square173205

Generally the function having zero points or poles with non-integer order such as f(z) = (z-a)^(1.5+i0.3) must be dealt with on appropriate Riemann surface. I tried to extend the argument principle for such functions on a single sheet of Riemann surface and got a formula similar to that of ordinary argument principle. Using that formula the winding number of f(z) = (z-a)^(1.5+i0.3) around the origin is expressed as 1.5+i0.3. For details, visit: http://hecoaustralia.fortunecity.com/argument/argument.htm

Newton's gravitational potential is a very accurate approximation of that derived from general relativity when the gravity is not extremely strong. Therefore the quantization of Newton's gravitational potential might give efficient approximation for that of actual gravitational field under appropriate conditions. I tried to describe the gauge field derived from Newton's gravitational potential and its quantization in the following site; http://hecoaustralia.fortunecity.com/gravity/gravity.htm

Usually a periodic force is used for the forced vibration. However when the potential is not harmonic, we often get insufficient amplification by the periodic force. In that case the aperiodic pulse excitation might be effective. I tried to simulate how the non-linear oscillator is excited by pulsed external force in the follwoing site; http://hecoaustralia.fortunecity.com/forced/forced.htm

The ultraviolet divergence in QED is an old problem. The renormalisation removes almost every divergence appeared in the perturbation calculations, however the divergent integral before renormalisation should be finite if the physical model is correct. This might imply it is necessary to quantise special relativity itself, for as long as concerning point-like particle in Lorentz space the above problem seems not to be mended. I tried to quantise Lorentz space in order to deal with the ultraviolet divergence in the following site; http://hecoaustralia.fortunecity.com/relativity/relativity3.htm

Gauss's law describes the electric field created by the point charge. This law is applied when the point charge is isotropic only. Do you think it's a trivial matter? I don't think so. I tried to extend Gauss's law to the case when the point charge is not isotropic in the following site; http://hecoaustralia.fortunecity.com/gauss/gauss.htm

The complex field is regarded as the extension of real field adding the root of the equation: x^2+1 = 0. Then does the extension of rational field adding some irrational square root represent the space having Lorentz metric? I tried to construct such space in the following site; http://hecoaustralia.fortunecity.com/complex/complex.htm

Surely, but even in the case of quadratic polynomial there exists a pair of domain and region where the mapping is one-to-one.

Isn't it a regulator?

