 appears as a promising technology. Access and use of Gridbased resources is thus an integral part of the design of the infrastructure.

There are a number of other projects that aim at developing Gridbased infrastructure for postgenomic cancer clinical trials, the most advanced of which are NCI's caBIG 8 (Cancer Biomedical Informatics Grid) in the USA and CancerGrid 8 in the UK. The overall approach in those projects is somewhat different from the one in ACGT. In caBIG, the bottomup, technologyoriented, approach was chosen, in which the focus was put on the integration of a large number of analysis tools but with weak concern on data privacy issues. CancerGrid on the other hand addresses the very needs of the British clinical community. As a result some aspects of the project may not fully overlap with the European and international scope of ACGT.

In ACGT, with two ongoing international clinical trials actually conducted in the framework of the project, the approach is topdown, with clinicians' and biomedical data analysts' needs at the heart of all technical decisions, considering data privacy issues as central as data analysis needs.

Finally it should be mentioned although these aspects are not described in the present text that the ACGT project addresses legal and ethical issues related to clinical trials in distributed computing environment. Defining a sound legal and ethical framework for the use of clinical data is essential to gain public acceptance and support for such initiatives.

R in clinico-genomic context

The present article describes the initial implementation of GridR, one of the important analysis tools to be used in the ACGT environment. GridR is based on the opensource statistical package R 8. The R environment provides a broad range of stateoftheart statistical, graphical techniques and advanced data mining methods (including comprehensive packages for linear and nonlinear modelling, cluster analysis, prediction, hypothesis tests, resampling, survival analysis, timeseries analysis), it is easy extensible and turned out as the de facto standard for statistical research and many applied statistics project, especially in the biomedical field. (R itself is based on the S language 8 which is also implemented in the commercially available Splus system.)

The associated project BioConductor 8 addresses the needs of the biomedical and biostatisticians community for genomic dataanalysis oriented R packages.

Numerous methods available as R/BioConductor packages and considered experimental a few years ago have been stabilized and became accepted standard in the analysis of highthroughput genomic data. Integrating R/BioConductor in an opensource clinical dataanalysis environment is thus completely meaningful.

In the ACGT analysis environment, R is used as a user interface and as an analysis tool. R as user interface is supposed to serve as programming language interface to the ACGT environment. Used as analysis tool, the goal is to achieve a seamless integration of the functionality of R and the ACGT semantic data services in a grid environment, hiding complexity of the grid environment as the user might not want to or is not capable to deal with.

GridR, developed in the context of the ACGT project, is a grid enabled R environment providing a large spectrum of statistical tools and services in the grid.

The rest of the paper is organized as follows. In Sec. 2 we sketch the ACGT architecture and give an overview over the key components. Sec. 3 describes the general scenario of statistical analysis in the context of clinico genomic trials and introduces a use case. In Sec. 4 we describe the grid enabled R environment GridR in ACGT, which will be evaluated in Sec. 5, based on the use case "Farmer scenario". Finally, in Sec. 6 we describe the relation to other project from the same area, discuss the results of the paper and give a future outlook.

2.ACGT Architecture

The ACGT requirements in terms of data management, efficient utilization of computational resources, and security can be matched by the adoption of a Grid infrastructure. The adopted architecture builds upon the Grid fabric and it is further enhanced by the deployment of Web Services and Semantic Web technologies. These technologies, although initially separated, are currently converging in a complementary way. The adopted architecture for ACGT is shown in more detail in Figure 1. A layered approach has been followed for providing different levels of abstraction and a classification of functionality into groups of homologous software entities 8. In this approach we consider the security services and components to be pervasive throughout ACGT so as to provide both for the user management, access rights management and enforcement, and trust bindings that are facilitated by the Grid and domain specific security requirements like pseudonymization and anonymization. Apart from the security requirements, the Grid infrastructure and other services are located in the first (lower) two layers: the Common Grid Layer and the Advanced Grid Middleware Layer. The upper layer is where the user access services, such as the portal and the visualization tools, reside. Finally, the Bioinformatics and Knowledge Discovery Services are the "workhorse" of ACGT and the corresponding layer is where the majority of ACGT specific services lie. Some characteristic examples of these services are:

• Mediator Services that offer uniform access to distributed and heterogeneous clinical and biomedical databases.

• Ontology

Services that provide a conceptualization of the domain through the Master Ontology for Cancer and the "domain of discourse" for constructing complex queries for the mediator services.

• Workflow Enactment Services that support the efficient management and execution of complex biomedical workflows.

• Metadata Repositories and the corresponding services for the persistence and management of services' metadata descriptions.

• An assortment of data mining and knowledge discovery tools and services that fulfill the data analysis requirements of ACGT. In essence the GridR service is the most prominent tool for performing these data analyses in ACGT.

In the following sections the description of GridR is related to the current status of the integration into the ACGT environment, which is currently focused on the access to the ACGT grid services.

3.Scenario

In order to illustrate the working principles of GridR, we have selected the article by Farmer et al. 8, a simple clinical research project available from the literature and for which all data were available online 8. This also provides a validation of GridR in a realistic usage.

In the "Farmer scenario", microarray data (Affymetrix U133A gene expression microarrays) obtained from breast cancer tumor samples of 49 patients are used to associate subtypes of breast cancer to patterns of gene expression and molecular signatures. R and BioConductor packages are used to load, normalize and analyze the data. The present work will be validated by showing that the results of the original paper can be reproduced using GridR.

The validation of GridR actually implements only a subset of the analysis steps presented in the original article, namely the principal component analysis (PCA). On the other hand steps related to the quality control of the arrays are shown, which were not presented in the original paper. Figure 2 shows the plots related to those steps; in particular the plot illustrating the PCA is seen to be identical to that in 8.

4.Use case

The given scenario can be described from the technical point of view as follows. A researcher wants to perform interactive, gridenabled data mining in the R environment. On his local machine (client) he develops algorithms using R as user interface. In clinical context, the data to be analyzed are usually so large (~800MB in the present, limited, scenario) that a transfer to the client might be ineffective or not possible. Besides, execution machines in the grid environment might have bigger computational power as client machines. It would be more efficient to just ship the algorithm to the execution machine (the best would be if the execution machine is the machine where the data is located in order to minimize transfer time), execute it remotely on this machine and transfer the results back to the client.

The R environment is used both as a user interface (client) on the client side and as tool in the grid environment. GridR provides the execution of a single R function or a whole R script (wrapped as function) in the grid. The following subsections give a detailed description on the current implementation of GridR.

1.1.R as tool

In ACGT the R environment, namely GridR, is used as a tool for the remote execution of R code in the grid. More specifically, the task of the execution of the R code is submitted as a grid job to a remote grid machine. The current implementation of the server side GridR components that are related to the grid environment is based on several external software components, namely the Globus Toolkit 4 grid middleware 8, an installation of the R environment on the grid machines which will execute the functions remotely and a GRMSServer installation from the Gridge toolkit 8 on a central machine in the grid environment that is responsible, for instance, for resource management. On the client side, GridR consists of a set of R functions and involves the Cog kit 8, which is responsible for proxy generation and data transfer and a GRMSClient 8.

The client side part is structured around the components "RemoteExecution" (JobSubmission and JobDescription Generator) and "Locking". The RemoteExecution component is responsible for the execution of R code as a job in the grid environment by transforming the R code to execute into a set of files, creating a job description file in the respective job description language and submitting the job to the resource management system by the GRMSclient. During this process, the locking component takes care of the consistency of files/variables. These components are based on R functions (see Table 1 andTable 2), so no changes in the core R implementation are necessary. The functions are based, among other, on the following R functionalities:

• callbacks functions that are executed automatically by R after the user has issued a command (this is used when checking for results)

• active bindings a variable is replaced by a function call which is handling the locking system and allows working interactively with that variable. When the variable is read, the predefined function is called and returns the value associated to the variable (or an error code if the variable is locked). When a value is assigned to the variable the function is called with the value as parameter for storage in an internal structure.

R as client

An R programming language interface that supports the access to the ACGT services is provided in the ACGT environment. This means that R users and developers will have access to distributed resources in a transparent fashion, as if those resources were local. The complexity of the grid is thus hidden from the user.

Again, accessing the ACGT grid environment requires no changes in the core R implementation. In practice grid access is performed through the call of predefined R functions loaded from a package. R users can make use of the grid technology in a transparent way by passing the functions to be executed in the grid as input to one of those predefined functions (grid.apply) in their local code.

Figure 3 shows the execution of a simple sum function with the GridR client side components. Details on the steps of execution will be described in the following section.

5.Evaluation and Testing

The implementation of GridR was validated on the basis of the "Farmer scenario", implementing some typical analysis steps of a microarray experiment.

In the present case, R was used in conjunction with the BioConductor packages affy, affyPLM and marray, which are specialized packages for microarray analysis, to build the individual modules of validation.

Besides loading the expression data matrix and associated clinical data, those modules contribute in:

• Producing some figures required for the quality control of the chips (e.g. RNA degradation plots) • Producing "MvA plots" to obtain an overall view of the fraction of differentially expressed genes.

• Using a variance filter to pick unique probeset per gene and performing a principal component analysis, to verify that samples with similar subtypes group together.

• Extracting symbols of genes most correlated to molecular markers relevant to the analysis (androgen receptor, AR, and estrogen receptor, ESR1).

The analysis steps are wrapped into functions for remote execution with GridR. In the following the process of executing a single R function in the grid is described. The different steps of execution are briefly shown in Figure 4 (see also Figure 3 for a view from the R client).

• Function loading. The GridR functions are loaded from a file or package into the workspace of the R client.

• initialization. The grid environment is initialized by calling the function grid.init. This function sets the paths to the cogkit and the grmsclient as well as the remote host that will be taken as execution machine.

• Code writing. The R code which is to be executed in the grid is written and wrapped as single R function in the local R environment

• Grid submission. The grid.apply function is called, that launches the process of submission. At first, the function to be executed in the grid and the needed parameters are written into a file (fx.bin). Then the R script which is actually executed on the remote machine is generated (script.R), which is followed by the creation of the job description for the gridjobsubmission by GRMS (job.xml). The job description file information for the resource management system, e.g. which application to execute in the grid (R in the present case), a dedicated execution machine, which files to stagein (out) from the grid to (from) the execution machine etc. Next a shell file is created, which specifies the "workflow" which is performed on the client side (shell.sh). After that, the R client executes the created shell file, which creates a new proxy if necessary, performs the file copy from the client to the grid (to a GridFTP server later used for file stagein), submits the job to the grid system and, after the execution is finished, performs the file copy of the result files (staged out by the grid system) to the client.

During the remote execution the created R script is executed on the remote machine, which reads the parameters and the function from fx.bin, executes y=f(x) and writes the result into a file (y.dat).

• Waiting for result. While the remote execution is active and the R client waits for result (by checking if the file y.dat is created) the variable y is locked.

