Example for mathematical induction – Mathematics Stack Exchange

Currently I am on mathematical induction and I’ve faced problem that I simply don’t know where to even start and I can’t find any examples that I can go on with, so just to clarify I am asking you for example with solution of similar task.

The task that I should solve is:
$$1053|3^{2n+2}*5^{2n}-3^{3n+2}*2^{2n}$$

If you possibly give me solution of this:
$$3|(n^{3}-n)$$
since I believe this one is easier to solve

P.S: I don’t want you to solve my problem, because I want to try to solve it by myself, I just want an example to see how you can solve these type of tasks. Thank you!

mathematical optimization – Continuous Min-Max problem

Being new to Mathematica, I tried my best to find some biult-in functions or guides on how to solve the classical min-max problem

$$ min_{x} max_{k} f(x,k,params) $$

with some additional variables $params$ and some simple constraints on the variables (e.g., $xin (x_{min},x_{max})$ and $kin (k_{min},k_{max})$) in the Mathematica language. Finding none (giving a link would be much appreciated), my approach was to first define function computing
$$ max_{k} f(x,k) $$
e.g.,

fMax(x_,params_) := 
  FindMaximum({f(x,k,params_), k > kmin, k < kmax}, {k, kinit});

with a parameter $x$ and then minimize fmax, e.g.,

fMinMax(x_,params_) := 
  FindMinimum({fMax(x_,params_), x > xmin, x < xmax}, {x, xinit});

However, the following error is consistently raised.

FindMaximum::nrnum: The function value -((9.27923*10^11-2.95367*10^10 p)/(5.15531*10^17+1.64099*10^16 p)) is not a real number at {k} = {10.}.

although upon evaluating the function at that given point, the value is indeed real. I would be glad for any help. To give the full setting $f$ amounts to

$$ f(x,k,a,b,alpha) = frac{frac{kpi}{b} cosh left(frac{kpi}{b} (a-alpha)right) + x sinh left(frac{kpi}{b} (a-alpha)right)}{frac{kpi}{b} cosh left(frac{kpi}{b} (a+alpha)right) + x sinh left(frac{kpi}{b} (a+alpha)right)} $$

where $a,b,alpha$ are positive parrameters such that $a>alpha>0,b>0$.

mathematical optimization – How to maximize an expression within an interval?

I have tried to maximize this expression but in the interval -1<= x <=1 , but I can’t get the correct syntax

Maximize({(4 - 3 x - Sqrt(16 - 24 x + 9 x^2 - x^3))^(1/3) + (4 - 3 x + Sqrt(16 - 24 x + 9 x^2 - x^3))^(1/3), x, -1 <=x <= 1)

help me with the syntax

functional programming – Simplifying the following mathematical expression using a computer?

I have this following beastly expression typed up very nicely in LaTeX formatting, as you can see. What is the easiest way that I can get a computer to simplify this expression for me? I have zero programming experience. I installed sagemath but it seems pretty complicated.

$W_{(1,1)}(t,v)=frac{-t^{-2k}v^k}{3}(frac{v^{frac{3}{2}}-v^{frac{-3}{2}}}{t^{frac{3}{2}}-t^{frac{-3}{2}}})(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})+frac{t^{-2k}v^k}{4}(frac{v-v^{-1}}{t-t^{-1}})^2+frac{t^{-2k}v^k}{12}(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})-frac{t^{-k}v^k}{4}(frac{v^2-v^{-2}}{t^2-t^{-2}}) + frac{t^{-k}v^k}{8}(frac{v-v^{-1}}{t-t^{-1}})^2+frac{t^{-k}v^k}{4}(frac{v-v^{-1}}{t-t^{-1}})(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})^2-frac{t^{-k}v^k}{8}(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})^4+frac{-v^kt^{k}}{4}(frac{v^2-v^{-2}}{t^2-t^{-2}})+frac{v^kt^{k}}{3}(frac{v^{frac{3}{2}}-v^{frac{-3}{2}}}{t^{frac{3}{2}}-t^{frac{-3}{2}}})(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})+frac{v^kt^{k}}{8}(frac{v-v^{-1}}{t-t^{-1}})^2-frac{v^kt^{k}}{4}(frac{v-v^{-1}}{t-t^{-1}})(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})^2+frac{v^kt^{k}}{24}(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})^4$

algebra precalculus – Simplifying the following mathematical expression using a computer?

I have this following beastly expression typed up very nicely in LaTeX formatting, as you can see. What is the easiest way that I can get a computer to simplify this expression for me? I have zero programming experience. I installed sagemath but it seems pretty complicated.

