## algebra – Simplify simple symbolic expression with KroneckerDelta

Suppose I have an expression like

``````A(b) B(d) KroneckerDelta(b, d + 1)
``````

I would like to simplify this expression to get

``````A(d + 1) B(d)
``````

(or vice versa). In my case, lowercase letters are integers $${0, …,infty }$$.

EDIT 1: I know the two expressions seems incompatible, but the fact is that the are inside a double summations over $$b,d$$. I do not want to deal with this summation explicitly, I just want the expression for a single (complicated) term after summing over e.g. $$n$$.

## symbolic – Meaning of the following output?

I got the following as an eigenvalue:

``````Root(-γ^4 ν^5 σ^3 -
4 γ^3 ν^6 σ^3 -
6 γ^2 ν^7 σ^3 -
4 γ ν^8 σ^3 - ν^9 σ^3 -
4 γ^4 ν^4 ξ σ^3 -
16 γ^3 ν^5 ξ σ^3 -
24 γ^2 ν^6 ξ σ^3 -
16 γ ν^7 ξ σ^3 -
4 ν^8 ξ σ^3 -
6 γ^4 ν^3 ξ^2 σ^3 -
24 γ^3 ν^4 ξ^2 σ^3 -
36 γ^2 ν^5 ξ^2 σ^3 -
24 γ ν^6 ξ^2 σ^3 -
6 ν^7 ξ^2 σ^3 -
4 γ^4 ν^2 ξ^3 σ^3 -
16 γ^3 ν^3 ξ^3 σ^3 -
24 γ^2 ν^4 ξ^3 σ^3 -
16 γ ν^5 ξ^3 σ^3 -
4 ν^6 ξ^3 σ^3 - γ^4 ν ξ^4
σ^3 - 4 γ^3 ν^2 ξ^4 σ^3 -
6 γ^2 ν^3 ξ^4 σ^3 -
4 γ ν^4 ξ^4 σ^3 - ν^5 ξ^4
σ^3 + β γ^3 ν^4 σ^4 -
4 γ^4 ν^4 σ^4 +
3 β γ^2 ν^5 σ^4 -
16 γ^3 ν^5 σ^4 +
3 β γ ν^6 σ^4 -
24 γ^2 ν^6 σ^4 + β ν^7 σ^4 -
16 γ ν^7 σ^4 - 4 ν^8 σ^4 +
4 β γ^3 ν^3 ξ σ^4 -
13 γ^4 ν^3 ξ σ^4 +
12 β γ^2 ν^4 ξ σ^4 -
55 γ^3 ν^4 ξ σ^4 +
12 β γ ν^5 ξ σ^4 -
87 γ^2 ν^5 ξ σ^4 +
4 β ν^6 ξ σ^4 -
61 γ ν^6 ξ σ^4 -
16 ν^7 ξ σ^4 +
6 β γ^3 ν^2 ξ^2 σ^4 -
15 γ^4 ν^2 ξ^2 σ^4 +
18 β γ^2 ν^3 ξ^2 σ^4 -
69 γ^3 ν^3 ξ^2 σ^4 +
18 β γ ν^4 ξ^2 σ^4 -
117 γ^2 ν^4 ξ^2 σ^4 +
6 β ν^5 ξ^2 σ^4 -
87 γ ν^5 ξ^2 σ^4 -
24 ν^6 ξ^2 σ^4 +
4 β γ^3 ν ξ^3 σ^4 -
7 γ^4 ν ξ^3 σ^4 +
12 β γ^2 ν^2 ξ^3 σ^4 -
37 γ^3 ν^2 ξ^3 σ^4 +
12 β γ ν^3 ξ^3 σ^4 -
