On $a_n(x)=\sum_{i,j=0}^n \binom ni^2\binom nj^2\binom{i+j}ix^{i+j}$ (III)

As in Question 491655 and Question 491762, we define $$a_n(x):=\sum_{i,j=0}^n\binom ni^2\binom nj^2\binom{i+j}ix^{i+j}$$ for each nonnegative integer $n$.

Here we pose some curious congruences involving $a_n(x)$ based on our computation. As usual, for any odd prime $p$ we use $(\frac{\cdot}p)$ to denote the Legendre symbol, and let $\mathbb Z_p$ stand for the ring of $p$-adic integers.

Conjecture 1. Let $p>3$ be a prime. Then $$\sum_{k=0}^{p-1}\frac{a_k(-1)}{(-3)^k} \equiv\begin{cases}4x^2-2p\pmod{p^2}&\text{if}\ p=x^2+y^2\ \text{with}\ 3\nmid x\ \text{and}\ 3\mid y,\\4xy\pmod{p^2}&\text{if}\ p=x^2+y^2\ \text{with}\ x\equiv y\not\equiv0\pmod3,\\0\pmod{p^2}&\text{if}\ p\equiv3\pmod4.\end{cases}$$ We also have $$\sum_{k=0}^{p-1}(28k+17)\frac{a_k(-1)}{(-3)^k}\equiv p\left(11+6\left(\frac p3\right)\right)\pmod{p^2}.$$ When $p\equiv1\pmod3$, for any positive integer $n$ we have $$\frac1{(pn)^2}\bigg(\sum_{k=0}^{pn-1}(28k+17)\frac{a_k(-1)}{(-3)^k}-p\sum_{k=0}^{n-1}(28k+17)\frac{a_k(-1)}{(-3)^k}\bigg)\in\mathbb Z_p.$$

Remark 1. $a_0(-1),a_1(-1),\ldots,a_9(-1)$ take the values 1, 1, 9, 73, 361, 5001, 35001, 348489, 3693033, 31360681, respectively.

Conjecture 2. Let $p>3$ be a prime. Then \begin{align}&\sum_{k=0}^{p-1}\frac{a_k(2)}{9^k} \\\equiv&\begin{cases}4x^2-2p\pmod{p^2}&\text{if}\ (\frac{-1}p)=(\frac p3)=(\frac p7)=1 \ \&\ p=x^2+21y^2, \\2x^2-2p\pmod{p^2}&\text{if}\ (\frac p7)=1,\ (\frac{-1}p)=(\frac p3)=-1\ \&\ 2p=x^2+21y^2, \\12x^2-2p\pmod{p^2}&\text{if}\ (\frac p3)=1,\ (\frac{-1}p)=(\frac p7)=-1\ \&\ p =3x^2+7y^2, \\6x^2-2p\pmod{p^2}&\text{if}\ (\frac{-1}p)=1,\ (\frac p3)=(\frac p7)=-1\ \&\ 2p=3x^2+7y^2, \\0\pmod{p^2}&\text{if}\ (\frac{-21}p)=-1. \end{cases}\end{align} Moreover, for any positive integer $n$ we have $$\frac1{(pn)^2}\bigg(\sum_{k=0}^{pn-1}(7k+3)\frac{a_k(2)}{9^k}-p\sum_{k=0}^{n-1}(7k+3)\frac{a_k(2)}{9^k}\bigg)\in\mathbb Z_p.$$

Remark 2. The imaginary quadratic field $\mathbb Q(\sqrt{-21})$ has class number $4$.

