## plotting – Reaching Maximum Number of Steps Using ParametricNDSolveValue with a Conditional Function

I’m trying to create a manipulatable SIR model with variable infection rate in Mathematica. To (very roughly) imitate the effect of government policies, I want the infection rate to be a certain constant if dI/dt is less than a certain value and another constant if dI/dt is above a certain value. This is what I have currently:

``````
g1(t_) =
If(
i(t) < 0.2, b0, b1
)
p1(t_) = 1;
(Beta)1(t_) = g1(t)*p1(t);

simpleSirModel1 = {
{
s'(t) == -(Beta)1(t)*s(t)*i(t),
i'(t) == (Beta)1(t)* s(t) *i(t) - (Gamma)*i(t),
r'(t) == (Gamma)*i(t)
},
s(0) == 1, i(0) == 0.001, r(0) == 0}
simpleSirSolution1 =
ParametricNDSolveValue(
simpleSirModel1, {s, i, r}, {t, 0, 300}, {(Gamma), b0, b1},
Method -> {"DiscontinuityProcessing" -> False})

Manipulate(
Plot(Evaluate@Through(simpleSirSolution1((Gamma), b0, b1)(t)), {t,
0, 100}, PlotLegends -> {"S", "I", "R"}, TicksStyle -> Medium,
AxesLabel -> {Style("Days After 3/2", Black, FontSize -> 10),
Style("Fraction of Population", Black, FontSize -> 10)},
PlotStyle -> {{Darker(Green), Thickness(0.005)}, {Red,
Thickness(0.005)}, {Magenta, Thickness(0.005)}},
ImageSize -> Large, PlotRange -> {{0, 100}, {0, 1}}),
{{b0, 0.2}, 0, 0.5},
{{b1, 0.1}, 0, 0.5},
{{(Gamma), 0.07}, 0, 0.5}
)

``````

When I execute the manipulate, I receive an error message:

``````ParametricNDSolveValue::mxst: Maximum number of 97466 steps reached at the point t == 46.10884123355165`.
``````

Is there a way to increase the maximum number of steps that ParametricNDSolveValue can handle? Is this the best way to approach my problem? I’m not sure how to otherwise approach it, since the function `(Beta)1(t)` changes value depending on the system it’s in.

## Are there any random variable \$X\$ and measurable function \$g\$ such that \$mathbb{E} [g(X)]=mathbb{E}[Xg(X)]=0\$ while \$g(X)ne 0\$?

As titled. I was going through some problems related to the relation between uncorrelatedness and indepedence, and reached a situation where the existence of such RV and function would be nice

## plugins – Calling PHP function with AJAX

I am creating a plugin for wordpress for the backend.
I when the admin press a button a js function is being called. In this js function, I am calling an API, and then I want to call a PHP with the result and the order id as a parms.
so I have done this:
`action.php`

``````function openpopup(\$order_id)
{

?>
<button id="shipping360_create_delivery" data-id= "<?php echo \$order_id?>"
style="background-color: #2e4453; color:#fff; border-radius: 5px;text-align: center;width:100px;height:auto;font-size:10px; line-height: 1.5em;  border: none;-webkit-appearance: none; outline: none; padding:10px;cursor: pointer;">create</button>
<?php
popup(\$order_id);

}

function update_order()
{

\$order_id = isset(\$_POST('id')) ? \$_POST('id') : 0;
\$shipping_number = isset(\$_POST('shipping_num')) ? \$_POST('shipping_num') : "";
\$order = wc_get_order(\$order_id);

update_post_meta(\$order_id, 'shipping_number', \$shipping_number);
die();

}

