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The phase 3 RATIONALE-303 trial (NCT03358875) investigated the efficacy and safety of tislelizumab versus docetaxel in pretreated patients with advanced NSCLC. Here, we report the efficacy and safety results and describe the exploratory biomarker analyses.A total of 805 patients aged more than or equal to 18 years with locally advanced or metastatic squamous or nonsquamous NSCLC were randomized 2:1 to intravenous tislelizumab 200 mg or docetaxel 75 mg/mAt the prespecified interim analysis (August 10, 2020), the co-primary end point of OS in the ITT population was met, with a statistically significant and clinically meaningful improvement in OS with tislelizumab versus docetaxel (median 17.2 versus 11.9 mo, respectively; hazard ratio [HR] = 0.64, p0.0001). At the final analysis (July 15, 2021), the other co-primary end point of OS in the PD-L1 tumor cell greater than or equal to 25% population was further met (median 19.3 versus 11.5 mo, respectively; HR = 0.53, p0.0001), and OS continued to improve in the ITT population (median 16.9 versus 11.9 mo, respectively, HR = 0.66). Exploratory biomarker analyses revealed the potential association of NOTCH1-4 mutations with improved tislelizumab efficacy for both OS and progression-free survival, whereas tissue tumor mutation burden correlated with progression-free survival benefit, but not OS benefit. No new safety signals were identified.Tislelizumab was found to have a significantly improved and long-term clinical benefit in OS versus docetaxel in pretreated patients with advanced NSCLC, regardless of PD-L1 expression.

495 Background: RATIONALE-301 (NCT03412773), a global Phase 3 study, comparing tislelizumab (TIS) to sorafenib (SOR) as 1L treatment in adult patients (pts) with unresectable HCC (uHCC), met its primary endpoint of OS non-inferiority. This analysis examined the HRQoL outcomes in pts in both arms. Methods: Systemic therapy-naïve adults with histologically confirmed uHCC were randomized 1:1 to receive TIS (200 mg IV Q3W, n=342) or SOR (400 mg PO BID, n=332). HRQoL was assessed using EORTC QLQ-C30, QLQ-HCC18 and EQ5D-5L. A mixed model for repeated measures using key pre-specified PRO endpoints of global health status/quality of life (GHS/QoL), physical functioning and fatigue scales of the QLQ-C30, and HCC18 index score, fatigue and pain scores at the key pre-specified clinical cycles 4 and 6 were performed. Time to deterioration was examined with the Kaplan-Meier method using the PRO endpoints. Results: At both cycles, TIS had better HRQoL outcomes than SOR, as indicated by LS mean differences in GHS/QoL, physical functioning, fatigue and HCC symptom index, but not for pain. TIS had a lower risk for deterioration of QLQ-C30 GHS/QoL (hazard ratio [HR] 0.68 [95% CI, 0.49-0.94]), physical functioning (0.53 [0.39-0.73]) and fatigue (0.48 [0.37, 0.63]) as well as for deterioration in the HCC18 index (0.53 [0.34-0.81]) and fatigue (0.60 [0.46-0.80]). Both arms had a similar risk for deterioration in pain (HR 0.78 [0.56-1.09]). TIS maintained while SOR declined EQ-5D-5L VAS (general health status) scores at cycle 4 (mean change from baseline =-0.4 [SD = 14.52] vs -4.3 [12.92]) and cycle 6 (-0.2 [17.03] vs -5.4 [13.09]). Conclusions: Pts with HCC treated with 1L TIS had better HRQoL outcomes compared with pts treated with SOR, particularly in terms of fatigue and physical functioning. These results, along with effects an overall survival, response rate, and a favorable safety profile, support the benefit of TIS as a potential 1L treatment option for uHCC. Clinical trial information: NCT03412773 .[Table: see text]

From a theoretical perspective, it is unclear whether hotels should offer a discount to customers who stay for longer periods, while it is rather clear that customers feel entitled to such length-of-stay (LOS)-based discounts. Accordingly, the major theoretical contribution of this study is to provide empirical evidence of hotels’ LOS-based pricing practices. The findings revealed that on average, hotels charge more per night when the guests stay longer. The finding that hotels increase their room rates as LOS increases, suggests that hotel revenue managers may want to address the customers’ misinformed expectations of getting lower prices for longer LOS and/or devise pricing or communication strategies to deal with it.

Les processus de Dirichlet matriciels, en reference a leur mesure reversible, apparaissent de maniere naturelle dans de nombreux modeles differents en probabilite. En utilisant la langage des operateurs de diffusion et la theorie des equations de bord, nous decrivont les processus de Dirichlet sur le simplexe matriciel et proposont deux modeles pour les processus de Dirichlet matriciels, qui peuvent etre realises par les projections diverses, par le movement brownien sur le groupe unitaire special et par les processus de Wishart.

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition, where $m\in [n, \infty)$ and $K\geq 0$ are two constants. Moreover, we introduce the $W$-entropy and prove the $W$-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition and on compact manifolds equipped with $(-K, m)$-super Ricci flows., To appear in Journal of Functional Analysis. This paper is an improved version of a part of our previous preprint [14] (arxiv:1412.7034, version1 (22 December 2014) and version 2 (7 February 2016))

In this paper, we prove the Li-Yau type Harnack inequality and Hamilton type dimension free Harnack inequality for the heat equation $\partial_t u=Lu$ associated with the time dependent Witten Laplacian on complete Riemannian manifolds equipped with a variant of the $(K, m)$-super Perelman Ricci flows and the $K$-super Perelman Ricci flows., To appear in the Special Issue of the Asian Journal of Mathematics in honor of Prof. N. Mok's 60th birthday