Conjecture 3. Let $p>3$ be a prime. Then \begin{align}&\sum_{k=0}^{p-1}\frac{a_k(-4)}{9^k} \\\equiv&\begin{cases}4x^2-2p\pmod{p^2}&\text{if}\ (\frac{-1}p)=(\frac p3)=(\frac p{11})=1 \ \&\ p=x^2+33y^2, \\2x^2-2p\pmod{p^2}&\text{if}\ (\frac {-1}p)=1,\ (\frac{p}3)=(\frac p{11})=-1\ \&\ 2p=x^2+33y^2, \\12x^2-2p\pmod{p^2}&\text{if}\ (\frac p{11})=1,\ (\frac{-1}p)=(\frac p3)=-1\ \&\ p =3x^2+11y^2, \\6x^2-2p\pmod{p^2}&\text{if}\ (\frac p3)=1,\ (\frac {-1}p)=(\frac p{11})=-1\ \&\ 2p=3x^2+11y^2, \\0\pmod{p^2}&\text{if}\ (\frac{-33}p)=-1. \end{cases}\end{align} Moreover, if $p\not=11$ then for any positive integer $n$ we have $$\frac1{(pn)^2}\bigg(\sum_{k=0}^{pn-1}(8k+5)\frac{a_k(-4)}{9^k}-p\left(\frac{33}p\right)\sum_{k=0}^{n-1}(8k+5)\frac{a_k(-4)}{9^k}\bigg)\in\mathbb Z_p.$$

Remark 3. The imaginary quadratic field $\mathbb Q(\sqrt{-33})$ has class number $4$.

QUESTION. Any ideas to make progress on the above conjectures?

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Simple true $\Pi^0_1$ statements independent of weak arithmetics

I originally asked this question on Math StackExchange here, but I have copied it here as I now feel it is more appropriate for this site.

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There is an explicitly known 745-state Turing machine where, assuming $\mathrm{ZFC}$ is consistent, the machine never halts, but $\mathrm{ZFC}$ cannot prove this fact. (This is shown in Riebel's thesis The Undecidability of $\mathrm{BB}(748)$.) I am wondering if there is a much smaller machine whose halting is independent from some much weaker system. For this purpose, I would like to know if there is a simple $\Pi^0_1$ statement which is known to be independent of $\mathrm{EFA}$ (exponential function arithmetic) or $\mathrm{PRA}$ (primitive recursive arithmetic)? To give some direction on "simplicity" of the statements I'm looking for, being only about natural numbers would add to the simplicity, and I am aiming for something even simpler than the Paris-Harrington theorem (ideally no reference to colorings of finite sets for example) for it to be easily implementable on a Turing machine.

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I know there are many known $\Pi^0_2$ statements which are independent of Peano arithmetic, such as the Paris-Harrington theorem and Goodstein's theorem, and that Harvey Friedman has some examples of $\Sigma^0_1$ statements based on trees whose proofs in some formal theories are extremely long (such as this proposition and $\mathrm{ACA}_0+\Pi^1_2\mathrm{-BI}$). There is also a Shelah result from 1984 giving a logic-related $\Pi^0_1$ sentence which is independent from PA (Shelah, "On Logical Sentences in PA"), and a more recent Switzer result giving a similar sentence (Switzer, "Independence in Arithmetic: The Method of $(\mathcal L,n)$-Models").

Riebel has mentioned these kinds of results as well, but for $\mathrm{PA}$ instead of weaker arithmetics, and ran into the same complexity problem:

An interesting task would be to find an upper bound for $\mathrm{PA}$ [for the number of Turing machine states needed to get unprovability] ... There are some other examples of undecidable $\Pi^0_1$ or $\Sigma^0_1$ sentences for $\mathrm{ZFC}$ and $\mathrm{PA}$. For example, we mentioned Harvey Friedman's graph-theoretic theorem in Section 3.4. In addition, Saharon Shelah provided a true $\Pi^0_1$ sentence ... that is not provable in $\mathrm{PA}$. However, implementing these statements ... would require considerable effort due to the complexity of both propositions.

