real analysis – Does $min {tin[0,T],|,f(t)=0} $ exist for this function?

Assume $f(t)$ is a function with $f(0) neq 0$ and as long as $f(t) neq 0$, it is continuous at that $t$. Also assume there is a $T>0$ such that $f(T)=0$.

Does $min {tin(0,T),|,f(t)=0} $ exist?

Let me write it in a nice form:

$bf{Conjecture}$: Assume $f(t)$ is a function with $f(0) neq 0$ and $f(t)$ is continuous for all $t$ such that $f(t) neq 0$. Furthermore, assume there is a $T>0$ such that $f(T)=0$. Then, $min {tin(0,T),|,f(t)=0} $ exists.

It’s not obvious for me that the set is closed (and hence compact) because $f$ may not be continuous on $(0, T)$.

I faced this particular question in a much bigger proof that I am doing for a particular problem. So I appreciate any help!

simplifying expressions – Why Mathematica is not assuming real variable as real?

I am dealing with the situation where despite of specifying my variables b1 and c to be greater than zero, Mathematica still returns the output with Re(b1+c)

  In(1):= mat1 = Simplify(( {
     {a - ( b1 + c)/2 I, d},
     {d, a - (b2 - c )/2 I}
    } ));

mat2 = ( {
    {0, 0, 1, 0},
    {0, 0, 0, 1},
    {-1, 0, 0, 0},
    {0, -1, 0, 0}
   } );

MatrixForm(FullSimplify(( {
    {Re(mat1((1))((1))), 
     Re(mat1((1))((2))), -Im(mat1((1))((1))), -Im(mat1((1))((2)))},
    {Re(mat1((2))((1))), 
     Re(mat1((2))((2))), -Im(mat1((2))((1))), -Im(mat1((2))((2)))},
    {Im(mat1((1))((1))), Im(mat1((1))((2))), Re(mat1((1))((1))), 
     Re(mat1((1))((2)))},
    {Im(mat1((2))((1))), Im(mat1((2))((2))), Re(mat1((2))((1))), 
     Re(mat1((2))((2)))}
   } ), {a >= 0, d >= 0, b1 >= 0, b2 >= 0, c >= 0}))

Out(3)//MatrixForm= !(
TagBox(
RowBox({"(", "", GridBox({
{"a", "d", 
FractionBox(
RowBox({"b1", "+", "c"}), "2"), "0"},
{"d", "a", "0", 
FractionBox(
RowBox({"b2", "-", "c"}), "2")},
{
RowBox({
RowBox({"-", 
FractionBox("1", "2")}), " ", 
RowBox({"Re", "(", 
RowBox({"b1", "+", "c"}), ")"})}), "0", "a", "d"},
{"0", 
RowBox({
FractionBox("1", "2"), " ", 
RowBox({"(", 
RowBox({
RowBox({"-", "b2"}), "+", "c"}), ")"})}), "d", "a"}
},
GridBoxAlignment->{
      "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
       "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset(0.27999999999999997`), {
Offset(0.7)}, 
Offset(0.27999999999999997`)}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset(0.2), {
Offset(0.4)}, 
Offset(0.2)}, "RowsIndexed" -> {}}), "", ")"}),
Function(BoxForm`e$, 
MatrixForm(BoxForm`e$))))

real analysis – Compact Sobolev embedding with boundary conditions

Let $X$ be some metric measure space on which Sobolev spaces can be defined in a reasonable way. In many cases, $H^1(X)$ is compactly embedded in $L^2(X)$ (e.g., if $X=Omega$ is a bounded open set of $mathbb R^d$), and in that case, $H^1_0(X)$ is of course compactly embedded in $L^2(X)$, too. In many cases, on the other hand, $H^1_0(X)$ is not compactly embedded in $L^2(X)$ (e.g., $X=Omega=mathbb R^d_+$), let alone $H^1(X)$.