In general solving the eigenvalue problem of three-dimensional Schroedinger equation is difficult unless the shape of potential is particularly simple. I tried to develop a generic method to solve that problem: In general solving Schroedinger equation is much harder than modeling quantum phenomenon with that. For example, the movement of electron around proton is quantised with the following Schroedinger equation. (ih/(2p))¶f/¶t = -{h2/((2p)22m)}Df-e2f/r To solve it as eigenvalue problem it is necessary to express the solution with the spherical function and Laguerre function. Apart from Laguerre function, calculating practically spherical function is laboured task. Are there any easier methods for solving the Schroedinger equation especially on the eigenvalue problem? One simple way is to specify the coordinates of electron on y- and z-axis. Then the Schroedinger equation becomes one-dimensional as, (ih/(2p))¶f(x,y0,z0)/¶t = -{h2/((2p)22m)}¶2f(x,y0,z0)/¶x2-e2f/r(x,y0,z0) As the eigenvalue problem, it is rewritten as, -{h2/((2p)22m)}¶2f(x,y0,z0)/¶x2-e2f/r(x,y0,z0) = W(y0,z0)f This is just ordinary differential equation. So we can solve it relatively easily (approximately in some cases, though.) However since the potential is the function of not only x but also y and z generally, the "eigenvalue" W depends on the coordinates of electron on y- and z-axis, i.e, W(y0,z0) ¹ W(y0',z0') where (y0,z0) ¹ (y0',z0') Is it impossible to obtain the three dimensional solution which has specific eigenvalue from one dimensional solution mentioned above? Let's go back to classical dynamics. The movement of point mass x = (x1(t),x2(t),x3(t)) under the potential U is described as, (1/2)m|dx/dt|2+U(x) = E In this case the total energy E doesn't change through time as you know. Now we describe the components x1, x2 and x3 of x with virtually independent three parameters as, x = (x1(t1),x2(t2),x3(t3)) Then the velocity vector v of x is, v = (dx1/dt1,dx2/dt2,dx3/dt3) It should be t1 = t2 = t3 = t in order to coincide v with ordinary velocity. The partial derivative of kinetic energy for tj is, (¶/¶tj)(1/2)m|dx/dt|2 = (¶/¶tj)(1/2)m(dxj/dtj)2 = m(d2xj/dtj2)(dxj/dtj) (j = 1,2,3) while the partial derivative of potential U for tj is, ¶U/¶tj = Sk(¶U/¶xk)(dxk/dtj) = (¶U/¶xj)(dxj/dtj) On the other hand, Newton's equation is expressed with the parameter tj as, m(d2xj/dtj2) = -¶U/¶xj Hence, (¶/¶tj){(1/2)m(dxj/dtj)2+U} = m(d2xj/dtj2)(dxj/dtj)+(¶U/¶xj)(dxj/dtj) = {m(d2xj/dtj2)+¶U/¶xj}(dxj/dtj) = 0 Namely when let Ej the j-component of kinetic energy of mass point, i.e. Ej = (1/2)m(dxj/dtj)2 (caution: not take the sum regarding the index j) then the above equation indicates that ((Ej + potential energy)) doesn't change through the time tj. Although the time tj is imaginary parameter, the preservation of that sum is not meaningless physically. For example, say j = 1, it can be rewritten with ordinary time t as, (d/dt){(1/2)m(dx/dt)2+U(x(t),y(t0),z(t0))} = 0 where the coordinate y and z of mass point are fixed as y(t0) and z(t0). The above preservation of the j-component kinetic energy plus potential energy is quantised as, (ih/(2p))¶fj/¶tj = -{h2/((2p)22m)}¶2fj/¶xj2+Ufj This is nothing other than ordinary one-dimensional Schroedinger equation. Therefore the eigenvalue problem on three-dimensional Schroedinger equation is decomposed into one-dimensional eigenvalue problem as, -{h2/((2p)22m)}¶2fj/¶xj2+Ufj = wjfj where wj corresponds to the j-component kinetic energy plus potential energy in the case of classical dynamics mentioned above. The solution fj is regarded as the function of coordinate xj and eigenvalue wj as, fj = fj(wj,xj) However as I mentioned previously, "eigenvalue" wj of one-dimensional Schroedinger equation depends on the other coordinates. Therefore we must take the convolution of wj for getting eigenfunction f(W,x,y,z) of Schroedinger equation on the three-dimensional space as, f(W,x,y,z) = òdw2dw3f1(W-w2-w3,x)f2(w2,y)f3(w3,z) where W is the eigenvalue of three-dimensional Schroedinger equation. Actually W is obtained from solving one-dimensional eigenvalue problem on Schroedinger equation for each component.

Is it possible to control servo system without integral action? Here is the method to control it with modified proportional action only; http://hecoaustralia.fortunecity.com/digital3/ftsc3.htm

If a phonon follows a certain gauge transform, it might have a mass following the spontaneous break of symmetry of the potential. I tried to interpret the high temperature superconductivity with the phonon having a mass in the following site; http://hecoaustralia.fortunecity.com/phonon/phonon.htm

Galois's theorem describes the relation between subfields and Galois groups on the field. I tried to apply Galois's theorem to the relation between subspaces and automorphisms on the topological space in the following site; http://hecoaustralia.fortunecity.com/topology/topology.htm

You don't allow using imaginary number, do you?

In general the reflecting optical telescope has a large-aperture concave mirror for gathering and focusing light from astronomical bodies. As you've known, larger the size of mirror becomes, more difficult making such mirror with sufficient accuracy. Here is an alternative method to avert such difficulty; http://hecoaustralia.fortunecity.com/telescope/reflect.htm

QWT, too:eyebrow: