The Reason for the Incompleteness of Some Theorems in Modern Mathematics: Non Numbers Enter Mathematical Analysis or the Concept Definition Is Incomplete

The concept of non-number and number is found, and the concept of non-number and number is defined by symbols. The basic theory of number theory is solved; it is proved that {Godel's Incomplete Theorem} are incomplete.  and it is also proved that the first number of natural number must be 1, not 0

Overview: How Does the Paper "The Continuity of Prime Numbers Can Lead to the Continuity of Even Numbers" Relate to the Goldbach Conjecture

The paper "The continuity of prime numbers can lead to the continuity of even numbers" proposes three clever methods to prove the Goldbach conjecture from another path that humans never anticipated and was also the correct path. "The continuity of prime numbers can lead to the continuity of even numbers" A preprint has previously been published (Xie ling. 2021).&nbsp

Enhanced quantum teleportation in the background of Schwarzschild spacetime by weak measurements

It is commonly believed that the fidelity of quantum teleportation in the gravitational field would be degraded due to the heat up by the Hawking radiation. In this paper, we point out that the Hawking effect could be eliminated by the combined action of pre- and post-weak measurements, and thus the teleportation fidelity is almost completely protected. It is intriguing to notice that the enhancement of fidelity could not be attributed to the improvement of entanglement, but rather to the probabilistic nature of weak measurements. Our work extends the ability of weak measurements as a quantum technique to battle against gravitational decoherence in relativistic quantum information.Comment: 9 pages, 5 figures, comments are welcom

Robust Spin Squeezing Preservation in Photonic Crystal Cavities

We show that the robust spin squeezing preservation can be achieved by utilizing detuning modification for an ensemble of N separate two-level atoms embedded in photonic crystal cavities (PCC). In particular, we explore the different dynamical behaviors of spin squeezing between isotropic and anisotropic PCC cases when the atomic frequency is inside the band gap. In both cases, it is shown that the robust preservation of spin squeezing is completely determined by the formation of bound states. Intriguingly, we find that unlike the isotropic case where steady-state spin squeezing varies smoothly when the atomic frequency moves from the inside to the outside band edge, a sudden transition occurs for the anisotropic case. The present results may be of direct importance for, e.g., quantum metrology in open quantum systems.Comment: 6 pages, 4 figures, accepted by Laser Physics Letter

5-(4-Bromo­phen­yl)-1,2,3,4-tetra­hydro­benzo[a]phenanthridine

The title compound, C23H18BrN, was synthesized by the reaction of 4-bromo­benzaldehyde, naphthalen-2-amine and cyclo­hexa­none in tetra­hydro­furan, catalyzed by iodine. The saturated six-membered ring adopts a half-chair conformation, and the four vicinal rings form a helical conformation, which results in a significant deviation from planarity for the pyridine ring. In the crystal, a weak C—H⋯π inter­action occurs, leading to inversion dimers