polynomials: Mathematica code for equations and mathematical symbols

Can anyone help me determine what the following Mathematica code is doing in terms of mathematical equations:

(I have trouble putting the code here, so I attach it in an image)

Is the code doing the equation in the following image:

(I'm having trouble putting Mathjax latex here)stieltjes

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

First, you should simply write Quit () and execute, then copy and paste the code you gave us into a new notebook. In general, when you get "protected errors," a variable has already been defined as something else.

If all else fails and you are not sure what is happening, 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})

The exact use of this code gives me a solution … although huge with conditionals.

$$ 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 } $$

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?

mathematical optimization: you need help finding a value to linearize the data

I have a two column data that is y vs x. I want to linearize the data with 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 write the following code to plot ln (y) vs 1 / (x-b) by changing the value b manually and looking at the graph ln (y) vs 1 / (x-b) to find the best linear behavior. Do you know a better way to do it 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)

Mathematical foundations: what math classes are relevant to machine learning?

I am a math and science student interested in machine learning, especially in the areas of reinforcement (deep) and NLP learning.

According to the current state of the research, what subjects of advanced mathematics (apart from the basic concepts such as calculus, linear algebra and probability) seem to be the most important for future theoretical work on ML? (Convex optimization? Game theory? Information theory?)

mathematical optimization: NMinimize displays the error message: NMinimize :: ivar: 0.28030267285728516` is not a valid variable

I have the minimization problem

data1 ={{-0.158842, 0.0427028}, 
        {-0.159004, 0.0299507}, 
        {-0.106839, 0.0341165}, 
        { 0.19894,  0.0157925}, 
        {-0.101634, 0.0395065},
        {........,..........}};

Clear(dist)
dist(vx_?NumericQ, vy_?NumericQ, px_?NumericQ, py_?NumericQ, pk_) := Module({lambda, d, v = {vx, vy}, p0 = {px, py}, 
big = 100000}, 
lambda = -v.(pk - p0)/v.v; 
If(lambda > 0, d = Sqrt((pk - p0 + lambda v).(pk - p0 + lambda v)), d = big); 
Return(d)
)

NMinimize({Sum(dist(vx, vy, px, py, data1((k))), {k, 1, Length(data1)}), 
vx^2 + vy^2 == 1}, {vx, vy, px, py}, Method -> "DifferentialEvolution")

An error message is issued. I don't have enough knowledge when I use a Module as a function to minimize / maximize, so I appreciate any information about it.

For “ Formulate & # 39; & # 39; and then retain / print mathematical expressions?

I have written a function myPfuntest,

myPfuntest(layer_) := 
    Table({-1, 1} /. 
        Abs(GroupBy(CoefficientList(
        Nest(x*D(#, x) &, Product(1 - x^(2^i), {i, 0, layer}), level)
        ,x), Sign)), {level, 1, layer}
)

myPfuntest(1) // MatrixForm
myPfuntest(2) // MatrixForm
myPfuntest(3) // MatrixForm
myPfuntest(4) // MatrixForm

enter the description of the image here

Hoping to get this kind of desired result

$ 1 + 2 = $ 3

Y

$
begin {aligned}
1 + 2 + 4 + 7 & = 3 + 5 + 6 \
1 ^ 2 + 2 ^ 2 + 4 ^ 2 + 7 ^ 2 & = 3 ^ 2 + 5 ^ 2 + 6 ^ 2
end {aligned}
$

Y

$
begin {aligned}
1 + 2 + 4 + 7 + 8 + 11 + 13 + 14 & = 3 + 5 + 6 + 9 + 10 + 12 + 15 \
1 ^ 2 + 2 ^ 2 + 4 ^ 2 + 7 ^ 2 + 8 ^ 2 + 11 ^ 2 + 13 ^ 2 + 14 ^ 2 & = 3 ^ 2 + 5 ^ 2 + 6 ^ 2 + 9 ^ 2 + 10 ^ 2 + 12 ^ 2 + 15 ^ 2 \
1 ^ 3 + 2 ^ 3 + 4 ^ 3 + 7 ^ 3 + 8 ^ 3 + 11 ^ 3 + 13 ^ 3 + 14 ^ 3 & = 3 ^ 3 + 5 ^ 3 + 6 ^ 3 + 9 ^ 3 + 10 ^ 3 + 12 ^ 3 + 15 ^ 3
end {aligned}
$

I can use Sqrt(), CubeRoot Y Surd to get the base numbers, or simplify the format based on the linear sets, but how can I prevent them from evaluating? For example

{1, 2, 4, 7}^2

would not give me

{1 ^ 2.2 ^ 2.4 ^ 2.7 ^ 2}

Language and its mathematical approach with respect to logical connections.

How do you express something that is beyond the limits of language? Does language infer pronunciation or is some form of non-auditory or stereo communication considered language?

Terminology: what is the mathematical name of a set that contains the domain and codomain of a function?

I am interested in this in order to name a type parameter in a program I am 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 for X, if any?

This is interesting from a program security perspective, since by setting X we can restrict the information so that it only flows within X.

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

Terminology: what is the mathematical name of a set that contains the domain and codomain of a function?

I am interested in this in order to name a type parameter in a program I am 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 for X, if any?

This is interesting from a program security perspective, since by setting X we can restrict the information so that it only flows within X.