Hölder Derivative of the Koch Curve

In this paper, we introduce a K Hölder p-adic derivative that can be applied to fractal curves with different Hölder exponent K. We will show that the Koch curve satisfies the Hölder condition with exponent and has a 4-adic arithmetic-analytic representation. We will prove that the Koch curve has exact -Hölder 4-adic derivative.

Lift-Off Effect of Koch and Circular Differential Pickup Eddy Current Probes

A flexible or planar eddy current probe with a differential structure can suppress the lift-off noise during the inspection of defects.However,the extent of the lift-off effect on differential probes,including different coil structures,varies.In this study,two planar eddy current probes with differential pickup structures and the same size,Koch and circular probes,were used to compare lift-off effects.The eddy current distributions of the probes perturbed by 0°and 90°cracks were obtained by finite element analysis.The analysis results show that the 90°crack can impede the eddy current induced by the Koch probe even further at relatively low lift-off distance.The peak-to-peak values of the signal output from the two probes were compared at different lift-off distances using finite element analysis and experimental methods.In addition,the effects of different frequencies on the lift-off were studied experimentally.The results show that the signal peak-to-peak value of the Koch probe for the inspection of cracks in 90°orientation is larger than that of the circular probe when the lift-off distance is smaller than 1.2 mm.In addition,the influence of the lift-off distance on the peak-to-peak signal value of the two probes was studied via normalization.This indicates that the influence becomes more evident with an increase in excitation frequency.This research discloses the lift-off effect of differential planar eddy current probes with different coil shapes and proves the detection merit of the Koch probe for 90°cracks at low lift-off distances.

CONSTANT DISTANCE BOUNDARIES OF THE t-QUASICIRCLE AND THE KOCH SNOWFLAKE CURVE

LetΓbe a Jordan curve in the complex plane and let Γ_(λ) be the constant distance boundary ofΓ.Vellis and Wu[10]introduced the notion of a(ζ,r_(0))-chordal property which guarantees that,whenλis not too large, Γ_(λ) is a Jordan curve whenζ=1/2 and Γ_(λ) is a quasicircle when 0<ζ<1/2.We introduce the(ζ,r_(0),t)-chordal property,which generalizes the(ζ,r_(0))-chordal property,and we show that under the condition thatΓis(ζ,r_(0),√t)-chordal with 0<ζ<r_(0)^(1−√t)/2,there existsε>0 such that Γ_(λ) is a t-quasicircle once Γ_(λ) is a Jordan curve when 0<λ<ε.In the last part of this paper,we provide an example:Γis a kind of Koch snowflake curve which does not have the(ζ,r_(0))-chordal property for any 0<ζ<1/2,however Γ_(λ) is a Jordan curve whenλis small enough.Meanwhile,Γhas the(ζ,r_(0),√t)-chordal property with 0<ζ<r_(0)^(1−√t)/2 for any t∈(0,1/4).As a corollary of our main theorem, Γ_(λ) is a t-quasicircle for all 0<t<1/4 whenλis small enough.This means that our(ζ,r_(0),t)-chordal property is more general and applicable to more complicated curves.

On the Lower Bound of the Hausdorff Measure of the Koch Curve

This paper gives a lower bound of the Hausdorff measure of the Koch curve bymeans of the mass distribution principle.

Design of low-pass filter with 3-interations Koch fractal Pi-shaped DGS

A novel asymmetrical Pi-shaped defected ground structure （DGS） with 3-interations Koch fractal curves is proposed to design a microstrip low-pass filter （LPF） with uhra-wide stop-band （SB）. The proposed LPFs with a single resonator and two cascaded resonators are both designed, simulated, manufactured and measured. Simulation and experiment results demonstrate that the designed LPF has a very sharp transition band （TB） and an ultra-wide SB performance compared with the existed similar symmetrical and asymmetrical DGS. The proposed LPF with two cascaded resonators is with a compact size of 36. 8 mm x 24.0 mm, a very low insertion loss of less than 0.7 dB under 1.9 GHz, and a wide SB from 2.2 GHz to 8 GHz with rejection of larger than 30 dB.

Two-dimensional Wave Equations with Fractal Boundaries

This paper focuses on two cases of two-dimensional wave equations with fractal boundaries. The first case is the equation with classical derivative. The formal solution is obtained. And a definition of the solution is given. Then we prove that under certain conditions, the solution is a kind of fractal function, which is continuous, differentiable nowhere in its domain. Next, for specific given initial position and 3 different initial velocities, the graphs of solutions are sketched. By computing the box dimensions of boundaries of cross-sections for solution surfaces, we evaluate the range of box dimension of the vibrating membrane. The second case is the equation with p-type derivative. The corresponding solution is shown and numerical example is given.