2d-RandomWalks with N steps and fixed distance between first and last point

community, I’m trying to figure out a way how to create a 2d list simulating a N-Step-Randomwalk with a fixed distance between first and last point. I already got the RandomWalk with the step size 1:


RW2D[n_] := Accumulate[Through[{Cos, Sin}[#]] & /@ RandomReal[{0, 2 [Pi]}, n]]

I have no idea how to connect the first and the last point. I’d be happy to receive some help. Thanks alot Greetings.

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High-order ODE from system of first order ODEs

In the case of a linear, first-order system with constant coefficients $ $ \mathbf{x}’=A \mathbf{x}, $ $ with $ \mathbf{x} \in \mathbf{R}^n$ , it is known that all of the components $ x_i$ of $ \mathbf{x}$ satisfy the same ODE of order $ n$ , called the secular equation [at least by Birkhoff and Rota]. It also known that even in the nonlinear case an ODE of high-order can be expressed as a system of first-order ODEs.

My question is: in the case of a nonlinear system of first-order ODEs, is there a method to get a high-order ODE satisfied by the components individually? Special cases, such as polynomial RHS’s are also of interest to me.

Thank you!

Precautions while redesigning site appearing in google search first page [duplicate]

This question already has an answer here:

  • How to retain/transfer ranking when updating/redesigning website? 4 answers
  • Site restyling affecting Google ranking? 3 answers
  • Redesigning an old site, structure change etc 2 answers

One of our clients site need to be re-designed. The updates would be mostly presentational including graphics, page layout updates (re-order page elements), color, icons, fonts, making it responsive etc. There will be minor content updates, mostly correcting old data and add some new.

Currently site pages appear on Google search first page for key business phrases/words.

What precautions should be taken while re-designing website without affecting the search ranking?

How in depth should a screen reader when it does the first read through of a page and a form

Take the following mockup as our example case:


Page Title

Page Instructions

Field 1*: [ text input ]

Field 2 : [ text input ]

Field 3*: [ text input ]

                            [Reset]    [Submit] 


We have noticed that when a screen reader (using default settings) first reads through the following page, it reads all the text content correctly. But the screen reader does not state if a given field is required on this first read through. When tabbing through to the fields after the fact, it will state if a field is required on first focus. Is this acceptable from an accessibility stand point?