* Correspondence: dominique.heymann@univ-nantes.fr (D.H.); frederic.lezot@univ-nantes.fr (F.L.); Tel.: +33-(0)2-4067-9841 (D.H.); +33-(0)2-4041-2846 (F.L.); Fax: +33-(0)2-4041-2860 (F.L.)
Received: 26 September 2018; Accepted: 19 October 2018; Published: 24 October 2018 Abstract: Background: Osteosarcoma is the most frequent form of malignant pediatric bone tumor. Despite the current therapeutic arsenal, patient life-expectancy remains low if metastases are detected at the time of diagnosis, justifying research into better knowledge at all stages of osteosarcoma ontogenesis and identification of new therapeutic targets. Receptor Activator of Nuclear factor κB (**RANK**)expression has been reported in osteosarcoma cells, raising the question of Receptor Activator of Nuclear factor κB Ligand (**RANKL**)/**RANK** signaling implications in these tumor cells (intrinsic), in addition to previously reported implications through osteoclast activation in the tumor microenvironment (extrinsic). Methods: Based on in vitro and in vivo experimentations using human and mouse osteosarcoma cell lines, the consequences on the main cellular processes of **RANK** expression in osteosarcoma cells were analyzed. Results: The results revealed that **RANK** expression had no impact on cell proliferation and tumor growth, but stimulated cellular differentiation and, in an immune-compromised environment, increased the number of lung metastases. The analysis of **RANKL**, **RANK** and osteoprotegerin (OPG) expressions in biopsies of a cohort of patients revealed that while **RANK** expression in osteosarcoma cells was not significantly different between patients with or without metastases at the time of diagnosis, the OPG/**RANK** ratio decreased significantly. Conclusion: Altogether, these results are in favor of **RANKL**-**RANK** signaling inhibition as an adjuvant for the treatment of osteosarcoma.

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bones as shown by radiological analyses (Figure 3c), than controls treated with empty adenovirus (Figure 3b; Figure 3a : tumour controls). In this case, **RANK**-Fc inhibits the tumour induced-osteolysis but is not sufficient to decrease the tumour burden. In a therapeutic approach, **RANK**-Fc may be associated with anti-tumour drugs to stop both the tumour proliferation and the tumour-associated osteolysis. Overall, these studies demonstrate the effectiveness of **RANK**-Fc in inhibiting bone resorption in different models of malignant osteolytic pathologies and the upside of using **RANK**-Fc, which cannot interfere with TRAIL- mediated cancer cell apoptosis. Another approach lies in using novel OPG-like peptidomimetics that restore bone loss in vivo by facilitating a defective **RANKL**-**RANK** receptor complex, thus modulating **RANK**-**RANKL** signalling pathways and altering the biological functions of **RANKL**-**RANK** receptor complex [67]. Therefore, these OPG derived small molecules can be used to develop more useful therapeutic agents in bone diseases.

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Initially considered to be a pro-metastatic factor, our vision of **RANKL** changed when the factor was linked to mammary gland development [ 119 ]. **RANKL** deficiency leads to a defect in the formation of the lobo-alveolar structures required for lactation [ 120 , 121 ]. In addition, **RANKL** is able to promote the survival and proliferation of epithelial cells simultaneously with the up- regulated expression of **RANK** during mammary gland develop- ment [ 119 – 121 ]. Disturbance in this coordinated mechanism can lead to the formation of pre-neoplasias and subsequently to that of tumour foci, as revealed by Gonzalez-Suarez et al. [ 122 ]. These authors established a mouse mammary tumour virus – **RANK** transgenic mice overexpressing the protein in mammary glands – and reported a high incidence of pre-neoplasia foci (multifocal ductal hyperplasias, multifocal and focally extensive mammary intraepithelial neoplasias), as well as the development of adeno- carcinoma lesions in these transgenic mice compared with the wild-type mice. Confirming the involvement of **RANKL** in the initial oncogenic process, administration of **RANK**-Fc decreased both mammary tumorigenesis and the development of lung meta- stases in MMTV-neu transgenic mice, a spontaneous mammary tumour model [ 122 ]. In a complementary work, this team demon- strated that the **RANKL**/**RANK** axis was pro-active in epithelial mesenchymal transition (EMT), promoted cell migration simul- taneously with neo-vascularization, and that their expression was significantly associated with metastatic tumours [ 123 ]. Overall, their data revealed that **RANK**/**RANKL** signalling promotes the initial stage in breast cancer development by inducing stemness and EMT in mammary epithelial cells. A similar process has been confirmed in head and neck squamous carcinoma [ 124 ], and in endometrial cancer [ 125 ], and **RANKL** expression has been as- sociated with the EMT and appears to be a new marker for EMT in prostate cancer cells [ 83 ].

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Summary. Primary and secondary bone tumors clearly deteriorate quality of life and the activity of daily living of patients. These undesirable diseases become a major social and economic burden. As both primary and secondary bone tumors develop in the unique bone tissue, it is therefore necessary to understand bone cell biology in tumor bone environment. Recent findings of the Receptor Activator of Nuclear Factor-κB ligand (**RANKL**)/**RANK**/osteoprotegerin (OPG) molecular triad, the key regulators of bone remodeling, opened new era of bone research. Although **RANK** is an essential receptor for osteoclast formation, activation and survival, functional **RANK** expression has been recently identified on several bone-associated tumor cells. When **RANK** is expressed on secondary bone tumor cells, it is implicated in tumor cell migration, whereas this is not the case for primary bone tumors. In any case, **RANK** is not involved in **RANK**-positive cell proliferation or death. In two models of bone metastases secondary to melanoma or prostate carcinoma, in vivo neutralization of **RANKL** by OPG resulted in complete protection from paralysis, due to metastases of vertebral body, and a marked reduction in tumor burden in bones, but not in other organs. OPG also decreased tumor formation and tumor burden in a mouse model of primary bone tumor, osteosarcoma. In all these models, tumor cells express **RANK**. These data revealed that local differentiation factors, such as **RANKL**, play an important role in cell migration in a metastatic tissue-specific manner. These findings substantiate the novel direct role of

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An increased **RANKL**/OPG ratio has been reported in periodontal tissues under pathological conditions, such as Pd ( Mogi et al., 2004 ). **RANKL** levels in periodontal fibroblasts are induced either by mechanical forces or bacterial challenge in periodontitis ( Bostanci et al., 2007 ), whereas OPG levels decrease under similar conditions ( César-Neto et al., 2007 ). The OPG null mutant mouse, which exhibits alveolar bone loss ( Koide et al., 2013 ) and early onset root resorption ( Liu et al., 2016 ), was recently reported as a model of Pd using the dental ligature procedure ( Mizuno et al., 2015 ). This study validated the importance of the **RANKL**/**RANK**/OPG signaling pathway in the physiopathology of Pd.

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Although there is a strong association between osteoporosis and skeletal muscle atrophy/dysfunction, the functional relevance of a particular biological pathway that regulates synchronously bone and skeletal muscle physiopathology is still elusive. Receptor-activator of nuclear factor κB (**RANK**), its ligand **RANKL** and the soluble decoy receptor osteoprotegerin (OPG) are the key regulators of osteoclast differentiation and bone remodelling. We thus hypothesized that **RANK**/**RANKL**/OPG, which is a key pathway for bone regulation, is involved in Duchenne muscular dystrophy (DMD) physiopathology. Our results show that muscle-specific **RANK** deletion (mdx-**RANK** mko ) in dystrophin deficient mdx mice improves significantly specific force [54% gain in force] of EDL muscles with no protective effect against eccentric contraction-induced muscle dysfunction. In contrast, full-length OPG-Fc injections restore the force of dystrophic EDL muscles [162% gain in force], protect against eccentric contraction-induced muscle dysfunction ex vivo and significantly improve functional performance on downhill treadmill and post-exercise physical activity. Since OPG serves a soluble receptor for **RANKL** and as a decoy receptor for TRAIL, mdx mice were injected with anti-**RANKL** and anti-TRAIL antibodies to decipher the dual function of OPG. Injections of anti-**RANKL** and/or anti-TRAIL increase significantly the force of dystrophic EDL muscle [45% and 17% gains in force, respectively]. In agreement, truncated OPG-Fc that contains only **RANKL** domains produces similar gains, in terms of force production, than anti-**RANKL** treatments. To corroborate that full-length OPG-Fc also acts independently of **RANK**/**RANKL** pathway, dystrophin/**RANK** double-deficient mice were treated with full-length OPG-Fc for 10 days. Dystrophic EDL muscles exhibited a significant gain in force relative to untreated dystrophin/**RANK** double-deficient mice, indicating that the effect of full-length OPG-Fc is in part independent of the **RANKL**/**RANK** interaction. The sarco/endoplasmic reticulum Ca 2+ ATPase (SERCA) activity is significantly depressed in dysfunctional and dystrophic muscles and full-length OPG-Fc treatment increased SERCA activity and SERCA-2a expression. These findings demonstrate the superiority of full-length OPG-Fc treatment relative to truncated OPG-Fc, anti-**RANKL**, anti-TRAIL or muscle **RANK** deletion in improving dystrophic muscle function, (Continued on next page)