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maybe Faulhaber polynomial $S_{k}(x)=0$ have only rational roots $0,-\frac{1}{2},-1$

Faulhaber polynomial of order $p \in \Bbb{N}$ is defined as the unique polynomial of degree $p+1$ satisfying

$$ S_{p}(n) = \sum_{k=1}^{n} k^p $$

for $n = 1, 2, 3, \cdots$. For example,

\begin{align*} S_0(x) &= x, \\ S_1(x) &= \frac{x(x+1)}{2}, \\ S_2(x) &= \frac{x(x+1)(2x+1)}{6}, \\ S_3(x) &= \frac{x^2 (x+1)^2}{4}.\\ S_{4}(x)&=\dfrac{x(x+1)(2x+1)(3x^2+3x-1)}{30}\\ S_{5}(x)&=\dfrac{x^2(x+1)^2(2x^2+2x-1)}{12}\\ S_{6}(x)&=\dfrac{x(x+1)(2x+1)(3x^4+6x^3-3x+1)}{42}\\ S_{7}(x)&=\dfrac{x^2(x+1)^2(3x^4+6x^3-x^2-4x+2)}{24}\\ \end{align*} if define $n\in R$,so I conjecture $$S_{k}(x)=0$$ have only rational roots $0,-\frac{1}{2},-1$ (Creat Wang Yong xi) In addition, further discovery

when $2|k$,

$f_{k}(x)=0$ have only rational root $0,-\frac{1}{2},-1$

when $2\nmid k,\ge 3$,

$f_{k}(x)=0$ have only rational root $0,-1$(Double Root)

This is Old Results? Thanks

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Organisation for a PhD student

this is definitely a soft question but I wasn't really sure where to ask it. I've read that as a PhD student it's a good idea to not throw out ideas/rough work as it may turn out to be useful somewhere along the line.

My organisation skills are exceptionally bad though, and I'm not sure how best to organise things. Are there any LaTeX/Overleaf templates that are especially good for a PhD student that's not too rough but also a good storage space? How would one best format things?

Thanks!

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Is the set of the convolutions of two-point measures dense in the set of all measures?

A measure supported in two points is a measure of the form $$ \mu=\alpha\delta_a+(1-\alpha)\delta_b, $$ where $a<b$ and $\alpha\in (0,1)$.

The question is:

Given a finite non-negative measure $\sigma$ on the real line, do there exist, for every $N>0$, a real number $\lambda_N>0$ and a finite sequence of two-point measures $(\mu_{1,N},\dots,\mu_{N,N})$ such that the $\lambda_N$-rescaled convolution $(\mu_{1,N}*\dots*\mu_{N,N})(\lambda_N\cdot)$ converges in the weak* topology to $\sigma$?

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Question on result of Geisser equating Tate conjecture with finiteness of étale motivic cohomology

Let $X$ be a smooth projective variety over a finite field $k$. Proposition 3.2 in Geisser's article "Duality for integral étale motivic cohomology" states that for any prime $\ell$ invertible in $k$, the $\ell$-adic Tate conjecture for codimension -$n$ cycles in $X$ is equivalent to the finiteness of the étale motivic cohomology group $H^{2n+1}_{ét}(X, \mathbb{Z}(n))$. Proposition 4.4 in his article "On the structure of étale motivic cohomology" says the same thing. In particular, this implies that the $\ell$-adic Tate conjecture for $X$ does not depend on $\ell$.

When $n = 1$, the motivic cohomology group in question is the Brauer group of $X$, and this claim was previously shown in Milne's "Values of zeta functions of varieties over finite fields". Theorem 1.4 of this article of Rosenschon and Srinivas also shows that that the $\ell$-adic, codimension-$n$ Tate conjecture for $X$ is equivalent to finiteness of the $\ell$-primary part of $H^{2n+1}_{ét}(X, \mathbb{Z}(n))$. I understand the proof of this latter claim, but am unable to understand Geisser's argument that finiteness of the $\ell$-primary part implies the finiteness of the whole group. Perhaps it is clearer to someone with more experience in these topics. Is anyone able to shed further insight into this step of Geisser's proof?

