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May 8

sin(sin t)?

What is the solution to t = sin(sin t)? In other words, how can I find all points common to the helices C_1 and C_2 where C_1(t) = (cos t, sin t, t), t is real; and C_2(s) = (cos s, s, sin s), s is real? 60.240.101.246 (talk) 03:23, 8 May 2010 (UTC)

t=0 is the obvious one... 69.228.170.24 (talk) 04:53, 8 May 2010 (UTC)

...and t=0 is the only solution (assuming t is in radians), because

$\frac{d}{dt}(t-\sin (\sin (t)))=1-\cos (t)\cos (\sin (t))$

which is 0 at t=0, and

$\frac{d^2}{d{t}^2}(t-\sin (\sin (t)))=\sin (t)\cos (\sin (t))+{\cos }^2(t)\sin (\sin (t))$

which is also 0 when t=0, but positive away from t=0, so slope of t-sin(sin(t)) increases as you move away from 0 in either direction. So helices only intersect at (1,0,0). Gandalf61 (talk) 09:26, 8 May 2010 (UTC)

(EC) And note that for t≠0 |sin(sin(t))|≤|sin(t)|<|t|, so there's no other solution.--84.221.69.102 (talk) 10:33, 8 May 2010 (UTC)

If we write

$\arcsin t=\sin t$

and take "arcsin" to mean the "multiple-valued" arcsine, and look at the two graphs superimposed on each other, it becomes obvious that they intersect only once. Therefore only the trivial solution t = 0 exists. Michael Hardy (talk) 03:37, 10 May 2010 (UTC)

The function $f(t)=t-\sin (\sin t)$ is an odd function: $(f(-t)=-f(t))$, and the function value of the complex conjugate argument is the complex conjugate function value of the argument: $(f(\bar{t})=\overline{f(t)})$. This implies that if $r$ is a root, $(f(r)=0)$, then so is $-r$ and $\bar{r}$ and $-\bar{r}$. Some nonzero roots are ±1.7856225020975647±2.8984466947375185i, ±2.2559594166866765±1.7316525254965243i, ±4.9222858483025655±3.1622510760997686i, and ±36.13956703186652±6.6383288953460431i. Bo Jacoby (talk) 13:29, 12 May 2010 (UTC).

The armed force of Ghana ...

                                                                  QUESTION    1

The armed forces of Ghana want an algorithm that can efficiently solve a particular decision problem T in the worst –case.Three algorithms are currently available .They are A,B and C with running times , T_A (n)={█(4T_A (n-1) + Ѳ(2^n) , n>0@6 , n=0 )┤ T_B (n)={█(Ѳ(1) if 1≤n<3@2T_B (⌊n/3⌋) + Ѳ(T_A (n) ) if n≥3)┤ T_C (n)={█(Ѳ(1) if 1≤n≤3@2T(⌈n/3⌉ ) + Ѳ(n) if n≥3)┤ (i) Explain why B should not be in the list (ii) Which program ( A,B or C ) would you recommend to the armed forces .Justify your answer.

                                                                   QUESTION 2

The population at time N U {0} of two nations Messi and Xavi , are noted by the recurrence relation ,

X(n)={█(rX(n-1) + g(n) if n>0@a if n=0)┤ and M(n) ={█(3T(⌈n/3⌉ ) + n√(n+1) if n>1@2 if 0≤n≤1 )┤respectively where a , r ∈ N and g is defined on the positive integers .Show that X(n) = r^n a + ⋋∑_(i=1)^n▒r^(n-1) g(i) for some constant ⋋ ∈ R . Hence or otherwise determine the value of lim┬(n⟶∞)⁡〖(M(n))/(X(n))〗 if r=4 a=6 and g(n)=2^(n ).

Find Big-oh expression for X(n) and M(n) . —Preceding unsigned comment added by Waslimp (talk • contribs) 12:00, 8 May 2010 (UTC)

Please explain what aspect of these problems you are needing help with. How far have you got with your analysis? BTW I see a big black square between the { and the ( in X(n)={█(rX(n-1). What is it meant to be? -- SGBailey (talk) 14:31, 8 May 2010 (UTC)

This post is just a repetition of a "Discrete_maths" question above dated May 7 (maybe both could be removed) --pma 07:04, 9 May 2010 (UTC)