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H 0 is empty. If H 0 is empty, **the** thesis holds. Otherwise, by Lemma 16, there exists S ⊆ E(H 0 ) such that for every vertex v ∈ V (H 0 ),
**1** ≤ dS(v) ≤ dH 0 (v) − 10 10 . (14)
Note that for every isolated edge uv **of** F 0 , one **of** its ends must belong to V (H 0 ) – we then arbitrarily choose one edge from S incident with this end **and** add it to F 0 provided that no other edge adjacent to uv was earlier added to F 0 . After repeating this procedure for every such isolated edge we obtain a graph F **of** F 0 ; note that **the** degeneracy **of** F is still less than 10 10 + 10 8 (as we may place **the** ends **of** **the** isolated edges **of** F 0 together with **the** vertices in V (F ) r V (F 0 ) at **the** end **of** **the** ordering witnessing **the** degeneracy **of** F , since these vertices induce a forest in F ). At **the** same time, by (14), **the** remaining subgraph **of** G, denoted by H (formed from H 0 by removing edges from S transferred to F 0 ), fulfills: δ(H) ≥ 10 10 .

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More directions for future works on neighbour-sum-**2**-distinguishing edge-weightings are also worth mentioning. Notably, we did not manage to improve **the** bounds given in Section **2** for many classes **of** graphs. Generally speaking, it does not seem obvious to us how to improve **the** bound in Corollary 2.2, **and** this would surely require new dedicated tools. Concerning particular classes **of** graphs, let us mention **the** case **of** subcubic graphs. Although we know that cubic graphs comply with Conjecture 1.1, **and** even Conjecture 5.1 (recall Corollary 2.4), we did not manage to prove that nice subcubic graphs, in general, also do. We believe this would be an appealing first case to consider towards **proving** Conjecture 1.1 for **3**-chromatic graphs, for which **the** **1**-**2**-**3** Conjecture holds.

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NMC electrode **and** after 5 cycles **of** charge-discharge performed in MESL + LiTFSI with or without FEC (5%) at room temperature **and** 40 °C (Fig. 7 ). Only **the** principal peaks (C 1s, F 1s, S 2p) corresponding to **the** electrolyte decomposition **and** formation **of** **the** passive layer are presented here. **The** Ni 2p, Mn 2p **and** Co 2p core level signals (not presented here) corresponding to **the** NMC electrode material show a small intensity decrease after cycling in MESL + LiTFSI without FEC **and** a signiﬁcant intensity decrease after cycling in MESL + LiTFSI + FEC. This signal attenuation **of** **the** Ni 2p, Mn 2p **and** Co 2p peaks can be attributed to SEI formation on **the** NMC electrode. XPS analyses show that there was no change in **the** oxidation state **of** nickel, manganese or cobalt. **The** small signal attenuation **of** Ni 2p, Mn 2p **and** Co 2p peaks observed after cycling NMC in MESL + LiTFSI without FEC is attributed to non- signiﬁcant surface modiﬁcations, which can be conﬁrmed by some negligible changes in **the** C 1s, F 1s **and** S 2p signals towards **the** non-cycled pristine NMC electrode (Fig. 7 ). **The** C 1s core level signal for **the** pristine NMC electrode (Fig. 7 a) presents six peaks, which can be attributed to: carbon black at 284.4 eV, –CH–CH– at 285.0 eV (PVDF), –C–O at 286.3 eV, –C=O at 288.5 eV, –CO **3** , at

