Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function. Let $H(x,p,u)$ be a continuous Hamiltonian which is strictly increasing in $u$, and is convex and coercive in $p$. For each parameter $\lambda>0$, we denote by $u^\lambda$ the unique viscosity solution of the H-J equation $$H( x,Du(x),\lambda u(x) )=c.$$ Under quite general assumptions, we prove that $u^\lambda$ converges uniformly, as $\lambda$ tends to zero, to a specific solution of the critical H-J equation $H(x,Du(x),0)=c.$ We also characterize the limit solution in terms of Peierls barrier and Mather measures

Double scaffold networks regulate edible Pickering emulsion gel for designing thermally actuated 4D printing

3D and 4D printing of emulsion gels can be achieved by controlling the continuous phase and the interface. Edible high internal phase water-in-oil Pickering emulsion gels (PEGs) with a tunable double scaffold network structure are designed and prepared by food-grade phytosterol nanoparticles (PPs). In PEGs, PPs through hydrogen bonding enable binding of emulsion droplets to form the first scaffold network, giving PEGs high viscoelasticity for 3D printing. In the continuous oil phase, palm kernel stearin (PKST) can crystallize forming the second scaffold network of crystals to reinforce 3D printed objects of PEGs. The PEG can be used as a biocompatible template to engineer edible and rigid porous materials with adjustable strength and pore size depending on the degree of curing. 4D printing of PEGs is achieved by the thermal response of the PKST crystal network, leading to the unlimited potential of highly biocompatible PEGs in many applications

Bubble attachment time and FTIR analysis of water structure in the flotation of sylvite, bischofite and carnallite

ManuscriptWater structure is a most important parameter that influences the flotation of soluble salts. In this paper bubble attachment time measurements and FTIR analyses were performed to investigate the effect of water structure on the flotation behavior of sylvite (KCl), bischofite (MgCl2•6H2O) and carnallite (KMgCl3•6H2O). The results from bubble attachment time measurements suggest that collector adsorption at the surface of KCl induces flotation with either the cationic collector, ODA or anionic collector, SDS. In contrast bubble attachment did not occur for bischofite (MgCl2•6H2O) or carnallite (KMgCl3•6H2O). Results show that the surface charge is not a determining factor in the flotation of soluble salts. Further, the interaction between water molecules and the three chloride salts dissolved in aqueous solution were studied by measuring the shift in the hydrogen bonding of water molecules. The results indicate that KCl is a structure breaker salt, while MgCl2•6H2O and KMgCl3•6H2O are structure maker salts. Viscosities for the brines of these three salts were determined. The results give additional evidence of differences in water structure and are in good agreement with the FTIR and bubble attachment results. The findings provide further evidence that water structure plays an important role in the flotation of soluble salts

Regret Analysis of Policy Gradient Algorithm for Infinite Horizon Average Reward Markov Decision Processes

In this paper, we consider an infinite horizon average reward Markov Decision Process (MDP). Distinguishing itself from existing works within this context, our approach harnesses the power of the general policy gradient-based algorithm, liberating it from the constraints of assuming a linear MDP structure. We propose a policy gradient-based algorithm and show its global convergence property. We then prove that the proposed algorithm has $\tilde{\mathcal{O}}({T}^{3/4})$ regret. Remarkably, this paper marks a pioneering effort by presenting the first exploration into regret-bound computation for the general parameterized policy gradient algorithm in the context of average reward scenarios

Provably Efficient Model-Free Algorithm for MDPs with Peak Constraints

In the optimization of dynamic systems, the variables typically have constraints. Such problems can be modeled as a Constrained Markov Decision Process (CMDP). This paper considers the peak constraints, where the agent chooses the policy to maximize the long-term average reward as well as satisfies the constraints at each time. We propose a model-free algorithm that converts CMDP problem to an unconstrained problem and a Q-learning based approach is used. We extend the concept of probably approximately correct (PAC) to define a criterion of $\epsilon$-optimal policy. The proposed algorithm is proved to achieve an $\epsilon$-optimal policy with probability at least $1-p$ when the episode $K\geq\Omega(\frac{I^2H^6SA\ell}{\epsilon^2})$, where $S$ and $A$ is the number of states and actions, respectively, $H$ is the number of steps per episode, $I$ is the number of constraint functions, and $\ell=\log(\frac{SAT}{p})$. We note that this is the first result on PAC kind of analysis for CMDP with peak constraints, where the transition probabilities are not known apriori. We demonstrate the proposed algorithm on an energy harvesting problem where it outperforms state-of-the-art and performs close to the theoretical upper bound of the studied optimization problem

Memory capacity and prioritization in female mice

Our brain's capacity for memory storage may be vast but is still finite. Given that we cannot remember the entirety of our experiences, how does our brain select what to remember and what to forget? Much like the triage of a hospital's emergency room, where urgent cases are prioritized and less critical patients receive delayed or even no care, the brain is believed to go through a similar process of memory triage. Recent salient memories are prioritized for consolidation, which helps create stable, long-term representations in the brain; less salient memories receive a lower priority, and are eventually forgotten if not sufficiently consolidated (Stickgold and Walker in Nat Neurosci 16(2):139-145, 2013). While rodents are a primary model for studying memory consolidation, common behavioral tests typically rely on a limited number of items or contexts, well within the memory capacity of the subject. A memory test allowing us to exceed an animal's memory capacity is key to investigating how memories are selectively strengthened or forgotten. Here we report a new serial novel object recognition task designed to measure memory capacity and prioritization, which we test and validate using female mice