Growing graphs with addition of communities

Paper proposes a model of large networks based on a random preferential attachment graph with addition of complete subgraphs (cliques). The proposed model refers to models of random graphs following the nonlinear preferential attachment rule and takes into account the possibility of {\guillemotleft}adding{\guillemotright} entire communities of nodes to the network. In the derivation of the relations that determine the vertex degree distribution, the technique of finite-difference equations describing stationary states of a graph is used. The obtained results are tested empirically (by generating large graphs), special cases correspond to known mathematical relations

Vertex degree distribution and arc endpoints degree distribution of graphs with a linear rule of preferential attachment and Pennock graphs

The article deals with two classes of growing random graphs following the preferential attachment rule with a linear weight function, L-graphs, and hybrid Pennock graphs. We determine the exact final vertex degree distribution and the exact final two-dimensional arcs degree distributions of graphs under consideration. The study proves that each hybrid Pennock graph is isomorphic to a certain L graph and that the converse does not hold since there are no Pennock graphs isomorphic to L graphs with negative displacements in the expression for the linear weight function. A formula is derived that makes it possible to determine the weight functions, which are used to generate graphs with the required asymptotic power-law vertex degree distribution. The reliability of the obtained results is confirmed by calculations using accurate numerical methods and simulation modeling, i.e. by direct generating of the graphs. The practical value of the results is demonstrated by an example of their effective application for accurate calibration of a growing graph that simulates a network of Internet at the level of autonomous systems.Comment: 12 pages, 8 figures, the article is prepared for the conference "Mechanical Science and Technology Update" (MSTU-2019

A generalized Ramsey excitation scheme with suppressed light shift

We experimentally investigate a recently proposed optical excitation scheme [V.I. Yudin et al., Phys. Rev. A 82, 011804(R)(2010)] that is a generalization of Ramsey's method of separated oscillatory fields and consists of a sequence of three excitation pulses. The pulse sequence is tailored to produce a resonance signal which is immune to the light shift and other shifts of the transition frequency that are correlated with the interaction with the probe field. We investigate the scheme using a single trapped 171Yb+ ion and excite the highly forbidden 2S1/2-2F7/2 electric-octupole transition under conditions where the light shift is much larger than the excitation linewidth, which is in the Hertz range. The experiments demonstrate a suppression of the light shift by four orders of magnitude and an immunity against its fluctuations.Comment: 5 pages, 4 figure

An elementary approach to toy models for D. H. Lehmer's conjecture

In 1947, Lehmer conjectured that the Ramanujan's tau function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ is the $m$-th Fourier coefficient of the cusp form $\Delta_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However, Bannai-Miezaki showed that none of the nonempty shells of the integer lattice \ZZ^2 in \RR^2 is a spherical 4-design, and that none of the nonempty shells of the hexagonal lattice $A_2$ is a spherical 6-design. Moreover, none of the nonempty shells of the integer lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for \QQ(\sqrt{-1}) and \QQ(\sqrt{-3}) is a spherical 2-design. In the proof, the theory of modular forms played an important role. Recently, Yudin found an elementary proof for the case of \ZZ^{2}-lattice which does not use the theory of modular forms but uses the recent results of Calcut. In this paper, we give the elementary (i.e., modular form free) proof and discuss the relation between Calcut's results and the theory of imaginary quadratic fields.Comment: 18 page

Perfect category-graded algebras

In a perfect category every object has a minimal projective resolution. We give a criterion for the category of modules over a categorygraded algebra to be perfect.Comment: A sufficient condition is replaced by a criterion. Several references added. 17 page