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Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400, Talence, France
January 25, 2021
Abstract
**The** **1**-**2**-**3** Conjecture asks whether almost all graphs can be (edge-)labelled with **1**, **2**, **3** so that no two adjacent vertices are incident to **the** same sum **of** labels. In **the** last decades, several aspects **of** this problem have been studied in literature, including more general versions **and** slight variations. Notable such variations include **the** List **1**-**2**-**3** Conjecture variant, in which edges must be assigned labels from dedicated lists **of** three labels, **and** **the** Multiplicative **1**- **2**-**3** Conjecture variant, in which labels **1**, **2**, **3** must be assigned to **the** edges so that adjacent vertices are incident to different products **of** labels. Several results obtained towards these two variants led to observe some behaviours that are distant from those **of** **the** original conjecture. In this work, we consider **the** list version **of** **the** Multiplicative **1**-**2**-**3** Conjecture, proposing **the** first study dedicated to this very problem. In particular, given any graph G, we wonder about **the** minimum k such that G can be labelled as desired when its edges must be assigned labels from dedicated lists **of** size k. Exploiting a relationship between our problem **and** **the** List **1**-**2**-**3** Conjecture, we provide upper bounds on k when G belongs to particular classes **of** graphs. We further improve some **of** these bounds through dedicated arguments.

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in favour **of** **the** number **of** assigned **2**’s, that is so huge that it cannot be caught up by **the** labelling freedom **of** G m **and** **the** copies **of** **the** corrector gadget C.
Once we know that **the** input **and** all outputs **of** G m must be assigned **1** by an equitable proper
**2**-labelling, **the** forcing mechanisms in **the** whole graph then become much easier to track, **and** it then becomes easier to design an equivalence with a **1**-in-**3** truth assignment φ satisfying F . Precise details. **The** construction **of** G is as follows. Let us start from **the** cubic bipartite graph G F modelling **the** structure **of** **the** 3CNF formula F . That is, for every variable x i **of** F we add a

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Keywords: **1**-**2**-**3** Conjecture; multiset version; product version; 4-chromatic graphs.
**1**. Introduction
This work takes place in **the** general context **of** distinguishing labellings, where **the** aim, given an undirected graph, is to label its edges so that its adjacent vertices get distinguished by some function computed from **the** labelling. Formally, a k-labelling ℓ : E(G) → {**1**, . . . , k} **of** a graph G assigns a label from {**1**, . . . , k} to each edge, **and**, for every vertex v, we can compute some function f (v) **of** **the** labels assigned to **the** edges incident to v. **The** goal is then to design ℓ so that f (u) 6= f (v) for every edge uv **of** G. As reported in a survey [**3**] by Gallian on **the** topic, there actually exist dozens **and** dozens types **of** distinguishing labelling notions, which all have their own particular behaviours **and** subtleties.

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16 6= **1**, where µ is a 16-th root **of**
unity such that µπ **2** is a primary element in Z[ζ 16 ].
**The** paper is organized as follows. In Section **2**, we introduce **the** high order power residue symbol **and** recall Eisenstein’s Reciprocity Laws, es- pecially for **the** Octic **and** Bioctic Reciprocity Laws. In Section **3**, we first state **the** facts we need from **the** arithmetic **of** **the** eighth **and** sixteenth cyclotomic fields, then we describe **the** main result **of** this paper. We prove this result in Section 4. Computational complexity **of** **the** generalized Lu- casian **primality** test related to our main result is analyzed in Section 5. We end this paper with an opened problem.

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Init ⇒ I (IV.**2**) [N ext] vars ∧ I ⇒ I 0 (IV.**3**)
I ⇒ P rop (IV.4)
where Init **and** N ext are defined in **the** Graham’s algorithm specification. **The** proofs **of** formula (IV.**2**) **and** (IV.4) are relatively intuitive, we don’t bother to detail on them. We lay **the** emphasis on **the** proof **of** **the** formula (IV.**3**) which is much more complicated due to **the** existence **of** numbers **of** predicates with quantifiers. Regarding this kind **of** proof obligation, we propose a structure decomposition rule **and** a quantifier decomposition rule correspondingly for **the** overall proof structure **and** **the** predicates, both **of** which can be implemented on computer to break a monolithic proof obligation into many simpler cases.

