FullSimplify of expressions with mathematical constants

L & # 39; writing:

{c1, c2, c3, c4, c5} = N({Tan(E), Sin(E), Tanh(E), E, Sinh(E)});

a = (c1 + c3) / 2;
b = Sqrt(c5^2 - (c2 - c4)^2) / 2;
c = 0;
d = (c2 + c4) / 2;
e = (c1 - c3) (c2 - c4) / (4 b);
f = c5 Sqrt(c5^2 - (c1 - c3)^2 - (c2 - c4)^2) / (4 b);
x = a + b Cos(t) + c Sin(t);
y = d + e Cos(t) + f Sin(t);

xmin = Minimize({x, 0 <= t <= 2π}, t)((1));
xmax = Maximize({x, 0 <= t <= 2π}, t)((1));
FullSimplify((xmin + xmax) / 2 == a)

ymin = Minimize({y, 0 <= t <= 2π}, t)((1));
ymax = Maximize({y, 0 <= t <= 2π}, t)((1));
FullSimplify((ymin + ymax) / 2 == d)

we have:

True

True

this is what is desired. However, by making a simple change:

{c1, c2, c3, c4, c5} = {Tan(E), Sin(E), Tanh(E), E, Sinh(E)};

we have:

True

...

that is to say, in the second case, there is no answer. So by defining the constants this other way:

SetAttributes(c1, Constant)
NumericQ(c1) = True;
N(c1, prec___) := N(Tan(E), prec)

SetAttributes(c2, Constant)
NumericQ(c2) = True;
N(c2, prec___) := N(Sin(E), prec)

SetAttributes(c3, Constant)
NumericQ(c3) = True;
N(c3, prec___) := N(Tanh(E), prec)

SetAttributes(c4, Constant)
NumericQ(c4) = True;
N(c4, prec___) := N(E, prec)

SetAttributes(c5, Constant)
NumericQ(c5) = True;
N(c5, prec___) := N(Sinh(E), prec)

we have:

True

Minimize :: infeas: There are no values ​​of {t} for which the constraints 0 <= t <= 2π are satisfied and the objective function (...) has a real value.

Maximize :: infeas: There are no values ​​of {t} for which the constraints 0 <= t <= 2π are satisfied and the objective function (...) has a real value.

Infinity :: indet: Indeterminate expression -∞ + ∞ encountered.

Undetermined == (c2 + c4) / 2

where, apparently, another problem arises. How to solve everything?

FullSimplify expression with mathematical constants

L & # 39; writing:

{c1, c2, c3, c4, c5} = N({Tan(E), Sin(E), Tanh(E), E, Sinh(E)});

a = (c1 + c3) / 2;
b = Sqrt(c5^2 - (c2 - c4)^2) / 2;
c = 0;
d = (c2 + c4) / 2;
e = (c1 - c3) (c2 - c4) / (4 b);
f = c5 Sqrt(c5^2 - (c1 - c3)^2 - (c2 - c4)^2) / (4 b);
x = a + b Cos(t) + c Sin(t);
y = d + e Cos(t) + f Sin(t);

xmin = Minimize({x, 0 <= t <= 2π}, t)((1));
xmax = Maximize({x, 0 <= t <= 2π}, t)((1));
FullSimplify((xmin + xmax) / 2 == a)

ymin = Minimize({y, 0 <= t <= 2π}, t)((1));
ymax = Maximize({y, 0 <= t <= 2π}, t)((1));
FullSimplify((ymin + ymax) / 2 == d)

we have:

True

True

this is what is desired. However, by making a simple change:

{c1, c2, c3, c4, c5} = {Tan(E), Sin(E), Tanh(E), E, Sinh(E)};

we have:

True

...

that is to say, in the second case, there is no answer. How to solve this problem?

mathematical optimization – I can't solve the system of equations, using the solve function?

First of all, you just need to type Quit () and run, then copy and paste the code you gave us into a new notebook. Generally, when you get "protected errors", a variable has already been defined as something else.

