# The polymath blog

## October 19, 2018

Filed under: discussion — Gil Kalai @ 9:10 am
Tags: , ,

Three short items:

### Progress on Rota’s conjecture (polymath12) by Bucić, Kwan, Pokrovskiy, and Sudakov

First, there is a remarkable development on Rota’s basis conjecture (Polymath12) described in the paper
Halfway to Rota’s basis conjecture, by Matija Bucić, Matthew Kwan, Alexey Pokrovskiy, and Benny Sudakov

Abstract: In 1989, Rota made the following conjecture. Given $n$ bases $B_{1},\dots,B_{n}$ in an $n$-dimensional vector space $V$, one can always find $n$ disjoint bases of $V$, each containing exactly one element from each $B_{i}$ (we call such bases transversal bases). Rota’s basis conjecture remains wide open despite its apparent simplicity and the efforts of many researchers (for example, the conjecture was recently the subject of the collaborative “Polymath” project). In this paper we prove that one can always find $\left(1/2-o\left(1\right)\right)n$ disjoint transversal bases, improving on the previous best bound of $\Omega\left(n/\log n\right)$. Our results also apply to the more general setting of matroids.

http://front.math.ucdavis.edu/1810.07462

Earlier the best result was giving $n/\log n$ disjoint transversal bases.

Here is a subsequent paper about the more general Kahn’s conjecture

https://arxiv.org/abs/1810.07464

### Polymath 16 is alive and kicking

Polymath 16 of the chromatic number of the plane is in its eleventh post. A lot of interesting developments and ideas in various directions!

### The polymath picture

I took some pictures which are a little similar to our logo picture (last picture below).

1. ,,

Comment by jeasu — March 1, 2019 @ 6:47 pm

2. Locomotion of a snake-like structure in accordance with the serpenoid curve, i.e. lateral undulation, is achieved if the joints of the robot move according to the reference joint trajectories in the form of a sinusoidal function with specified amplitude, frequency, and phase shift. In particular, using the foregoing defined new states, we define a constraint function for the ith joint of the snake robot by
Φi = αsin(η + (i − 1)δ) + ϕo
(45)
where i∈{1,…,N−1}, α denotes the amplitude of the sinusoidal joint motion, and δ is a phase shift that is used to keep the joints out of phase. Moreover, ϕo is an offset value that is identical for all of the joints. It was illustrated in [16] how the offset value ϕo affects the orientation of the snake robot in the plane. Building further on this insight, we consider the second-order time derivative of ϕo in the form of a dynamic compensator, which will be used to control the orientation of the robot. In particular, through this control term, we modify the orientation of the robot in accordance with a reference orientation. This will be done by adding an offset angle to the reference trajectory of each joint. We will show that this will steer the position of the CM of the robot towards the desired path. The constraint function (45) is dynamic, since it depends on the solution of a dynamic compensator.
In this subsection, we define a constraint function for the head angle of the robot. In particular, we use a line-of-sight (LOS) guidance law as the reference angle for the head link. LOS guidance is a much-used method in marine control systems (see, e.g. [27]). In general, guidance-based control strategies are based on defining a reference heading angle for the vehicle through a guidance law and designing a controller to track this angle [27]. Motivated by marine control literature, in [17] based on a simplified model of the snake robot, using cascade systems theory, it was proved that if the heading angle of the snake robot was controlled to the LOS angle, then also the position of the CM of the robot would converge to the desired path. We will show that a similar guidance-based control strategy can successfully steer the robot towards the desired path. However, we perform the model-based control design based on a more accurate model of the snake robot which does not contain the simplifying assumptions of [17] which are valid for small joint angles.
To define the guidance law, without loss of generality, we assign the global coordinate system such that the global x-axis is aligned with the desired path. Consequently, the position of the CM of the robot along the y-axis, denoted by py, defines the shortest distance between the robot and the desired path, often referred to as the cross-track error. In order to solve the path following problem, we use the LOS guidance law as a virtual holonomic constraint, which defines the desired head angle as a function of the cross-track error as observed prof dr hircea orasanu and prof drd horia orasanu and Gowers’s Weblog

here in connection with quasi must to precised that are observed by prof dr mircea orasanu and prof drd horia orasanu as followed in profound important situations

ΦN=−tan−1(pyΔ)
(46)
where Δ>0 is a design parameter known as the look-ahead distance. The idea is that steering the head angle of the snake robot such that it is headed towards a point located at a distance Δ ahead of the robot along the desired path will make the snake robot move towards the path and follow it.
Defining a constraint manifold
We collect all the foregoing defined constraint functions in the following vector-valued function
Φ=+ϕo,tan−1(pyΔ)]T∈RN[αsin(η)+ϕo,…,αsin(η+(N−1)δ)
(47)
For trajectory planning using virtual holonomic constraints

Comment by posogiu — March 1, 2019 @ 6:53 pm

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