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By **the** above both det N (G) (A) **and** det N (G) (B) are at least one, it follows that
det N (G) (A) = **1**.
Finally, if A is a square matrix over Q[G], then we can write A = r · B with r ∈ Q **and** B a matrix over Z[G]. It follows immediately from **the** aforementioned result **of** [ES05] **and** from **the** definitions that A is also **of** determinant class. Now we denote by G **the** class **of** all sofic groups G. To **the** best **of** our knowledge it is not known whether there exist finitely presented groups that are not sofic. Moreover, we do not know whether any matrix A over any real group ring is **of** determinant class.

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2.10 2.12
**1**-((1R,3S,5S)-**3**-(((tert-butyldiphenylsilyl)oxy)methyl)-**2**-oxabicyclo[3.1.0]hexan-**1**-yl)-4- (1H-1,2,4-triazol-**1**-yl)pyrimidin-**2**(1H)-one (2.12). To a flask containing 1,2,4-triazole (875.2
mg, 12.67mmol) **and** acetonitrile (24 mL) at 0 °C, phosphoryl chloride (332.16 mg, 0.20 mL, 2.12 mmol) was added over a period **of** **3** minutes using a syringe pump, followed by triethylamine (1.90 mL, 13.69 mmol). **The** heterogeneous mixture was stirred for **1** h at 0 °C. A solution **of** pyrimidinone 2.10 (189.10 mg, 0.41 mmol) in acetonitrile ( 5.10 mL) was transferred via cannula into **the** mixture, which was allowed to warm to room temperature **and** stirred for 2.5 h. **The** reaction was quenched by **the** addition **of** saturated aqueous NaHCO **3** . **The** aqueous phase was extracted with CH **2** Cl **2** (**3** × 50 mL). **The** combined organic extracts were dried over (Na **2** SO 4 ), filtered **and** concentrated under reduced pressure. **The** residue was purified by silica gel flash chromatography eluting with CH **2** Cl **2** containing methanol (0-**2**%) to give triazole 2.12 (174.00 mg, 83%) as white foam. **1** H NMR (400 MHz, CDCl

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Abstract
**The** eigen decomposition **of** covariance matrices is at **the** core **of** many data analysis tech- niques. **The** study **of** **2**-components or **3**-components vector fields typically requires comput- ing numerous eigen decompositions **of** **2** × **2** or **3** × **3** matrices. This is, for example, **the** case in **the** analysis **of** interferometric or polarimetric SAR images, see MuLoG algorithm (https://hal.archives-ouvertes.fr/hal-01388858). **The** closed-form expression **of** eigen- values **and** eigenvectors then provides a way to derive faster data processing algorithms. This note gives these expressions in **the** general case (special cases where some coefficients are zero, or **the** eigenvalues are not separated may not be covered **and** then require either to introduce a small perturbation **of** **the** initial matrix or to derive other expressions).

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Keywords Reference materials . Mass spectrometry/ LC-MS . Marine toxins
Introduction
Azaspiracids (AZAs) are a class **of** lipophilic polyether marine biotoxins that were first detected in harvested mussels (Mytilus edulis) from Killary Harbour on **the** west coast **of** Ireland in 1995. Symptoms resembling those **of** diarrhetic shellfish poisoning (DSP) were reported by those affected, including nausea, vomiting, stomach cramps, **and** severe diarrhea. A relationship between these incidents **and** a specific toxin could not be immediately determined because DSP **and** PSP toxins were only present in low levels **and** known toxin producing phytoplankton species were absent in **the** associated water samples [ **1** ]. A new toxic compound was soon identified as **the** causative agent **and** provisionally named Killary toxin-**3** (KT3) in recognition **of** **the** location where **the** mussels originated [ **2** ]. Following elucidation **of** **the** structure, it was renamed azaspiracid-**1** (AZA1) [ **3** ]. AZAs possess a unique spiral ring assembly, a cyclic amine **and** a carboxylic acid group (Fig. **1** ). Shortly after **the** initial discovery **of** AZA1, two further analogues, 22-desmethylazaspiracid (AZA3) **and** 8-methylazaspiracid (AZA2) were discovered [ 4 ]. Subsequently, further hydrox- ylated analogues were discovered by **the** use **of** mass

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flected by **the** different values obtained for α when **the** data is fitted over **the** temperature ranges 348 K–548 K **and** 548– 1173 K, which are summarized in Table **3** . **The** change in α over these two temperature regions is likely to be structural in origin **and** may have implications for cathode use. Furthermore, **the** anomaly at 548 K was found to manifest itself in other mea- surements such as thermogravimetric analysis, see Fig. **3** , as well as **the** electrical conductivity behavior, which is discussed in **the** following section.