If everything else fails and you're not sure what's going on, try restarting 🙂

f(x_) := a*Sin(b*(x + c)) + d;
g(x_) := -1/10125*Exp(4)*x^3 + 17/1350*Exp(4)*x^2 - 79/135*Exp(4)*x + 
   90 + Exp(1) + (1343/162 + Log(2) - Log(3))*Exp(4);
h := 38;
Solve({f(h) == g(h), f'(h) == g'(h), f''(h) == g''(h), 
  f'''(h) == g'''(h)}, {a, b, c, d})

Using this code gives me exactly a solution … albeit huge with conditions.

$$ left { left {a to text {ConditionalExpression} left ( frac {1} {250} (-21) i sqrt {33} e ^ 4, c_1 in mathbb {Z } right), b to text {ConditionalExpression} left (- frac {1} {3} i sqrt { frac {2} {21}}, c_1 in mathbb {Z} right) , c to text {ConditionalExpression} left ( frac {126 i pi c_1-38 sqrt {42} -126 tanh ^ {- 1} left ( frac {1} {3} left ( sqrt {42} – sqrt {33} right) right)} { sqrt {42}}, c_1 in mathbb {Z} right), d to text {ConditionalExpression} left ( frac {21} {250} e ^ 4 sqrt {33} sinh left ( frac {1} {3} sqrt { frac {2} {21}} left (38+ frac {126 i pi c_1-38 sqrt {42} -126 tanh ^ {- 1} left ( frac {1} {3} left ( sqrt {42} – sqrt {33} right) right) } { sqrt {42}} right) right) – frac {887 e ^ 4} {750} + e + 90 + e ^ 4 log (2) -e ^ 4 log (3), c_1 in mathbb {Z} right) right }, left {a to text {ConditionalExpression} left ( frac {1} {250 } (-21) i sqrt {33} e ^ 4, c_1 in mathbb {Z} right), b to text {ConditionalExpression} left ( frac {1} {3} i sqrt { frac {2} {21}}, c_1 in mathbb {Z} right), c to text {ConditionalExpression} left ( frac {-126 i pi c _1-38 sqrt {42} +126 tanh ^ {- 1} left ( frac {1} {3} left (- sqrt {33} – sqrt {42} right) right)} { sqrt {42}}, c_1 in mathbb {Z} right), d to text {ConditionalExpression} left (- frac {21} {250} sqrt {33} e ^ 4 sinh left ( frac {1} {3} sqrt { frac {2} {21}} left (38+ frac {-126 i pi c_1-38 sqrt {42} +126 tanh ^ {-1} left ( frac {1} {3} left (- sqrt {33} – sqrt {42} right) right)} { sqrt {42}} right) right) – frac {887 e ^ 4} {750} + e + 90 + e ^ 4 log (2) -e ^ 4 log (3), c_1 in mathbb {Z} right) right }, left {a to text {ConditionalExpression} left ( frac {21} {250} i sqrt {33} e ^ 4, c_1 in mathbb {Z} right), b to text {Conditiona lExpression} left (- frac {1} {3} i sqrt { frac {2} {21}}, c_1 in mathbb {Z} right), c to text {ConditionalExpression} left ( frac {126 i pi c_1-38 sqrt {42} -126 tanh ^ {- 1} left ( frac {1} {3} left ( sqrt {33} + sqrt {42} right) right)} { sqrt {42}}, c_1 in mathbb {Z} right), d to text {ConditionalExpression} left (- frac {21} {250} sqrt { 33} e ^ 4 sinh left ( frac {1} {3} sqrt { frac {2} {21}} left (38+ frac {126 i pi c_1-38 s qrt {42 } -126 tanh ^ {- 1} left ( frac {1} {3} left ( sqrt {33} + sqrt {42} right) right)} { sqrt {42}} right) right) – frac {887 e ^ 4} {750} + e + 90 + e ^ 4 log (2) -e ^ 4 log (3), c_1 in mathbb {Z} right ) right }, left {a to text {ConditionalExpression} left ( frac {21} {250} i sqrt {33} e ^ 4, c_1 in mathbb {Z} right) , b to text {ConditionalExpression} left ( frac {1} {3} i sqrt { frac {2} {21}}, c_1 in mathbb {Z} right), c to text {ConditionalExpression} left ( frac {-126 i pi c_1-38 sqrt {42} +126 tanh ^ {- 1} left ( frac {1} {3 } left ( sqrt {33} – sqrt {42} right) right)} { sqrt {42}}, c_1 in mathbb {Z} right), d to text {ConditionalExpression} left ( frac {21} {250} e ^ 4 sqrt {33} sinh left ( frac {1} {3} sqrt { frac {2} {21}} left (38+ frac {-126 i pi c_1-38 sqrt {42} +126 tanh ^ {- 1} left ( frac {1} {3} left ( sqrt {33} – sqrt {42} right) right)} { sqrt {42}} right) right) – frac {887 e ^ 4} {750} + e + 90 + e ^ 4 log (2) -e ^ 4 log (3), c_1 in mathbb {Z} right) right } right } $$

lo.logic – What types of mathematical proofs like Martin-Löf types are constructive proofs, and what is wrong?

