A proof of the four-colour theorem

The four-colour problem remained unsolved for more than a hundred years has played a role of the utmost importance in the development of graph theory. The four-colour theorem was confirmed in 1976, which is not completely satisfied due to: i) part of the proof using computers cannot be verified by hand; ii) even the part, supposedly hand-checkable, is extraordinarily complicated and tedious, and as far as we know, no one has entirely verified it. Seeking a hand-checkable proof of the four-colour theorem is one of world-interested problems, which is addressed in this paper. A necessary and sufficient condition for n-colour theorem in a space is: there exists a largest n-complete graph base in the same space. Examples are given to illustrate applications

Short time dynamics of molecular junctions after projective measurement

In this work, we study the short time dynamics of a molecular junction described by Anderson-Holstein model using full-counting statistics after projective measurement. The coupling between the central quantum dot (QD) and two leads was turned on at remote past and the system is evolved to steady state at time $t=0$, when we perform the projective measurement in one of the lead. Generating function for the charge transfer is expressed as a Fredholm determinant in terms of Keldysh nonequilibrium Green's function in the time domain. It is found that the current is not constant at short times indicating that the measurement does perturb the system. We numerically compare the current behaviors after the projective measurement with those in the transient regime where the subsystems are connected at $t=0$. The universal scaling for high-order cumulants is observed for the case with zero QD occupation due to the unidirectional transport at short times. The influences of electron-phonon interaction on short time dynamics of electric current, shot noise and differential conductance are analyzed

An investigation into natural vibrations of fluid-structure interaction systems subject to Sommerfeld’s radiation condition

A fluid-structure interaction system subject to Sommerfeld’s condition is defined as a Sommerfeld system which is divided into three categories: Fluid Sommerfeld (FS) System, Solid Sommerfeld (SS) System and Fluid Solid Sommerfeld (FSS) System of which Sommerfeld conditions are imposed on a fluid boundary only, a solid boundary only and both fluid and solid boundaries, respectively. This paper follows the previous initial results claimed by simple examples to further mathematically investigate the natural vibrations of generalized Sommerfeld systems. A new parameter representing the speed of radiation wave for generalized 3-D problems with more complicated boundary conditions is introduced into the Sommerfeld condition which allows investigation of the natural vibrations of a Sommerfeld system involving both free surface and compressible waves. The mathematical demonstrations and selected examples confirm and reveal the natural behaviour of generalized Sommerfeld systems defined above. These generalized conclusions can be used in theoretical or engineering analysis of the vibrations of various Sommerfeld systems in engineering

Distributed interaction between computer virus and patch: A modeling study

The decentralized patch distribution mechanism holds significant promise as an alternative to its centralized counterpart. For the purpose of accurately evaluating the performance of the decentralized patch distribution mechanism and based on the exact SIPS model that accurately captures the average dynamics of the interaction between viruses and patches, a new virus-patch interacting model, which is known as the generic SIPS model, is proposed. This model subsumes the linear SIPS model. The dynamics of the generic SIPS model is studied comprehensively. In particular, a set of criteria for the final extinction or/and long-term survival of viruses or/and patches are presented. Some conditions for the linear SIPS model to accurately capture the average dynamics of the virus-patch interaction are empirically found. As a consequence, the linear SIPS model can be adopted as a standard model for assessing the performance of the distributed patch distribution mechanism, provided the proper conditions are satisfied

Quark number scaling of hadronic pTp_T spectra and constituent quark degree of freedom in pp-Pb collisions at sNN=5.02\sqrt{s_{NN}}=5.02 TeV

We show that the experimental data of $p_T$ spectra of identified hadrons released recently by ALICE collaboration for $p$-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV exhibit a distinct universal behavior --- the quark number scaling. We further show that the scaling is a direct consequence of quark (re-)combination mechanism of hadronization and can be regarded as a strong indication of the existence of the underlying source with constituent quark degree of freedom for the production of hadrons in $p$-Pb collisions at such high energies. We make also predictions for production of other hadrons.Comment: 5 pages, 3 figure