where 
and 
defined as
is given by
denotes the number of points of intersection between the curves
as
is
Number of subsets of 
whose sum of elements is divisible by 
and 
,
,
.
define
(independent of 
and of 
)![$$
\lim_{x\to 1^{-}}
F_{a,\beta}^{(k,r)}(x)\,
e^{-\Lambda_{k,r,\lambda}\,(1-x)^{-\gamma}}
\;=\;
\begin{cases}
0, & a<c,\\[6pt]
\infty, & a>c.
\end{cases}
$$](http://latex.artofproblemsolving.com/e/3/8/e38c6b43d513bcc697587fff610a846b3a1b0e0a.png)
explicitly and verify that it coincides with the classical case 
,
.
(your answer may depend on 
but not on 
there exist nonnegative integers 
and integers 
such that 
Y by Euler_Gauss
Let 
be a 
-dimensional inner product space of column vectors, where for 
and 
, the inner product of 
and 
is defined as ![\[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]](//latex.artofproblemsolving.com/9/f/9/9f9b7ff2fb19fee7e935c1b18dea3ec80547b9f9.png)
For 
, define a linear transformation 
on as follows:
![\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]](//latex.artofproblemsolving.com/9/6/4/9646290e2b5e540b7f85d0d71e4d8992ccd8f542.png)
Given 
satisfying
![\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]](//latex.artofproblemsolving.com/d/8/0/d800fe48958cfee9ac01aa8bf11fe431ef45f8ef.png)
let 
. Then the dimension of the linear space formed by all linear transformations 
satisfying 
is 
be a 
-dimensional inner product space of column vectors, where for 
and 
, the inner product of 
and 
is defined as ![\[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]](http://latex.artofproblemsolving.com/9/f/9/9f9b7ff2fb19fee7e935c1b18dea3ec80547b9f9.png)
For 
, define a linear transformation 
on as follows:![\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]](http://latex.artofproblemsolving.com/9/6/4/9646290e2b5e540b7f85d0d71e4d8992ccd8f542.png)
Given 
satisfying![\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]](http://latex.artofproblemsolving.com/d/8/0/d800fe48958cfee9ac01aa8bf11fe431ef45f8ef.png)
let 
. Then the dimension of the linear space formed by all linear transformations 
satisfying 
is 
is the orthogonal symmetry with respect to 
.
;
and 
is a rotation with angle 
.
is diagonalizable over 
,
.
.
