11,417 research outputs found
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 , 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 . 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 spectra and constituent quark degree of freedom in -Pb collisions at TeV
We show that the experimental data of spectra of identified hadrons
released recently by ALICE collaboration for -Pb collisions at
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 -Pb collisions at
such high energies. We make also predictions for production of other hadrons.Comment: 5 pages, 3 figure
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