69 γ^2 ν^3 ξ^3 σ^4 +
4 β ν^4 ξ^3 σ^4 -
55 γ ν^4 ξ^3 σ^4 -
16 ν^5 ξ^3 σ^4 + β γ^3 ξ^4
σ^4 - γ^4 ξ^4 σ^4 +
3 β γ^2 ν ξ^4 σ^4 -
7 γ^3 ν ξ^4 σ^4 +
3 β γ ν^2 ξ^4 σ^4 -
15 γ^2 ν^2 ξ^4 σ^4 + β ν^3
ξ^4 σ^4 - 13 γ ν^3 ξ^4 σ^4 -
4 ν^4 ξ^4 σ^4 +
3 β γ^3 ν^3 σ^5 -
6 γ^4 ν^3 σ^5 +
9 β γ^2 ν^4 σ^5 -
24 γ^3 ν^4 σ^5 +
9 β γ ν^5 σ^5 -
36 γ^2 ν^5 σ^5 + 3 β ν^6 σ^5 -
24 γ ν^6 σ^5 - 6 ν^7 σ^5 +
9 β γ^3 ν^2 ξ σ^5 -
15 γ^4 ν^2 ξ σ^5 +
30 β γ^2 ν^3 ξ σ^5 -
69 γ^3 ν^3 ξ σ^5 +
33 β γ ν^4 ξ σ^5 -
117 γ^2 ν^4 ξ σ^5 +
12 β ν^5 ξ σ^5 -
87 γ ν^5 ξ σ^5 -
24 ν^6 ξ σ^5 +
9 β γ^3 ν ξ^2 σ^5 -
12 γ^4 ν ξ^2 σ^5 +
36 β γ^2 ν^2 ξ^2 σ^5 -
69 γ^3 ν^2 ξ^2 σ^5 +
45 β γ ν^3 ξ^2 σ^5 -
138 γ^2 ν^3 ξ^2 σ^5 +
18 β ν^4 ξ^2 σ^5 -
117 γ ν^4 ξ^2 σ^5 -
36 ν^5 ξ^2 σ^5 +
3 β γ^3 ξ^3 σ^5 -
3 γ^4 ξ^3 σ^5 +
18 β γ^2 ν ξ^3 σ^5 -
27 γ^3 ν ξ^3 σ^5 +
27 β γ ν^2 ξ^3 σ^5 -
69 γ^2 ν^2 ξ^3 σ^5 +
12 β ν^3 ξ^3 σ^5 -
69 γ ν^3 ξ^3 σ^5 -
24 ν^4 ξ^3 σ^5 +
3 β γ^2 ξ^4 σ^5 -
3 γ^3 ξ^4 σ^5 +
6 β γ ν ξ^4 σ^5 -
12 γ^2 ν ξ^4 σ^5 +
3 β ν^2 ξ^4 σ^5 -
15 γ ν^2 ξ^4 σ^5 -
6 ν^3 ξ^4 σ^5 +
3 β γ^3 ν^2 σ^6 -
4 γ^4 ν^2 σ^6 +
9 β γ^2 ν^3 σ^6 -
16 γ^3 ν^3 σ^6 +
9 β γ ν^4 σ^6 -
24 γ^2 ν^4 σ^6 + 3 β ν^5 σ^6 -
16 γ ν^5 σ^6 - 4 ν^6 σ^6 +
6 β γ^3 ν ξ σ^6 -
7 γ^4 ν ξ σ^6 +
24 β γ^2 ν^2 ξ σ^6 -
37 γ^3 ν^2 ξ σ^6 +
30 β γ ν^3 ξ σ^6 -
69 γ^2 ν^3 ξ σ^6 +
12 β ν^4 ξ σ^6 -
55 γ ν^4 ξ σ^6 -
16 ν^5 ξ σ^6 +
3 β γ^3 ξ^2 σ^6 -
3 γ^4 ξ^2 σ^6 +
21 β γ^2 ν ξ^2 σ^6 -
27 γ^3 ν ξ^2 σ^6 +
36 β γ ν^2 ξ^2 σ^6 -
69 γ^2 ν^2 ξ^2 σ^6 +
18 β ν^3 ξ^2 σ^6 -
69 γ ν^3 ξ^2 σ^6 -
24 ν^4 ξ^2 σ^6 +
6 β γ^2 ξ^3 σ^6 -
6 γ^3 ξ^3 σ^6 +
18 β γ ν ξ^3 σ^6 -
27 γ^2 ν ξ^3 σ^6 +
12 β ν^2 ξ^3 σ^6 -
37 γ ν^2 ξ^3 σ^6 -
16 ν^3 ξ^3 σ^6 +
3 β γ ξ^4 σ^6 -
3 γ^2 ξ^4 σ^6 +
3 β ν ξ^4 σ^6 -
7 γ ν ξ^4 σ^6 -
4 ν^2 ξ^4 σ^6 + β γ^3 ν
