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Receptor activator of nuclear factor-kappa B (RANK) ligand (RANKL) binds RANK on the surface of osteoclast precursors to trigger osteoclastogenesis. Recent studies have indicated that osteocytic RANKL has an important role in osteoclastogenesis during bone remodelling; however, the role of osteoblastic RANKL remains unclear. Here we show that vesicular RANK, which is secreted from the maturing osteoclasts, binds osteoblastic RANKL and promotes bone formation by triggering RANKL reverse signalling, which activates Runt-related transcription factor 2 (Runx2). The proline-rich motif in the RANKL cytoplasmic tail is required for reverse signalling, and a RANKL(Pro29Ala) point mutation reduces activation of the reverse signalling pathway. The coupling of bone resorption and formation is disrupted in RANKL(Pro29Ala) mutant mice, indicating that osteoblastic RANKL functions as a coupling signal acceptor that recognizes vesicular RANK. RANKL reverse signalling is therefore a potential pharmacological target for avoiding the reduced bone formation associated with inhibition of osteoclastogenesis. Osteoclasts secrete small extracellular vesicles that stimulate osteoblasts, promoting bone formation via receptor activator of nuclear factor-kappa B ligand (RANKL), thereby linking bone formation and resorption.

A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-$k$ tensor is the outer product of $k$ nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of which is symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz [J. Algebraic Geom., 4 (1995), pp. 201-222], is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most $r$ is not closed unless $r=1$.

The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consists of fixed-rank optimization, rank increase step and rank reduction step. We explore its performance applied to the low-rank matrix completion problem. Numerical experiments on synthetic and real-world datasets illustrate that the proposed rank-adaptive method compares favorably with state-of-the-art algorithms. In addition, it shows that one can incorporate each aspect of this rank-adaptive framework separately into existing algorithms for the purpose of improving performance.

The estimation of large covariance and precision matrices is fundamental in modern multivariate analysis. However, problems arise from the statistical analysis of large panel economic and financial data. The covariance matrix reveals marginal correlations between variables, while the precision matrix encodes conditional correlations between pairs of variables given the remaining variables. In this paper, we provide a selective review of several recent developments on the estimation of large covariance and precision matrices. We focus on two general approaches: a rank-based method and a factor-model-based method. Theories and applications of both approaches are presented. These methods are expected to be widely applicable to the analysis of economic and financial data.

Receptor activator of Nfkb ligand (RANKL) activates, while osteoprotegerin (OPG) inhibits, osteoclastogenesis. In turn a neutralizing Ab against RANKL, denosumab improves bone strength in osteoporosis. OPG also improves muscle strength in mouse models of Duchenne's muscular dystrophy (mdx) and denervation-induce atrophy, but its role and mechanisms of action on muscle weakness in other conditions remains to be investigated. We investigated the effects of RANKL inhibitors on muscle in osteoporotic women and mice that either overexpress RANKL (HuRANKL-Tg+), or lack Pparb and concomitantly develop sarcopenia (Pparb-/-). In women, denosumab over 3 years improved appendicular lean mass and handgrip strength compared to no treatment, whereas bisphosphonate did not. HuRANKL-Tg+ mice displayed lower limb force and maximal speed, while their leg muscle mass was diminished, with a lower number of type I and II fibers. Both OPG and denosumab increased limb force proportionally to the increase in muscle mass. They markedly improved muscle insulin sensitivity and glucose uptake, and decrease anti-myogenic and inflammatory gene expression in muscle, such as myostatin and protein tyrosine phosphatase receptor-γ. Similarly, in Pparb-/-, OPG increased muscle volume and force, while also normalizing their insulin signaling and higher expression of inflammatory genes in skeletal muscle. In conclusions, RANKL deteriorates, while its inhibitor improves, muscle strength and insulin sensitivity in osteoporotic mice and humans. Hence denosumab could represent a novel therapeutic approach for sarcopenia.

RANKL, the essential cue for osteoclast differentiation, is the membrane-bound factor expressed by osteoclastogenesis-supporting cells such as osteoblasts and osteocytes. In vivo evidence indicates that RANKL functions as the indispensable and irreplaceable in the program of osteoclast differentiation. The reason why RANKL plays a critical role in osteoclastogenesis is discussed from the viewpoint of the distinct signaling pathways mediated by co-stimulatory receptors and the key transcription factor NFATc1.

