Calculus and Analysis – Why does the kernel crash with this integral in V12?

With V12 under Windows 10

Any idea why the kernel crashes on this integral now?

ClearAll(x);
Integrate((1 + x^2)^3/(1 + x^2 + x^4)^(3/2), x)

Mathematica Graphics

No problem with V 11.3

Mathematica Graphics

Does this happen for other and other systems or only for Windows 10?

I notice that the V12 kernel crashes more than the V11.3 kernel, and in a strange way. This makes it very difficult to run a long script if the kernel crashes again and again.

ps. I hope I will not be rejected again because I asked for a potential problem in Mathematica, as in the last post on this bizarre kernel crash.

Analysis of the pdes wave equation with data on neutral surfaces

Consider the solid cone $ C $ as the region inside $ z = 1- sqrt {x ^ 2 + y ^ 2} $ and limited by $ 0 leq z leq 1 $, Let us now define $$ omega = {z = 0 } cap C quad text {and} quad Sigma = ( partialC cap {z leq frac {1} {2} }) setminus Omega $$,

We want to solve the wave equation $$ Box u = partial ^ 2_z u- partial ^ 2_x u – partial ^ 2_y u = 0 $$
within the region $$ C cap {z leq frac {1} {2} } $$
be subject to $$ u | _ { Omega} = f, quad partial_z u | _ { Omega} = g, quad u | _ { Sigma} = h. $$

Is that possible, we say for everyone $ (f, g, h) in H ^ 1 ( Omega) times L ^ 2 ( Omega) times H ^ 1 ( Sigma) $ with the natural compatibility conditions for $ f $ and $ h $ on $ partial omega $?

Singapore cuts its leverage cap – News & Analysis

Singapore has cut its leverage cap by more than half. This means that forex traders in the country now have access to a leverage of 1:20, as opposed to 1:50 as before the changes.

However, the decision was taken by the Monetary Authority of Singapore, as there are certain gaps due to the regulatory changes proposed by the European Securities and Markets Authority, allowing more leverage under stringent conditions.

Accredited investors in Singapore have access to the original debt capital, but this, as in Europe, would mean meeting strict capital requirements.

The requirements include a personal fortune of more than 2 million Singapore dollars (1.5 million US dollars) or proof of an annual income of more than 300,000 Singapore dollars. Traders with more than $ 1 million in cash in cash can also qualify for higher leverage.

real analysis – I seem to have a simple problem with job statistics …

I've been struggling with this problem for a while. I'll get right to it. Suppose that $ X $ will be delivered $ N (0,1) $. $ Y $ is distributed normally with positive mean and given variance, and $ Z $ is normally distributed with a positive mean and given variance. All three are independent. I am interested in the following calculation:

$ P (X> min (Y, Z)) $,

What I want to show is that if I increase the variance of either $ Y $ or the $ Z $ variable, this probability increases.

Graphically it seems to work: Imagine your three normal distributions on the same axis. We are interested in when the values ​​of the variables with the fixed distribution are furthest to the right $ N (0,1) $ is bigger than one of the two on the right. By increasing the variances from either of the two to the right, this distribution becomes flatter and "smoother," and hence the likelihood that $ X $ is larger than this variable seems to be increasing.

The following reference (https://www.untruth.org/~josh/math/normal-min.pdf) is a good way to get an overview of these probabilities. However, it is difficult to prove that their variance increases.

Any help would be appreciated.

Calculus and Analysis – The most bizarre kernel crash of all time. Kernel crash in Integrate under simple different setups. Cause can not be found

V 12 under Windows 10 64 bit. Note: this problem do not show up in V 11.3. Only in V12.

In the last 2 hours I've been trying to figure out why the V12 kernel crashes when invoked Integrate to this problem.

This is part of a much larger test suite. When this simple integration is performed, the kernel crashes and only one beep sounds without messages.

I tried to make the MWE as easy as possible.

First, the basic MWE is displayed, where it crashes, and then simple changes are displayed to fix the crash. In all of these MWE below, I always go out of the fresh kernel.

To run this MWE, the integration problem must be read from a file. The file contains a line. It is a pure text file. I will put a link to the file at the end. Also add a link to a small zip file where the notebook and the input file are in the same folder to make the job easier.