σ^7 - γ^4 ν σ^7 +
3 β γ^2 ν^2 σ^7 -
4 γ^3 ν^2 σ^7 +
3 β γ ν^3 σ^7 -
6 γ^2 ν^3 σ^7 + β ν^4 σ^7 -
4 γ ν^4 σ^7 - ν^5 σ^7 + β
γ^3 ξ σ^7 - γ^4 ξ σ^7 +
6 β γ^2 ν ξ σ^7 -
7 γ^3 ν ξ σ^7 +
9 β γ ν^2 ξ σ^7 -
15 γ^2 ν^2 ξ σ^7 +
4 β ν^3 ξ σ^7 -
13 γ ν^3 ξ σ^7 -
4 ν^4 ξ σ^7 +
3 β γ^2 ξ^2 σ^7 -
3 γ^3 ξ^2 σ^7 +
9 β γ ν ξ^2 σ^7 -
12 γ^2 ν ξ^2 σ^7 +
6 β ν^2 ξ^2 σ^7 -
15 γ ν^2 ξ^2 σ^7 -
6 ν^3 ξ^2 σ^7 +
3 β γ ξ^3 σ^7 -
3 γ^2 ξ^3 σ^7 +
4 β ν ξ^3 σ^7 -
7 γ ν ξ^3 σ^7 -
4 ν^2 ξ^3 σ^7 + β ξ^4 σ^7 -
γ ξ^4 σ^7 - ν ξ^4 σ^7 + (γ^3
ν^2 ξ σ^2 + 3 γ^2 ν^3 ξ σ^2 +
3 γ ν^4 ξ σ^2 + ν^5 ξ σ^2
+ 2 γ^3 ν ξ^2 σ^2 +
6 γ^2 ν^2 ξ^2 σ^2 +
6 γ ν^3 ξ^2 σ^2 +
2 ν^4 ξ^2 σ^2 + γ^3 ξ^3 σ^2 +
3 γ^2 ν ξ^3 σ^2 +
3 γ ν^2 ξ^3 σ^2 + ν^3 ξ^3
σ^2 + β γ^2 ν^2 σ^3 +
3 β γ ν^3 σ^3 +
2 β ν^4 σ^3 +
2 β γ^2 ν ξ σ^3 + γ^3 ν
ξ σ^3 + 7 β γ ν^2 ξ σ^3 +
4 γ^2 ν^2 ξ σ^3 +
5 β ν^3 ξ σ^3 +
6 γ ν^3 ξ σ^3 +
3 ν^4 ξ σ^3 + β γ^2 ξ^2
σ^3 + γ^3 ξ^2 σ^3 +
5 β γ ν ξ^2 σ^3 +
6 γ^2 ν ξ^2 σ^3 +
4 β ν^2 ξ^2 σ^3 +
11 γ ν^2 ξ^2 σ^3 +
6 ν^3 ξ^2 σ^3 + β γ ξ^3
σ^3 +
2 γ^2 ξ^3 σ^3 + β ν ξ^3
σ^3 + 5 γ ν ξ^3 σ^3 +
3 ν^2 ξ^3 σ^3 + β γ^2 ν
σ^4 + 4 β γ ν^2 σ^4 +
3 β ν^3 σ^4 + β γ^2 ξ
σ^4 +
7 β γ ν ξ σ^4 + γ^2 ν
ξ σ^4 + 7 β ν^2 ξ σ^4 +
4 γ ν^2 ξ σ^4 +
3 ν^3 ξ σ^4 +
3 β γ ξ^2 σ^4 + γ^2 ξ^2
σ^4 + 5 β ν ξ^2 σ^4 +
6 γ ν ξ^2 σ^4 +
6 ν^2 ξ^2 σ^4 + β ξ^3 σ^4 +
2 γ ξ^3 σ^4 +
3 ν ξ^3 σ^4 + β γ ν σ^5
+ β ν^2 σ^5 + β γ ξ σ^5 +
2 β ν ξ σ^5 + γ ν ξ
σ^5 + ν^2 ξ σ^5 + β ξ^2 σ^5 +
γ ξ^2 σ^5 +
2 ν ξ^2 σ^5 + ξ^3 σ^5) #1 +
(γ^2 ν σ + 3 γ ν^2 σ +
2 ν^3 σ + γ^2 ξ σ +
4 γ ν ξ σ +
3 ν^2 ξ σ + γ ξ^2 σ + ν
ξ^2 σ + γ^2 σ^2 + β ν σ^2 +
4 γ ν σ^2 +
3 ν^2 σ^2 + β ξ σ^2 +
2 γ ξ σ^2 +
4 ν ξ σ^2 + ξ^2 σ^2 + γ
σ^3 + ν σ^3 + ξ σ^3) #1^2 + #1^3 &,
1)/(σ (γ ν + ν^2 + γ ξ + ν
ξ + γ σ + ν σ + ξ σ))
``````