• Result processing. If file y.dat was created on the remote machine and, together with the result files, transferred back to the client, the file is loaded. The exit status is checked and -if the job was successful -the value is assigned to y and the variable is unlocked. Further more, if e.g. one of the result files is a Postscript file, ghostview is started for result visualization. In this paper we have presented the integration of R in a Grid environment. In the biomedical and biostatisticians community R is widely used and turned out as de facto standard. GridR, as one of the important analysis tools in the ACGT environment, enables users to run experiments in the grid and profit from the advantages of grid technology by provides a grid enabled environment for R. The presented use case showed the execution of a complex analysis scenario. It should be considered a proof of concept, as measurements and statistics about the practical gains in terms of computational power still need to be done in the context of large clinical trials. However, it becomes clear that the availability of GridR will be of great use to clinicians and clinicaldata analysts interested in computationally heavy datamining, such as resampling techniques, full crossvalidation of classifiers or metaanalyses.

The expected benefits for the users are twofold based on the duality of using the R as a client tool and executing the R scripts on the Grid. Firstly, we argue that the R environment is a popular tool among biostaticians and something that a lot of users are familiar with. Therefore using R as a user access environment to interact with the ACGT platform is beneficial for the acceptance and the active use of the project's infrastructure. Secondly, the "gridification" of the R execution layer implies that the users' analysis tasks are executed in an efficient and secure way relieving the client side of the computational burden and the need to download all the input data sets. This architecture also leaves room for many possible optimizations in the future, e.g. the implementation of a scheduler that in collaboration with the Grid scheduling facilities determines the best place to execute a specific R task based on the data that it requires.

To our knowledge there has not been so far any attempt for grid enabling the R environment itself and using it as user interface for accessing the Grid resources and invoking the Grid services.

Analytical services in the Grid have been the focus of work in various previous projects. In particular the Grid Weka 8 toolkit was an effort to extend the Weka toolkit 8 so that it uses the Grid computational resources for data analysis tasks. The Weka4WS toolkit 8 is a similar work to support the data analysis on the Grid by offering a WSRF compliant interface to the machine learning algorithms provided by Weka. The DataMiningGrid project 8 aims to deliver Grid interfaces and middleware components to facilitate the discovery and execution of data mining tools in distributed Grid environments. One of the results of this project is a "Data Mining Application Enabler" which gridenables existing data mining applications. Nevertheless, the focus is on generic, mostly command line, applications (e.g. Bash or Python scripts, C applications, etc.) although R scripts could be also handled in the same way: as inputs to the command line R application.

In 8 a short overview over support for concurrent computation in R is given. Packages such as snow (Simple Network Of Workstations), based on the adoption of an article about benefits and problems with parallel processes in R 8, and rpvm provide interfaces in order to support the parallel computation on clusters. In contrast to the messagepassing interface (MPI) 8 or the parallel virtual machine (PVM) 8, the snow package provides a higher level of abstraction that is independent of the communication technology. Some rudimentary support for building client server applications that use R in the server side has been offered by toolkits like Rserve 8 or Rweb 8 but these efforts do not offer a seamless integration with the Grid technologies.

As future work we plan to extend the functionality offered by GridR to support the interfacing with the rest of the ACGT services, e.g. the mediator. This will enable the use of GridR as an allencompassing user interface to the whole ACGT environment. Additionally the provision of a WSRF compliant interface to the GridR service functionality will give the ability to compose higher level workflows that include GridR and will also make it more compatible with the service oriented view of the Grid.

Figure 1 :

 1 Figure 1: The ACGT layered initial architecture.

Figure 2 :

 2 Figure 2: Farmer Scenario plots

Figure 3 :

 3 Figure 3: Simple GridR Example

Figure 4 :

 4 Figure 4: Steps of execution

Table 1 :

 1 List of functions for the users

	• parsing of error code checks for missing
	values, variables and functions in the code
	which is executed remotely
	Name	Action
	grid.init	Initializaion of all variables
		necessary for the grid execution
	grid.exit	Unlocks	all	variables
		(grid.unlockAll) and removes all
		callbacks
	grid.printJobs	Prints a summary of running jobs
	grid.getLibraries	Returns the remote list of libraries
		(through grid.apply)
	grid.getVersion	Returns the remote R version
		(through grid.apply)
	grid.apply	Performs a remote execution of R
		functions; waits (callback) or sets

Table 2 :

 2 List of internal used functions

7.Acknowledgments

The authors gratefully acknowledge the financial support of the European Commission for the Project ACGT, FP6/2004/IST026996.


Hugo Lingua

François Vibert

Dominique Mouysset

Didier Siri

Michèle P Bertrand
email: michele.bertrand@univ-amu.fr

Laurence Feray
email: laurence.feray@univ-amu.fr

Iron(II)/copper(I)-mediated stereoselective carbozincation of ynamides. One-pot synthesis of a-allyl-tetrasubstituted-enamides

Keywords:

The iron(II) chloride-and copper(I) iodide-mediated carbozincation of a panel of substituted ynamides is described in this article. The reaction is totally regio-and stereoselective. Experiments showed that the reaction mediated with Fe(II) was more substrate dependent than the reaction performed with Cu(I). Interestingly, in the presence of allylbromide, stereoselective carboallylation can be achieved for the first time in a one-pot procedure, leading to skipped dienamides.

.

Introduction

Intermolecular carbometalation 1 of ynamides is one of the most flexible approaches for the enamide synthesis. [START_REF] Evano | For general reviews on ynamides[END_REF] Marek and coworkers were the first to describe the carbocupration and the copper-catalyzed carbomagnesiation of ynamides. 3 Oshima and Yorimitsu have applied this methodology to prepare ynamides bearing an allyl group to perform subsequent aza-Claisen rearrangement. 4 Both carbozincation and carboboration of ynamides were achieved by Lam and coworkers, using a rhodium catalyst in the presence of dialkylzincs and arylboronic acids, respectively. 5 This process enables access to a range of multisubstituted enamides in a stereo-and regio-controlled fashion. This methodology was extended by Hou and coworkers to the alkylative carboxylation of ynamides in the presence of a CO 2 atmosphere, 6 in which a (NHC)-copper catalyst was used in place of the rhodium catalyst. The Pd-catalyzed addition of boronic acids has also been reported. 7 Recently, a cobaltcatalyzed carbozincation of ynamides was described by Gosmini and Gillaizeau. 8 Intramolecular carbometalation of ynamides were also investigated to prepare indole and pyridine derivatives. [START_REF] Couty | For the synthesis of indole derivatives see: a)[END_REF] We have investigated FeCl 2 as a potential mediator for intermolecular ynamides carbozincation. 10 Experiments conducted with Cu(I), a catalyst previously used for the carbozincation of cyclopropenes 11 and alkynyl sulfoximines and sulfones 12 were carried out in parallel. This led us to propose a novel one-pot procedure for the regio-and stereoselective carboallylation of ynamides leading to skipped dienamides.

Results and Discussion

In a blank experiment, ynamide 1a was reacted with diethylzinc without any additive. After one night at room temperature the substrate was recovered and only a trace amount of the alkylated product 2a was detected. [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF] The use of 2 equivalents of iron(II) chloride allowed the formation of the expected compound in 91% yield (Table 1, entry 2). 14 In the presence of 0.5 equivalent of FeCl 2 the reaction was still efficient and the substrate was totally consumed to give 2a in 73% isolated yield (Table 1, entry 3).

When CuI was used as the mediator, good results were obtained in the presence of either 2 or 0.5 equivalents of this transmetalation agent (Table 1, entries 4 and 5). The reaction was also conducted with 10 mol % of CuI. The substrate was totally consumed to give the desired enamide 2a in 74 % yield (Table 1, entry 6). In both cases, the monitoring of the reaction showed that the consumption of the substrate needed much more than 6 h to be completed at room temperature, for the sake of homogeneity, all the experiments were carried out over 18 h in the presence of 0.5 equivalents of the mediator. In each case carbometalation occurred with high regio-and stereoselectivity leading to a unique stereoisomer among the 4 possible ones. The regio-and stereoselectivity were the same as that observed by Marek and coworkers, and Lam and coworkers. 3,5 NOE and HMBC NMR experiments were performed to prove structural assignments. Iron(II) chloride was found as efficient as copper(I) iodide to perform this transformation.

Scheme 1. Mechanism.

Since the use of 0.1 equivalent of the mediator was sufficient to perform the reaction, a possible catalytic cycle for these transformations is presented in Scheme 1. This mechanism is based on that proposed for copper-catalyzed carbomagnesiation. 3 The reaction of iron(II) chloride with diethylzinc would lead to an ethyliron species 3. The carboironation would be directed by the coordination of iron(II) with the oxygen atom of the carbonyl group to regioselectively form the alkenyliron species 4. Finally, transmetalation would occur between diethylzinc and 4 to give the alkenylzinc species 5 precursor of 2a through hydrolysis. The observed reactivity relies on the ability of the substrate to form a five-membered chelated vinyl metal intermediate.

Scheme 2.

Ynamides 1a-g. Et 2 Zn (2 equiv.)

DCM, r.t., 18 h N R 2 Et R 3 H 1 2 O R 1 O R 1 additive (0.5 equiv.)
The scope of the process was explored with a panel of ynamides and dimethylzinc was also used in place of diethylzinc. The results are described in Tables 2 and3. Ynamides 1 (Scheme 2) were prepared by slightly modifying known procedures. 15,16 , 17 Ethylzincation was first applied to the oxazolidinone-derived ynamides 1a-d. The reaction tolerated both alkyl (1a, 1c-d) and aryl (1b) groups on the distal carbon of the ynamide. It led to the corresponding enamides 2a-d in moderate to good yields with a total regioselectivity. Moreover, only one stereoisomer of the enamide was detected using either iron(II) or copper(I) (Table 2, entries 1-12). However, the reaction involving FeCl 2 seemed to be more dependant on the nature of the substrate than that mediated with copper(I).

Ynamide 1e derived from phthalimide showed to be the least reactive substrate and 4 equivalents of diethylzinc were necessary to obtain the corresponding ethylated enamide in 80% yield in the presence of copper(I) (Table 3, entry 5). In the presence of iron(II), the yield was lower whether using a large excess of diethylzinc or just 2 equivalents (entry 11). Ynamide 1f derived from γ-lactam prove to be a good partner in these transformations (entries 6, 12). Finally, ynamide 1g derived from Nbenzyltosylamine failed to react either using CuI or FeCl 2 . Up to 80% of the starting material was recovered after workup. Ynamides are p-nucleophiles. Their nucleophilicity increases with the delocalization of the nitrogen electron pair towards the triple bond p-system. However, this electron-donating property is counterbalanced by the electron-withdrawing substituent at nitrogen, which improves their stability. [START_REF] Evano | For general reviews on ynamides[END_REF] According to the scale established by Mayr, 18 ynamide 1g is a better nucleophile than 1f however, it is unreactive in carbozincation processes. Conversely, N-tosylynamides were shown to be reactive in carbomagnesiation. 3c The scale established from the kinetics of reactions with carbocations cannot be applied to the multistep carbometalation process. Rationalization is not straightforward. B3LYP/6-31+G(d,p) calculations were performed on 1b, 1f and 1g for the sake of collecting reactivity quantitative parameters (see: supplementary content). According to the global electrophilicity index defined by Parr, 19 the electrophilicity is low, it varies in the following order: 1g (w=1.59 eV) > 1f (w=1.30 eV) > 1b (w=1.28 eV), even though the gap between 1f and 1b is tiny. Not surprisingly, this sequence based on w correlates with the relative energies of the LUMOs, 1g having the lowest LUMO in the series (LUMOs energies in eV: 1g (-1.54) < 1f (-1.15) < 1b (-1.09). The nucleophilicity sequence should be the reversal (according to HOMOs energies in eV, the position of 1f and 1g should be reversed: 1f (-5.85) > 1b (-6.01) > 1g (-6.10)). Condensed Fukui's functions were also calculated using NPA charges. 20 The maps of Fukui's dual descriptors [START_REF] Morell | [END_REF] (𝟐) .

