Publications by Li,Zhi
Projection, Tangent Space, Covariant Derivative and Geodesic Equations
Projection onto the tangent plane https://math.stackexchange.com/questions/2275019/geodesic-curvature-and-projection-onto-the-tangent-plane https://math.stackexchange.com/questions/578680/projection-of-a-curve-on-a-plane Let’s have a unit speed parameters sphere surface \[\bar r = \{r \cos(\theta/r),r\sin(\theta/r), \sqrt{a^2 - r^2}\}\] Sphere ...
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Leverage Space Martingale
The book, The Leverage Space Trading Model by RALPH VINCE Page 173, applied Martingale, an introduction Suppose we found a set of f, i.e. leverages for some financial instruments. This can be done by controlling the leverages to allow the RD probability of drawdown at a low level, and choose leverages that produce the best GHPR among low probab...
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The Leverage Space Trading Model C H A P T E R 4
The book, The Leverage Space Trading Model by RALPH VINCE Mostly these are only organizing data, listing statistics and producing data tables. Please focusing on computing HPR for multiple financial instruments and summing them together. This case study is long, I better say more on few things. 1.Put each instrument data into 5 bins, compute bin ...
917 sym R (9828 sym/75 pcs)
Leverage Space and Drawdown
The book, The Leverage Space Trading Model by RALPH VINCE Page 131, binned data and joint probabilities MktSysA=c(-150.00,-45.33,-45.33,13.00,13.00,13.00,13.00,13.00, 79.67,79.67,79.67,136) MktSysB=c(253.00,-1000.00,-64.43,-64.43,-64.43,253.00 ,253.00, 448.00,-64.43,-64.43,-64.43,253) MktSysC=c(533.00,220.14,220.14,-500.0...
326 sym R (2151 sym/8 pcs)
Christoffel symbols second kind
Christoffel symbols of unit sphere – second kind Tangent vector \(e_i\) is a degree 1 partial derivative on the surface point. We have done it before, we let \(x^i\) be the surface parameters, \(\bar r\) be the surface point. Define \[e_i=\frac{\partial \bar r}{\partial x^i}.\] We also have done the degree 2 partial derivative, \[\frac{\partial...
2049 sym R (3664 sym/31 pcs)
Ricci tensor
Ricci tensor We need to start with 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="*"),collapse="+") ) } g=matrix(c("1","0","0","sin(theta)^2"),nrow=2) inv_g=matrix(c("1","0","0","1/sin(the...
384 sym R (1946 sym/7 pcs)
Surface Normal, Tangent Plane
Surface Normal, Tangent Plane This is a worked out example from Paul’s online notes, can be found here. https://tutorial.math.lamar.edu/Classes/CalcIII/ParametricSurfaces.aspx A surface is made of points in 3D space, and points are vectors of coordinates \(\{x,y,z\}\). We need 3 separate functions to define a point vector in 3D space. The \(u\)...
1909 sym R (935 sym/13 pcs)
Einstein Notation
Einstein Notation Superscript and subscript are for contravariant and covariant components, we do not use them so far. We only use subscript as indices. Vector dot product. \(a\cdot b=a_ib_i\) a=c(1,2,3) b=c(4,5,6) a%*%b ## [,1] ## [1,] 32 Dyadic vector product \(a⊗b=a_ib_j\) a=c(1,2,3) b=c(4,5,6) a%*%t(b) ## [,1] [,2] [,3] ##...
699 sym R (761 sym/16 pcs)
Line of Simultaneity
Line of Simultaneity Line of Simultaneity is basically a line of events which happen at the same time. If we have a frame that time t is the vertical axis and length x is the horizontal axis, so the line of simultaneity is just a horizontal line that parallel to the x axis. Because points on the line all have same height t, they all happen at the...
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Time is imaginary
Time is imaginary Time is imaginary, therefore, time squared is a negative number. Pythagorean theorem, in relativity theory, is called spacetime. And spacetime is invariant in all frames of reference. \[ict\] \[(ict)^2=-c^2t^2\] Here are the well known Lorentz Transforms, I coded them in to functions. #Lorentz Transforms x.prime <- function(x,v...
915 sym R (380 sym/5 pcs)