SetDirectory(NotebookDirectory());
ClearAll("Global`*");
test1(nameOfTestFile_) := Module({n, res, lst},(*crash*)
   lst = ReadList(nameOfTestFile);
   lst = DeleteCases(lst, Null);
   Do(
    Print("Before calling Integrate");
    res = AbsoluteTiming(TimeConstrained(Integrate(lst((n, 1)), lst((n, 2))), 60*3))
    ,
    {n, 1, Length@lst}  
    );
   );

Do(
 test1("Timofeev_Problems.m");
 Print("Finished")
 , {myCounter, 1, 1}
 )

 (*crash*)

You only need Timofeev_Problems.m File in same folder as a notebook, the above MWE runs. The file contains only one line

   {Sin(x)/(1 + Sin(x)), x, 2, x + Cos(x)/(1 + Sin(x))}

Here is a link to the above file in case Timofeev_Problems.m

Here is a link to the ZIP file that contains the notebook and the input file in a folder: strage_bug.zip

Now, after the crash, the craziness really starts.

The crash disappears when you do one of these simple things (which makes no sense whatsoever, why they eliminate the crash).

Removing a print statement in test1 eliminates the crash

Remove that Print Before in the test1the crash goes away. Like this

SetDirectory(NotebookDirectory());
ClearAll("Global`*");
test1(nameOfTestFile_) := Module({n, res, lst},(*crash*)
   lst = ReadList(nameOfTestFile);
   lst = DeleteCases(lst, Null);
   Do(
    res = AbsoluteTiming( TimeConstrained(Integrate(lst((n, 1)), lst((n, 2))), 60*3))
    ,
    {n, 1, Length@lst}  
    );
   );

Do(
 test1("Timofeev_Problems.m");
 Print("Finished")
 , {myCounter, 1, 1}
 )

 (*no crash*)

Removing the print statement in the loop that call test1 resolves the crash

SetDirectory(NotebookDirectory());
ClearAll("Global`*");
test1(nameOfTestFile_) := Module({n, res, lst},(*crash*)
   lst = ReadList(nameOfTestFile);
   lst = DeleteCases(lst, Null);
   Do(
    Print("Before calling Integrate");
    res = AbsoluteTiming(TimeConstrained(Integrate(lst((n, 1)), lst((n, 2))), 60*3))
    ,
    {n, 1, Length@lst}  
    );
   );

Do(
 test1("Timofeev_Problems.m")
 , {myCounter, 1, 1}
 )

 (*no crash*)

Maintaining all printing instructions, but removing AbsoluteTiming, eliminates the crash

SetDirectory(NotebookDirectory());
ClearAll("Global`*");
test1(nameOfTestFile_) := Module({n, res, lst},(*crash*)
   lst = ReadList(nameOfTestFile);
   lst = DeleteCases(lst, Null);
   Do(
    Print("Before calling Integrate");
    res = TimeConstrained(Integrate(lst((n, 1)), lst((n, 2))), 60*3)
    ,
    {n, 1, Length@lst}  
    );
   );

Do(
 test1("Timofeev_Problems.m");
 Print("Finished")
 , {myCounter, 1, 1}
 )

 (*no crash*)

Calling test1 out of a loop eliminates the crash

SetDirectory(NotebookDirectory());
ClearAll("Global`*");
test1(nameOfTestFile_) := Module({n, res, lst},(*crash*)
   lst = ReadList(nameOfTestFile);
   lst = DeleteCases(lst, Null);
   Do(
    Print("Before calling Integrate");
    res = 
     AbsoluteTiming(TimeConstrained(Integrate(lst((n, 1)), lst((n, 2))), 60*3))
    ,
    {n, 1, Length@lst}  
    );
   );

test1("Timofeev_Problems.m");
(*no crash, since call is not made from inside a loop !*)

The question is: Why integrate crash up? How can you eliminate it? To find messages from the kernel, what happened, as only one beep is emitted.

Functional Analysis – $ L ^ 2 $ norm of $ L ^ 2 $ norm less than $ L ^ 4 $ norm?

Can I estimate that? $ L ^ 2 $standard of $ L ^ 2 $standard of the $ L ^ 4 $-Standard? In other words, I can do something like that
$$ | | u | _2 ^ 2 | _2 lesssim | u | _4 ^ 2? $$
We would have to show that explicitly
$$ left ( int left | int | u | ^ 2dx right | ^ 2dx right) ^ {1/2} lesssim left ( int | u | ^ 4dx right) ^ {1 / 2}. $$
Is that possible?