To date, the best rationale regarding the regioselectivity of the process has been proposed by Marek, 3 who pointed the major role of the oxygen atom in coordinating the metal and directing the metalation process. This is in agreement with the above discussed values of 𝑓 " condensed Fukui's function at oxygen. Additional calculations of the intermediate vinylzincs evidenced the stabilizing role of the carbonyl oxygen atom as an intramolecular ligand, which favors the transmetalation step and controls the overall enthalpy of the process. The electron density at the oxygen atom has a large impact on the stability of the putative vinylzinc intermediate. The overall reaction enthalpies for the formation of the vinylzinc intermediate from 1b, 1f, 1g and Et 2 Zn were calculated at the B3LYP/6-31+G(d,p) level. They are -44.9, -44.3 and -33.8 kcal mol -1 , respectively (see, supplementary content).

Methylzincation was achieved with dimethylzinc. In the presence of CuI, methylzincation and ethylzincation led to similar yields for all the ynamides investigated. Quite good yields and selectivity were observed with ynamides 1a-f. Substrate 1g failed to react. When the methylzincation was conducted with FeCl 2 , the results were significantly different. In most cases yields were far lower, except in the case of 1f. An additional experiment was conducted in presence of diisopropylzinc with ynamide 1f and CuI. It led to the expected enamide 7f in 67% yield (Scheme 3).

Scheme 3. Isopropylzincation of 1f.

The trapping of the vinylzinc intermediate was achieved with ND 4 Cl, which led to a mixture of deuterated 10b and non deuterated 2b in a 89:11 ratio (60% yield). Similarly, it could be trapped by other electrophiles like PhSeCl. The corresponding vinylselenide was isolated in 50% isolated yield.

The synthesis of tetrasubstituted enamides was also envisaged by adding allylbromide in a sequential step. Substrate 1b was reacted in the presence of CuI (0.5 equiv.) and diethylzinc for 18h. Allylbromide was then added to give after workup the desired allylenamide in moderate yield. The proton NMR spectrum of the crude reaction showed a mixture of allylated and non-allylated products (8b:2b, 73:27). [START_REF] Comparatively ; Frisch | benzaldehyde or benzoylchloride led to complex mixtures. 23 Gaussian 09[END_REF] To our knowledge, the carboallylation of ynamides has only been reported in sequential procedures. 3,5 We have shown that far better yields could be obtained when the carboallylation was achieved in a one-pot procedure, by introducing directly allylbromide into the initial mixture (Table 4). Interestingly, the copper-mediated ethylallylation of ynamide 1b led to the desired product 8b with a quantitative yield, no competing side reaction was observed (Table 4, entry 1). The reaction was still efficient in the presence of 10 mol % of CuI (entry 2). Only one stereoisomer was detected from the analysis of the proton NMR spectrum of the crude reaction mixture. The same behavior was observed for the methylallylation of 1b conducted in the presence of dimethylzinc (entry 4). Moving from copper(I) to iron(II) led to half the yield of 8b, despite the use of 2 equivalents of iron(II) (entry 3). To widen the scope of this one-pot procedure, ynamides 1a, 1d and 1f were submitted to the same experimental conditions. They led to the corresponding dienamides in good yields with a total stereoselectivity (entries 5-7).

Scheme 5. Mechanism for the carboallylation of ynamides 1

Two pathways can be envisaged to explain the formation of the allylated products (Scheme 5): the allylation of a vinylzinc intermediate (path B) or the direct allylation of a vinylcopper intermediate leading to the formation of CuBr that can participate in the catalytic cycle (path A).

Conclusions

In conclusion, the carbozincation of ynamides could be mediated either by CuI and FeCl 2 , two common and inexpensive reagents, to prepare regio-and stereoselectively trisubstituted enamides. The use of CuI proved to be more efficient and less substrate dependent in most cases. It enabled to develop an original one-pot procedure for the direct stereocontrolled formation of tetrasubstituted allylenamides.

Experimental Section

General.

Commercially available solvents were used as purchased. CH 2 Cl 2 (GC grade) was used as solvent; it was stored on molecular sieves. Analytical thin layer chromatography was performed on pre-coated silica gel plates. NMR spectra were recorded at 300 MHz and 400 MHz ( 1 H) and 75 MHz and 100 MHz ( 13 C) using CDCl 3 as the solvent. Chemical shifts (δ) are reported in ppm. Signals due to residual protonated solvent or to the deuterated solvent served as the internal standards to calibrate the spectra ( 1 H NMR, CHCl 3 , 7.26 ppm; [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF] C NMR, CDCl 3 , 77.16 ppm). One or more of the following descriptors indicate multiplicity: s (singlet), d (doublet), t (triplet), q (quartet), quint (quintet), sext (sextuplet), sept (septet), m (multiplet), br (broad). The J values are given in Hz.

Computational details

All the calculations were performed with Gaussian 09 Rev. D01 package. 23 The geometries of 1b, 1f and 1g were fully optimized at the B3LYP/6-31+G(d,p) level of theory. Vibrational frequency calculations were performed to insure that the geometries are minima (no imaginary frequency). The Fukui dual descriptors

N R 2 Me R 3 H O R 1 6 N R 3 R 2
Me 2 Zn (2 equiv.) DCM, r.t., 18 h 1 O R 1 additive (0.5 equiv.)

Ph N 1f O 7f i-Pr 2 Zn (2 equiv.) DCM, r.t., 18 h CuI (0.5 equiv.) N O H Ph 67% N R 2 Et R 3 O R 1 8 N R 3 R 2 R 2 Zn (2 equiv.) DCM, 0 °C to r.t., 18 h 1 O R 1 additive (0.5 equiv.) allylBr (9 equiv.) N R 2 Me R 3 O R 1 9 Et 2 Zn CuX XZnEt N Et R 3 Cu R 1 R 2 O N Et R 3 Allyl R 1 R 2 O 1 allylBr 8 EtCu CuI Et 2 Zn EtCu IZnEt Et 2 Zn 1 N Et R 3 Cu R 1 R 2 O N Et R 3 Et Zn R 1 R 2 O allylBr 8 Path A Path B
were calculated using Multiwfn software [START_REF] Lu | [END_REF] with the neutral, cation and anion wavefunctions (isodensity value=0.002).

General procedure for ynamides synthesis.

In a two-neck round-bottom flask equipped with a stir-bar, CuCl 2 (0.2 equiv), oxazolidin-2-one (5.0 equiv), Na 2 CO 3 or Cs 2 CO 3 (2.0 equiv) and molecular sieve (4Å) were combined. The reaction flask was purged with oxygen for 15 minutes. A solution of pyridine (2.0 equiv) in dry toluene (0.2M) was added to the reaction flask via a syringe at room temperature. Two balloons filled with oxygen were connected to the reaction flask via a needle. The flask was placed in an oil-bath and heated to 70 °C. A solution of alkyne (1.0 equiv) in dry toluene was added over 4 h using a syringe pump. After the addition was completed, the reaction mixture was stirred at 70 ºC for 8 h and then cooled to room temperature. The crude mixture was concentrated under vacuum and the residue was purified by flash chromatography on silica gel.

3-{4-[(2-Methoxyethoxy)methoxy]but-1-yn-1-yl}-1,3oxazolidin-2-one (1a

3-(2-Phenylethynyl

)-1,3-oxazolidin-2-one (1b). 16 The reaction was conducted according to the general procedure in the presence of CuCl 2 (0.2 mmol, 27.0 mg), oxazolidin-2-one (5.0 mmol, 435 mg), Na 2 CO 3 (2.0 mmol, 212 mg), a solution of pyridine (2.0 mmol) in dry toluene (5.0 mL) and phenylacetylene (1.0 mmol, 101 mg) was added in dry toluene (5.0 mL). Purification by flash chromatography on silica gel (pentane/ethyl acetate (9/1 to 6/4)) led to ynamide 1b (170 mg, 91% yield) as a white powder. 1 H NMR (300 MHz, CDCl 3 ) δ: 4.01 (pseudo t, J = 8.2, 2H), 4.50 (pseudo t, J = 8.2, 2H), 7.25-7.30 (m, 3H), 7.42-7.46 (m, 2H).

3-[3-(Benzyloxy)

prop-1-yn-1-yl]-1,3-oxazolidin-2-one (1c). 16 The reaction was conducted according to the general procedure in the presence of CuCl

3-(Hex-1-yn-1-yl)-1,3-oxazolidin-2-one (1d)

. 25 The reaction was conducted according to the general procedure in the presence of CuCl 2 (0.2 mmol, 27.0 mg), oxazolidin-2-one (5.0 mmol, 435 mg), Na 2 CO 3 (2.0 mmol, 212 mg), a solution of pyridine (2.0 mmol) in dry toluene (5.0 mL) and butylacetylene (1.0 mmol, 0.114 mL) was added in dry toluene (5.0 mL). Purification by flash chromatography on silica gel (pentane/ethyl acetate (9/1 to 6/4)) led to ynamide 1d (98 mg, 58% yield) as a colorless oil. 1 H NMR (400 MHz, CDCl 3 ) δ: 0.90 (t, J = 7.3, 3H), 1.39 (sext, J = 7.1, 2H), 1.49 (quint, J = 7.1, 2H), 2.29 (t, J = 7.1, 2H), 3.86 (pseudo t, J = 8.0, 2H), 4.40 (pseudo t, J = 8.0, 2H).

2-(2-Phenylethynyl)-2,3-dihydro-1H-isoindole-1,3-dione (1e

). 26 The reaction was conducted according to the general procedure in the presence of Cu(OAc) 2 (3.9 mmol, 710 mg, 0.2 equiv), phtalimide (97.7 mmol, 14.38 g, 5.0 equiv), Na 2 CO 3 (39.1 mmol, 4.14 g, 2.0 equiv), a solution of pyridine (39.1 mmol, 3.15 mL, 2.0 equiv) in dry toluene (100 mL) and phenylacetylene (19.5 mmol, 2.15 mL, 1.0 equiv) was added in dry toluene (100 mL). Purification by flash chromatography on silica gel (pentane/ethyl acetate (100/0 to 95/5)) led to ynamide 1e (860 mg, 18% yield) as colorless oil. 1 H NMR (400 MHz, CDCl 3 ) δ: 7.31-7.40 (m, 3H), 7.58 (dd, J = 6.6 and 3.0, 2H), 7.82 (dd, J = 5.5, 3.1, 2H), 7.96 (dd, J = 5.5 and 3.1, 2H).