What I don’t understand is the following:

``````#1 + (γ^2 ν σ + 3 γ ν^2 σ +
2 ν^3 σ + γ^2 ξ σ +
4 γ ν ξ σ +
3 ν^2 ξ σ + γ ξ^2 σ + ν
ξ^2 σ + γ^2 σ^2 + β ν σ^2 +
4 γ ν σ^2 +
3 ν^2 σ^2 + β ξ σ^2 +
2 γ ξ σ^2 +
4 ν ξ σ^2 + ξ^2 σ^2 + γ
σ^3 + ν σ^3 + ξ σ^3) #1^2 + #1^3 &,
1)/(σ (γ ν + ν^2 + γ ξ + ν
ξ + γ σ + ν σ + ξ σ))
``````

What is the meaning of the hashtags?

They didn’t appear in any other solution to other models I considered.

## Calculating number of nonzero elements in any symbolic matrix

``````ClearAll(p, d)
Format(p(a_, b_)) := Subscript(p, Row@{a, b})
d = 2
p(a_, a_) := 0;
p(a_, b_) := -p(b, a) /; (a > b)
pMat = Array(p, {2*d, 2*d})
MatrixForm(pMat)
Length(Select(tMat, # != 0 &))
``````

How to calculate number of nonzero elements in any symbolic matrix. I have given an example of one symbolic matrix p above.

## regionmeasure – Symbolic computation of a region boundary

Consider the three disk regions plotted here, where of course I had to set a single radius (scale) to make an actual plot:

But in my actual problem, the scale is not set. For instance, we can set the radius of the large green disk to be $$x>0$$, and without loss of generality place its center at the origin.

I’m interested in the area of the dark orange wedge, as well as the length of its perimeter.

Thus I define the regions symbolically as follows:

``````(ScriptCapitalR)left = Disk({0, x/2}, x/2);
(ScriptCapitalR)top = Disk({x/2, x}, x/2);
(ScriptCapitalR)big = Disk({0, 0}, x);
(ScriptCapitalR)goal =
RegionDifference(
RegionIntersection((ScriptCapitalR)top, (ScriptCapitalR)big),
(ScriptCapitalR)left);
Assuming(x > 0, RegionMeasure((ScriptCapitalR)goal))
``````

I get the RegionMeasure (i.e., area) just as I seek… in terms of $$x$$:

$$frac{1}{8} x^2 left(-2+pi -tan ^{-1}left(frac{44}{117}right)right)$$

Fine.