Real Analysis – Dominated Convergence Theorem

I have trouble understanding the proof in the paper Learning about the temporal evolution of spatial dependence
Generalized spatiotemporal Gaussian process models
,

Theorem 2.1 on page 33 uses the dominated convergence theorem (DCT) to show that the following series converges $ L ^ 1 ( mathcal {Z} times mathcal {Z}) $ feel where $ mathcal {Z} = mathbf {X} times mathcal {T} $
$$ sum_ {l = 1} ^ infty lambda_l (t) mathcal {C} _ { mathcal {T}} (t, t & # 39;) lambda_l (t & # 39;) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39;) $$
by showing that
$$ sum_ {l = 1} ^ infty Bigg vert int _ { mathcal {Z}} int _ { mathcal {Z}} lambda_l (t) mathcal {C} _ { mathcal {T}} (t, t)) lambda_l (t)) phi_l ( mathbf {x}) phi_l ( mathbf {x}) d mathbf {z} d mathbf {z} & # 39; Bigg vert < infty $$
Where $ mathbf {z} = ( mathbf {x}, t) $ and $ mathbf {z} = ( mathbf {x} & # 39 ;, t & # 39;) $,

For me, however, the direct application of DCT would be the limits of
$$ sum_ {l = 1} ^ infty int _ { mathcal {Z}} int _ { mathcal {Z}} Big vert lambda_l (t) mathcal {C} _ { mathcal {T}} (t, t #) lambda_l (t #) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39;
Big vert d mathbf {z} d mathbf {z} & # 39; < infty $$

with a monotonous sequence of dominating functions
$$ sum_ {l = 1} ^ L Big vert lambda_l (t) mathcal {C} _ { mathcal {T}} (t, t & # 39;) lambda_l (t & # 39;) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39;
Big vert ge Bigg vert sum_ {l = 1} ^ L lambda_l (t) mathcal {C} _ { mathcal {T}} (t, t & # 39;) lambda_l (t & # 39;) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39;
Bigg vert quad forall L in mathbf {N} $$

have the following condition
$$ int _ { mathcal {Z}} int _ { mathcal {Z}} sum_ {l = 1} ^ L Big vert lambda_l (t) mathcal {C} _ { mathcal { T}} (t, t)) lambda_l (t)) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39;
Big vert d mathbf {z} d mathbf {z} & # 39; $$

$$ xrightarrow () {L rightarrow infty} sum_ {l = 1} ^ infty int _ { mathcal {Z}} int _ { mathcal {Z}} large vert lambda_l ( t) mathcal {C} _ { mathcal {T}} (t, t & # 39;) lambda_l (t & # 39;) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39)
Big vert d mathbf {z} d mathbf {z} & # 39; < infty $$

Do I miss something?

Thank you in advance.

Calculus and Analysis – The definite integral of a broken polynomial contains sin (x) and x ^ n

I meet a certain integral as $ int _ {- infty} ^ {+ infty} frac { sin ( text {x0} omega) – sin (x omega)} {( sin (x omega) – sin ( text {x0} omega)) ^ 2+ (x- text {x0}) ^ 2} , dx $, then I try it

Integrate((- Sin(x (Omega)) + 
    Sin(x0 (Omega)))/((x - x0)^2 + (Sin(x (Omega)) - 
     Sin(x0 (Omega)))^2), {x, -(Infinity), +(Infinity)}) // 
Simplify

The Mathematica, however, can find the result.
Could you help me?
Thanks.

Calculus and Analysis – Real and imaginary part of the complex logarithm

I have to use Mathematica to get the real and imaginary parts of the following expression:

$ (- i a – m ^ 2) ln ( frac {i m ^ 2} {2 a}) $.

Where $ a $ and $ m $ are real.

So we have:

Refine(Re((-I a - m^2) Log((I m^2)/(2 a))), {Element(a, Reals), Element(m, Reals)})

However, Mathematica returns the command again. How can I proceed?

Functional Analysis – Is $ { phi in mathcal C ([0,1], mathbb R) mid forall k in mathbb N ^ +: int ^ 1_0 x ^ k phi (x) = 0 } $ a singleton?

To let $$ A = { phi in mathcal C ((0,1), mathbb R) mid forall k in mathbb N ^ +: int ^ 1_0 x ^ k phi (x) = 0 } $$

Clear is the function $ (0,1) to mathbb R, , x mapsto 0 $ belongs $ A $, I would like to ask if $ A $ contains any other element.

Thank you for your help!