Piecewise linear iterated function systems on the line of overlapping construction

In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.

Connecting Homomorphism and Separating Cycles

We discuss the construction of a long semi-exact Mayer–Vietoris sequence for the homology of any finite union of open subspaces. This sequence is used to obtain topological conditions of representation of the integral of a meromorphic n-form on an n-dimensional complex manifold in terms of Grothendieck residues. For such a representation of the integral to exist, it is necessary that the cycle of integration separates the set of polar hypersurfaces of the form. The separation condition in a number of cases turns out to be a sufficient condition for representing the integral as a sum of residues. Earlier, when describing such cases (in the works of Tsikh, Yuzhakov, Ulvert, etc.), the key was the condition that the manifold be Stein. The main result of this article is the relaxation of this condition

The scenery flow of self-similar measures with weak separation condition

We show that self-similar measures on $\mathbb R^d$ satisfying the weak separation condition are uniformly scaling. Our approach combines elementary ergodic theory with geometric analysis of the structure given by the weak separation condition.

Double layered solutions to the extended Fisher–Kolmogorov P.D.E.

We construct double layered solutions to the extended Fisher–Kolmogorov P.D.E., under the assumption that the set of minimal heteroclinics of the corresponding O.D.E. satisfies a separation condition. The aim of our work is to provide for the extended Fisher–Kolmogorov equation, the first examples of two-dimensional minimal solutions, since these solutions play a crucial role in phase transition models, and are closely related to the De Giorgi conjecture.

Study on the separation of silicon from refining slag of industrial silicon

During the process of production and refining of industrial silicon, the discharged slag usually contains more than 15 mass% of metallic silicon. However, the separation of silicon from slag is very difficult due to the close density of silicon and slag as well as the high viscosity of slag, which results in the waste of resources. In the present work, the effect of Na2O addition on the separation of silicon from slag is investigated in detail. It is found that the optimum separation condition of slag and silicon is 1723 K reacting for 60 min by adding 10 mass% Na2O to the slag. Viscosity and density are two important factors affecting the separation effect of silicon from slag. The addition of Na2O reduces the viscosity of slag and promotes the separation of silicon from slag. Even if the addition of Na2O will decrease the density of slag which is detrimental to the separation of silicon, the density variation is not the determining factor affecting the separation relative to viscosity. The separation and extraction of metallic silicon from silicon slag is of great significance for improving utilization of resources and reducing environmental pollution.

Gas-Liquid Flow in a Vertical Pipe Equipped with a Double Helical Swirl Separator Element

Two different gas-liquid flow behavior downstream a double helical swirl element inside a vertical pipe was observed in our preliminary experiment. The present Computational Fluid Dynamics (CFD) study confirms that the dynamics of gas-liquid flows inside the swirl separator is highly influenced by the liquid superficial velocity. The separation behavior in this work at a liquid superficial velocity of 0.1 m/s was the worst both axially and radially since the gas core cannot be sustained up to the outlet. The separation condition was improved by the increase of the liquid superficial velocity. The best separation condition in this study was achieved at the liquid superficial velocity of 1.0 m/s where the dense gas core can be maintained up to the outlet.

Normal Families of Möbius Maps

Several necessary and sufficient conditions for a family of Möbius maps to be a normal family in the extended complex plane $$\mathbb {C}_\infty $$ C ∞ are established. Each of these conditions involves collections of two or three points which may vary with the Möbius maps in the family, provided the points satisfy a uniform separation condition. In addition, we derive a sufficient condition for the normality of a family of Möbius maps in terms of the average value of the reciprocal of the chordal derivative.

Characterization of the Optimal Damping Coefficient in the Continuous Contact Model

This paper presents an analytical formula to characterize the damping coefficient as a function of system's parameters in a continuous force model of impact. The contact force element consists of a linear damper which is in a parallel connection to a spring with Hertz force-deformation characteristic. Unlike the existing models in which the separation condition is assumed to be at the time at which both zero penetration (deformation) and zero force occur, in this study, only zero contact force is considered as the separation condition. To ensure that the continuous contact model obtains the desired restitution, an optimization process is performed to find the equivalent damping coefficient. The analytical and numerical investigations show that the resulting damping coefficient can be expressed as a function of system's parameters such as the effective mass, penetration speed at the start of the impact, Hertz spring constant, and the coefficient of restitution.