The question is motivated by this surprising sentence from The QED Manifesto Revisited by Freek Wiedijk.

I agree that the QED type systems that exist today are not good enough
to start developing a library as described in the QED manifesto.

Surprising for me, that is, who had never heard of the QED project before.

I understand the Curry-Howard correspondence as s / th in the sense of "Any computer program is constructive proof that the habitation of its input type implies the habitation of its output type"; when i first presented it, my very next question was: "OK then, what types should we limit ourselves to, if the programs they type must be mathematical proofs in classical logic ? ". I assumed it had to be existential types, guarded subtypes (i.e., types restricted by an assertion expressed in classical logic), or such others; then I put it aside and turned to more urgent matters.

It seems to me that the natural way to tackle the QED challenge would be to formulate an answer to the above question, build the corresponding proof wizard, and then watch what is going on. Given the Wiedijk document, the real story had to be very different. Obviously, the people who tried the approach quickly encountered a problem and started to walk on their heads: instead of developing proof assistants for classical logic, they just rewrote the math classics for a constructive logic. With easily predictable results: if John Random Mathgeek hadn't hated the constructive logic of all his guts, he would have chosen computing first, not math.

Hence my question: which type systems are the most natural for expressing assertions in classical logic? Qu & # 39; have they in practice who makes their theory so disreputable?

mathematical optimization – Need help finding a value to linearize data

I have two column data which is y vs x. I want to linearize the data by this formula 1 / (x-b) but I don't know how to determine the value b so that ln (y) vs 1 / (x-b) becomes a linear line. So I wrote the following code to plot the ln (y) vs 1 / (xb) by changing the value b manually and looking at the graph ln (y) vs 1 / (xb) to find the best linear behavior . Do you know a better way to do this instead of changing the b value manually?

x = {331,334,335,336};
y = {10,50,100,1000};
b = 290;
Xinv = 1/(x - b)
lnY = N(Log(y));
(data = Transpose({Xinv, lnY})) // MatrixForm;
ListPlot(data, PlotMarkers -> {"O"}, 
 PlotStyle -> {Darker@Green, PointSize(3)})
nlm = NonlinearModelFit(data, a*x + b, {a, b}, x);
Show(ListPlot(data, PlotMarkers -> "!(*
StyleBox("O",nFontWeight->"Plain"))"), Plot(nlm(x), {x, -1, 2}),
  Frame -> True)

soft question – What is the mathematical branch on which one should focus more to better understand the other mathematical branches?

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mathematical writing – What does the abbreviation "p.p." say in the context of convergence

What does the abbreviation "p.p." does it mean when we refer to convergence? For example. in the next article by Harry Pollard

THEOREM. Yes $ f in L ^ p $ for some people $ p $ in the perimeter $ tfrac {4} {3} <p < infty $, then his Legendre
the series converges p.p. The result fails if $ 1 <p < tfrac {4} {3} $.

Almost everywhere the convergence of the Legendre series, Harry Pollard

I know the convergence a.s., a.e., in probability, pending, etc. Does this mean punctually?

terminology – What is the mathematical name of a set that contains the domain and codomain of a function?

I'm interested in this so that I can name a type parameter in a program that I'm writing.

There is a function that has three parameters.

D, domain

C, Codomain

X, where D is a subset of X and C is a subset of X

What is the special name of X, if there is one?

This is interesting from the point of view of the security of the program because by defining X, we can force the information to circulate only in X.

Maybe I should rather use category theory names here? If yes, what would these names be?

fonts – Problem with mathematical symbols in the display

I'm having a strange problem on my Android device. Oh, it's mathematical:

On my Android device, using Brave, Opera or DDG browsers, I don't see many symbols in the math pages of the wiki. Symbols do not appear or appear in white. It seems like these are just a few symbols, but it can make it very difficult to keep up with what's going on. The symbol p is generally missing (but q shows sup).

for example, https://en.wikipedia.org/wiki/Neighbourhood_(mathematics)

All p and V are missing (but they take up space).

I have no idea what's going on and I don't know where to look to ask. Is this a wiki problem? MathJax? These browsers? My device? This makes it very painful to use the device to read wikipedia. I haven't seen it on other sites but I don't go to many other sites for math.

Ideas?

Theory of Mathematical Numbers – Math Stack Exchange

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