$W_{(1,1)}(t,v)=frac{-t^{-2k}v^k}{3}(frac{v^{frac{3}{2}}-v^{frac{-3}{2}}}{t^{frac{3}{2}}-t^{frac{-3}{2}}})(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})+frac{t^{-2k}v^k}{4}(frac{v-v^{-1}}{t-t^{-1}})^2+frac{t^{-2k}v^k}{12}(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})-frac{t^{-k}v^k}{4}(frac{v^2-v^{-2}}{t^2-t^{-2}}) + frac{t^{-k}v^k}{8}(frac{v-v^{-1}}{t-t^{-1}})^2+frac{t^{-k}v^k}{4}(frac{v-v^{-1}}{t-t^{-1}})(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})^2-frac{t^{-k}v^k}{8}(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})^4+frac{-v^kt^{k}}{4}(frac{v^2-v^{-2}}{t^2-t^{-2}})+frac{v^kt^{k}}{3}(frac{v^{frac{3}{2}}-v^{frac{-3}{2}}}{t^{frac{3}{2}}-t^{frac{-3}{2}}})(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})+frac{v^kt^{k}}{8}(frac{v-v^{-1}}{t-t^{-1}})^2-frac{v^kt^{k}}{4}(frac{v-v^{-1}}{t-t^{-1}})(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})^2+frac{v^kt^{k}}{24}(frac{v^{frac{1}{2}}-v^{frac{-1}{2}}}{t^{frac{1}{2}}-t^{frac{-1}{2}}})^4$

mathematical optimization – Maximize an expression with respect to a variable and minimize it with respect to other variables

I started to use Mathematica a few time ago. I want to minimize the following expression (function of $l,p,q,r,c$) with respect to variables $l, p, q, r$ and then maximize the result obtained with respect to variable $c$. However, when I try to obtain an expression function of $c$ to maximize later using Minimize, I do not get any result because it takes too long. How can I solve this issue?

Minimize({(((l^2/2)*(1-(1/4-c))+(l*p)*(1-1/4)+(l*q)*(1-(1/4+c))+(l*r)*(1-1/2)+(p^2/2)*(1-c)+(p*q)*(1-2*c)+(p*r)*(1-(1/4+c))+(q^2/2)*(1-c)+(q*r)*(1-1/4)+(r^2/2)*(1-(1/4-c)))/((l^2/2)*(1-(1/4-c))+(l*p)*(1-1/4)+(l*q)*(1-(1/4-c))+(l*r)*(1-0)+(p^2/2)*(1-c)+(p*q)*(1-(1/4-c))+(p*r)*(1-0)+(q^2/2)*(1-(1/4-c))+(q*r)*(1-1/4)+(r^2/2)*(1-0))), l >= 1, p >= 0, q >= 0, r >= 0, l + p + q + r == 1000000, 1/7<c<1/5}, {l, p, q, r})

mathematical programming – What is the difference between a fraction and a float?

I understand any fraction to be a quotient of integers besides 0, but after coming across the term “float” in various programming languages (such as JavaScript) I misunderstand why it is even needed and we don’t say a fraction instead.

What is the difference between a fraction and a float?

mathematical optimization – Maximize: x in Vectors[5, R] is not a valid variable

I am learning how to do optimization with a variable number of variables in Mathematica. I tried to write one experiment that I need to do but I have failed to get even a stripped-down version working. The code is the following

y = {1, 2, 3, 4, 5}
Maximize({Sum(Indexed(y, i)/Indexed(x, i), {i, 1, 5}), 
    ForAll(i,Indexed(y, i)^(1/4) <= Indexed(x, i) <= Indexed(y, i)^(1/2))}, 
    x (Element) Vectors(5, Reals))

But it says

Maximize::ivar: x(Element)Vectors(5,(DoubleStruckCapitalR)) is not a valid variable.

I have tried to re-write the optimization problems in different ways but nothing helped. Any advice on what I am doing wrong would be greatly appreciated.

mathematical optimization – Speeding up a Table with NMinimize

I have a parametric system of differential equations (a and b are the parameters). For every value of the parameters in a given range I want to find the right initial conditions to fit some observations. With this aim I defined a table which runs over the parameters a and b, and for every parameter I use NMinimize to minimize a chi-square variable over the initial conditions space {x0,vx0,y0,vy0}. I wonder if there is a way to speed up this process. The main part of the code is this:

Table({a, b,  NMinimize(chisquare(x0, vx0, y0, vy0)(a)(b), {{800 < x0 < 1100}, {400 < vx0 < 900},{1350 < y0 < 
1750}, {10 < vy0 < 110}}, {x0, vx0, y0, vy0})((1))}, {a,arange}, {b, orange})

NMinimize takes 146 seconds for a given couple of parameters {a,b}. Do you think there is a more efficient way to do this?

mathematical optimization – SemidefiniteProgramming for operator norms: Stuck at the edge of dual feasibility

I’m trying to calculate operator norms of linear transformations over space of matrices. For instance, find norm of $f(A)=XA$ by optimizing following:

$$max_{|A|=1} |XA|$$

This looks like a semidefinite programme, but I’m having trouble solving it with SemidefiniteOptimization. Simplest failing example is to find operator norm of $f(A)=5A$ in 1 dimension. It fails with Stuck at the edge of dual feasibility. Any suggestions?

Constraints
$$
A succ 0\
Isucc A \
x I succ -5 A
$$

Objective

$$
text{min}_{A,x} x
$$

d = 1;
ii = IdentityMatrix(d);
(* Symbolic symmetric d-by-d matrix *)
ClearAll(a);
X = 5*ii;
A = Array(a(Min(#1, #2), Max(#1, #2)) &, {d, d});
vars = DeleteDuplicates(Flatten(A));

cons0 = VectorGreaterEqual({A, 0}, {"SemidefiniteCone", d});
cons1 = VectorGreaterEqual({ii, A}, {"SemidefiniteCone", d});
cons2 = VectorGreaterEqual({x ii, -X.A}, {"SemidefiniteCone", d});
SemidefiniteOptimization(x, cons0 && cons1 && cons2, {x}~Join~vars)
```