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We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength\beta\in\mathbb{R}. This equation exhibits a transition from pulled to pushed front behavior at\beta_{c}=2. We prove convergence of the solutions to a traveling wave in a reference frame centered at a positionm_{\beta}(t)and study the asymptotics of the front locationm_{\beta}(t). When\beta < 2, it has the same form as for the standard Fisher-KPP equation established by Bramson:m_{\beta}(t) = 2t - (3/2)\log t + x_{\infty} + o(1)ast\to\infty. This form is typical of pulled fronts. When\beta > 2, the front is located at the position m_{\beta}(t)=c_{*}(\beta)t+x_{\infty}+o(1)with c_{*}(\beta)=\beta/2+2/\beta, which is the typical form of pushed fronts. However, at the critical value \beta_{c} = 2, the expansion changes tom_{\beta}(t) = 2t - (1/2)\log t + x_{\infty} + o(1), reflecting the “pushmi-pullyu” nature of the front. The arguments for\beta<2rely on a new weighted Hopf–Cole transform that allows one to control the advection term, when combined with additional steepness comparison arguments. The case \beta>2relies on standard pushed front techniques. The proof in the case\beta=\beta_{c}is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at\beta_{c}=2and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights. 

We propose a novel method for establishing the convergence rates of solutions to reaction–diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function introduced in An et. al. [Arch. Ration. Mech. Anal. 247 (2023), no. 5, article no. 88]. It turns out that the convergence rate is controlled by the distance between thephantom front locationfor the shape defect function and the true front location of the solution. Curiously, the convergence to a traveling wave has a pulled nature, regardless of whether the traveling wave itself is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach dramatically simplifies the proof in the Fisher–KPP case and gives a unified, succinct explanation for the known algebraic rates of convergence in the Fisher–KPP case and the exponential rates in the pushed case. 

Abstract We present probabilistic interpretations of solutions to semi-linear parabolic equations with polynomial nonlinearities in terms of the voting models on the genealogical trees of branching Brownian motion (BBM). These extend McKean’s connection between the Fisher–KPP equation and BBM (McKean 1975Commun. Pure Appl. Math.28323–31). In particular, we present ‘random outcome’ and ‘random threshold’ voting models that yield any polynomial nonlinearityfsatisfying $f\left(0\right)=f\left(1\right)=0$and a ‘recursive up the tree’ model that allows to go beyond this restriction onf. We compute several examples of particular interest; for example, we obtain a curious interpretation of the heat equation in terms of a nontrivial voting model and a ‘group-based’ voting rule that leads to a probabilistic view of the pushed-pulled transition for a class of nonlinearities introduced by Ebert and van Saarloos. 

We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank- bounded matrices. The Hessian of this loss at low-rank matrices can theoretically blow up, which creates challenges to analyze convergence of gradient optimization methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss as well as convergence results for finite step size gradient descent under certain assumptions on the initial weights. 

We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank-bounded matrices. The Hessian of this loss at low-rank matrices can theoretically blow up, which creates challenges to analyze convergence of gradient optimization methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss as well as convergence results for finite step size gradient descent under certain assumptions on the initial weights. 

Engineered nanomaterials interfaced with plant seeds can improve stress tolerance during the vulnerable seedling stage. Herein, we investigated how priming seeds with antioxidant poly(acrylic acid)-coated cerium oxide nanoparticles (PNC) impacts cotton ( Gossypium hirsutum L.) seedling morphological, physiological, biochemical, and transcriptomic traits under salinity stress. Seeds primed with 500 mg L −1 PNC in water (24 h) and germinated under salinity stress (200 mM NaCl) retained nanoparticles in the seed coat inner tegmen, cotyledon, and root apical meristem. Seed priming with PNC significantly ( P < 0.05) increased seedling root length (56%), fresh weight (41%), and dry weight (38%), modified root anatomical structure, and increased root vitality (114%) under salt stress compared with controls (water). PNC seed priming led to a decrease in reactive oxygen species (ROS) accumulation in seedling roots (46%) and alleviated root morphological and physiological changes induced by salinity stress. Roots from exposed seeds exhibited similar Na content, significantly decreased K (6%), greater Ca (22%) and Mg content (60%) compared to controls. A total of 4779 root transcripts were differentially expressed by PNC seed priming alone relative to controls with no nanoparticles under non-saline conditions. Under salinity stress, differentially expressed genes (DEGs) in PNC seed priming treatments relative to non-nanoparticle controls were associated with ROS pathways (13) and ion homeostasis (10), indicating that ROS and conserved Ca 2+ plant signaling pathways likely play pivotal roles in PNC-induced improvement of salinity tolerance. These results provide potential unifying molecular mechanisms of nanoparticle-seed priming enhancement of plant salinity tolerance.