Now I try to find its perimeter. First I define its boundary:

``````(ScriptCapitalR)goalBoundary =
Assuming(x > 0, RegionBoundary((ScriptCapitalR)goal));
``````

Now, I try to find the length of its perimeter:

``````ArcLength((ScriptCapitalR)goalBoundary)
``````

or

``````RegionMeasure((ScriptCapitalR)goalBoundary)
``````

However, if I set a particular value for $$x$$, I get the perimeter fine:

``````ArcLength((ScriptCapitalR)goalBoundary) /. x -> 1
``````

$$frac{1}{4} left(pi +2 sin ^{-1}left(frac{3}{5}right)+4 sin ^{-1}left(frac{4}{5}right)right)$$

(Of course I can plot and visualize the boundary only if I set a value for $$x$$, as in: `RegionPlot((ScriptCapitalR)goalBoundary /. x -> 1)`. Again, fine.)

How do I compute the perimeter symbolically for arbitrary $$x>0$$?

Of course I could solve for $$x to 1$$ and then exploit my knowledge that the perimeter scales as $$x$$, but I’m looking for a more general solution (applicable to other problems) that relies on symbolic computation, not my “human understanding” of dimensional scaling.

## calculus and analysis – Get derivation of symbolic matrix equations

I want to use Mathematica to infer kinematics equations (derivate position equations to speed equations).

$$begin{equation} vec{c}=vec{t}_{m}+R_{b m} vec{c^{m}} end{equation}$$
$$begin{equation} vec{d}=vec{t}_{m}+R_{b m} vec{t_{s}}+R_{b m} R_{m s} vec{d^{s}} end{equation}$$

where $$R_i$$ means the rotation matrix, the $$vec{c^m},vec{d^s}$$ are constant, and others are relative to time $$t$$

I want to use Mathematica to get the symbolic form of $$frac{text{d}vec{c}}{text{d}t}frac{text{d}vec{d}}{text{d}t}$$. The answer I get by hand is this (for convenience, I don’t write the vector hat $$vec{}$$ in the equations below):

$$begin{array}{c} v_{c}=frac{d c}{d t}=R_{b m}^top dot{t_{m}}+R_{b m}^top Sleft(w_{m}right) R_{b m} c^{m}=R_{b m}^top dot{t_{m}}+Sleft(R_{b m}^top w_{m}right) c^{m} \ =R_{b m}^top dot{t_{m}}-Sleft(c^{m}right) R_{b m}^top w_{m} \ v_{d}=frac{d d}{d t}=dot{t_{s}}+R_{b m}^top dot{t_{m}}+R_{b m}^top Sleft(w_{m}right) R_{b m} t_{s}+R_{b m}^top Sleft(w_{m}right) R_{b m} R_{m s} d^{s} \ quad+R_{b m}^top R_{b m}left(Sleft(w_{s m}right) R_{m s} d^{s}right) \ =dot{t_{s}}+R_{b m}^top dot{t_{m}}-Sleft(t_{s}right) R_{b m}^top w_{m}-Sleft(R_{m s} d^{s}right) R_{b m}^top w_{m}+left(Sleft(w_{s m}right) R_{m s} d^{s}right) \ left.=dot{t_{s}}+R_{b m}^top dot{t_{m}}-Sleft(t_{s}right)right) R_{b m}^top w_{m}-Sleft(R_{m s} d^{s}right) R_{b m}^top w_{m}-left(Sleft(R_{m s} d^{s}right) w_{s m}right) end{array}$$
where $$S(vec{a})$$ means the (antisymmetric) skew matrix form of vector $$vec{a}$$. How could I get the result?

By the way, what bothers me most is how could I represent a symbolic matrix with a number?

The problem source:
(1) Y. Cai, S. Zheng, W. Liu, Z. Qu和J. Han, 《Model Analysis and Modified Control Method of Ship-Mounted Stewart Platforms for Wave Compensation》, IEEE Access, 卷 9, 页 4505-4517, 2021, doi: 10.1109/ACCESS.2020.3047063.