1-(2-Phenylethynyl)pyrrolidin-2-one (1f)

. 27

N-Benzyl-4-methyl-N-(2-phenylethynyl)benzene-1sulfonamide (1g

). 28 The reaction was conducted under argon in the presence of K 2 CO 3 (2.0 mmol, 276 mg, 2 equiv), N-benzyl-4methylbenzene-1-sulfonamide (1.0 mmol, 261 mg, 1 equiv) and FeCl 3 .6H

General procedure for carbozincation reactions

Method A: Copper Iodide or Iron(II) chloride (0.5-2 equiv) was added under argon at room temperature to a solution of ynamide 1 (1 equiv, 0.2 M) in non-degassed dichloromethane. Dialkylzinc (1 M in hexane, 2 equiv) was added and the reaction was stirred for 18 h and quenched with saturated NH 4 Cl. The layers were separated and the aqueous layer was extracted with CH 2 Cl 2 (x3). The combined organic phases were dried over MgSO 4 , filtered on silica gel and concentrated under vacuum. Enamides 2/6/7 were purified by flash chromatography on silica gel (pentane/ethyl acetate).

Method B:

Copper Iodide or Iron (II) chloride (0.5-2 equiv) and allylbromide (9 equiv) were added under argon at 0 °C to a solution of ynamide 1 (1 equiv, 0.2 M) in non-degassed dichloromethane. Dialkylzinc (1 M in hexane, 2 equiv) was added and the reaction was stirred at room temperature for 18 h and quenched with saturated NH 4 Cl. The layers were separated and the aqueous layer was extracted with CH 2 Cl 2 (x3). The combined organic phases were dried over MgSO 4 , filtered on silica gel and concentrated under vacuum. Enamides 8/9 were purified by flash chromatography on silica gel (pentane/ethyl acetate).

3-[(1Z)-2-Ethyl-4-[(2-methoxyethoxy)methoxy]but-1-en-1-yl]-1,3-oxazolidin-2-one (2a).

The reaction was conducted on 3-{4-[(2-methoxyethoxy)methoxy]but-1-yn-1-yl}-1,3-oxazolidin-2one (0.44 mmol, 107 mg), 1a according to method A with diethylzinc (880 µL) and CuI (0.22 mmol, 42 mg, 0.5 equiv) to give 2a as colorless oil (0.37 mmol, 100 mg, 83%). The reaction was conducted on 3-{4-[(2-methoxyethoxy) methoxy]but-1-yn-1-yl}-1,3-oxazolidin-2-one (0.45 mmol, 110 mg) 1a according to method A with diethylzinc (900 µL) and FeCl 2 (0.23 mmol, 29 mg, 0.5 equiv) to give 2a as a colorless oil (0.33 mmol, 90 mg, 73%). 1 H NMR (400 MHz, CDCl 3 ) δ: 1.04 (t, J = 7.4, 3H), 2.09 (qd, J = 7.4 and 1.1, 2H), 2.42 (t, J = 7.0, 2H), 3.37 (s, 3H), 3.51-3.56 (m, 2H), 3.59 (t, J = 7.0, 2H), 3.61-3.68 (m, 2H), 3.83 (pseudo t, J = 8.6, 2H), 4.36 (pseudo t, J = 8.6, 2H), 4.68 (s, 2H), 5.99 (br s, 1H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF] C NMR (100 MHz, CDCl 3 ) δ: 12.9 (CH 3 ), 27

3-[(1Z)-2-Phenylbut-1-en-1-yl]-1,3-oxazolidin-2-one (2b). 5a

The reaction was conducted on 3-(2-phenylethynyl)-1,3oxazolidin-2-one (0.27 mmol, 50 mg) 1b according to method A with diethylzinc (540 µL) and CuI (0.14 mmol, 27 mg, 0.5 equiv) to give 2b as a colorless oil (0.19 mmol, 41 mg, 69%). The reaction was conducted on 3-(2-phenylethynyl)-1,3oxazolidin-2-one (0.54 mmol, 100 mg) 1b according to method A with diethylzinc (2.16 mL, 4 equiv) and FeCl 2 (0.27 mmol, 34 mg, 0.5 equiv) to give 2b as a colorless oil (0.14 mmol, 30 mg, 25%). 1 H NMR (400 MHz, CDCl 3 ) δ: 0.99 (t, J = 7.4, 3H), 2.37 (q, J = 7.4, 2H), 3.03 (pseudo t, J = 7.9, 2H), 4.12 (pseudo t, J = 7.9, 2H), 6.55 (s, 1H), 7.19-7.22 (m, 2H), 7.25-7.37 (m, 3H).

3-[(1Z)-2-[(Benzyloxy)methyl]but-1-en-1-yl]-1,3-oxazolidin-2one (2c).

The reaction was conducted on 3-[3-(benzyloxy)prop-1-yn-1-yl]-1,3-oxazolidin-2-one (0.45 mmol, 103 mg) 1c according to method A with diethylzinc (900 µL) and CuI (0.22 mmol, 42 mg, 0.5 equiv) to give 2c as a colorless oil (0.29 mmol, 76 mg, 65%). The reaction was conducted on 3-[3-(benzyloxy)prop-1-yn-1-yl]-1,3-oxazolidin-2-one (0.37 mmol, 85 mg) 1c according to method A with diethylzinc (740 µL) and FeCl 2 (0.19 mmol, 23 mg, 0.5 equiv) to give 2c as a colorless oil (0.06 mmol, 16 mg, 17%). 1 H NMR (300 MHz, CDCl 3 ) δ: 1.07 (t, J = 7.4, 3H), 2.18 (qd, J = 7.4 and 0.9, 2H), 3.75 (pseudo t, J = 7.5, 2H), 4.04 (s, 2H), 4.26 (pseudo t, J = 7.5, 2H), 4.49 (s, 2H), 6.17 (br s, 1H), 7.27-7.40 (m, 5H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

3-[(1Z)-2-Methylhex-1-en-1-yl]-1,3-oxazolidin-2-one (2d).

The reaction was conducted on 3-(hex-1-yn-1-yl)-1,3-oxazolidin-2one (0.51 mmol, 85 mg) 1d according to method A with diethylzinc (1.02 mL) and CuI (0.25 mmol, 48 mg, 0.5 equiv) to give 2d (0.47 mmol, 92 mg, 91%) as a colorless oil. The reaction was conducted on 3-(hex-1-yn-1-yl)-1,3-oxazolidin-2-one (0.38 mmol, 64 mg) 1d according to method A with diethylzinc (760 µL) and FeCl 2 (0.19 mmol, 24 mg, 0.5 equiv) to give 2d as a colorless oil (0.23 mmol, 45 mg, 60%). 1 H NMR (300 MHz, CDCl 3 ) δ: 0.91 (t, J = 7.0, 3H), 1.04 (t, J = 7.4, 3H), 1.21-1.47 (m, 4H), 1.99-2.15 (m, 4H), 3.75 (pseudo t, J = 7.4, 2H), 4.36 (pseudo t, J = 7.4, 2H), 5.87 (s, 1H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF] C NMR (75 MHz, CDCl 3 ) δ: 12.9 (CH 3 ), 14.0 (CH 3 ), 23.0 (CH 2 ), 27.5 (CH 2 ), 29.4 (CH 2 ), 30.3 (CH 2 ), 47.0 (CH 2 ), 62.2 (CH 2 ), 118.3 (=CH), 138.0 (=C), 157.6 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 11 H 20 NO 2 : 198.1489, found 198.1490.

2-[(1Z)-2-Phenylbut-1-en-1-yl]-2,3-dihydro-1H-isoindole-1,3dione (2e).

The reaction was conducted on 2-(2-phenylethynyl)-2,3-dihydro-1H-isoindole-1,3-dione 1e (0.28 mmol, 70 mg) according to method A with diethylzinc (756 µL) and CuI (0.19 mmol, 36 mg, 0.5 equiv) to give 2e as a colorless oil (0.19 mmol, 53 mg) with 65% yield. The reaction was conducted on 2-(2phenylethynyl)-2,3-dihydro-1H-isoindole-1,3-dione 1e (0.28 mmol, 70 mg) according to method A with diethylzinc (810 µL, 2.0 equiv) and FeCl 2 (0.14 mmol, 18 mg, 0.5 equiv) to give 2e as a colorless oil (0.11 mmol, 30 mg) with 39% yield. 1 H NMR (400 MHz, CDCl 3 ) δ : 0.98 (t, J = 7.4, 3H), 2.59 (q, J = 7.4, 2H), 6.12 (s, 1H), 7.10-7.26 (m, 5H), 7.55-7.75 (m, 4H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

1-[(1Z)-2-Phenylbut-1-en-1-yl]pyrrolidin-2-one (2f).

The reaction was conducted on 1-(2-phenylethynyl)pyrrolidin-2-one (0.38 mmol, 70 mg) 1f according to method A with diethylzinc (760 µL) and CuI (0.19 mmol, 36 mg, 0.5 equiv) to give 2f as a colorless oil (0.25 mmol, 53 mg, 65%). The reaction was conducted on 1-(2-phenylethynyl)pyrrolidin-2-one (0.27 mmol, 50 mg) 1f according to method A with diethylzinc (540 µL) and FeCl 2 (0.14 mmol, 17 mg, 0.5 equiv) to give 2f as a colorless oil (0.21 mmol, 45 mg, 77%). 1 H NMR (400 MHz, CDCl 3 ) δ: 0.98 (t, J = 7.4, 3H), 1.78 (quint, J = 7.7, 2H), 2.33 (t, J = 8.2, 2H), 2.39 (q, J = 7.4, 2H), 2.87 (t, J = 7.2, 2H), 6.72 (s, 1H), 7.13-7.20 (m, 2H), 7.22-7.36 (m, 3H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

3-[(1Z)-4-[(2-Methoxyethoxy)methoxy]-2-methylbut-1-en-1yl]-1,3-oxazolidin-2-one (6a).

