mathematical optimization: find the minimum with the positive definition matrix constraint

Let's say I want to find the minimum value of the determinant of a matrix under the condition that the matrix is ​​positively defined. Then I try:

M = {{a,0},{0,b}}

FindMinimum[{Det[M],a>=1,b>=1,PositiveDefiniteMatrixQ[M]},{a,b}]

This returns an error that Constraints in {False} are not all equality or inequality constraints..., suggesting that the PositiveDefiniteMatrixQ is being evaluated immediately by arbitrary a,b and not evaluated every iteration for a,b values.

Then I could try to delay the evaluation of PositiveDefiniteMatrixQ with Delayed, which returns a similar error Constraints in {Delayed[PositiveDefiniteMatrixQ[M]],a>=1,b>=1} are not all equality or inequality constraints.

How can I impose such a restriction on the FindMinimum function?

arithmetic geometry: equivalent definition of the ring $ B_ {cris} $

I am reading Laurie's note on the Fargues-Fontaine curve and I think it uses a different definition of $ B_ {cris} $generally when $ R $ It is a perfect ring of features $ p $, $ B + cris (R) define as $ p $-Full enclosure of the power split envelope of the map $ W (R) a R $ Y $ B_ {cris} = B + {cris} $.

but in these notes when R is the valuation ring of an algebraically closed perfect field $ B $ defined as the completion of $ frac (W (R)) $ with respect to all gauss standards and the Fargues-Fontaine curve defined by it.

I want to know the relationship between $ B $ Y $ B_ {cris} $ In general, is it true that they are isomorphic if R is the titration ring of a prefective field?

automaton: definition of a regular grammar but without $ Q rightarrow varepsilon $

I defined a regular grammar (FSM), which begins with $ ab $ and ends with $ ba $ as follows:

  1. $ S rightarrow aS $
  2. $ S rightarrow bS $
  3. $ S rightarrow aT $
  4. $ T rightarrow bR $
  5. $ R rightarrow aQ $
  6. $ Q rightarrow aQ $
  7. $ Q rightarrow bQ $
  8. $ Q rightarrow epsilon $

, where $ S $ is the initial element $ epsilon $ It is the empty element (null) and the rest are only variables.

Rules 6, 7 and 8 are there, so we can finish a word. However, I am trying to rewrite my grammar but without the $ Q rightarrow epsilon $. I cannot use the empty element.

Can be done? I am not sure how.

Thank you

Probability – Definition of a permutation $ r- $

I am trying to understand the definition of a $ r- $permutation. Suppose you have $ 7 $ aligned seats and $ 7 $ different people, there are $ 7! $ Different ways to settle them.

Suppose I am trying to sit down $ 7 $ different people in $ 9 $ seats, is that when I use the formula $ P (n, r) = frac {n!} {(N-r)!} $?

My opinion is that there are two same seats (empty), so the number of distinguishable permutations is $ 9! / two! $ what is that previous formula So my question is, is $ r- $ permutation a method to discover the permutation of $ r $ objects in $ n $ boxes when $ n ge k $?

Follow up if you have time: If the seats formed a circle, we would divide by $ 7 $ How are the seats equivalent to the correct rotation?

syntax – Definition domain for the following variables: is it possible to derive in Mathematica?

Consider 4 4 vectors
$$
P_ {0} = (E_ {0}, 0,0, sqrt {E_ {0} ^ {2} -m_ {0} 2}}, P_ {i} = (E_ {i} , p_ {i} s ( theta_ {i}) c ( phi_ {i}), p_ {i} s ( phi_ {i}) s ( theta_ {i}), p_ {i} c ( theta_ {i})),
$$

with $ c equiv cos, s equiv sin $, $ p_ {i} equiv sqrt {E_ {i} ^ {2} -m_ {i} ^ {2}} $ and scalar products
$$
P_ {i} cdot P_ {j} equiv P_ {i} ^ {0} P_ {j} ^ {0} – sum_ {k = 1} ^ {3} P_ {i} ^ {k} P_ { j} k
$$