In this paper, we prove the characterization of the $(K, \infty)$-super Perelman Ricci flows by various functional inequalities and gradient estimate for the heat semigroup generated by the Witten Laplacian on manifolds equipped with time dependent metrics and potentials. As a byproduct, we derive the Hamilton type dimension free Harnack inequality on manifolds with $(K, \infty)$-super Perelman Ricci flows. Based on a new second order differential inequality on the Boltzmann-Shannon entropy for the heat equation of the Witten Laplacian, we introduce a new $W$-entropy quantity and prove its monotonicity for the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(K, \infty)$-condition and on compact manifolds with $(K, \infty)$-super Perelman Ricci flows. Our results characterize the $(K, \infty)$-Ricci solitons and the $(K, \infty)$-Perelman Ricci flows. We also prove a second order differential entropy inequality on $(K, m)$-super Ricci flows, which can be used to characterize the $(K, m)$-Ricci solitons and the $(K, m)$-Ricci flows. Finally, we give a probabilistic interpretation of the $W$-entropy for the heat equation of the Witten Laplacian on manifolds with the $CD(K, m)$-condition., Comment: We remove Section 5 from the previous version and add two new results in Section 5. arXiv admin note: text overlap with arXiv:1412.7034

In this survey paper, we give an overview of our recent works on the study of the $W$-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the $W$-entropy formula for the heat equation associated with the Witten Laplacian on $n$-dimensional complete Riemannian manifolds with the $CD(K, m)$-condition, and the $W$-entropy formula for the heat equation associated with the time dependent Witten Laplacian on $n$-dimensional compact manifolds equipped with a $(K, m)$-super Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$. Furthermore, we prove an analogue of the $W$-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result recaptures an important result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two $W$-entropy formulas, we introduce the Langevin deformation of geometric flows on the cotangent bundle over the Wasserstein space and prove an extension of the $W$-entropy formula for the Langevin deformation. Finally, we make a discussion on the $W$-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory., Comment: Survey paper. Submitted to Science China Mathematics. arXiv admin note: text overlap with arXiv:1604.02596, arXiv:1706.07040

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We consider Brownian motion on symmetric matrices of octonions, and study the law of the spectrum. Due to the fact that the octonion algebra is nonassociative, the dimension of the matrices plays a special role. We provide two specific models on octonions, which give some indication of the relation between the multiplicity of eigenvalues and the exponent in the law of the spectrum.

The onset of puberty is initiated by an increase in the release of the gonadotropin-releasing hormone (GnRH) from GnRH neurons in the hypothalamus. However, the precise mechanism that leads to the activation of GnRH neurons at puberty remains controversial. Spines are small protrusions on the surface of dendrites that normally receive excitatory inputs. In this study, we analyzed the number and morphology of spines on GnRH neurons to investigate changes in synaptic inputs across puberty in rats. For morphological estimation, we measured the diameter of the head (DH) of each spine and classified them into small-type (DH 0.65 μm), large-type (DH 0.65 μm) and giant-type (DH 0.9 μm). The greatest number of spines was observed at the proximal dendrite within 50 μm of the soma. At the soma and proximal dendrite, the number of spines was greater in adults than in juveniles in both male and female individuals. Classification of spines revealed that the increase in spine number was due to increases in large- and giant-type spines. To further explore the relationship between spines on GnRH neurons and pubertal development, we next analyzed adult rats neonatally exposed to estradiol benzoate, in which puberty onset and reproductive functions are disrupted. We found a decrease in the number of all types of spines. These results suggest that GnRH neurons become to receive more and greater excitatory inputs on the soma and proximal dendrites as a result of the changes that occur at puberty and that alteration to spines plays a pivotal role in normal pubertal development.

We study the generalized Dyson Brownian motion (GDBM) of an interacting $N$-particle system with logarithmic Coulomb interaction and general potential $V$. Under reasonable condition on $V$, we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures of GDBM is tight on $\mathcal {C}([0,T],\mathscr{P}(\mathbb{R}))$ and all the large $N$ limits satisfy a nonlinear McKean-Vlasov equation. Inspired by previous works due to Biane and Speicher, Carrillo, McCann and Villani, we prove that the McKean-Vlasov equation is indeed the gradient flow of the Voiculescu free entropy on the Wasserstein space of probability measures over $\mathbb{R}$. Using the optimal transportation theory, we prove that if $V"\geq K$ for some constant $K\in \mathbb{R}$, the McKean-Vlasov equation has a unique weak solution. This proves the Law of Large Numbers and the propagation of chaos for the empirical measures of GDBM. Finally, we prove the longtime convergence of the McKean-Vlasov equation for $C^2$-convex potentials $V$., Comment: Update revised version of the previous version. arXiv admin note: text overlap with arXiv:1303.1240

In this paper, we develop a new approach to prove the $W$-entropy formula for the Witten Laplacian via warped product on Riemannian manifolds and give a natural geometric interpretation of a quantity appeared in the $W$-entropy formula. Then we prove the $W$-entropy formula for the Witten Laplacian on compact Riemannian manifolds with time dependent metrics and potentials, and derive the $W$-entropy formula for the backward heat equation associated with the Witten Laplacian on compact Riemannian manifolds equipped with Lott's modified Ricci flow. We also extend our results to complete Riemannian manifolds with negative $m$-dimensional Bakry-Emery Ricci curvature, and to compact Riemannian manifolds with $K$-super $m$-dimensional Bakry-Emery Ricci flow. As application, we prove that the optimal logarithmic Sobolev constant on compact manifolds equipped with the $K$-super $m$-dimensional Bakry-Emery Ricci flow is decreasing in time., This is an extended version of the previous version. Section $6$ and Remark $2.2$ are added