Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model

<p style='text-indent:20px;'>One of the most impressive findings in chemotaxis is the aggregation that randomly distributed bacteria, when starved, release a diffusive chemical to attract and group with others to form one or several stable aggregates in a long time. This paper considers pattern formation within the minimal Keller–Segel chemotaxis model with a focus on the stability and dynamics of its multi-spike steady states. We first show that any steady-state must be a periodic replication of the spatially monotone one and they present multi-spikes when the chemotaxis rate is large; moreover, we prove that all the multi-spikes are unstable through their refined asymptotic profiles, and then find a fully-fledged hierarchy of free entropy energy of these aggregates. Our results also complement the literature by finding that when the chemotaxis is strong, the single boundary spike has the least energy hence is the most stable, the steady-state with more spikes has larger free energy, while the constant has the largest free energy and is always unstable. These results provide new insights into the model's intricate global dynamics, and they are illustrated and complemented by numerical studies which also demonstrate the metastability and phase transition behavior in chemotactic movement.</p>

A random matrix analysis of random Fourier features: beyond the Gaussian kernel, a precise phase transition, and the corresponding double descent*

This article characterizes the exact asymptotics of random Fourier feature (RFF) regression, in the realistic setting where the number of data samples n, their dimension p, and the dimension of feature space N are all large and comparable. In this regime, the random RFF Gram matrix no longer converges to the well-known limiting Gaussian kernel matrix (as it does when N → ∞ alone), but it still has a tractable behavior that is captured by our analysis. This analysis also provides accurate estimates of training and test regression errors for large n, p, N. Based on these estimates, a precise characterization of two qualitatively different phases of learning, including the phase transition between them, is provided; and the corresponding double descent test error curve is derived from this phase transition behavior. These results do not depend on strong assumptions on the data distribution, and they perfectly match empirical results on real-world data sets.

Precise Synthesis and Thermoresponsive Property of Poly(ethyl glycidyl ether) and Its Block and Statistic Copolymers with Poly(glycidol)

In this paper, we describe a comprehensive study of the thermoresponsive properties of statistic copolymers and multiblock copolymers synthesized by poly(glycidol)s (PG) and poly(ethyl glycidyl ether) (PEGE) with different copolymerization methods. These copolymers were first synthesized by ring-opening polymerization (ROP), which was initiated by tert-butylbenzyl alcohol (tBBA) and 1-tert-butyl-4,4,4-tris(dimethylamino)-2,2-bis[tris(dimethylamino)phosphoranylidenamino]-2Λ5,4Λ5-catenadi(phosphazene) (t-Bu-P4) as the catalyst, and then the inherent protective groups were removed to obtain the copolymers without any specific chain end groups. The thermoresponsive property of the statistic copolymer PGx-stat-PEGEy was compared with the diblock copolymer PGx-b-PEGEy, and the triblock copolymers were compared with the pentablock copolymers. Among them, PG-stat-PEGE, PG-b-PEGE-b-PG-b-PEGE-b-PG, and PEGE-b-PG-b-PEGE-b-PG-b-PEGE, and even the specific ratio of PEGE-b-PG-b-PEGE, exhibited LCST-type phase transitions in water, which were characterized by cloud point (Tcp). Although the ratio of x to y affected the value of the Tcp of PGx-stat-PEGEy, we found that the disorder of the copolymer has a decisive effect on the phase-transition behavior. The phase-transition behaviors of PG-b-PEGE, part of PEGE-b-PG-b-PEGE, and PG-b-PEGE-b-PG copolymers in water present a two-stage phase transition, that is, firstly LCST-type and then the upper critical solution temperature (UCST)-like phase transition. In addition, we have extended the research on the thermoresponsive properties of EGE homopolymers without specific α-chain ends.

Second-order Phase Transition Behavior in a Polymer above the Glass Transition Temperature

<p>Glass transition was primarily considered to be not phase transition; however, it has similarity to the second-order phase transition. Recent single-molecule spectroscopy developments have prompted re-investigating glass transition at the microscopic scale, revealing that glass transition includes phenomena similar to second-order phase transition. They are characterized by microscopic collective polymer motion and discontinuous changes in temperature dependent relaxation times, later of which is similar to critical slowing down, within a temperature window that includes the polymer calorimetric glass transition temperature. Considering that collective motion and critical slowing down are accompaniments to critical phenomena, second-order phase transition behavior was identified in polymer glass transition.</p>