## Symbolic Computation with Summation – Mathematica Stack Exchange

With the help of the community, I have writen the following formula $$a,d$$ into mathematica. However I want to do symbolic computation with respect to $$T,n$$.

$$a,d$$ is defined as follows:

$$begin{multline*} a=sum_{t=0}^{T-1}f_1^2(t)+11sum_{t=0}^{T-1}f_2^2(t)+\ sum_{t=0}^Tmathbb{I}(n_{t+1}>0)sum_{tau =0}^{n_{t+1}-1}g_1^2(t,tau)+11sum_{t=0}^{T}mathbb{I}(n_{t+1}>0)sum_{tau=0}^{n_{t+1}-1}g_2^2(t,tau), end{multline*}$$

$$begin{multline*} d=10Tcdot operatorname{logit}^2left(frac{alpha}{alpha+beta}right)+2cdot operatorname{logit}left(frac{alpha}{alpha+beta}right)cdot left(sum_{t=0}^{T-1}f_3(t)right)+\10cdot operatorname{logit}^2left(frac{alpha}{alpha+beta}right)cdot left(sum_{t=0}^{T}n_{t+1}right) +2cdot operatorname{logit}left(frac{alpha}{alpha+beta}right)cdot left(sum_{t=0}^{T}n_{t+1}cdot f_2(t)right), end{multline*}$$

where $$mathbb{I}(cdot)$$ is the indicator function, $$n$$ is a vetor, $$n_{t}$$ means the t-th element of $$n$$, which starts from 0, and $$T$$ is a integer.

$$f_1(t), f_2(t), g_1(t,tau), g_2(t,tau)$$ are defined below:

begin{align*} f_1(t) &= operatorname{logit}left(frac{alpha+t}{alpha+beta+(sum_{tau=0}^{t}n_{tau+1})+t}right)\ f_2(t) &= operatorname{logit}left(frac{alpha+t}{alpha+beta+t}right)\ g_1(t,tau)&=operatorname{logit}left(frac{alpha+t}{alpha+beta+(sum_{t’=1}^{t}n_{t’})+t+tau}right)\ g_2(t, tau)&=operatorname{logit}left(frac{alpha}{alpha+beta+(sum_{t’=0}^{t}n_{t’})+tau}right), end{align*}

where $$alpha, beta$$ are integer, $$n_0$$ is defined as 0 to make the formula more clean, and $$operatorname{logit}(x)=operatorname{log}(frac{x}{1-x})$$.

Code is as follows:

``````logit(x_) := Log(x/(1 - x));

f1(t_, (Alpha)_, (Beta)_, n_List) :=
logit(((Alpha) + t)/((Alpha) + (Beta) +
Sum(n(((Tau) + 1)), {(Tau), 0, t}) + t));

f2(t_, (Alpha)_, (Beta)_, n_List) :=
logit(((Alpha) + t)/((Alpha) + (Beta) + t));
g1(t_, (Tau)_, (Alpha)_, (Beta)_, n_List) :=
logit(((Alpha) + t)/((Alpha) + (Beta) +
Sum(n((tt)), {tt, 1, t}) + t + (Tau)));

g2(t_, (Tau)_, (Alpha)_, (Beta)_, n_List) :=
logit((Alpha)/((Alpha) + (Beta) +
Sum(n((tt)), {tt, 1, t}) + (Tau)));

a(T_, n_List) := (Sum(f1(t, (Alpha), (Beta), n)^2, {t, 0, T - 1}) +
11 Sum(f2(t, (Alpha), (Beta), n)^2, {t, 0, T - 1}) +
Sum(If(n((t + 1)) > 0,
Sum(g1(t, (Tau), (Alpha), (Beta), n)^2, {(Tau), 0,
n((t + 1)) - 1}), 0), {t, 0, T}) +
11 Sum(If(n((t + 1)) > 0,
Sum(g2(t, (Tau), (Alpha), (Beta), n)^2, {(Tau), 0,
n((t + 1)) - 1}), 0), {t, 0, T}));

d(T_, n_List) :=
10 T*logit((Alpha)/((Alpha) + (Beta)))^2 +
2*logit((Alpha)/((Alpha) + (Beta)))*
Sum(f3(t, (Alpha), (Beta), n), {t, 0, T - 1}) +
10*logit((Alpha)/((Alpha) + (Beta)))^2*
Sum(n((t + 1)), {t, 0, T}) +
2*logit((Alpha)/((Alpha) + (Beta)))*
Sum(n((t + 1))*f2(t, (Alpha), (Beta), n), {t, 0, T})
``````

However, when I do

``````a(T, 9, 1, n)*d(T, 9, 1, n) // Simplify
``````

get

``````a(T, 9, 1, n) d(T, 9, 1, n)
``````

, which is surely not what I expect.

