Publications by Li,Zhi
Document
Inventory Based Market Making, from page 172, OPTION TRADING by Euar Sinclair 2010 Welcome to my algorithmic trading and quantitative finance pages on rpub. In Euar Sinclair’s book, there is a great material of market making techniques. Here I wrote it in R language, and got similar results. There are 2 types of orders buy/sell. To buy we need ...
1905 sym R (995 sym/14 pcs) 2 img
Geometry of Relativity
Geometry of Relativity E has the coordinates x=4, t=5 in the statinary frame, and x’=1.73, t’=3.46 in the other frame. In relativity theory, axises need to tilt an angle. Let \(\theta\) be the tilting angle, and \(v\) be the relative speed, \(c\) is the light speed. \(v/c\) is tangent of the angle \(\theta\), then we have \[\theta=tan^{-1}(...
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Time jump
Time Jump This was a result from studying the twin paradox. The key idea is that one can start a new line of simultaneity by reversing velocity, just like jumping over time points. A pair of twins, they have a relative speed \(v\) to each other. We want to know which one of the twins is younger or in other words comparing their times. Because of ...
2417 sym R (135 sym/2 pcs) 1 img
Contravariant and covariant components
Contravatiant Covariant Components In a previous example, https://rpubs.com/lizhi1800/Geometry_of_Relativity, we have seen that to get observations \(\{1.732051,3.464102\}\)vector in the traveling frame, we need to use a transformation matrix multiplying vector \(E=\{4,5\}\). Vector \(E=\{4,5\}\) is called a contravariant vector. And we can also ...
1447 sym R (617 sym/18 pcs)
Christoffel symbols of unit sphere – first kind
Christoffel symbols of unit sphere – first kind Christoffel symbols of first kind are derived from the second kind. \[\Gamma_{lij}=g_{lk}\Gamma^k_{ij}\] Notation for first kind is \(\Gamma_{kij}\) or \([ij,k]\). Choosing \(k\) or \(l\) as indices does not matter. We need to remember indices \(ij\) is a 2D matrix, we make separate 2D matrices in...
3646 sym R (1649 sym/9 pcs)
Riemann curvature tensor second kind
Riemann curvature tensor – second kind I followed these papers, very detailed references. We need metric \(g\) and \(g\)-inverse to get Christoffel symbols of second kind. require(Deriv) ## Loading required package: Deriv ## Warning: package 'Deriv' was built under R version 4.0.5 dot_prod<-function(a,b){ Simplify(paste(paste(a,b,sep="*"),col...
561 sym R (3005 sym/13 pcs)
Differential Geometry 6
Curvature decomposition on surface https://rpubs.com/lizhi1800/Surface_Normal_Tangent_Plane There is a surface \(S\) and curve \(C\) the on surface. At point \(P\), the surface has a unit normal vector \(N\) and tangent plane \(\bar r_u,\bar r_v\). At point \(P\), the curve has a unit tangent vector \(t\) and unit normal vector \(n\). The curve�...
6943 sym R (3242 sym/51 pcs) 5 img
Differential Geometry 2
Space curve By xyz coordinate definition, note we are using \(t\), not arc length \(s\), as an input: \[\bar r=\{x=f(t),y=g(t),z=h(t)\}\] In our example, \(\bar r=\{t,t^2,2/3t^3\}\). require(Deriv) ## Loading required package: Deriv ## Warning: package 'Deriv' was built under R version 4.0.5 #rbar x="t" y="t^2" z="2/3*t^3" rbar=c(x,y,z) t=1...
3788 sym R (4717 sym/17 pcs)
Differential Geometry 1
This is my notes from reading solitaryroad.com documents. I wanted to focus on programming geometrical functions and symbolic math derivations. But eventually I realized I have missed out many important concepts. R programming may or may not be a good way of doing these. I am a reader, I have no choice. Derivative of arc length In the x-y plan, ...
1894 sym R (2216 sym/28 pcs)
Schwarzschild solution in R
Schwarzschild solution in R The plan is: 1. To have a generalized form of Ricci tensor and Ricci scalar. 2. Substitute into Einstein’s equation to find solution. To begin, Schwarzschild solution is spherical, we can have the spherical spacetime metric \[g_{ij}=\left| {\begin{array} *{1}&{0}&{0}&{0}\\ {0}&{-1}&{0}&{0}\\ {0}&{0}&{-r^2}&{0}\\ {0}&...
2871 sym R (8969 sym/46 pcs)