Bone protein link to fever The protein RANK (receptor-activator of nuclear factor κB) and its ligand RANKL are essential bone marrow regulators, and antibodies against RANKL are being developed as therapeutics in osteoporosis. RANKL and RANK are also expressed in the central nervous system, though their function there has been unclear. Studies in rats and mice now show that RANKL/RANK are expressed in astrocytes in the brain and that surprisingly, animals injected with RANKL develop severe fever, whereas genetically engineered mice with astrocytes lacking RANK are fever-resistant. Other data are consistent with a role for RANKL/RANK in both the central fever response in inflammation and in the control of female body temperature. Interestingly, clinical observations of two children with osteoporosis associated with RANK mutations revealed an absence of fever during bouts of pneumonia. It is possible that RANKL/RANK are factors in the hot flashes or flushes sometimes experienced by women during the menopause. Receptor-activator of NF-κB ligand (RANKL) and its receptor RANK are known to be essential regulators of bone remodelling, lymph node organogenesis and formation of a lactating mammary gland, but the functional relevance of their expression in the brain has been unclear. RANKL and RANK are now reported to have an essential role in the brain, with the central injection of RANKL into mice and rats triggering severe fever, and a further potential role in the control of thermoregulation in females. Receptor-activator of NF- κ B ligand (TNFSF11, also known as RANKL, OPGL, TRANCE and ODF) and its tumour necrosis factor (TNF)-family receptor RANK are essential regulators of bone remodelling, lymph node organogenesis and formation of a lactating mammary gland 1 , 2 , 3 , 4 . RANKL and RANK are also expressed in the central nervous system 5 , 6 . However, the functional relevance of RANKL/RANK in the brain was entirely unknown. Here we report that RANKL and RANK have an essential role in the brain. In both mice and rats, central RANKL injections trigger severe fever. Using tissue-specific Nestin-Cre and GFAP-Cre rank floxed deleter mice, the function of RANK in the fever response was genetically mapped to astrocytes. Importantly, Nestin-Cre and GFAP-Cre rank floxed deleter mice are resistant to lipopolysaccharide-induced fever as well as fever in response to the key inflammatory cytokines IL-1β and TNFα. Mechanistically, RANKL activates brain regions involved in thermoregulation and induces fever via the COX2-PGE 2 /EP3R pathway. Moreover, female Nestin-Cre and GFAP-Cre rank floxed mice exhibit increased basal body temperatures, suggesting that RANKL and RANK control thermoregulation during normal female physiology. We also show that two children with RANK mutations exhibit impaired fever during pneumonia. These data identify an entirely novel and unexpected function for the key osteoclast differentiation factors RANKL/RANK in female thermoregulation and the central fever response in inflammation.

We analyse the identifiability of the number of components in k‐variate, M‐component finite mixture models in which each component distribution has independent marginals, including models in latent class analysis. Without making parametric assumptions on the component distributions, we investigate how one can identify the number of components from the distribution function of the observed data. When k≥2, a lower bound on the number of components (M) is non‐parametrically identifiable from the rank of a matrix constructed from the distribution function of the observed variables. Building on this identification condition, we develop a procedure to estimate a lower bound on the number of components consistently.

We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field; the value of the nuclear norm of a real 33-tensor depends on whether we regard it as a real 33-tensor or a complex 33-tensor with real entries. We show that every tensor has a nuclear norm attaining decomposition and every symmetric tensor has a symmetric nuclear norm attaining decomposition. There is a corresponding notion of nuclear rank that, unlike tensor rank, is lower semicontinuous. We establish an analogue of Banach’s theorem for tensor spectral norm and Comon’s conjecture for tensor rank; for a symmetric tensor, its symmetric nuclear norm always equals its nuclear norm. We show that computing tensor nuclear norm is NP-hard in several ways. Deciding weak membership in the nuclear norm unit ball of 33-tensors is NP-hard, as is finding an ε\\varepsilon-approximation of nuclear norm for 33-tensors. In addition, the problem of computing spectral or nuclear norm of a 44-tensor is NP-hard, even if we restrict the 44-tensor to be bi-Hermitian, bisymmetric, positive semidefinite, nonnegative valued, or all of the above. We discuss some simple polynomial-time approximation bounds. As an aside, we show that computing the nuclear (p,q)(p,q)-norm of a matrix is NP-hard in general but polynomial-time if p=1p=1, q=1q = 1, or p=q=2p=q=2, with closed-form expressions for the nuclear (1,q)(1,q)- and (p,1)(p,1)-norms.