## equation solving – How to locate the real maximizer of a nonlinear symbolic function?

I have a symbolic function:

`f(a_)=(((-1+a)^2 b^2+(-1+b) b (-2+a+a b) n-(-1+b)^2 (-1+a b) n^2) ((-2+a) c (-n+b (-1+a+n))-2 (-1+a) (b (-1+n)-n) v))/(2 (-1+a) (-1+b) (b (-1+n)-n) (n-b (-1+a+n))^2)`

where `0<a<1, n>1, 0<b<1, v>c>0`.

I need to find its maximizer. Numerically, I can see that sometimes `a=0` is the maximizer and sometimes `a^*` is a number between 0 and 1. How can I analytically prove these observations? I examined the first-order condition, but no luck.

## How to write a really complicated formula into mathematica and do symbolic computation?

I want write the following formula $$a$$ into mathematica and do symbolic computation. $$a$$ is defined as follows:
$$begin{multline*} a=sum_{t=0}^{T-1}f_1^2(t)+11sum_{t=0}^{T-1}f_2^2(t)+\ sum_{t=0}^Tmathbb{I}(n_{t+1}>0)sum_{tau =0}^{n_{t+1}-1}g_1^2(t,tau)+11sum_{t=0}^{T}mathbb{I}(n_{t+1}>0)sum_{tau=0}^{n_{t+1}-1}g_2^2(t,tau), end{multline*}$$

where $$mathbb{I}(cdot)$$ is the indicator function, $$n$$ is a vetor, $$n_{t}$$ means the t-th element of $$n$$, and $$T$$ is a integer.

$$f_1(t), f_2(t), g_1(t,tau), g_2(t,tau)$$ are defined below:

begin{align*} f_1(t) &= operatorname{logit}left(frac{alpha+t}{alpha+beta+sum_{tau=0}^{t}n_{tau+1}+t}right)\ f_2(t) &= operatorname{logit}left(frac{alpha+t}{alpha+beta+t}right)\ g_1(t,tau)&=operatorname{logit}left(frac{alpha+t}{alpha+beta+sum_{t’=1}^{t}n_{t’}+t+tau}right)\ g_2(t, tau)&=operatorname{logit}left(frac{alpha}{alpha+beta+sum_{t’=0}^{t}n_{t’}+tau}right), end{align*}

where $$alpha, beta$$ are integer, and $$operatorname{logit}(x)=operatorname{log}(frac{x}{1-x})$$.

An example is shown below: when $$T=2,alpha=9,beta=1, n=(0,0,0,1)^T$$, $$a=180.41$$.

I will really appreciate it if anybody could help me out of this problem, which troubles me for a long time. Thanks a lot!

I want to create a shadow directory of a directory D. The shadow directory D1 should have the same structure as D, but should not share any nodes with D. Each file in D should be represented by a symlink to that file in D1. The idea is that operations on D1 should never modify D, so that I can freely delete anything in D1 without affecting D. How can I achieve this? I could of course do a simple tree copy of D but that would make duplicate copies of the files.

## differential equations – Replace in a Symbolic Derivative doesnt work with Pi/2

I was writing some small functions for GR applications, and I was defining a Function that gives me the Geodesic equations. When testing if those worked with the Schwarzschild Metric I came upon a problem when trying to replace the angle $$theta$$ with $$frac{pi}{2}$$ this then didn’t properly simplify it when there are derivatives of $$theta$$.

I have made a simple example to showcase what my problem is:

``````Sum(D(xx((i))((Tau)), (Tau)), {i, 4}) /. t -> Pi/2
``````

This produces the following output:

$$left(frac{pi }{2}right)'(tau )+x'(tau )+y'(tau )+z'(tau )$$

What can I do to either prevent this from happening or resolve this issue?