Search Results for ""

Linear algebra is the study of linear sets of equations and their transformation properties. Linear algebra allows the analysis of rotations in space, least squares fitting, ...

Square triangle picking is the selection of triples of points (corresponding to endpoints of a triangle) randomly placed inside a square. n random triangles can be picked in ...

The q-analog of the Pochhammer symbol defined by (a;q)_k={product_(j=0)^(k-1)(1-aq^j) if k>0; 1 if k=0; product_(j=1)^(|k|)(1-aq^(-j))^(-1) if k<0; ...

A q-series is series involving coefficients of the form (a;q)_n = product_(k=0)^(n-1)(1-aq^k) (1) = product_(k=0)^(infty)((1-aq^k))/((1-aq^(k+n))) (2) = ...

A method for finding roots of a polynomial equation f(x)=0. Now find an equation whose roots are the roots of this equation diminished by r, so (1) The expressions for f(r), ...

The difference between the sum of face angles A_i at a polyhedron vertex of a polyhedron and 2pi, delta=2pi-sum_(i)A_i.

The involute of the astroid is a hypocycloid involute for n=4. Surprisingly, it is another astroid scaled by a factor (n-2)/n=2/4=1/2 and rotated 1/(2·4)=1/8 of a turn. For ...

The parametric equations for a catenary are x = t (1) y = cosht, (2) giving the involute as x_i = t-tanht (3) y_i = secht. (4) The involute is therefore half of a tractrix.

product_(k=1)^(n)(1+yq^k) = sum_(m=0)^(n)y^mq^(m(m+1)/2)[n; m]_q (1) = sum_(m=0)^(n)y^mq^(m(m+1)/2)((q)_n)/((q)_m(q)_(n-m)), (2) where [n; m]_q is a q-binomial coefficient.

If a_1>=a_2>=...>=a_n (1) b_1>=b_2>=...>=b_n, (2) then nsum_(k=1)^na_kb_k>=(sum_(k=1)^na_k)(sum_(k=1)^nb_k). (3) This is true for any distribution.