On Plastic Dislocation Density Tensor

This article attempts to clarify an issue regarding the proper definition of plastic dislocation density tensor. This study shows that the Ortiz's and Berdichevsky's plastic dislocation density tensors are equivalent with each other, but not with Kondo's one. To fix the problem, we propose a modified version of Kondo's plastic dislocation density tensor.Comment: 3 pages, 1 figur

Note on Divergence of the Chapman-Enskog Expansion for Solving Boltzmann Equation

Within about a year (1916-1917) Chapman and Enskog independently proposed an important expansion for solving the Boltzmann equation. However, the expansion is divergent or indeterminant in the case of relaxation time $\tau \geq 1$. Even since this divergence problem has puzzled this subject for a century. By using a modified M\"obius series inversion formula, this paper proposes a modified Chapman-Enskog expansion with a variable upper limit of the summation. The new expansion can give not only a convergent summation but also provide the best-so-far explanation on some unbelievable scenarios occurred in previous practice.Comment: 4 pages, 0 figures, 2 table

Kepler's third law of n-body periodic orbits in a Newtonian gravitation field

This study considers the periodic orbital period of an n-body system from the perspective of dimension analysis. According to characteristics of the n-body system with point masses $(m_1,m_2,...,m_n)$, the gravitational field parameter, $\alpha \sim Gm_im_j$, the n-body system reduction mass $M_n$, and the area, $A_n$, of the periodic orbit are selected as the basic parameters, while the period, $T_n$, and the system energy, $|E_n|$, are expressed as the three basic parameters. Using the Buckingham $\pi$ theorem, We obtained an epic result, by working with a reduced gravitation parameter $\alpha_n$, then predicting a dimensionless relation $T_n|E_n|^{3/2}=\text{const} \times \alpha_n \sqrt{\mu_n}$ ($\mu_n$ is reduced mass). The const$=\frac{\pi}{\sqrt{2}}$ is derived by matching with the 2-body Kepler's third law, and then a surprisingly simple relation for Kepler's third law of an n-body system is derived by invoking a symmetry constraint inspired from Newton's gravitational law: $T_n|E_n|^{3/2}=\frac{\pi}{\sqrt{2}} G\left(\frac{\sum_{i=1}^n\sum_{j=i+1}^n(m_im_j)^3}{\sum_{k=1}^n m_k}\right)^{1/2}$. This formulae is, of course, consistent with the Kepler's third law of 2-body system, but yields a non-trivial prediction of the Kepler's third law of 3-body: $T_3|E_3|^{3/2}= \frac{\pi}{\sqrt{2}} G \left[\frac{(m_1m_2)^3+(m_1m_3)^3+(m_2m_3)^3}{m_1+m_2+m_3}\right]^{1/2}$. A numerical validation and comparison study was conducted. This study provides a shortcut in search of the periodic solutions of three-body and n-body problems and has valuable application prospects in space exploration.Comment: 6 pages, 11 figue

Proving Redundancy In Decentralized Storage Networks

Proof of Storage (PoS) is a scheme that proves the data is stored honestly. PoS is the general term for a collection of related proofs. Such as proof of retrievability and proof of data possession. In recent years, various PoS variants have been proposed, each with advantages. However, there is no protocol for auditing between peers without involving third-party auditors. This thesis proposes a protocol which allows auditing between peers without third-part involved. The protocol is fully implemented with Go coding language and deployed on the Swarm test network. The designed protocol is mainly based on proof of retrievability, proof of data possession and proof of replication. The proposed protocol uses a challenge-response protocol to send messages between nodes in a decentralised file storage system. Experiments show that our idea is feasible. After experimenting with different test nodes, the results show that the efficiency of our proposed protocol is related to the number of redundant data chunks owned by the challenger and prover in the same network. If either challenger or prover holds a relatively small amount of data chunks, our proposed protocol will have better performance

Cosmology in a universe with Bose-Einstein-condensed scalar field dark matter

We consider an alternative cold dark matter candidate, ultralight bosons (m>10^{-22} eV/c^2) described by a complex scalar field (SFDM) with global U(1) symmetry, with comoving particle number density conserved after particle production during standard reheating. We allow for repulsive self-interaction. In a Lambda-SFDM universe, SFDM starts relativistic, evolving from stiff (w=1) to radiation-like (w=1/3), becoming nonrelativistic (w=0) at late times. Thus, a stiff-SFDM-dominated era precedes the familiar radiation-dominated era. SFDM particle mass m and quartic self-interaction strength lambda, are therefore constrained by cosmological observables, N_{eff}, the effective number of neutrino species during BBN, and z_{eq}, the matter-radiation equality redshift. Since the stochastic gravitational wave background (SGWB) from inflation is amplified during the stiff-SFDM-dominated era, it can also contribute a radiation-like component large enough to affect these observables. Remarkably, this amplification makes this SGWB detectable by current GW experiments, e.g., aLIGO/Virgo and LISA, for Lambda-SFDM models satisfying cosmological constraints, for a range of reheat temperatures T_{re} and currently allowed values of tensor-to-scalar ratio r. For given r and lambda/(mc^2)^2, the marginally-allowed Lambda-SFDM model for each T_{re} has the smallest m that satisfies cosmological constraints. For example, for marginally-allowed models with r=0.01 and lambda/(mc^2)^2=10^{-18} eV^{-1} cm^3, null detection by the aLIGO O1 run excludes 8.75*10^3<T_{re} (GeV)<1.7*10^5 at 95% confidence, demonstrating that GW experiments already place a new kind of cosmological constraint on SFDM. A wider parameter range should be accessible to aLIGO/Virgo O5, with potential to detect this signature of Lambda-SFDM. For this same illustrative family, 3-sigma detection is predicted for 600<T_{re} (GeV)<10^7.Astronom