On the Chow groups of certain cubic fourfolds

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.     

On the Number of 5‐Cycles in a Tournament

We find a formula for the number of directed 5‐cycles in a tournament in terms of its edge scores and use the formula to find upper and lower bounds on the number of 5‐cycles in any n‐tournament. In particular, we show that the maximum number of 5‐cycles is asymptotically equal to , the expected number 5‐cycles in a random tournament (), with equality (up to order of magnitude) for almost all tournaments.     

Hamiltonicity of edge chromatic critical graphs

Vizing conjectured that every edge chromatic critical graph contains a 2-factor. Believing that stronger properties hold for this class of graphs, Luo and Zhao (2013) showed that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{6n}{7}$ is Hamiltonian. Furthermore, Luo et al. (2016) proved that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{4n}{5}$ is Hamiltonian. In this paper, we prove that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{3n}{4}$ is Hamiltonian. Our approach is inspired by the recent development of Kierstead path and Tashkinov tree techniques for multigraphs.     

Center problem and bifurcation behavior for a class of quasi analytic systems

Results about the study of nonanalytic systems’ center-focus and bifurcations of limit cycles are hardly seen in published references up till now. In this paper, we investigated the problems of determining center or focus and bifurcations for a class of planar quasi cubic analytic systems. The recursive formula to figure out generalized focal values is given, ulteriorly the conditions for four limit cycles from the origin or the point at infinity are obtained and center problems are considered. What is worth pointing out is that we offer a kind of interesting phenomenon that the exponent parameter λ control the non-analyticity of studied system (3.8) in this paper. In terms of nonanalytic differential systems, our work is new.     

Properly colored paths and cycles

In an edge-colored graph, let d^c(v) be the number of colors on the edges incident to v and let δ^c(G) be the minimum d^c(v) over all vertices v∈G. In this work, we consider sharp conditions on δ^c(G) which imply the existence of properly edge-colored paths and cycles, meaning no two consecutive edges have the same color.     

Bifurcation of limit cycles in fluid film bearings

Rotors supported by journal bearings may become unstable due to self-excited vibrations when a critical rotor speed is exceeded. Linearised analysis is usually used to determine the stability boundaries. Non-linear bifurcation theory or numerical integration is required to predict stable or unstable periodic oscillations close to the critical speed. In this paper, a dynamic model of a short journal bearing is used to analyse the bifurcation of the steady state equilibrium point of the journal centre. Numerical continuation is applied to determine stable or unstable limit cycles bifurcating from the equilibrium point at the critical speed. Under certain working conditions, limit cycles themselves are shown to disappear beyond a certain rotor speed and to exhibit a fold bifurcation giving birth to unstable limit cycles surrounding the stable supercritical limit cycles. Numerical integration of the system of equations is used to support the results obtained by numerical continuation. Numerical simulation permitted a partial validation of the analytical investigation.     

Color-critical graphs have logarithmic circumference

A graph G is k-critical if every proper subgraph of G is (k−1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least , improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed . We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.     

ON THE NUMBER OF LIMIT CYCLES OF A POLYNOMIAL DIFFERENTIAL SYSTEM

In this paper, we employ qualitative analysis and methods of bifurcation theory to study the maximum number of limit cycles for a polynomial system with center in global bifurcation.     

Complex dynamics of epidemic models on adaptive networks

There has been a substantial amount of well mixing epidemic models devoted to characterizing the observed complex phenomena (such as bistability, hysteresis, oscillations, etc.) during the transmission of many infectious diseases. A comprehensive explanation of these phenomena by epidemic models on complex networks is still lacking. In this paper we study epidemic dynamics in an adaptive network proposed by Gross et al., where the susceptibles are able to avoid contact with the infectious by rewiring their network connections. Such rewiring of the local connections changes the topology of the network, and inevitably has a profound effect on the transmission of the disease, which in turn influences the rewiring process. We rigorously prove that the adaptive epidemic model investigated in this paper exhibits degenerate Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation. Our study shows that adaptive behaviors during an epidemic may induce complex dynamics of disease transmission, including bistability, transient and sustained oscillations, which contrast sharply to the dynamics of classical network models. Our results yield deeper insights into the interplay between topology of networks and the dynamics of disease transmission on networks.     

On a motivic interpretation of primitive,variable and fixed cohomology

If M is a smooth projective variety whose motive is Kimura finite‐dimensional and for which the standard Lefschetz Conjecture B holds, then the motive of M splits off a primitive motive whose cohomology is the primitive cohomology. Under the same hypotheses on M, let X be a smooth complete intersection of ample divisors within M. Then the motive of X is the sum of a variable and a fixed motive inducing the corresponding splitting in cohomology. I also give variants with group actions.