$ m_ {0-3}, E_ {0} $ play the role of real parameters, with $ E_ {0}> m_ {0}> m_ {1} + m_ {2} + m_ {3} $ Y $ E_ {i} geqslant m_ {i} $, While $ E_ {i}, theta_ {i}, phi_ {i} $ they are variable

The implicit region of the definition of $ E_ {i}, theta_ {i}, phi_ {i} $ is given by
$$
tag 1 P_ {3} = P_ {0} -P_ {1} -P_ {2},
$$

$$
tag 2 s_ {12, text {min}} (s_ {23}) <s_ {12} <s_ {12, text {max}} (s_ {23}), quad s_ {23, text {min}} <s_ {23} <s_ {23, text {max}},
$$

where $ s_ {ij} = m_ {i} 2 + m_ {j} 2 + 2P_ {i} cdot P_ {j} $Y
$$
tag 3 s_ {12, text {min} / text {max}} = m_ {1} ^ {2} + m_ {2} ^ {2} – frac {1} {2s_ {23}} bigg (s_ {23} -m_ {0} 2 + m_ {1} 2) (s_ {23} -m_ {2} 2 -m_ {3} {2}) pm \ pm sqrt { lambda (s_ {23}, m_ {0} 2, m_ {1} 2) lambda (s_ {23}, m_ {2} ^ {2} , m_ {2})} bigg),
$$

$$
tag 4 s_ {23, text {min}} = (m_ {2} + m_ {3}) ^ {2}, s_ {23, text {max}} = (m_ {0} -m_ {1} 2,? (A, b, c) = (abc) 2 -4bc
$$

I need to integrate a function $ f (E_ {i}, theta_ {i}, phi_ {i}) $ about the domain of the definition $ (1) – (4) $ of the mentioned variables. Is it possible to derive the definition domain in Mathematica, at least implicitly, to perform the integration? There are so many variables …

abstract algebra: show that an identity element does not exist with the definition

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What it means: automatic in a function definition

I've been trying to solve it, without success. When reading someone else's code, I find a function definition, where it says something like this

UserDefinedFunction[a_Integer, b_Integer, c_Integer, d_, k_:Automatic]/;Abs[a]>b:=0;

I understand that it means that the function should return 0 if Abs [a]> b, but what does the label mean? Automatic in k_:Automatic half? What is the widest use? How does it relate when I draw something and tell Mathematica to assign PlotRange -> Automatic?

Information theory – Definition of collision entropy

Collision entropy is defined as Renyi's entropy for that matter. $ alpha = 2 $. It is given by

$$ mathrm {H} _ {2} (X) = – log sum_ {i = 1} ^ {n} p_ {i} ^ {2} tag {1} $$

Take two random variables $ X $ Y $ X & # 39; $ that follow the same probability distribution. The probability of a collision is simply $ P _ { rm coll} = sum_ {i = 1} ^ {n} p_ {i} 2 $. Then I would expect us to say that collision entropy is just $ H (P rm coll) that is to say

$$ – left ( sum_ {i = 1} ^ {n} p_ {i} ^ {2} right) log left ( sum_ {i = 1} ^ {n} p_ {i} ^ { 2} right) – left (1 – sum_ {i = 1} ^ {n} p_ {i} ^ {2} right) log left (1- sum_ {i = 1} ^ {n } p_ {i} ^ {2} right) $$

This is in analogy with binary entropy but with the probability replaced by the probability of a collision.

What is the motivation behind choosing $ (1) $ be the definition of collision entropy?

How to find f (x) y (a) as f & # 39; (A) is equal to the definition of the limit given above

Consider lim h → 0 frac{sqrt(4){16+h}-2}{h}

a) find a and f (x) such that f & # 39; (a) is equal to the limit given above
b) Use your answer to part a to evaluate the li>
c) Find the equation of the line tangent to the graph of f (x) in (a, f (a)).

8 – How to replace the definition of fields in a form by twig template?

In a custom module, I have a multi-step form (inspired by this).
The last step looks like this:
enter the description of the image here
This form is made with different types of field: inline_template, checkbox, checkboxes, radios.
The pieces in blue are values ​​entered in previous steps.

I would like to replace all these field definitions with a twig template.
It's possible?
If so, how?