The reaction was conducted on 3-{4-[(2-methoxyethoxy)methoxy]but-1-yn-1-yl}-1,3-oxazolidin-2-one 1a according to method A with dimethylzinc (800 µL) and CuI (0.21 mmol, 40 mg, 0.5 equiv) to give 6a as a colorless oil (0.14 mmol, 37 mg, 62%). 1 H NMR (400 MHz, CDCl 3 ) δ: 1.77 (s, 3H), 2.42 (t, J = 6.7, 2H), 3.39 (s, 3H), 3.54-3.57 (m, 2H), 3.65 (t, J = 6.9, 2H), 3.65-3.71 (m, 2H), 3.82 (pseudo t, J = 8.0, 2H), 4.37 (pseudo t, J = 8.0, 2H), 4.70 (s, 2H), 5.98 (s, 1H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

3-[(1Z

)-2-Phenylprop-1-en-1-yl]-1,3-oxazolidin-2-one (6b). 29 The reaction was conducted on 3-(2-phenylethynyl)-1,3oxazolidin-2-one (0.53 mmol, 106 mg) 1b according to method A with dimethylzinc (1.1 mL) and CuI (0.27 mmol, 51 mg, 0.5 equiv) to give 6b as colorless oil (0.41 mmol, 84 mg, 77%). The reaction was conducted on 3-(2-phenylethynyl)-1,3-oxazolidin-2one (0.27 mmol, 50 mg) 1b according to method A with dimethylzinc (540 µL) and FeCl 2 (0.54 mmol, 63 mg, 2.0 equiv) to give 6b as a colorless oil (0.07 mmol, 19.6 mg, 25%). 1 H NMR (400 MHz, CDCl 3 ) δ: 2.05 (s, 3H), 3.09 (t, J = 7.9, 2H), 4.14 (t, J = 7.9, 2H), 6.55 (s, 1H), 7.23-7.36 (m, 5H).

3-[(1Z

)-3-(benzyloxy)-2-methylprop-1-en-1-yl]-1,3-oxazolidin-2-one (6c). The reaction was conducted on 3-[3-(benzyloxy)prop-1-yn-1-yl]-1,3-oxazolidin-2-one (0.30 mmol, 70 mg) 1c (according to method A with dimethylzinc (600 µL) and CuI (0.15 mmol, 28 mg, 0.5 equiv) to give 6c as a colorless oil (0.22 mmol, 53.3 mg, 72%). The reaction was conducted on 3-[3-(benzyloxy)prop-1-yn-1-yl]-1,3-oxazolidin-2-one (0.26 mmol, 60 mg) 1c according to method A with dimethylzinc (520 µL) and FeCl 2 (0.13 mmol, 17 mg, 0.5 equiv) to give 6c as a colorless oil (0.14 mmol, 35 mg, 54%). 1 H NMR (300 MHz, CDCl 3 ) δ: 1.84 (s, 3H), 3.70 (pseudo t, J = 8.0, 2H), 4.04 (s, 2H), 4.28 (pseudo t, J = 8.0, 2H), 4.50 (s, 2H), 6.10 (s, 1H), 7.29-7.34 (m, 5H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

3-[(1Z)-2-methylhex-1-en-1-yl]-1,3-oxazolidin-2-one (6d).

The reaction was conducted on 3-(hex-1-yn-1-yl)-1,3-oxazolidin-2one (0.68 mmol, 114 mg) 1d according to method A with dimethylzinc (1.26 mL) and CuI (0.34 mmol, 65 mg, 0.5 equiv) to give 6d (0.51 mmol, 94 mg, 75%) as a colorless oil. The reaction was conducted on 3-(hex-1-yn-1-yl)-1,3-oxazolidin-2one (0.60 mmol, 100 mg) 1d according to method A with dimethylzinc (1.2 mL) and FeCl 2 (0.30 mmol, 38 mg, 0.5 equiv) to give 6d as a colorless oil (0.014 mmol, 25 mg, 23%). 1 H NMR (400 MHz, CDCl 3 ) δ: 0.91 (t, J = 7.2, 3H), 1.29-1.42 4H), 1.73 (s, 3H), 2.09 (t, J = 8.2, 2H), 3.74 (pseudo t, J = 8.0, 2H), 4.37 (pseudo t, J = 8.0, 2H), 5.84 (s, 1H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF] C NMR (100 MHz, CDCl 3 ) δ: 14.0 (CH 3 ), 20.

2-[(1Z)-2-Phenylprop-1-en-1-yl]-2,3-dihydro-1H-isoindole-1,3-dione (6e).

The reaction was conducted on 2-(2phenylethynyl)-2,3-dihydro-1H-isoindole-1,3-dione 1e (0.4 mmol, 100 mg) according to method A with dimethylzinc (810 µL) and CuI (0.2 mmol, 38 mg, 0.5 equiv) to give 6e as a colorless oil (0.32 mmol, 85 mg) with 80% yield. The reaction was conducted on 2-(2-phenylethynyl)-2,3-dihydro-1Hisoindole-1,3-dione 1e (0.4 mmol, 100 mg) according to method A with dimethylzinc (810 µL) and FeCl 2 (0.2 mmol, 26 mg, 0.5 equiv) to give 6e as a colorless oil (0.05 mmol, 12.8 mg) with 12% yield. 1 H NMR (400 MHz, CDCl 3 ) δ: 2.25 (d, J = 1.6, 3H), 6.15 (q, J = 1.4, 1H), 7.15-7.25 (m, 5H), 7.65-7.67 (m, 2H), 7.74-7.76 (m, 2H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

1-[(1Z)-2-Phenylprop-1-en-1-yl]pyrrolidin-2-one (6f).

The reaction was conducted on 1-(2-phenylethynyl)pyrrolidin-2-one (0.22 mmol, 40 mg) 1f according to method A with dimethylzinc (430 µL) and CuI (0.11 mmol, 21 mg, 0.5 equiv) to give 6f as a colorless oil (0.19 mmol, 38 mg, 87%). The reaction was conducted on 1-(2-phenylethynyl)pyrrolidin-2-one (0.27 mmol, 50 mg) 1f according to method A with dimethylzinc (540 µL) and FeCl 2 (0.14 mmol, 17 mg, 0.5 equiv) to give 6f as a colorless oil (0.24 mmol, 49 mg, 91%). 1 H NMR (400 MHz, CDCl 3 ) δ: 1.74 (quint, J = 7.6, 2H), 1.99 (s, 3H), 2.28 (t, J = 8.1, 2H), 2.93 (t, J = 8.1, 2H), 6.60 (s, 1H), 7.13-7.25 (m, 5H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF] C NMR (100 MHz, CDCl 3 ) δ: 18.9 (CH 2 ), 24

1-[(1Z)-3-methyl-2-phenylbut-1-en-1-yl]pyrrolidin-2-one (7f).

The reaction was conducted on 1-(2-phenylethynyl)pyrrolidin-2one (0.22 mmol, 40 mg) 1f according to method A with diisopropylzinc (440 µL) and CuI (0.11 mmol, 21 mg, 0.5 equiv) to give 7f as a colorless oil (0.15 mmol, 32 mg, 67%). 1 H NMR (300 MHz, CDCl 3 ) δ: 1.03 (d, J = 6.9, 6H), 1.74 (quint, J = 7.6, 2H), 2.30 (t, J = 8.2, 2H), 2.60 (sept, J = 6.9, 1H), 2.74 (t, J = 8.2, 2H), 6.75 (s, 1H), 7.09-7.15 (m, 2H), 7.24-7.32 (m, 3H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

3-[(4Z)-5-Ethyl-7-[(2-methoxyethoxy

)methoxy]hepta-1,4dien-4-yl]-1,3-oxazolidin-2-one (8a). The reaction was conducted on 3-{4-[(2-methoxyethoxy) methoxy] but-1-yn-1-yl}-1,3-oxazolidin-2-one (0.21 mmol, 50 mg) 1a according to method B with diethylzinc (620 µL) and CuI (0.31 mmol, 59 mg, 1.5 equiv) to give 8a as a colorless oil (0.10 mmol, 31 mg, 47%). 1 H NMR (400 MHz, CDCl 3 ) δ: 1.05 (t, J = 7.6, 3H), 2.16 (q, J = 7.6, 2H), 2.40 (t, J = 7.0, 2H), 3.06 (d, J = 6.4, 2H), 3.39 (s, 3H), 3.52-3.58 (m, 2H), 3.60-3.74 (m, 6H), 4.33 (t, J = 7.7, 2H), 4.69 (s, 2H), 5.04-5.19 (m, 2H), 5.79 (ddt, J = 16.6, 9.9 and 6.5, 1H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

3-[(4Z)-5-Phenylhepta-1,4-dien-4-yl]-1,3-oxazolidin-2-one (8b).

The reaction was conducted on 3-(2-phenylethynyl)-1,3oxazolidin-2-one (0.27 mmol, 50 mg) 1b according to method B with diethylzinc (540 µL) and CuI (0.14 mmol, 27 mg, 0.5 equiv) to give 8b as a colorless oil (0.27 mmol, 69 mg, 100%). The reaction was conducted on 3-(2-phenylethynyl)-1,3oxazolidin-2-one (0.27 mmol, 50 mg) 1a according to method B with diethylzinc (540 µL) and FeCl 2 (0.54 mmol, 68 mg, 2 equiv) to give 8b as a colorless oil (0.12 mmol, 32 mg, 73%). 1 H NMR (400 MHz, CDCl 3 ) δ: 0.94 (t, J = 7.5, 3H), 2.46 (q, J = 7.5, 2H), 3.10 (pseudo t, J = 8.5, 2H), 3.29 (d, J = 6.6, 2H), 3.94 (pseudo t, J = 8.5, 2H), 5.10 (dq, J = 10.0 and 1.5, 1H), 5.19 (dq, J = 16.7 and 1.5, 1H), 5.85 (ddt, J = 16.7, 10.0 and 6.7, 1H), 7.17-7.36 (m, 5H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

3-[(4Z)-5-Ethylnona-1,4-dien-4-yl]-1,3-oxazolidin-2-one (8d).

The reaction was conducted on 3-(hex-1-yn-1-yl)-1,3-oxazolidin-2-one (0.30 mmol, 50 mg) 1d according to method B with diethylzinc (600 µL) and CuI (0.6 mmol, 114 mg, 2.0 equiv) to give 8d as a colorless oil (0.23 mmol, 54 mg, 76%). 1 H NMR (300 MHz, CDCl 3 ) δ: 0.89 (t, J = 7.1, 3H), 1.02 (t, J = 7.6, 3H), 1.23-1.44 (m, 4H), 2.04 (distorsed t, J = 7.7, 2H), 2.12 (q, J = 7.6, 2H), 3.04 (d, J = 6.6, 2H), 3.63 (pseudo t, J = 8.0, 2H), 4.31 (pseudo t, J = 8.0, 2H), 5.05 (dq, J = 10.0 and 1.7, 1H), 5.12 (dq, J = 16.6 and 1.7, 1H), 5.79 (ddt, J = 16.6, 10.0 and 6.6, 1H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF] C NMR (75 MHz, CDCl 3 ) δ: 13.2 (CH 3 ), 14.0 (CH 3 ), 23.2 (CH 2 ), 24.1 (CH 2 ), 30.2 (CH 2 ), 30.9 (CH 2 ), 34.

  C NMR (100 MHz, CDCl 3 ) δ: 13.3 (CH 3 ), 18.8 (CH 2 ), 30.6 (CH 2 ), 31.1 (CH 2 ), 48.4 (CH 2 ), 120.0 (=CH), 127.2 (=CH), 128.0 (=CH), 129.1 (=CH), 129.4 (=C), 139.8 (=C), 175.1 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 14 H 18 NO: 216.1383, found 216.1382.