``````

the `popup` function opens a pop-up with the order details.
inside the `popup` function I have a button with the id “create_order” (this function has a lot of HTML and doesn’t really matter – the point is the button).
Now for the `app.js` file

``````(function (\$) {
\$('#close').click(function () {
//this is closing the pop-up
document.body.style = "overflow: scroll";

\$('#order-model').hide();
});
\$("#createorder").click(function () {
/// this is the call for the main function
createDelivery();
});

\$("#shipping360_create_delivery").click(function () {
// this is opening the pop-up
\$('#order-model').show();
document.body.style = "overflow: hidden;margin: 0;";

})
})
}(jQuery));

function createDelivery() {
// creates new delivery

const URL = BASE + "/shipping";

const distributor = document.getElementById("distributor").value;
const token = document.getElementById("ordertoken").innerHTML;
const companyid = document.getElementById("ordercompanyid").innerHTML;
const direction = "1";
const type = document.getElementById("ordertype").innerHTML;
const order_id = document.getElementById("ordernumber").innerHTML;

if (distributor == "") {
}

else {
senderinfo = "";//have data -- does not really matte
receiverinfo = ""; // have data -- does not really matter
const Data = {
senderinfo: senderinfo,
type: type,
distributor: distributor,
companyid: companyid,
token: token,
direction: direction
}
const otherPram = {
headers: { "Content-Type": "application/json; charset=UTF-8" },
body: JSON.stringify(Data),
method: 'POST'
};
console.log(URL)
fetch(URL, otherPram)
.then(data => {
return data.json()
})
.then(res => {
body = res("body");
//////////////////-----------------------///////////////////////////////
up to this point everything is working fine
(function (\$) {
\$.ajax({
type: "POST",
url: ajaxurl,
cache: false,
data: {
'action': 'update_order',
'id': order_id,
'shipping_num': body,
},
success: function (response) {

// not popping up
},
fail: function () {

// I have put this and the one below just for checking
// not popping up
},
complete: function () {
// when I call my function this is that only on the pops up
}
});
})(jQuery);

}).catch(function (error) {

});

}

}

``````

Thanks for the help

## c# – How to properly visualize Physics2D.CircleCast function in Unity?

I have a `Physics2D.CircleCast()` function in order to set `true` or `false` if the character is on the ground.

Based on this function, I want to draw a circle, mostly for debugging. I think this method could be quite useful to understand distances between different objects. such as between the character and the platform. The following image shows the variable set to `true` when the character is on the ground.

In order to draw the circle, my approach was to create an empty object with a Line Renderer component. Then, I created the following script (removing the section in which variables were declared):

``````public class DrawCircleCast : MonoBehaviour
{
// Call Game Object
public GameObject player;

// Start is called before the first frame update
void Start()
{
sizeValue = (2.0f * Mathf.PI) / 0.01f;
size = (int)sizeValue;
size++;

lr = GetComponent<LineRenderer>();

lr.material.color = Color.cyan;
lr.startWidth = 0.03f;
lr.endWidth = 0.03f;
lr.positionCount = size;
}

// Update is called once per frame
void Update()
{
DrawCircle();
}

void DrawCircle()
{
playerPosition = player.transform.position;
transform.position = playerPosition;

theta = 0f;
thetaScale = 0.01f;

for (int i = 0; i < size; i++)
{
theta += (2.0f * Mathf.PI * thetaScale);

x += gameObject.transform.position.x;
y += gameObject.transform.position.y;

pos = new Vector3(x, y, 0);
lr.SetPosition(i, pos);
}
}
}
``````

This piece of code creates a circle around the character, as follows:

So, my next approach consisted on change the circle’s position from the character to around the CircleCast measure, in order to have a better position. So I added new variables to the function and changed the circle’s position:

``````void DrawCircle()
{
playerPosition = player.transform.position;
platformPosition = platform.transform.position;

X = playerPosition.x;
Y = (playerPosition.y + (platformPosition.y)) / 2;

transform.position = new Vector3(X, Y, 0);

theta = 0f;
thetaScale = 0.01f;

for (int i = 0; i < size; i++)
{
theta += (2.0f * Mathf.PI * thetaScale);

x += gameObject.transform.position.x;
y += gameObject.transform.position.y;

pos = new Vector3(x, y, 0);
lr.SetPosition(i, pos);
}
}
``````

The result changed as follows:

Although the circle’s position is now located around the CircleCast measure, I am missing the distance value that I also added to the `Physics2D.CircleCast()` function.

So my question is how can I improve the circle’s position to have a better sense of how CircleCast is measuring the distance between two objects (i.e. character and platform). I couldn’t relate the distance, which is a parameter of the `Physics2D.CircleCast()` function, to the values I used to draw the circle. How the distance affects CircleCast() and how can I use this value to improve the circle’s position.

## real analysis – Prove the existence of local minimum of a multi-variable absolute value function.

I have the following function
$$H(mathbf{x}) = a_{1}|x_{1}-A_{1}|+a_{2}|x_{2}-A_{2}|+cdots+a_{n}|x_{n}-A_{n}| +b_{1}|x_{1}-B_{1}|+b_{2}|x_{2}-B_{2}|+cdots+a_{n}|x_{n}-B_{n}| +c_{1}|x_{1}-x_{0}|+c_{2}|x_{2}-x_{1}|+cdots+c_{n}|x_{n}-x_{n-1}|+c_{n+1}|x_{n+1}-x_{n}|$$
where all coefficients are positive.