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User Facility operated for **the** US DOE Office **of** Science. LANL, an affirmative-action equal opportunity employer, is operated by Los Alamos National Security for **the** National Nuclear Security Administration **of** **the** US DOE under contract DE-AC52-06NA25396. C.K. **and** B.T. acknowledge high-performance computing resources from Grand Equipment National de Calcul Intensif (CINES/IDRIS, grant 2016-[x2016097682]). DFT calculations were performed at **the** Institut des Sciences Chimiques de Rennes, which received funding from **the** European Union’s Horizon 2020 Programme for Research **and** Innovation under grant 687008. This work made use **of** **the** SPID (confocal microscopy) **and** EPIC (scanning electron microscopy) facilities **of** Northwestern University’s NUANCE Center, which has received support from **the** Soft **and** Hybrid Nanotechnology Experimental Resource (NSF ECCS-1542205), **the** Materials Research Science **and** Engineering Centers (NSF DMR-1121262), **the** International Institute for Nanotechnology (IIN), **the** Keck Foundation, **and** **the** State **of** Illinois through **the** IIN. Use **of** **the** Advanced Photon Source at Argonne National Laboratory was supported by **the** Basic Energy Sciences program **of** **the** US DOE Office **of** Science under contract DE-AC02-06CH11357. **3**

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38 ]. A particular case is **the** now well-documented **1**:**2** resonance relationship, where two eigenfrequencies
(ω **1** , ω **2** ) are such that ω **2** 2ω **1** . This second-order
internal resonance involves quadratic nonlinearity, **and** is now classical, since **the** first report **of** its effect on **the** response **of** a ship system by Froude [ 10 , 22 ]. Refer- ences [ 11 , 21 , 23 , 24 , 35 ] provide a complete picture **of** analytical solutions **and** experimental observations. Note that we use here **the** terminology “**1**:**2**” resonance to name that case whereas it is often denoted **2**:**1** res- onance in other studies. Complications to **the** classical **1**:**2** case have already been considered as it appears in many physical systems such as strings, cables, plates, **and** shells. Lee **and** Perkins [ 16 ] reported a study on a **1**:**2**:**2** resonance occurring in suspended cables be- tween in-plane **and** out-**of**-plane modes, **and** denoted that resonance as a **2**:**1**:**1** case. In their study, only one **of** **the** two high-frequency modes was excited, **and** **the** coupling with **the** two other modes was studied. In **the** field **of** nonlinear vibrations **of** shells, multiple cases involving different combinations **of** **1**:**1** **and** **1**:**2** reso- nances have been found to occur frequently. Chin **and** Nayfeh studied **the** case **of** a **1**:**1**:**2** resonance in a cir- cular cylindrical shell, where only one **of** **the** two low- frequency modes were excited [ 8 ]. Thomas et al. stud- ied theoretically **and** experimentally **the** **1**:**1**:**2** reso- nance occurring in shallow spherical shells, where **the** driven mode is **the** high-frequency one [ 33 , 34 ]. **The** case **of** a **1**:**1**:**1**:**2** internal resonance occurring in closed circular cylindrical shells was also tackled by Amabili, Pellicano, **and** Vakakis [ 5 , 28 ]. In that case, only one **of** **the** low-frequency modes was excited, **and** solutions to a particular case for **the** parameter values was analyt- ically **and** numerically exhibited. Finally, a **1**:**2**:4 res- onance has been studied by Nayfeh et al. [ 25 ], where **the** excitation frequency was selected in **the** vicinity **of** **the** high-frequency mode.

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