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cells participates in the development of osteolytic lesions observed in our POS-1 model.
The OPG/**RANK**/**RANKL** triad is an important therapeutic axis in pathologies involving a dysregulation in bone re- modelling, including tumor-associated osteolysis (4). A number of studies provide evidence for the direct production of **RANKL** by tumor cells themselves, as reported in multiple myeloma (25), prostate cancer (26), carcinoma cell lines (6) or human neuroblastoma (27). **RANKL** can then bind to its cognate receptor, **RANK**, at the surface of osteoclast precursors acting directly on osteoclast differentiation and activation. In the present POS-1 osteosarcoma model, POS-1 cells express **RANK**, not **RANKL**. **RANK** is known to be predominantly present at the surface of osteoclasts and some immune cells (28), but its expression has also been revealed in marrow stromal cells and osteoblasts and was strongly up-regulated when activated by T-cell conditioned medium (21). Another publication from Miyamoto et al reported that human osteosarcoma-derived cell lines expressed both **RANK** and **RANKL** mRNAs but the functionality of the receptor, **RANK**, was not investigated (20). However, the presence of a functional receptor, **RANK**, at the surface of cancer cells is in agreement with the results of Tometsko et al who recently reported the direct effects of **RANKL** on **RANK**-expressing human breast cancer cells, MDA-MB-231, and prostate PC3 (29). They demonstrated that **RANKL** treatment of both MDA-231 and PC3 cells led to the activation of signal trans- duction pathways (p38 MAPK, p42/44 MAPK, NF- κ B) and upregulated the expression of 194 mRNA as assessed by micro-array. From these data, we can hypothesize that, while neither cell line (MDA-231, PC3 and POS-1) expresses **RANKL** in vitro, it is probable that the locally increased **RANKL** within the bone microenvironment could activate tumor cell-expressed **RANK** in a paracrine manner.

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It is known that the lower bound is often accurate but the upper bound is not tight [16]. Furthermore, exact results are known in the case of binary quantics (n = 2) and ternary cubics (k = 3) [22, 16, 47, 35].
7.1. Alexander-Hirschowitz Theorem. It was not until the work [1] of Alexan- der and Hirschowitz in 1995 that the generic symmetric **rank** problem was completely settled. Nevertheless, the relevance of their result has remained largely unknown in the applied and computational mathematics communities. One reason is that the connection between our problem and the interpolating polynomials discussed in [1] is not at all well-known in the aforementioned circles. So for the convenience of our readers, we will state the result of Alexander and Hirschowitz in the context of the symmetric outer product decomposition below.

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the subset of d ∈ F (T ) such that L 0 (E d , 1) = 0, which is really surprising
compared to the even-**rank** case. The numerical data seems to support this fact. On the other hand, extensive numerical computations by Watkins [Wat] seem to indicate otherwise. Indeed we want to emphasize that one has always to be careful with deducing too strong of statements from numerical investigations.

(Brachat et al., 2009), (Comas, Seiguer, 2001), (Landsberg, Teitler, 2009) and (Sylvester, 1886)); σ 2 (X n,d ), σ 3 (X n,d ) (any n,d, see Section 4); σ r (X 2,4 ), for r ≤ 5. In the first three
cases we also give an algorithm to compute the symmetric **rank**. Some of these results were known or partially known, with different approaches and different algorithms, e.g in (Landsberg, Teitler, 2009) bounds on the symmetric **rank** are given for tensors in σ 3 (X n,d ), while the possible values of the symmetric **rank** on σ 3 (X 2,3 ) can be found in

When solving large scale semidefinite programs that admit a low-**rank** solution, a very efficient heuristic is the Burer-Monteiro factorization: Instead of optimizing over the full matrix, one optimizes over its low-**rank** factors. This strongly reduces the number of variables to optimize, but destroys the convexity of the problem, thus possibly introducing spurious second-order critical points which can prevent local optimization algorithms from finding the solution. Boumal, Voroninski, and Bandeira [2018] have recently shown that, when the size of the factors is of the order of the square root of the number of linear constraints, this does not happen: For almost any cost matrix, second-order critical points are global solutions. In this article, we show that this result is essentially tight: For smaller values of the size, second-order critical points are not generically optimal, even when considering only semidefinite programs with a **rank** 1 solution.

