To Prove Four Color Theorem

In this paper, we give a proof for four color theorem(four color conjecture). Our proof does not involve computer assistance and the most important is that it can be generalized to prove Hadwiger Conjecture. Moreover, we give algorithms to color and test planarity of planar graphs, which can be generalized to graphs containing $K_x(x>5)$ minor. There are four parts of this paper: Part-1: To Prove Four Color Theorem Part-2: An Equivalent Statement of Hadwiger Conjecture when $k=5$ Part-3: A New Proof of Wagner's Equivalence Theorem Part-4: A Geometric View of Outerplanar GraphComment: The paper is further reduced, and each part is more self-contained, is the fina

TSONN: Time-stepping-oriented neural network for solving partial differential equations

Deep neural networks (DNNs), especially physics-informed neural networks (PINNs), have recently become a new popular method for solving forward and inverse problems governed by partial differential equations (PDEs). However, these methods still face challenges in achieving stable training and obtaining correct results in many problems, since minimizing PDE residuals with PDE-based soft constraint make the problem ill-conditioned. Different from all existing methods that directly minimize PDE residuals, this work integrates time-stepping method with deep learning, and transforms the original ill-conditioned optimization problem into a series of well-conditioned sub-problems over given pseudo time intervals. The convergence of model training is significantly improved by following the trajectory of the pseudo time-stepping process, yielding a robust optimization-based PDE solver. Our results show that the proposed method achieves stable training and correct results in many problems that standard PINNs fail to solve, requiring only a simple modification on the loss function. In addition, we demonstrate several novel properties and advantages of time-stepping methods within the framework of neural network-based optimization approach, in comparison to traditional grid-based numerical method. Specifically, explicit scheme allows significantly larger time step, while implicit scheme can be implemented as straightforwardly as explicit scheme

The role of binding site on the mechanical unfolding mechanism of ubiquitin.

We apply novel atomistic simulations based on potential energy surface exploration to investigate the constant force-induced unfolding of ubiquitin. At the experimentally-studied force clamping level of 100 pN, we find a new unfolding mechanism starting with the detachment between β5 and β3 involving the binding site of ubiquitin, the Ile44 residue. This new unfolding pathway leads to the discovery of new intermediate configurations, which correspond to the end-to-end extensions previously seen experimentally. More importantly, it demonstrates the novel finding that the binding site of ubiquitin can be responsible not only for its biological functions, but also its unfolding dynamics. We also report in contrast to previous single molecule constant force experiments that when the clamping force becomes smaller than about 300 pN, the number of intermediate configurations increases dramatically, where almost all unfolding events at 100 pN involve an intermediate configuration. By directly calculating the life times of the intermediate configurations from the height of the barriers that were crossed on the potential energy surface, we demonstrate that these intermediate states were likely not observed experimentally due to their lifetimes typically being about two orders of magnitude smaller than the experimental temporal resolution

Dimeth­yl(2-oxo-2-phenyl­eth­yl)sulfanium bromide

Single crystals of the title compound, C10H13OS+·Br−, were obtained from ethyl acetate/ethyl ether after reaction of acetophenone with hydro­bromic acid and dimethyl­sulfoxide. The carbonyl group is almost coplanar with the neighbouring phenyl ring [O—C—C—C = 178.9 (2)°]. The sulfanium group shows a trigonal–pyramidal geometry at the S atom. The crystal structure is stabil­ized by C—H⋯Br hydrogen-bonding inter­actions. Weak π–π inter­actions link adjacent phenyl rings [centroid–centroid distance = 3.946 (2) Å]