  C NMR (75 MHz, CDCl 3 ) δ: 18.7 (CH 2 ), 22.1 (CH 3 ), 30.6 (CH 2 ), 35.7 (CH), 48.4 (CH 2 ), 119.8 (=CH), 127.1 (=CH), 127.7 (=CH), 129.9 (=CH), 133.1 (=C), 139.1 (=C), 175.1 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 15 H 20 NO: 230.1539, found 230.1538.

  C NMR (100 MHz, CDCl 3 ) δ: 12.8 (CH 3 ), 26.8 (CH 2 ), 34.5 (CH 2 ), 46.8 (CH 2 ), 62.4 (CH 2 ), 117.20 (=CH 2 ), 127.5 (=CH), 128.0 (=CH), 128.6 (=CH), 129.0 (=C), 134.7 (=CH), 140.2 (=C), 142.3 (=C), 157.0 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 16 H 20 NO 2 : 258.1489, found: 258.1487.

Table 1 .

 1 Ethylzincation of ynamide 1a with FeCl 2 and CuI

	Entry	Additive (equiv.)	2a (isolated yield %)
	1	-	Traces
	2	FeCl2 (2)	91
	3	FeCl2 (0.5)	73
	4	CuI (2)	76
	5	CuI (0.5)	83
	6	CuI (0.1)	74

Table 2 .

 2 Ethylzincation of ynamides 1

	Entry	1	Additive	2 (isolated yield %)
	1	1a	CuI	2a (83)
	2	1b	CuI	2b (69)
	3	1c	CuI	2c (65)
	4	1d	CuI	2d (91)
	5	1e	CuI	2e (80) [a]
	6	1f	CuI	2f (65)
	7	1a	FeCl2	2a (73)
	8	1b	FeCl2	2b (25)
	9	1c	FeCl2	2c (17)
	10	1d	FeCl2	2d (60)
	11	1e	FeCl2	2e (39)
	12	1f	FeCl2	2f (77)
	[a] Et2Zn (4 equiv.)

are shown in Erreur ! La référence est incorrecte pour un signet.. The

		1b			1f
			1g
	1b	𝑓 "		𝑓 $	𝑓 (()
	O	0.073	0.051	-0.022
	Ca	0.126	0.107	-0.019
	Cb	0.120	0.000	-0.120
	1f	𝑓 "		𝑓 $		𝑓 (()
	O	0.093	0.085	-0.008
	Ca	0.116	0.112	-0.004
	Cb	0.125	0.020	-0.105
	1g	𝑓 "		𝑓 $		𝑓 (()
	O	0.033	0.035	0.002
	Ca	0.077	0.044	-0.033
	Cb	0.191	0.031	-0.160
	Figure 1. Maps of Fukui's dual descriptors for 1b, 1f and 1g
	(more Nu than E zones in turquoise blue; more E than Nu
	zones in lavender-blue). Table of values of condensed Fukui
	functions	𝒇 " and	𝒇 $	and	dual	descriptor
	𝒇
	values of condensed Fukui's function 𝑓 " , gives the following
	sequence of nucleophilicity for O, Cα and Cβ sites:
	O : 1f > 1b > 1g
	Cα : 1b > 1f >> 1g
	Cβ : 1g > 1f > 1b
	According to condensed Fukui's function 𝒇 $ Cα is the most
	electrophilic site in 1f and 1b. Comparatively, this value is low in
	1g. The two components neutralize each other in 𝒇 (𝟐) but the
	residual nucleophilicity is mainly localized at Cβ as shown in
	Figure 1. No clean conclusion can be drawn from the
	examination of these quantitative parameters. The reactivity
	seems to increase as Cβ nucleophilicity decreases.

Table 3 .

 3 Methylzincation of ynamides 1

	Entry	1	Additive	6 (isolated yield %)
	1	1a	CuI	6a (62)
	2	1b	CuI	6b (77)
	3	1c	CuI	6c (72)
	4	1d	CuI	6d (74)
	5	1e	CuI	6e (80)
	6	1f	CuI	6f (87)
	7	1a	FeCl2	6a (-) [a]
	8	1b	FeCl2	6b (13)(25) [b]
	9	1c	FeCl2	6c (54)
	10	1d	FeCl2	6d (23)
	11	1e	FeCl2	6e (35) [c]
	12	1f	FeCl2	6f (91)
	[a] The substrate was fully recovered, [b] 2 equivalents of FeCl2 were used,
	[c] Me2Zn (5 equiv.)

Table 4 .

 4 One-pot procedure for stereoselective carboallylation of ynamides 1

	Entry	1	Additive	R	8/9 (isolated yield %)
	1	1b	CuI	Et	8b (100)
	2	1b	CuI (0.1)	Et	8b (70)
	3	1b	FeCl2	Et	8b (46)
	4	1b	CuI	Me	9b (93)
	5	1d	CuI	Et	8d (76)
	6	1a	CuI	Et	8a (47)
	7	1f	CuI	Et	8f (53)

  ). The reaction was conducted according to the general procedure in the presence of CuCl 2 (1.0 mmol, 134.5 mg), oxazolidin-2-one (25.0 mmol, 2.18 g) Na 2 CO 3(10.

	0 mmol,
	1.27 g), a solution of pyridine (10.0 mmol, 0.95 ml) in dry
	toluene (30 mL) and 1-[(but-3-yn-1-yloxy) methoxy]-2-
	methoxyethane (5.0 mmol, 1.0 g) was added in dry toluene (20
	mL). Purification by flash chromatography on silica gel
	(pentane/ethyl acetate (10/0 to 8/2)) led to ynamide 1a (990 mg,
	81% yield) as an oil.

1

H NMR (400 MHz, CDCl 3 ) δ: 2.58 (t, J = 6.8, 2H), 3.36 (s, 3H), 3.55-3.57 (m, 2H), 3.69 (t, J = 6.8, 2H), 3.66-3.71 (m, 2H), 3.87 (t, J = 7.8, 2H), 4.40 (t, J = 7.8, 2H), 4.74 (s, 2H).

[START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

C NMR (100 MHz, CDCl 3 ) δ: 20.0 (CH 2 ), 46.9 (CH 2 ), 59.1 (CH 3 ), 63.0 (CH 2 ), 66.1 (CH 2 ), 66.9 (CH 2 ), 68.1 (≡C), 71.0 (≡C), 71.8 (CH 2 ), 95.5 (CH 2 ), 156.6 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 11 H 18 NO 5 : 261.1145, found 261.1444.

  .8 (CH 2 ), 30.2 (CH 2 ), 46.8 (CH 2 ), 59.1 (CH 3 ), 62.3 (CH 2 ), 65.8 (CH 2 ), 67.1 (CH 2 ), 71.9 (CH 2 ), 95.6 (CH 2 ), 120.2 (=CH), 132.6 (=C), 157.6 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 13 H 24 NO 5 : 274.1649, found 274.1648.

  C NMR (75 MHz, CDCl 3 ) δ: 13.1 (CH 3 ), 27.3 (CH 2 ), 46.3 (CH 2 ), 62.3 (CH 2 ), 66.8 (CH 2 ), 72.6 (CH 2 ), 121.9 (=CH), 128.0 (=CH), 128.1 (=CH), 128.6 (=CH), 129.1 (=C), 138.2 (=C), 157.1 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 15 H 20 NO 3 : 262.1438, found 262.1437.

  C NMR (100 MHz, CDCl 3 ) δ: 12.6 (CH 3 ), 29.8 (CH 2 ), 112.5 (=CH), 123.5 (=CH), 127.2 (=CH), 127.6 (=CH), 128.4 (=CH), 132.0 (=C), 134.1 (=CH), 138.6 (=C), 149.0 (=C), 167.2 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 18 H 16 NO 2 : 278.1176, found 278.1176.

  5 (CH 3 ), 22.9 (CH 2 ), 29.8 (CH 2 ), 31.2 (CH 2 ), 47.0 (CH 2 ), 62.2 (CH 2 ), 118.8 (=CH), 135.5 (=C), 157.6 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 10 H 18 NO 2 : 184.1332, found 184.1332.

  .1 (CH 3 ), 30.6 (CH 2 ), 48.3 (CH 2 ), 120.7 (=CH), 124.2 (=C), 127.3 (=CH), 128.2 (=CH), 128.4 (=CH), 140.8 (=C), 175.3 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 13 H 16 NO: 202.1226, found: 202.1227.

  C NMR (100 MHz, CDCl 3 ) δ: 13.2 (CH 3 ), 24.4 (CH 2 ), 31.6 (CH 2 ), 34.3 (CH 2 ), 47.0 (CH 2 ), 59.2 (CH 3 ), 62.3 (CH 2 ), 66.0 (CH 2 ), 67.1 (CH 2 ), 72.0 (CH 2 ), 95.7 (CH 2 ), 117.1 (=CH 2 ), 128.4 (=C), 135.0 (=CH), 139.4 (=C), 160.7 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 16 H 28 NO 5 : 314.1962, found: 314.1963.

  3 (CH 2 ), 47.1 (CH 2 ), 62.1 (CH 2 ), 116.8 (=CH 2 ), 126.3 (=C), 135.3 (=CH), 142.9 (=C), 157.0 (C=O). HRMS (ESI): m/z: calcd for [M+H + ] C 14 H 24 NO 2 : 238.1802, found 238.1802

1-[(4Z)-5-Phenylhepta-1,4-dien-4-yl]pyrrolidin-2-one (8f). The reaction was conducted on 1-(2-phenylethynyl)pyrrolidin-2one (0.25 mmol, 46 mg) 1f according to method B with diethylzinc (500 µL) and CuI (0.12 mmol, 24 mg, 0.5 equiv) to give 8f as a colorless oil (0.13 mmol, 34 mg, 53%). 1 H NMR (400 MHz, CDCl 3 ) δ: 0.95 (t, J = 7.5, 3H), 1.63 (quint, J = 7.5, 2H), 2.16 (t, J = 8.0, 2H), 2.46 (q, J = 7.5, 2H), 2.94 (t, J = 6.9, 2H), 3.25 (d, J = 6.7, 2H), 5.08 -5.04 (dq, J = 10.0 and 1.6 ,1H), 5.14 (dq, J = 17.1 and 1.6, 1H), 5.83 (ddt, J = 16.8, 10.0 and 6.7, 1H), 7.34 -7.16 (m, 5H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF] C NMR (100 MHz, CDCl 3 ) δ:

3-[(2Z)-2-Phenylhexa-2,5-dien-3-yl]-1,3-oxazolidin-2-one (9b).

