# Haut PDF Distributed primality proving and the primality of (2 +1)/3

### Distributed primality proving and the primality of (2 +1)/3

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### Proving the safety of highly-available distributed objects

1. for all states σ, σ 0 ∈ S, (σ, σ 0 )  Inv conc , and 2. for any state σ ∈ S, σ  Inv. Corollary 1. The soundness proposition ( 1 ) is a direct consequence of Lemma 5 . We remark at this point that there are numerous program logic approaches to proving invariants of shared-memory concurrent programs, with Rely/Guar- antee [ 15 ] and concurrent separation logic [ 6 ] underlying many of them. While these approaches could be adapted to our use case (propagating-state distributed systems), this adaptation is not evident. As an indication of this complexity: one would have to predicate about the different states of the different replicas, re- state the invariant to talk about these different versions of the state, encode the non-deterministic behaviour of merge, etc. Instead, we argue that our specialised rules are much simpler, allowing for a purely sequential and modular verification that we can mechanise and automate. This reduction in complexity is the main theoretical contribution of this paper.
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### Decomposability of graphs into subgraphs fulfilling the 1-2-3 Conjecture

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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### Neighbour-Sum-2-Distinguishing Edge-Weightings: Doubling the 1-2-3 Conjecture

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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### Sulfone Based-Electrolytes for Lithium-Ion Batteries: Cycling Performances and Passivation Layer Quality of Graphite and LiNi 1/3 Mn 1/3 Co 1/3 O 2 Electrodes

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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### Prediction of a New Layered Polymorph of FeS 2 with Fe 3+ S 2– (S 2 ) 1/2 2- Structure

and mainly differ in how the S 2 2- dimers are arranged around the Fe 2+ c T c c - b 1 f 1 ) 26 1 1 ) 26 for marcasite structures. The relative errors of predicted equilibrium volume of the unit cell with respect to experimental results are as low as 0.5% (159.6 3 1 3 1 3 1 3 for pyrite and marcasite, respectively). Both the structural and energetic properties are in very good agreement with the experimental and theoretical data, which validates our methodological approach. Finally, note that the third structure in energy ranking (therefore before the C2/m-3D that is ranked at the fourth position), to be denoted P-1-3D, displays 5-coordinated iron atoms and contains both sulfides and persulfides moieties (see Figure 3 c). In this layered material, each S 2 dimer links two
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### On a List Variant of the Multiplicative 1-2-3 Conjecture

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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### 1-2-3 Conjecture in Digraphs: More Results and Directions

The proof is by induction on the number of vertices of D. We focus on proving the general case. Let v be any vertex of D. According to the induction hypothesis, the digraph D 0 obtained by removing v from D has an L-arc-weighting which is as desired. We wish to extend it to all arcs (outgoing and incoming) incident to v. For every neighbour u of v in D, it is possible that, when assigning a weight w from L(− vu) (resp. L(− → uv)) to − → vu (resp. − → uv), → u is now involved in a conflict (regarding Γ) with one of its neighbours different from v. As explained earlier, this only occurs when σ − (u) + w = σ + (u) (resp. σ + (u) + w = σ − (u)). From that perspective, we say that w is unsafe in L(− vu) (resp. L(− → uv)). A consequence → is that, when assigning the second weight from the list to − vu (resp. − → uv), such a conflict → involving u cannot occur. That second weight of the list is thus called safe (with respect to u). Thus, some of the neighbours of v in D are fragile, in the sense that the arc joining them to u has its list including an unsafe weight. This is because the weight to be assigned to the arc between v and a fragile vertex is somehow forced (the safe one must be assigned). To every arc joining v and a fragile neighbour, let us thus assign the safe weight. Then:
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### Further Results on an Equitable 1-2-3 Conjecture

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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### Proving the safety of highly-available distributed objects (Extended version)

1. for all states σ, σ 0 ∈ S, (σ, σ 0 )  Inv conc , and 2. for any state σ ∈ S, σ  Inv. Corollary 1. The soundness proposition ( 1 ) is a direct consequence of Lemma 5 . We remark at this point that there are numerous program logic approaches to proving invariants of shared-memory concurrent programs, with Rely/Guar- antee [ 15 ] and concurrent separation logic [ 6 ] underlying many of them. While these approaches could be adapted to our use case (propagating-state distributed systems), this adaptation is not evident. As an indication of this complexity: one would have to predicate about the different states of the different replicas, re- state the invariant to talk about these different versions of the state, encode the non-deterministic behaviour of merge, etc. Instead, we argue that our specialised rules are much simpler, allowing for a purely sequential and modular verification that we can mechanise and automate. This reduction in complexity is the main theoretical contribution of this paper.
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### Further Evidence Towards the Multiplicative 1-2-3 Conjecture

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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### Explicit primality criteria for $h\cdot 2^n\pm 1$

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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### Proving Computational Geometry Algorithms in TLA+2

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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### The L^2 -Alexander torsion of 3-manifolds

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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### Synthesis of 1,2-methano-tetrahydrofuran derivatives and 1´,2´-methano-2´,3´-dideoxynucleosides as potential antivirals

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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### Closed-form expressions of the eigen decomposition of 2 x 2 and 3 x 3 Hermitian matrices

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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### The preparation of certified calibration solutions for azaspiracid-1, -2, and -3, potent marine biotoxins found in shellfish

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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### A comparative study of the Ruddlesden-Popper series, Lan+1NinO3n+1 (n=1, 2 and 3), for solid-oxide fuel-cell cathode applications

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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### New Type of 2D Perovskites with Alternating Cations in the Interlayer Space, (C(NH 2 ) 3 )(CH 3 NH 3 ) n Pb n I 3 n +1 : Structure, Properties, and Photovoltaic Performance

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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### Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

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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