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Most inequalities used in commercial softwares are split cuts
Question : what is the split **rank** of the 2 row-inequalities ?
In how many rounds of split cuts only can we generate the inequalities ?
The Cook-Kannan-Schrijver has infinite **rank** and we prove that the other triangles have finite **rank**.

goulart@gipsa-lab.fr , pierre.comon@gipsa-lab.fr ). This work is supported by the European Research Council under the European Programme FP7/2007-2013, Grant AdG-2013-320594 “DECODA.”
1 Though (R
1 , . . . , R N ) is the mrank of ˆ X rather than its **rank**, we employ the usual terminology

When L(t, x) = f 0 (1 − t)U (x), with f a convex distortion satisfying f (0) = 0, f (1) = 1 and U a concave utility index then V is a **rank**-dependent utility (RDU), in the linear case U (x) = x then V is a Yaari utility. In the case of a Yaari utility, V is the support function of the core of the distortion of the underlying probability by f , hence differentiablity properties of V are tightly linked to the geometry of the core. The differentiability properties of RDU functionals have been studied in [1] using a characterization of the core of convex distortions of a probability. For a more general L, the previous approach is not adapted and different arguments have to be developed to compute the superdifferential of V and the set of random variables where V is Gˆateaux-differentiable.

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sr(F ) = min{lengthΓ | Γ ⊂ P(T 1 ) smoothable, dimΓ = 0, I Γ ⊂ F ⊥ }
and the **rank** r(F ) is defined as
r(F ) = min{lengthΓ | Γ ⊂ P(T 1 ) smooth, dimΓ = 0, I Γ ⊂ F ⊥ }.
Clearly cr(F ) ≤ sr(F ) ≤ r(F ). A separate notion of border **rank**, br(F ), often considered, is not defined by apolarity. The border **rank** is rather the minimal r, such that F is the limit of polynomials of **rank** r. Thus br(F ) ≤ sr(F ). These notions of **rank** coincide with the no- tions of length of annihilating schemes in Iarrobino and Kanev book [Iarrobino, Kanev 1999, Definition 5.66]: Thus cactus **rank** coincides with the scheme length, cr(F ) = lsch(F ), and smoothable **rank** coincides with the smoothable scheme length, sr(F ) = lschsm(F ), while border **rank** coincides with length br(F ) = l(F ). In addition they consider the differential length ldiff(F ), the maximum of the dimensions of the space of k-th order partials of F as k varies between 0 and degF . This length is the maximal **rank** of a catalecticant or Hankel matrix at F , and is always a lower bound for the cactus **rank**: ldiff(F ) ≤ cr(F ).

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In the first section, we sketch the background on Drinfeld modules/mod- ular forms and introduce notation.. It doesn’t contain any new material.[r]

As before, let α be the goal inequality such that L α is triangle of type T 2B .
Disjunction Sequence 7.2.
1. Initialization step: First consider the two-variable problem P ((r 1 , r 3 ), f ). By definition the triangle C := f w 1 w 3 does not contain any integer point in its interior. Therefore φ(C) is a valid inequality for conv(P ((r 1 , r 3 ), f )). By Proposition 5.1, there also exists > 0 such that, denoting u [2] := f +r 2 , we obtain that β [2] := φ(4(w 1 w 3 u [2] )) is a valid inequality for P (R, f ). By Proposition 5.1 and Proposition 6.1, we also know that this inequality has a split **rank** at most two. Let q [2] be the intersection point of (4(w 1 w 3 u [2] )) with the line {x ∈ R 2 | x 2 = 1}. We then directly proceed to step 2 in the inductive process.

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The first two parts of the thesis are dedicated to two practical problems of estimation of low-**rank** matrices: the matrix completion problem and quan- tum state tomography, where the objective is to estimate the so-called density matrix, that is often assumed to be low-**rank** by physicists. For matrix com- pletion, we show that a quasi-Bayesian estimator satisfies an optimal oracle inequality, and thus reaches the minimax-optimal rates (up to log terms). The strong point of our results is that it holds without any assumption on the sampling distribution - this is the first result without such an assumption up to our knowledge. For the quantum state tomography problem, we build a pseudo-Bayesian estimator. Note that in most previous works, the definition of a prior probability distribution was only tackled in the case of the 1 qubit problem (the smallest possible instance of the problem, where the matrix to be estimated is 2 × 2). Inspired by the prior used for matrix completion, we propose a prior distribution that can be used to estimate density matrices of any dimension. We show that our pseudo-Bayesian estimator reaches the best up-to-date known rate of convergence while its numerical performance was tested on simulated and real data sets.

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n(n 2 −1) .
4.2 RSim Quality Measure
The observed similarity measures do not cover sufficiently the fine grained re- quirements arising, comparing top-k rankings in the Web search context. So we need a new similarity metric taking into consideration: a)The absolute difference between the predicted and actual position for each Webpage as large difference indicates a less accurate prediction and b)The actual ranking position of a Web page, because failing to predict a highly ranked Webpage is more important than a low-ranked. Based on these observations, we introduce a new measure, named RSim. Every inaccurate prediction made incurs a certain penalty depending on the two noted factors. If prediction is 100% accurate (same predicted and actual **rank**), the penalty is equal to zero. Let B i be the predicted **rank** position for

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