The Koch curve as a smooth manifold

2008 ◽  
Vol 38 (2) ◽  
pp. 334-338 ◽  
Author(s):  
Marcelo Epstein ◽  
Jędrzej Śniatycki
Keyword(s):  
Smooth Manifold ◽  
Koch Curve

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

scholarly journals Missing the point in noncommutative geometry

Synthese ◽  
2021 ◽  
Author(s):  
Nick Huggett ◽  
Fedele Lizzi ◽  
Tushar Menon
Keyword(s):  
Scalar Field ◽  
Noncommutative Geometry ◽  
Smooth Manifold ◽  
Operational Definition ◽  
Spectral Triple ◽  
Fundamental Quantity

AbstractNoncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.

scholarly journals Pseudoinversion of degenerate metrics

2003 ◽  
Vol 2003 (55) ◽  
pp. 3479-3501 ◽  
Author(s):  
C. Atindogbe ◽  
J.-P. Ezin ◽  
Joël Tossa
Keyword(s):  
Differential Geometry ◽  
Large Class ◽  
Smooth Manifold ◽  
Differential Operators ◽  
Type Operator ◽  
Lorentzian Space ◽  
Lightlike Hypersurface ◽  
Definition Of ◽  
Lightlike Hypersurfaces ◽  
Theoretical Results

Let(M,g)be a smooth manifoldMendowed with a metricg. A large class of differential operators in differential geometry is intrinsically defined by means of the dual metricg∗on the dual bundleTM∗of 1-forms onM. If the metricgis (semi)-Riemannian, the metricg∗is just the inverse ofg. This paper studies the definition of the above-mentioned geometric differential operators in the case of manifolds endowed with degenerate metrics for whichg∗is not defined. We apply the theoretical results to Laplacian-type operator on a lightlike hypersurface to deduce a Takahashi-like theorem (Takahashi (1966)) for lightlike hypersurfaces in Lorentzian spaceℝ1n+2.

Fractal Koch curve for UHF band application

2007 ◽  
Author(s):  
Mohd Nazri A. Karim ◽  
Mohamad Kamal A. Rahim ◽  
Mohamad Irfan ◽  
Thelaha Masri
Keyword(s):  
Koch Curve ◽  
Uhf Band

scholarly journals Modified Koch Fractal Antenna for Multi and Wideband Wireless Applications

2021 ◽  
Vol 19 (2) ◽  
pp. 182-189
Author(s):  
Shweta Rani ◽  
Sushil Kakkar
Keyword(s):  
Numerical Simulations ◽  
Impedance Matching ◽  
Optimization Procedure ◽  
Design Parameters ◽  
Impedance Bandwidth ◽  
Fractal Antenna ◽  
Bacterial Foraging Optimization ◽  
Koch Curve ◽  
Wireless Applications ◽  
Koch Fractal

This paper focuses on the design and development of modified Koch fractal antenna. Compared to traditional Koch curve antenna, the presented antenna possesses a greater number of frequency bands and better impedance matching. Furthermore, the bacterial foraging optimization (BFO) approach is implemented to enhance the impedance bandwidth. The developed technique has been verified by employing various numerical simulations. The design parameters generated from the optimization procedure have been utilized to manufacture the antenna and the respective experimental and simulated results compared. The measured results show that the designed antenna exhibits multi and wideband behavior, covering WLAN, WIMAX, and various other wireless applications.

scholarly journals Dirac structures on Hilbert spaces

1999 ◽  
Vol 22 (1) ◽  
pp. 97-108 ◽  
Author(s):  
A. Parsian ◽  
A. Shafei Deh Abad
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Smooth Manifold ◽  
Real Hilbert Space ◽  
Dirac Structure ◽  
Dirac Structures ◽  
Basic Properties ◽  
Hilbert Subspace ◽  
Densely Defined

For a real Hilbert space(H,〈,〉), a subspaceL⊂H⊕His said to be a Dirac structure onHif it is maximally isotropic with respect to the pairing〈(x,y),(x′,y′)〉+=(1/2)(〈x,y′〉+〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures onHare in one-to-one correspondence with isometries onH, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structureLonH, everyz∈His uniquely decomposed asz=p1(l)+p2(l)for somel∈L, wherep1andp2are projections. Whenp1(L)is closed, for any Hilbert subspaceW⊂H, an induced Dirac structure onWis introduced. The latter concept has also been generalized.

The Koch Curve: A Geometric Proof

2007 ◽  
Vol 114 (1) ◽  
pp. 61-66 ◽  
Author(s):  
Šime Ungar
Keyword(s):  
Geometric Proof ◽  
Koch Curve

A NOVEL 3D MICROMIXER WITH QUARTIC KOCH CURVE FRACTAL

2021 ◽  
pp. 2150049
Author(s):  
SIYUE XIONG ◽  
XUEYE CHEN
Keyword(s):  
Numerical Simulation ◽  
Deflection Angle ◽  
Reynolds Numbers ◽  
Mixing Efficiency ◽  
Flow State ◽  
Koch Curve ◽  
Mixing Performance ◽  
Chaotic Convection ◽  
The Deflection Angle ◽  
High Application

In this paper, we mainly study the mixing performance of the micromixer with quartic Koch curve fractal (MQKCF) by numerical simulation. Changing the structure of the microchannel based on the fractal principle can significantly improve the fluid flow state in the microchannel and improve the mixing efficiency of the micromixer. This paper discussed the effects of different fractal deflection angles, microchannel heights and different fractal times on the mixing efficiency under four different Reynolds numbers (Re). It is found that changing the deflection angle of the fractal can bring extremely high benefits, which makes the fluid deflect and fold in the microchannel, enhancing the chaotic convection in the microchannel, and improve the mixing efficiency of the fluid. Under the reasonable arrangement of the quartic Koch curve fractal principle, it can give the micro-mixture more than 99% mixing efficiency. Based on the excellent mixing performance of MQKCF, it also has extremely high application value in the biochemical neighborhood.

Jacobi Matrices and the Spectrum of the Neumann Operator on a Family of Riemann Surfaces

1993 ◽  
Vol 45 (4) ◽  
pp. 709-726
Author(s):  
Julian Edward
Keyword(s):  
Harmonic Function ◽  
Riemann Surfaces ◽  
Smooth Manifold ◽  
Hypergeometric Functions ◽  
Normal Derivative ◽  
Jacobi Matrices ◽  
Uniform Bounds ◽  
Neumann Operator ◽  
Direct Sum ◽  
Asymptotics Of The Eigenvalues

AbstractThe Neumann operator is an operator on the boundary of a smooth manifold which maps the boundary value of a harmonic function to its normal derivative. The spectrum of the Neumann operator is studied on the curves bounding a family of Riemann surfaces. The Neumann operator is shown to be isospectral to a direct sum of symmetric Jacobi matrices, each acting on l2(ℤ). The Jacobi matrices are shown to be isospectral to generators of bilateral, linear birth-death processes. Using the connection between Jacobi matrices and continued fractions, it is shown that the eigenvalues of the Neumann operator must solve a certain equation involving hypergeometric functions. Study of the equation yields uniform bounds on the eigenvalues and also the asymptotics of the eigenvalues as the curves degenerate into a wedge of circles.

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