Mathematics

We introduce and analyze a model for the dynamics of flocking and steering of a finite number of agents. In this model, each agent's acceleration consists of flocking and steering components. The flocking component is a generalization of many of the existing models and allows for the incorporation of many real world features such as acceleration bounds, partial masking effects and orientation bias. The steering component is also integral to capture real world phenomena. We provide rigorous sufficient conditions under which the agents flock and steer together. We also provide a formal singular perturbation study of the situation where flocking happens much faster than steering. We end our work by providing some numerical simulations to illustrate our theoretical results.

In a minimal flow, the hitting time is the exponent of the power law, as r goes to zero, for the time needed by orbits to become r-dense. We show that on the so-called Ornithorynque origami the hitting time of the flow in an irrational slope equals the diophantine type of the slope. We give a general criterion for such equality. In general, for genus at least two, hitting time is strictly bigger than diophantine type.

Different classes of the whorl fingerprint are discussed. A general dynamical system with a parameter theta is created using differential equations to simulate these classes by varying the value of theta. The global dynamics is studied, and the existence and stability of equilibria are analyzed. The Maple is used to visualize fingerprint orientation image as a smooth deformation of the phase portrait of a planar dynamical system.

We formulate a general statement of the problem of defining invariant measures with certain properties and suggest an ergodic method of perturbations for describing such measures.

We construct a model of the cubic connectedness locus.

Let a<b be coprime positive integers, both at least 2 , and let A,B be closed subsets of [0,1] that are forward invariant under multiplication by a , b respectively. Let C=A×B . An old conjecture of Furstenberg asserted that any planar line L not parallel to either axis must intersect C in Hausdorff dimension at most max{dimC,1}−1 . Two recent works by Shmerkin and Wu have given two different proofs of this conjecture. This note provides a third proof. Like Wu's, it stays close to the ergodic theoretic machinery that Furstenberg introduced to study such questions, but it uses less substantial background from ergodic theory. The same method is also used to re-prove a recent result of Yu about certain sequences of sums.

We conduct a detailed analysis of the Assouad type dimensions and spectra in the context of limit sets of geometrically finite Kleinian groups and Julia sets of parabolic rational maps. Our analysis includes the Patterson-Sullivan measure in the Kleinian case and the analogous conformal measure in the Julia set case. Our results constitute a new perspective on the Sullivan dictionary between Kleinian groups and rational maps. We show that there exist both strong correspondences and strong differences between the two settings. The differences we observe are particularly interesting since they come from dimension theory, a subject where the correspondence described by the Sullivan dictionary is especially strong.

In this paper a new type of chaotic system based on sin and logistic systems is introduced. Also the behavior of this new system is studied by using various tests. The results of these tests indicate the appropriate behavior for the proposed new system.

We consider SISI epidemic model with discrete-time. The crucial point of this model is that an individual can be infected twice. This non-linear evolution operator depends on seven parameters and we assume that the population size under consideration is constant, so death rate is the same with birth rate per unit time. Reducing to quadratic stochastic operator (QSO) we study the dynamical system of the SISI model.

Let C 1 and C 2 be two Cantor sets with convex hull [0,1] . Newhouse proved if τ( C 1 )⋅τ( C 2 )≥1 , then the arithmetic sum C 1 + C 2 is an interval, where τ( C i ),1≤i≤2 denotes the thickness of C i . In this paper, we generalize this thickness theorem as follows. Let K i ⊂R,i=1,⋯,d , be some Cantor sets (perfect and nowhere dense) with convex hull [0,1] . Suppose f( x 1 ,⋯, x d−1 ,z)∈ C 1 is a continuous function defined on R d . Denote the continuous image of f by f( K 1 ,⋯, K d )={f( x 1 ,⋯ x d−1 ,z): x i ∈ K i ,z∈ K d ,1≤i≤d−1}. If for any ( x 1 ,⋯, x d−1 ,z)∈[0,1 ] d , we have (τ( K i ) ) −1 ≤ ∣ ∣ ∣ ∂ x i f ∂ z f ∣ ∣ ∣ ≤τ( K d ),1≤i≤d−1 then f( K 1 ,⋯, K d ) is a closed interval. We give two applications. Firstly, we partially answer some questions posed by Takahashi. Secondly, we obtain various nonlinear identities, associated with the continued fractions with restricted partial quotients, which can represent real numbers.