For two dimensional case, I can plot $$H(x_{1},x_{2}) = a_{1}|x_{1}-A_{1}|+a_{2}|x_{2}-A_{2}| +b_{1}|x_{1}-B_{1}|+b_{2}|x_{2}-B_{2}| +c_{1}|x_{1}-x_{0}|+c_{2}|x_{2}-x_{1}|+c_{3}|x_{3}-x_{2}|$$
and it indeed has only one minimum,

However, I cannot prove it in a more general case. Does anyone once had encountered such problems?

## Viewing Old iOS App Function

I have an old photo editing 32-bit iOS app from around 2013/14. It no longer works on iOS since the transition to 64-bit but I wondered if there was a way to look into the apps code to discover some of the photo editing settings? There’s a few photo filters I would love to replicate in Photoshop.

## research – Relationship between the prime counting function of some number and the sum of prime numbers less than the square root of that number

A little bit more than a year ago I was able to prove that the sum of the prime numbers less than the square root of some number was asymptotically equivalent to the number of primes up that number (see http://vixra.org/abs/1911.0316).

In this same pre-print, I studied the numbers for which the number of prime numbers up to some number was exactly equal to the sum of the prime numbers less than the square root of that number. As a result, I derived the set of prime numbers that you can see here (http://oeis.org/A329403) and conjectured that there existed infinitely many of such prime numbers. This would imply that the value of the prime counting function of a given number would be infinitely often equal to the sum of prime numbers up to the square root of that number, and in other cases the sum described would be a very good estimator of the prime counting function.

A little bit more than a year after, I am unable to prove or disprove this conjecture. Unfortunately, as an amateur mathematician I feel uncapable of bringing this interesting research further, but it would be great if someone were able to prove or disprove the conjecture.

I find the relationship exposed between sums of prime numbers and the prime counting function really appealling and misterious, and I feel that there might be some hidden and profound mathematics there that I am not able to grasp.

I would like to hear your thoughts and ideas (if any) of how to bring the research to a happy ending. Thanks in advance for your time and effort!

## matrix – Function representing a forest fire after one time step

How can I create a function that would have as its argument an array mat of 0s, 1s, and 2s where the 2s are burning trees and the function would give an array that represents the forest after one time step?

I have tried using

``````mat=RandomChoice[{0,1,2},{10,10}]
nextstep[mat_]:=Sequence[mat2 + {1, 0}, mat2 + {0, 1}, mat2 + {-1, 0}, mat2 + {0,-1}]
``````

but that doesn’t seem to work. After one time step, every tree, represented by 1s, that is in the von Neumann neighborhood of a burning tree would catch fire, but the 0s, which means there are no trees, would remain the same.

## Problems in plotting a function of two variables

``````Clear("Global`*")

a = Re(Sqrt(x^2 - 2 x y + y^2 - 1^2));
``````

You cannot take the derivative of a function which contains `Re`. Use `ComplexExpand` to avoid use of `Re`

``````mat = {{0, -a/Sqrt((x - y)^2 + Abs(a)^2)}, {(x - y)/
Sqrt((x - y)^2 + Abs(a)^2), 0}} //
ComplexExpand(#, TargetFunctions -> {Re, Im}) & // Simplify;

d1 = D(mat, x);
d2 = Simplify(mat . d1, {x >= 0, y >= 0});
fun(x_, y_) = Simplify(
Tr(d1 . d1) + (1/Det(mat)) Tr(d2 . d2),
{x >= 0, y >= 0});

Plot3D(
fun(x, y), {x, 0, 5}, {y, 0, 5},
PlotPoints -> 75,
MaxRecursion -> 5,
ClippingStyle -> None) // Quiet
``````

## Harmonic function is the real part of a holomorphic function

Let $$u: Bbb R^2 to Bbb R$$ be an harmonic function, then there exists an holomorphic function $$f: Bbb C to Bbb C$$ s.t. $$Re(f)=u$$. The only hint I have are the Cauchy–Riemann equations:
$$u_x=v_y \ u_y=-v_x$$
I don’t see how I can use it… Can someone pls helps me?