The reaction was conducted on 3-(2-phenylethynyl)-1,3oxazolidin-2-one (0.27 mmol, 50 mg) 1b according to method B with dimethylzinc (540 µL) and CuI (0.14 mmol, 27 mg, 0.5 equiv) to give 9b as a colorless oil (0.25 mmol, 62 mg, 93%). 1 H NMR (400 MHz, CDCl 3 ) δ: 2.09 (s, 3H), 3.12 (pseudo t, J = 8.6, 2H), 3.30 (d, J = 6.6, 2H), 3.98 (pseudo t, J = 8.6, 2H), 5.12 (dq, J = 10.0 and 1.6, 1H), 5.19 (dq, J = 16.7 and 1.6, 1H), 5.86 (ddt, J = 16.7, 10.0 and 6.6, 1H), 7.24-7.36 (m, 5H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF]

3-[(1Z)-2-phenyl(1-²H)but-1-en-1-yl]-1,3-oxazolidin-2-one (10b).

The reaction was conducted on 3-(2-phenylethynyl)-1,3oxazolidin-2-one (0.27 mmol, 50 mg) 1b according to method A with diethylzinc (540 µL) and CuI (0.14 mmol, 27 mg, 0.5 equiv). The reaction was stirred for 18 h and quenched with saturated ND 4 Cl to give 10b as a colorless oil (0.16 mmol, 35 mg, 60%). 1 H NMR (400 MHz, CDCl 3 ) δ: 0.98 (t, J = 7.4, 3H), 2.36 (q, J = 7.4, 2H), 3.03 (pseudo t, J = 7.9, 2H), 4.12 (pseudo t, J = 7.9, 2H), 7.16-7.22 (m, 2H), 7.24-7.36 (m, 3H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF] C NMR (100 MHz, CDCl 3 ) δ: 13.3 (CH 3 ), 30.9 (CH 2 ), 45.

(E)-3-(2-phenyl-1-(phenylselanyl)but-1-en-1-yl)oxazolidin-2-one (11b).

The reaction was conducted on 3-(2phenylethynyl)-1,3-oxazolidin-2-one (0.27 mmol, 50 mg) 1b according to method A with diethylzinc (540 µL) and CuI (0.14 mmol, 27 mg, 0.5 equiv). The reaction was stirred for 18 h and PhSeCl (0.108 mmol, 207 mg) in DCM (0.4 ml) was added at 0 °C. reaction mixture was stirred for 10 h and quenched with saturated NH 4 Cl. The layers were separated and the aqueous layer was extracted with CH 2 Cl 2 (x3). The combined organic phases were dried over MgSO 4 , filtered on silica gel and concentrated under vacuum and purified by flash chromatography on silica gel (pentane/ether) to give 11b as colorless oil (0.134 mmol, 50%). NMR (400 MHz, CDCl 3 ) δ: 0.97 (t, J = 7.5 Hz, 3H), 2.76 (q, J = 7.5 Hz, 2H), 3.15 (t, J = 7.7 Hz, 2H), 3.15 (pseudo t, J = 7.7 Hz, 2H), 3.66 (pseudo t, J = 7.6 Hz, 2H), 7.36 -7.20 (m, 8H), 7.70 -7.63 (m, 2H). [START_REF] Buchwald | The possible role of contaminants in FeCl3-catalyzed coupling processes has been emphasized[END_REF] C NMR (100 MHz, CDCl 3 ) δ: 12.7 (CH 3 ), 30.9 (CH 2 ), 46.6 (CH 2 ), 61.9 (CH 2 ), 123.2 (=C), 125.7 (CH), 127.7 (CH), 128.0 (=C), 128.4 (CH), 128.5 (CH), 129.4 (CH), 134.8 (CH), 139.9 (=C), 149.9 (=C), 155.7 (C=O). Supplementary content. NMR spectra for all new componds. Calculations of 1b, 1g, 1f, the resulting vinylzincs and Et 2 Zn at the B3LYP/6-31+G(d,p) level of theory. Condensed Fukui's functions 𝒇 " , 𝒇 $ and dual descriptor 𝒇 (𝟐) : table of their values at O, Ca, and Cb.


M Nazaikinskii

Rouleux

Maslov canonical operator on a pair of Lagrangian manifolds and asymptotic solutions of stationary

equations with localized right-hand sides

Maslov's canonical operator on a pair of Lagrangian manifolds and asymptotic solutions of stationary equations with localized right-hand sides 2017 Рi. A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, M. Rouleux

Аннотация

The problem of constructing the asymptotics of the Green's function for the Helmholtz operator h 2 ∆ + n 2 (x), x ∈ R n with the small positive parameter h and smooth n 2 (x) was studied in many papers, e.g. [1,2,4]. In the case of variable coefficients, the asymptotics was constructed by matching the asymptotics of the Green's function for the equation with frozen coefficients and WKB-type asymptotics or more generally Maslov's canonical operator. In our work we present a different method for calculating Green's function without assuming any knowledge of exact Green's function for the operator with frozen variables. Our approach also works for a wider class of operators, and also if the right hand side is a smooth localized function rather than δ-function. In particular, the method works for the linearized water waves equation.

Introduction.

The problem of short-wave asymptotics for Green's function of Helholtz operator in R n was studied in several works, e.g. [1], [2,3], [4] (other references could be fount in book [5]). If the refractive index does not depend on x ∈ R n , then the Green's functions can be calculated exactly (e.g. for n = 2 it is expressed via Hankel function). If the refractive index is variable, the asympotics for Green's function was constructed in above mentioned papers by pasting the exact solution of the equation with constant refractive index ('frozen' in the point of the source) with the WKB-asymptotics (see [2]) or more generally with Maslov"s canonical operator [4], when focal points appear. This approach can not be straightforwardly generalized to different types of operators, because it is necessary to know the exact Green's function for operators with frozen coefficients, and also to have in one"s disposal effective asymptotic formulas for large |x|, and finally be able to paste the latter with WKBtype solutions.

In this problem a geometric object called a 'pair of Lagrangian manifolds' natutally appears. It was first observed by Melrose and Uhlmann [6] (who introduced the term 'Lagrangian intersection') in a close problem of constructing parametrix for operators with real characterstics by methods of Fourier integral operators. In the problem of short-wave asyptotics for right inverse of Helmholtz-type operators the analogous construction was proposed in [7]. However, in that work a Lagrangian manifold with boundary was used rather than pair of Lagrangian manifolds. As a result some terms in their formula have 'parasitic' singularities (which cancel out in sum), making the proposed formula less effecient in concrete calculations.

The problem of Green's function is a limit case of a more physically interesting problem with localized near a point ξ ∈ R n right-hand side instead of Dirac"s δ-function.This problem in turn is a special case of a right-hand side represented with help of Maslov's canonical operator. We construct a formal asymptotic solution of the latter generalized problem, and then study this solution in case of a localized right-hand side. Our approach works for a wide class of operators, and does not appeal to exact solutions of equation with frozen coefficients, and does not lead to 'parasitic' singularities in a formula for solution.

We will use the following terminology. Let f (x, h) be a smooth function of x = (x 1 , . . . , x n ) and parameter h ∈ (0, 1]. It will be said that f = O(h m ) in domain U ⊂ R n , which may depend on parameter h, if the function f is defined for x ∈ U for all h ∈ (0, 1], moreover, for any measurable compact set K R n , not depending on h, and all integer s ≥ 0 it follows that f s,K∩U ≤ Ch m for some constant C (depending on K, s, but not h), where

f s,Ω = s |α|=0 h |α| f (α) 2 L 2 (Ω) 1/2 . If domain U is not specified explicitly, it is assumed that U = R n . Let f depend on additional parameters. It is said that f = O(h m )
uniformly with respect to these parameters, if the constants in the above bounds do not depend on the parameters. Finally, we say that

f = O(h ∞ ), if f = O(h m ) for all m.
2. Statement of the problem. Let H = H( 2x, -i 1 h∂/∂x, h) be an hpseudo-differential operator with the symbol H(x, p, h) ∈ S ∞ being a smooth function of variables x, p ∈ R n and parameter h ∈ [0, 1] that satisfies for some m the bounds

∂ |α|+|β|+k H(x, p, h) ∂x α ∂p β ∂h k ≤ C αβk (1 + |x| + |p|) m , |α|, |β|, k = 0, 1 , 2, . . .
(see [START_REF] Maslov | Operator methods[END_REF] for details). Consider in R n the inhomogeneous equation:

Hu = f (x, h). (1)
Definition. A function u(x, h), x ∈ R n , h ∈ (0, 1] is called an asymptotic solution of equation (1) with (asymptotic) limit absorbtion condition, if there exists a family of functions u ε (x, h) depending on parameter ε ∈ (0, 1)

such that ( H -iε)u ε = f + O(h ∞ )
uniformly in ε, and u -u ε s,Ω → 0 uniformly in h for ε → 0 for any bounded domain Ω ⊂ R n and any integer s ≥ 0.

Remark. The condition stated in this definition is none else than the asymptotic analog of the well-known limit absorbtion principle for Helmholtz equation

We will construct asymptotic solutions of the equation ( 1) in a sense of the above definition in case, when the right-hand side f (x, h) is represented as follows:

f (x, h) = [K h (Λ,dµ) A](x, h), (2)
where

K h (Λ,dµ)
is a Maslov's canonical operator [START_REF] Maslov | [END_REF], [START_REF] Dobrokhotov | [END_REF] on a Lagrangian manifold Λ ⊂ R 2n , satisfying the quantization conditions, and equipped with a smooth measure (volume form) dµ, and A ∈ C ∞ (Λ) is a compactly supported function in Λ. (Recall that [START_REF] Dobrokhotov | [END_REF] the canonical operator is defined uniquely up to arbitrary unimodular factor modulo higher-order terms as h → 0. To define this factor, it is necessary to choose an initial point on Λ and also choose the argument of some Jacobian; we don't reflect these objects in our notation to avoid clumsity.) Particularly, if the Lagrangian manifold and measure are as follows:

Λ = {(x, p) ∈ R 2n : x = ξ, p is arbitrary}, dµ = dp 1 ∧ • • • ∧ dp n , (3)
the right-hand side as h → 0 near the point x = ξ has a form

f (x, h) = 1 h n/2 V x -ξ h , where V (y) = e iπn/4 (2π) n/2 R n e ipy A(p) dp 1 • • • dp n (4
) is an inverse Fourier transform of the function A(p).

We will describe construction of asymptotic solution for a general righthand side (2), and then study the equation with right-hand side (4), which is the most important in applications and admits far more constructive solution formula. 3. General case. First of all, we introduce the assumptions: C0. The principal symbol H(x, p) = H(x, p, 0) of the operator H is real-valued.

