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

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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!