The factorization method and Capon’s method for random source identification in experimental aeroacoustics

Experimental aeroacoustics is concerned with the estimation of acoustic source power distributions, which are for instance caused by fluid structure interactions on scaled aircraft models inside a wind tunnel, from microphone array measurements of associated sound pressure fluctuations. In the frequency domain aeroacoustic sound propagation can be modelled as a random source problem for a convected Helmholtz equation. This article is concerned with the inverse random source problem to recover the support of an uncorrelated aeroacoustic source from correlations of observed pressure signals. We show a variant of the factorization method from inverse scattering theory can be used for this purpose. We also discuss a surprising relation between the factorization method and a commonly used beam-forming algorithm from experimental aeroacoustics, which is known as Capon’s method or as the minimum variance method. Numerical examples illustrate our theoretical findings

Path spaces of higher inductive types in homotopy type theory

The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman '13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16). We provide a formalization of the main technical results in the proof assistant Lean.Comment: v1: 23 pages; v2: 24 pages, small reformulations and reorganization

The factorization method and Capon’s method for random source identification in experimental aeroacoustics

Experimental aeroacoustics is concerned with the estimation of acoustic source power distributions, which are for instance caused by fluid structure interactions on scaled aircraft models inside a wind tunnel, from microphone array measurements of associated sound pressure fluctuations. In the frequency domain aeroacoustic sound propagation can be modeled as a random source problem for a convected Helmholtz equation. This article is concerned with the inverse random source problem to recover the support of an uncorrelated aeroacoustic source from correlations of observed pressure signals. We show that a variant of the factorization method from inverse scattering theory can be used for this purpose. We also discuss a surprising relation between the factorization method and a commonly used beamforming algorithm from experimental aeroacoustics, which is known as Capon\u27s method or as the minimum variance method. Numerical examples illustrate our theoretical findings

A Rewriting Coherence Theorem with Applications in Homotopy Type Theory

Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.Comment: 30 pages. arXiv admin note: text overlap with arXiv:2001.0765

A rewriting coherence theorem with applications in homotopy type theory

Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution Coherence via Wellfoundedness by laying out the construction in the language of higher-dimensional rewriting

Mont Blanc and Aiguilles Rouges geology of their polymetamorphic basement (external massifs, Westerns Alps, France-Switzerland)

Les massifs du Mont Blanc et des Aiguilles Rouges appartiennent aux massifs dits cristallins externes de la chaîne alpine occidentale. Ils sont constitués de roches pré-mésozoïques et dessinent des nappes de socle dans le bâti alpin. Ces massifs ont enregistré une longue histoire géologique comprenant le dépôt de sédiments néoprotérozoïques à cambriens, la mise en place de roches magmatiques basiques et ultrabasiques au paléozoïque inférieur, ainsi que l’intrusion de granitoïdes ordoviciens en contexte de marge active. Ces roches sont considérées appartenir à un ensemble de blocs continentaux originaires de la marge septentrionale du Gondwana et accrétés à la marge sud-européenne après leur détachement du Gondwana et leur dérive vers le nord consécutif à l’ouverture de la Paléotéthys. Cet épisode d’accrétion correspond à l’orogenèse varisque (hercynienne), bien documentée dans les massifs du Mont Blanc et des Aiguilles Rouges, par une évolution tectono-métamorphique polyphasée essentiellement carbonifère avec formation de migmatites et intrusion de granitoïdes de types variés. Une érosion active, liée à une forte exhumation, est enregistrée au carbonifère supérieur dans les dépôts détritiques continentaux de bassins d’effondrement de type graben. Ce mémoire présente des cartes géologiques inédites et des suggestions d’excursions dans ces secteurs nouvellement cartographiés. Les lithologies sont abondamment illustrées et décrites en détail du point de vue structural, pétrologique et géochimique. Les analyses chimiques sont fournies en annexe.The Aiguilles Rouges and Mont Blanc external massifs belong to the pre-Mesozoic basement areas of the external domain of the Alps. Before their involvement into the Alpine building (basement nappes) they registered a multiple geological evolution comprising the deposition of Neoproterozoic to Cambrian sediments and emplacement of granitoid and metabasic to ultramafic magmatic rocks of Early Palaeozoic age at the Gondwanan border. After rifting and drifting (formation of Palaeotethys) all rocks underwent polyphase metamorphic and structural transformations during the Variscan orogeny, and were intruded by late Variscan granitoids. The resulting polymetamorphic basement was eroded during formation of Upper Carboniferous sedimentary troughs. New geological maps are presented in this volume, together with structural, petrological and geochemical characteristics of all lithologies. The geochemical data are presented in annexes

Bimodal magmatism as a consequence of the post-collisional readjustment of the thickened Variscan continental lithosphere (Aiguilles Rouges-Mont Blanc Massifs, Western Alps)

ABSTRACT: High Precision U-Pb zircon and monazite dating in the Aiguilles Rouges-Mont Blanc area allowed discrimination of three short-lived bimodal magmatic pulses: the early 332 Ma Mg-K Pormenaz monzonite and associated 331 Ma peraluminous Montées Pélissier monzogranite; the 307 Ma cordierite-bearing peraluminous Vallorcine and Fully intrusions; and the 303 Fe-K Mont Blanc syenogranite. All intruded syntectonically along major-scale transcurrent faults at a time when the substratum was experiencing tectonic exhumation, active erosion recorded in detrital basins and isothermal decompression melting dated at 327-320 Ma. Mantle activity and magma mixing are evidenced in all plutons by coeval mafic enclaves, stocks and synplutonic dykes. Both crustal and mantle sources evolve through time, pointing to an increasingly warm continental crust and juvenile asthenospheric mantle sources. This overall tectono-magmatic evolution is interpreted in a scenario of post-collisional restoration to normal size of a thickened continental lithosphere. The latter re-equilibrates through delamination and/or erosion of its mantle root and tectonic exhumation/erosion in an overall extensional regime. Extension is related to either gravitational collapse or back-arc extension of a distant subduction zon