thank you for your help! antonio

thank you for your help! antonio

Please ignore it. I have found the answer. msolve(x^2 = 670, 961)

x^2 = 670 (mod 961). or x^2 - 961*y = 670.

sorry it's not (mod 291), it's (mod 961). is there any other way apart from checking all squares from 1,2,..., 961 manually? many thanks!

thank you for your response! i have found this in Maple help. > with(simplex); cnsts := {3*x+4*y-3*z <= 23, 5*x-4*y-3*z <= 10, 7*x+4*y+11*z <= 30}; obj := -x+y+2*z; maximize(obj, `union`(cnsts, {0 <= z, 0 <= x, 0 <= y})); / 49 1\ { x = 0, y = --, z = - } \ 8 2/

Thank you!

thank you!

thank you!

Thank you all and Happy New Year!

thank you, I tried putting it into Maple and got the following: > ode := {diff(y(t), t) = -3*y(t), y(0) = 2}; / d \ { --- y(t) = -3 y(t), y(0) = 2 } \ dt / / d \ { --- y(t) = -3 y(t), y(0) = 2 } \ dt / > soln := dsolve(ode, numeric, type = foreuler, stepsize = .2); Error, (in dsolve/numeric/an_args/SC) keyword was type, optional keyword must be one of 'range', 'abserr', 'relerr', 'initstep', 'maxfun', 'initial', 'steppast', 'output', 'procedure', 'procvars', 'start', 'number', 'startinit', 'implicit', 'interpolate', 'stop_cond', 'optimize', 'complex' > soln(.2); soln(0.2) > (plots[odeplot])(soln, [t, y(t)], t = 0 .. 5); Error, (in plots/odeplot) input is not a valid dsolve/numeric solution The output is 2 error messages for soln and plots commands...

yes, thank you!

n*(n+1) - meant to type n/(n+1) - sorry.

thanx a lot! I tried the following in Maple: lim(n-> infinity) a(n+1)/a(n) and it seems to work. Thanx again for your help!

Thanks a lot! The upper bound (maximum number of comparisons) is in O(n*ln(n)) and the lower bound (minimum number of comparisons) is in omega(n*ln(n)). Just wondered if there is a step-by-step procedure for different levels of algorithm efficiency for n number of elements in a heap. In other words, if, say, we have 12 elements in a heap: how it would be possible to re-arrange the heap into a list showing the max and min number of comparisons. I could do it manually, but for large n, it is perhaps best to use Maple (if such procedure exists). Many thanks in advance. Antonio