Categoricity in multiuniversal classes

The third author has shown that Shelah's eventual categoricity conjecture holds in universal classes: class of structures closed under isomorphisms, substructures, and unions of chains. We extend this result to the framework of multiuniversal classes. Roughly speaking, these are classes with a closure operator that is essentially algebraic closure (instead of, in the universal case, being essentially definable closure). Along the way, we prove in particular that Galois (orbital) types in multiuniversal classes are determined by their finite restrictions, generalizing a result of the second author.     

Curvature identities derived from the integral formula for the first Pontrjagin number

We give an integral formula for the first Pontrjagin number of a compact almost Hermitian surface and derive curvature identities from the integral formula based on the fundamental fact that the first Pontrjagin number in the deRham cohomology group is a topological invariant. Further, we provide some applications of the identities.     

Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

Lovász and Schrijver (SIAM J. Optim. 1:166–190, 1991) have constructed semidefinite relaxations for the stable set polytope of a graph G = (V,E) by a sequence of lift-and-project operations; their procedure finds the stable set polytope in at most α(G) steps, where α(G) is the stability number of G. Two other hierarchies of semidefinite bounds for the stability number have been proposed by Lasserre (SIAM J. Optim. 11:796–817, 2001; Lecture Notes in Computer Science, Springer, Berlin Heidelberg New York, pp 293–303, 2001) and by de Klerk and Pasechnik (SIAM J. Optim. 12:875–892), which are based on relaxing nonnegativity of a polynomial by requiring the existence of a sum of squares decomposition. The hierarchy of Lasserre is known to converge in α(G) steps as it refines the hierarchy of Lovász and Schrijver, and de Klerk and Pasechnik conjecture that their hierarchy also finds the stability number after α(G) steps. We prove this conjecture for graphs with stability number at most 8 and we show that the hierarchy of Lasserre refines the hierarchy of de Klerk and Pasechnik.      

Extremal metrics for the first eigenvalue of the Laplacian in a conformal class

Let be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on to be extremal for with respect to conformal deformations of fixed volume. In particular, these conditions show that for any lattice of , the flat metric induced on from the standard metric of is extremal (in the previous sense). In the second part, we give, for any , an upper bound of on the conformal class of and exhibit a class of lattices for which the metric maximizes on its conformal class.

     

On the number of conjugacy classes of finite nilpotent groups

We establish the first super-logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group. In particular, we obtain that for any constant c there are only finitely many finite p-groups of order pm with at most c⋅m conjugacy classes. This answers a question of L. Pyber.     

Quasiconvex functions and null Lagrangians in the stability problems of classes of mappings

We obtain stability theorems for classes of solutions to the differential equations constructed by means of quasiconvex functions and null Lagrangians.     

Description of stability for linear time-invariant systems based on the first curvature

This paper focuses on using the first curvature κ(t) of trajectory to describe the stability of linear time-invariant system. We extend the results for two and three-dimensional systems (Wang, Sun, Song et al, arXiv:1808.00290) to n-dimensional systems. We prove that for a system $\text{◂=▸}\dot{r}(t)=\text{◂⋅▸}Ar(t)$, (a) if there exists a measurable set whose Lebesgue measure is greater than zero, such that $\text{◂≠▸}\underset{t\to +\infty }{\lim }\kappa (t)\ne 0$ or $\underset{t\to +\infty }{\lim }\kappa (t)$ does not exist for any initial value in this set, then the zero solution of the system is stable; (b) if the matrix A is invertible, and there exists a measurable set whose Lebesgue measure is greater than zero, such that $\text{◂=▸}\underset{t\to +\infty }{\lim }\kappa (t)=+\infty$ for any initial value in this set, then the zero solution of the system is asymptotically stable.     

On the stability of sets defined by a finite number of equalities and inequalities

Let a set be defined by a finite number of equalities and inequalities. For smooth data, the condition of Mangasarian and Fromovitz is known to be equivalent to the local stability—in a strong sense—of the set. We study here weaker forms of stability. Namely, we state a condition generalizing the one of Mangasarian and Fromovitz that, for some weak form of stability, is necessary. If the gradients of the equality constraints are linearly independent or if there is no equality constraint, this condition is also sufficient.     

A quantitative study on the stability of delay discrete singular systems

For quadratic delay discrete singular systems, an algebraic criterion on the stability is established, and the size of the uniform stability region and asymptotic stability region around zero is estimated. Hence, the criterion is both qualitative and quantitative. With the computer techniques, the criterion dependent of delay is easy test and applies to the application in the practice. An illustrative simulation is given to illustrate the application of the obtained result.     

On the number of non-dominated points of a multicriteria optimization problem

This work proposes an upper bound on the maximal number of non-dominated points of a multicriteria optimization problem. Assuming that the number of values taken on each criterion is known, the criterion space corresponds to a comparability graph or a product of chains. Thus, the upper bound can be interpreted as the stability number of a comparability graph or, equivalently, as the width of a product of chains. Standard approaches or formulas for computing these numbers are impractical. We develop a practical formula which only depends on the number of criteria. We also investigate the tightness of this upper bound and the reduction of this bound when feasible, possibly efficient, solutions are known.     

Characterization of the stability radius via bifurcation techniques

Robustness of stability of linear time-invariant systems using the relationship between the structured complex stability radius and a parametrized algebraic Riccati equation is analysed. Our approach is based on the observation that the algebraic Riccati equation can be viewed as a bifurcation problem. It is proved that the stability radius is, under certain assumptions, associated with a turning point of the bifurcation problem given by the parametrized algebraic Riccati equation. As a byproduct, the stability radius can be computed via path following. Some numerical examples are presented.     

Classroom note: The number c in Cauchy's average value theorem

The location of the number c arising from Cauchy's Average Value Theorem is described when the size of the interval is small.     

On global stability of the intra-host dynamics of malaria and the immune system

In this paper we consider an intra-host model for the dynamics of malaria. The model describes the dynamics of the blood stage malaria parasites and their interaction with host cells, in particular red blood cells (RBC) and immune effectors. We establish the equilibrium points of the system and analyze their stability using the theory of competitive systems, compound matrices and stability of periodic orbits. We established that the disease-free equilibrium is globally stable if and only if the basic reproduction number satisfies R₀?1 and the parasite will be cleared out of the host. If R₀>1, a unique endemic equilibrium is globally stable and the parasites persist at the endemic steady state. In the presence of the immune response, the numerical analysis of the model shows that the endemic equilibrium is unstable.     

Modelling and stability of epidemic model with free-living pathogens growing in the environment

To understand the impact of free-living pathogens (FLP) on the epidemics, an epidemic model with FLP is constructed. The global dynamics of our model are determined by the basic reproduction number $R_0$. If $R_0<1$, the disease free equilibrium is globally asymptotically stable, and if $R_0>1$, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to illustrate our analytical results.