Wikipedia:Reference desk/Archives/Mathematics/2010 May 8

= 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
 * $$\frac{d^2}{dt^2}(t-\sin(\sin(t))) = \sin(t)\cos(\sin(t)) + \cos^2(t)\sin(\sin(t))$$
 * 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 $$\scriptstyle f(t)=t-\sin(\sin t)$$ is an odd function: $$\scriptstyle (f(-t)=-f(t))$$, and the function value of the complex conjugate argument is the complex conjugate function value of the argument: $$\scriptstyle  (f(\overline t)=\overline {f(t)})$$. This implies that if $$\scriptstyle  r$$ is a root, $$\scriptstyle (f(r)=0)$$, then so is $$\scriptstyle  -r$$ and $$\scriptstyle  \overline r$$ and  $$\scriptstyle  -\overline r$$. 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) --pm a 07:04, 9 May 2010 (UTC)