C1. The Hamiltonian vector field

V H ≡ d dt = (H p , -H x ) is tranversal to Λ everywhere in L 0 = {(x, p) ∈ Λ : H(x, p) = 0}.
C2. Every trajectory (X(x 0 , p 0 , t), P (x 0 , p 0 , t)) = g t H ((x 0 , p 0 )) of the vector field V H issueing from (x 0 , p 0 ) ∈ L 0 (a) does not go back to L 0 , (b) satisfy the condition lim t→T |X(x 0 , p 0 , t)| = ∞, where [0, T ), T ≤ +∞ is the maximal existence interval of the trajectory. (Here the assumption (b) is more essential, sufficient condition for it can be found e.g. in [4]. The condition (Р • ) has a technical nature, it can be dropped out but theorems would be a bit more complicated.) Proposition 1. If conditions C0-C2 hold, then:

1. The set L 0 is a smooth (n -1)-dimensional submanifold in Λ.

2. The set Λ + = t≥0 g t H (L 0 ) is a Lagrangian manifold with boundary L 0 in the space R 2n1 .

3. The pair (Λ, Λ + ) defines a Lagrangian intersection in sense of [6]:

T (x 0 ,p 0 ) Λ ∩ T (x 0 ,p 0 ) Λ + = T (x 0 ,p 0 ) L 0 for (x 0 , p 0 ) ∈ L 0 .
4. Let Λ ⊂ Λ be sufficiently small neighborhood of the submanifold L 0 in Λ, and t 0 > 0 is sufficiently small. Then for all t ∈ [0, t 0 ] the following Lagrangian manifolds Λ t = g t H ( Λ) are well-defined. 5. If the manifold Λ meets the quantization conditions, then so do the manifolds Λ + and Λ t , t ∈ [0, t 0 ].

Let us define some other objects that are necessary to construct a canonical operator on Λ + and Λ t .

Without loss of generality, we will assume that the initial point α * on the manifold Λ is chosen on L 0 ; let us also choose the same point as initial on Λ + . Finally, on Λ t we choose the point g t H (α * ). Next, we define a measure on Λ t as follows: dµ t = (g t * H ) -1 dµ, where dµ is measure on Λ, and the measure dµ + on Λ + in the following way. At points of the submanifold L 0 ⊂ Λ, the measure dµ may written out as dµ = dH ∧dσ, where dσ is some (uniquely determinted) measure on L 0 . At last, we define the measure dµ + on submanifold g t H (L 0 ) ⊂ Λ + for all t > 0, by setting dµ + = dt ∧ (g t * H ) -1 dσ. For trajectories of the vector field V H , issuing from every point (x 0 , p 0 ) ∈ Λ, we determine a function A as a solution of the transport equation:

d A dt
+ H sub (X(x 0 , p 0 , t), P (x 0 , p 0 , t)) A = 0, A t=0 = A(x 0 , p 0 ), (5) where A ∈ C ∞ 0 (Λ) is the amplitude in formula (2) for the right-hand side, and

H sub (x, p) = i ∂H ∂h (x, p, 0) - 1 2 n j=1 ∂ 2 H ∂x j ∂p j (x, p)
is a sub-principal symbol of the operator H. Let us denote the restrictions of the function A onto Lagrangian manifolds Λ + and Λ t by A + and A(t) respectively. Let ρ ∈ C ∞ (Λ) and θ ∈ C ∞ 0 (R 1 ) be cutoff functions such that

ρ(α) = 1 in a neighborhood of L 0 , supp ρ ⊂ Λ; θ(t) = 1 for |t| ≤ t 0 /2, θ(t) = 0 for |t| ≥ t 0 . (6)
The function θ(t) may be interpretted as a function on Λ + , if we think of t as a time along a trajectory of the field V H . Similarly, ρ may be thought of as a function on Λ t , if we assume it to be constant along the trajectories of V H .

Theorem 1. If conditions C0-C2 are fullfilled, then the equation (1) with right-hand side (2) has an asymptotic solution u = u(x, h), satisfying the asymptotic limit absorbtion principle and (if the arguments of Jacobians in definition of canonical operators are chosen appropriately) 2 with the principal term as follows:

u = K h (Λ,dµ) (1 -ρ(α))A H + 2π h 1/2 e iπ/4 K h (Λ + ,dµ + ) [(1 -θ(t))A + ] + i h t 0 0 exp g t H (α * ) α * p dx K h (Λ t ,dµ t ) ρ(α)θ(t)A(t) dt + O(h 1/2 ), (7)
where the integral in the exponent is calculated along the trajectory of the field V H .

4. The case of localized right-hand side. Let us study in more detail the particular case of ( 1), ( 2), when the right-hand side is the localized function (4). Then the Lagrangian manifold and the measure are (3), and the condition C1 takes the form:

H p (ξ, p) = 0 РїСЂРё H(ξ, p) = 0.
Theorem 2. If the right-hand side of (1) has the form (4), then the solution u(x, h) constructed in Theorem 1 u(x, h) has the principle term as follows:

u(x, h) = e iπn/4 (2πh) n/2 lim ε→+0 R n A(p)e i h p(x-ξ) H(ξ, p) -iε dp 1 • • • dp n + O(h 2β-3/2 ) (8)
in domain |x -ξ| < Ch β , and

u(x, h) = 2π h 1/2 e iπ/4 [K h (Λ + ,dµ + ) (1-θ(Lt/h β ))A + ](x, h)+O(h 2β-3/2 ) (9)
in domain |x -ξ| > Ch β for all constant C > 0, β ∈ (1/2, 1), and sufficiently large L > 0.

The equality [START_REF] Maslov | Operator methods[END_REF] means that the principal term of the asymptotic solution for small |x -ξ| coincides with the solution u 0 (x, h) given by the limit absorbtion principle of the equation ( 1) with coefficients frozen in the point x = ξ. This solution can be calculated without passage to the limit as ε → 0 (which is difficult from the numerical viewpoint, since the integrand contains a singularity). To do this, one should use the following formula:

u 0 (x, h) = e iπ(n+1)/4 (2πh) n/2 h 1/2 t 0 0 R n e i h (p(x-ξ)-tH(ξ,p)) θ(t)ρ(p)A(p) dp 1 . . . dp n dt + i 2πh n/2 R n (1 -ρ(p))A(p)e i h p(x-ξ) H(ξ, p) dp 1 • • • dp n + O(h 1/2 ).
Formula ( 8) describes a part of the solution associated with the Lagrangian manifold Λ, which becomes singular in a limit A(p) → 1 (if it exists). Formula ( 9) describes rapidly oscillating 'wave' (or 'hyperbolic') part of the solution. One of the key results of our work is that we have an explicit formula for the function [START_REF] Sveshnikov | [END_REF] in terms of Fourier transform A of the righthand side (4). This function according to general theory [START_REF] Maslov | [END_REF], [START_REF] Dobrokhotov | [END_REF] as h → +0 is localized near the projection of the support of A + onto R n x . The canonical operator admits different representation in different points x, depending on whether they are located on caustics, i.e. a projection onto R n

x of the singularity cycle on the manifold Λ + . Let φ be the coordinates on L 0 . Then we can choose as coordinates on Λ + the pair (τ, φ), where τ ≥ 0 is the Hamiltonian time along V H . Thus, the manifold Λ + is given by the equations: (x, p) = (X (τ, φ), P(τ, φ)). The point (τ, φ) on Λ + is said to be regular, if the Jacobian J(τ, φ) = det ∂X ∂µ + (τ, φ) does not vanish in it. If the point x belongs to the domain |x -ξ| > Ch β , and its pre-image on the manifold Λ + is the unique and regular point (τ, φ), then the principal part of the solution (9) near x may be written out as

u(x, h) = √ 2πA + (τ, φ) h|J(τ, φ)| e i h S(τ,φ)-iπ 2 (m-1 2 ) , S(τ, φ) = τ 0 p • H p x=X (φ,η) p=P(φ,η) dη, m = ind γ + s - 0 H * p (X (0, φ 0 ), ξ) H p (X (0, φ 0 ), ξ) H pp (X (0, φ 0 ), ξ)
.

Here φ 0 is the point on L 0 , where the determinant of the described above self-adjoint matrix A does not vanish. Let s -(A) be its negative inertia index, and ind γ the Maslov's index [START_REF] Maslov | [END_REF], [START_REF] Dobrokhotov | [END_REF] of the path γ, connecting in Λ + two regular points: (+0, φ 0 ) and (τ, φ).

If points φ 0 with mentioned earlier properties do not exist, then the formula for m becomes more complicated. This formula along with a solution expression near caustics (С ЃРj. [START_REF] Dobrokhotov | [END_REF]) will be presented in a detailed paper.

Remark. Under additional assumptions in formulas ( 8) and ( 9) one can set A = 1, then one obtains the Green's function asymptotics. And vice versa, if one knows the asymptotics for Green's function, then one can write out a solution of (1) as a convolution with the right-hand side. However, the direct way of finding the asymptotic formulas from Theorems 1 and 2 in our opinion is simpler and more naturally from the viewpoint of applications.

Examples.

Let n = 2. Consider the Helmholtz equation H(x, p, h) = H(x, p) = p 2 -n 2 (x), n 2 (x) > 0, and the linear water waves equation [12] (H(x, p) = |p| tanh(|p|D(x)) -ω 2 and H sub (x, p) = 0). Here the smooth function D(x) determines the depth, and ω > 0 is the frequency. Let us study two forms of right-hand sides (4), in which

V (y) = exp - 1 2 y 2 1 a 2 + y 2 2 b 2 or (a 1 y 1 + b 1 y 2 ) exp - 1 2 y 2 1 a 2 + y 2 2 b 2
and, respectively,

A(p) = ab exp - a 2 p 2 1 + b 2 p 2 2 2 or iab(a 2 a 1 p 1 + b 2 b 1 p 2 ) exp - a 2 p 2 1 + b 2 p 2 2 2 .
Let ξ = 0. Then L 0 = {x = 0, p = ρ(cos φ, sin φ)}, where ρ = |n(0)| for Helmholtz equation, and ρ is the root of the equation ρ tanh(ρD(0)) = ω 2 for the water waves problem. As coordinates on Λ + we choose (φ, t), where t ≥ 0 is the Hamiltonian time along V H , and φ is the initial point on L 0 . The amplitude A + has the following form: respectively for the first and second form of the right-hand side. In case (a) the maximal amplitude is attained for smaller values of ρ, and in case (b) this maximum will be at the point ρ = 1/ a 2 cos 2 φ + b 2 sin 2 φ, tan φ = b 2 b 1 a 2 a 1 . Thus, by varying the parameters ω, a and b, we can increase or decrease the amplitude in a given domain of the space R 2

x . For the Helmholtz equation with constant refractive index n 2 (x), and A = 1, theorem gives the Green's function as a Hankel function, along with its asymptotics for large x. At last, we remark that the Maupertuis-Jacobi principle the initial Hamiltonian for Helmholtz and water waves equation can be replaced with C(x)|p|, which simplifies the computer implementation of canonical operator [13].

The work was supported by RFBR (grant no. 14-01-00521). The authors thank A.I. Shafarevich for useful discussions.

A

  + (φ, t) = abE(φ), where E(φ) = exp -ρ 2 (a 2 cos 2 φ + b 2 sin 2 φ 2 ,(a)A + (φ, t) = iabρ(a 2 a 1 cos φ + b 2 b 1 sin φ)E(φ) (b)

In case of Green's function for the Helmholtz equation, the manifold Λ + was introduced in [4].

The choice of arguments of Jacobians will be described in a detailed version of the paper.