diff --git a/content/examples/6Rlinkrobot.md b/content/examples/6Rlinkrobot.md index cdb7d86..8b199b8 100644 --- a/content/examples/6Rlinkrobot.md +++ b/content/examples/6Rlinkrobot.md @@ -29,9 +29,7 @@ Let us denote by $z_1,\ldots,z_6$ the unit vectors that point in the direction o for $\alpha=(\alpha_1\ldots, \alpha_5)$ and $a=(a_1,\ldots,a_9)$ and $p=(p_1,p_2,p_3)$. The $\alpha_i$ are the "twist angle" between joints, the $a_i$ are the "link length" between joint axes -and the $p$ encodes the position of the hand. Here $\times$ is the cross product in $\mathbb{R}^3$. - -see the above reference for a detailed explanation on how these numbers are to be interpreted). Here $\times$ is the cross product in $\mathbb{R}^3$. +and the $p$ encodes the position of the hand (see the above reference for a detailed explanation on how these numbers are to be interpreted). Here $\times$ is the cross product in $\mathbb{R}^3$. In this notation the forward problem consists of computing $(\alpha,a)$ given the $z_i$ and $p$ and the backward problem consists of computing $z_2,\ldots,z_5$ that realize some fixed $(\alpha,a,p,z_1,z_6)$ (knowing $z_1,z_6$ means that the position where the robot is attached to the ground and the position where its hand should be are fixed). @@ -70,7 +68,7 @@ Result with 16 solutions ``` -We find 16 solutions, which is the correct number of solutions for these type of systems. +We find 16 solutions, which is the correct number of solutions for this type of systems. But if we study the problem a little bit closer, we can see that the equations are bi-homogeneous with respect to the variable groups $\\{z_2, z_4\\}$ and $\\{z_3, z_5\\}$. The multi-homogeneous Bezout number with respect ot this variable group is diff --git a/content/examples/bacillus-subtilis.md b/content/examples/bacillus-subtilis.md index ab3751f..42b53eb 100644 --- a/content/examples/bacillus-subtilis.md +++ b/content/examples/bacillus-subtilis.md @@ -110,7 +110,7 @@ F = System(SteadyStates, Now, we solve `F=0` for the parameter values `p`. ```julia -julia> S = solve(F, target_parameters = p) +S = solve(F, target_parameters = p) Result with 44 solutions ======================== • 76 paths tracked @@ -122,7 +122,7 @@ Result with 44 solutions Only real positive zeros are physically meaningful. Using our implementation we can certify that there are 12 real zeros: ```julia -julia> cert = certify(F, S, target_parameters = p) +cert = certify(F, S, target_parameters = p) CertificationResult =================== • 44 solution candidates given @@ -133,16 +133,16 @@ CertificationResult We can also certify that among them there is a unique positive one. ```julia -julia> c = certificates(cert) -julia> pos_real = c[findall(is_positive.(c))] -julia> length(pos_real) +c = certificates(cert) +pos_real = c[findall(is_positive.(c))] +length(pos_real) 1 ``` The positive real solution has the following values: ```julia -juila> certified_solution_interval(pos_real[1]) +certified_solution_interval(pos_real[1]) 10×1 Arblib.AcbMatrix: [0.00406661084 +/- 5.50e-12] + [+/- 2.45e-12]im [0.0557971948 +/- 5.02e-11] + [+/- 2.14e-11]im