1. ## Introduction

I am currently working on an emergency license for math. I am trying to pass the Math Praxis 0061 to receive my certification. Need some help on some of the questions.

Thanks,
karies4083

2. ## Re: Introduction

Hi! I am relatively new as well - what is an emergency license for maths? I have never heard of it, but it sounds kind of unusual!

3. ## Re: Introduction

I am a teacher working on an additional licensure. .. . I am currently certified to teach in one area .. and working on another subject area.

Hope this helps

4. ## Re: Introduction

Oh I see - well good luck with it!

5. ## Re: Introduction

Dear team
Good afternoon.....
recently i have completed my BBA>.
now am planning for IAS officer..
so here i just want to clear my all concept of Maths

6. ## Re: Introduction

Well, the only way we can help is if you post specific problems. What kind of problems do you have difficulty with?

7. ## Re: Introduction

Hello, I have graduated from math -uhm- maybe 20 years ago and now, after some turmoil in ICT field, I have scrubbing a job from here and there (part time teaching and re-warming the research career.)

My current problem is this: Given a unit hypersphere S_3 (in R^4) and four normalized vectors n1,n2,n3,n4 (ni<>nj), how to calculate the -uhm- space angle defined by the vectors (subvolume of the sphere)?

I am familiar of the normal unit sphere S_2 which have a space angle Phi defined by 3 vectors n1,n2,n3 as: Phi= a12+a23+a31 -pi where a_{ij} = cos^{-1}(ni . nj). (E.g. the familiar rectangular corner slice of the sphere which has the area Phi = 3 pi/2 -pi = 4pi / 8 .)

Is there a nice equation for S_3 case? Preferably one using a_{ij} angles. Or a recursive one? Actually my hunger goes up to S_11 (12 vectors in R^12), so I need references if the problem is
not trivial. The intended use is non-commercial, so I am not trying to milk you for my profit.

8. ## Re: Introduction

Problem (almost) solved. I stumbled upon Heinz Hopf, Selected Chapters of Geometry, 1940, which answers my problem with the first incoming breeze. (Let's see if the computation is too expensive next... And I had a mistake in the definition of cos -terms, have to snip away the components along nk from ni and nj... )