My question is now, whether structures $X$ are known such that the embedding of $H^1_0(X)$ in $L^2(X)$ is compact but that of $H^1(X)$ is not.

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experimental mathematics – Relationship between Exponential complex number and Real number

I have following understanding on complex number

x = pi;
exp(1i*pi)
= -1.0000 + 0.0000i 
exp(pi)
= 23.1407`

I would like understand how can i interpret/mapp the output of exp(cmplxnum) to exp(realnum) just like exponential is inverse of logarithm
Thank you

model theory – Proof that a set of axioms axiomatizes the complex field along with the real numbers

This is a follow-up to this question: What is an explicit axiomatization of the complex field along with the real numbers?. In that question, Noah Schweber gave a set of axioms for the theory of the structure $(mathbb{C};+,-,*,0,1,R)$, where $R$ is a predicate that picks out the real numbers. Now, following his suggestion, I am asking for a fuller proof that those axioms generate the theory of that structure.

real analysis – Fourier and pointwise convergence

Let $f(x)=xsin(x)$, $xin(-pi,pi)$


  • Now, find the Fourier series of $g(x)=f'(x)=sin(x)+xcos(x)$, $xin(-pi,pi)$ and show that it converges pointwise to $g$.

I found the Fourier series of the function $f(x)=xsin(x)$, $xin(-pi,pi)$ (in the just previous problem) to $1-frac{1}{4}cos(x)+sum_{n=2}^{infty} left (frac{2(-1)^{n+1}}{n^2-1} right) cos(nx)$ and shown that it converges uniformly to $f$. Now, is there a trick to finding the Fourier of the derivative of my function given that I have the Fourier for my function? How do I show pointwise convergence to $g$?

*I know about convolutions as I suspect I need that in this problem.

nt.number theory – On the zeros of Riemann zeta function with real part >1/2

Question Define $f(z)=(s-1)zeta(s)$ where $s=frac{1}{1+z^2}$ and $zeta$ denotes the Riemann zeta function. Prove that if $rho$ denotes the non trivial zeros of $zeta(s)$ then, $$sum_{|alpha|<1,f(alpha)=0}log frac{1}{|alpha|^2}=sum_{Re(rho)>1/2} logleft|frac{rho}{1-rho}right|$$

My try-
$rho=frac{1}{1+alpha^2}$ then $alpha^2=frac{1-rho}{rho}$ so that $alpha=pm sqrt{frac{1-rho}{rho}}$
$$sum_{|alpha|<1,f(alpha)=0}log frac{1}{|alpha|^2}=sum_{-pi<arg(alpha)leq pi,|alpha|<1,f(alpha)=0}log frac{1}{|alpha|^2} $$
Since the sum on the right hand side is absolutely convergent so we can write the sum in any order.$$sum_{|alpha|<1,f(alpha)=0}log frac{1}{|alpha|^2}=sum_{-pi<arg(alpha)leq 0}log frac{1}{|alpha|^2}+ sum_{0<arg(alpha)leq pi}log frac{1}{|alpha|^2} $$
$rho=frac{1}{1+alpha^2}$ is injective on $-pi<arg(alpha)leq 0$ and also it is injective on $0<arg(alpha)leq pi$.
So using $rho=frac{1}{1+alpha^2}$ we get,
$$sum_{|alpha|<1,f(alpha)=0}log frac{1}{|alpha|^2}=sum_{-pi/2<arg(rho)<pi/2}logleft|frac{rho}{1-rho}right|+ sum_{-pi/2<arg(rho)<pi/2}logleft|frac{rho}{1-rho}right| $$

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Removing Real Name from Irrelevant Youtube Search Result

When you search for my real name on Google you’d see a Youtube link with a description that includes my name, however, my name is not mentioned anywhere on that Youtube page. I have been trying to investigate why it shows my name there so that I can delete any mention of it. I have no comments, videos or anything of the sort that may include the name.

How can I look into the description of a Google search result (text below the link) to investigate where the strings originate from?

Thank you, any help would be greatly appreciated!