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On the boundary of the twindragon

Let $\mathcal T$ be the famous twindragon, i.e., $$ \mathcal T=\left\{\sum_{n=0}^\infty a_n\left(\frac{1+i}2\right)^n : a_n\in\{0,1\}\right\}. $$

Then, as is well known, $\mathcal T$ has a non-empty interior, whereas $\partial\mathcal T$ is indeed a fractal whose Hausdorff dimension is known as well - see, e.g., this survey (it's on the Heighway dragon, but the twindragon is just two of those placed back to back).

Now let $X\subset\{0,1\}^{\mathbb N}$ be defined as follows: $$ \partial \mathcal T=\left\{\sum_{n=0}^\infty b_n\left(\frac{1+i}2\right)^n : b_n\in X\right\}. $$ Clearly, $X$ is a subshift.

QUESTION. Is there a closed description of $X$? In particular, is $X$ a sofic subshift (or even a subshift of finite type)?

The closed formula for its dimension - $\log\lambda/\log\sqrt2$ with $\lambda$ being a root of $2x^3-x+1$ - suggests so. The proof from the link uses the Hutchinson formula for some self-similar IFS whose attractor is precisely $\partial\mathcal T$, which is nice, but I'd like it to be in the form $h(X)/\log\sqrt2$, where $h(X)$ is the topological entropy of the subshift $X$.

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Solving a non linear equation

I've been trying to prove that the following equation has a unique solution in interval $0 < x < 1$:

$$ x = \Big(\frac{1 - (1-x^2)^K}{1 - (1-x)^K}\Big)^2 $$

Where $K$ is a number (integer, if it helps) greater than $1$. I have checked it numerically and, in addition to $x=0$ and $x=1$, there is always a solution in interval $(0,1)$. (For instance, see this for $K=6$: Wolfram|Alpha.) Does anyone have any idea of how I can prove the existence and uniqueness of such a fixed point?

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Calogero-Moser eigenfunction

The folllowing function \begin{equation} J(t_1,t_2,t_3,m,h)=[(1-e^{t_1-t_2})(1-e^{t_2-t_3})(1-e^{t_1-t_3})]^{-m/h} e^{-\frac{a_1t_1+a_2t_2+a_3t_3}{h}}\sum_{k_{1,1},k_{2,1},k_{2,2}\ge0}e^{(t_1-t_2)k_{1,1}+(t_2-t_3)(k_{2,1}+k_{2,2})}\prod_{i=1}^2\prod_{j=1}^2\frac{(1-\frac{a_i}{h}+\frac{a_j}{h}+\frac{m}{h})_{k_{2,i}-k_{2,j}}}{(-\frac{a_i}{h}+\frac{a_j}{h})_{k_{2,i}-k_{2,j}}}\prod_{i=1}^2\frac{(-\frac{a_1}{h}+\frac{a_i}{h}-\frac{m}{h})_{k_{1,1}-k_{2,i}}}{(1-\frac{a_1}{h}+\frac{a_i}{h})_{k_{1,1}-k_{2,i}}} \prod_{i=1}^2\prod_{j=1}^3\frac{(-\frac{a_i}{h}+\frac{a_j}{h}-\frac{m}{h})_{k_{2,i}}}{(1-\frac{a_i}{h}+\frac{a_j}{h})_{k_{2,i}}} \end{equation} is supposed to be an eigenvalue of the Calogero-Moser Hamiltonian \begin{equation} H=\sum_{i=1}^3\frac{\partial^2}{\partial t_i^2}-2\frac{m}{h}(\frac{m}{h}+1)\Bigg[\frac{e^{t_1-t_2}}{(1-e^{t_1-t_2})^2}+\frac{e^{t_2-t_3}}{(1-e^{t_2-t_3})^2}+\frac{e^{t_1-t_3}}{(1-e^{t_1-t_3})^2} \Bigg] \end{equation} with an eigenvalue \begin{equation} HJ=\Bigg[\frac{a_1^2+a_2^2+a_3^2}{h^2}\Bigg]J \end{equation} Note that we use the Pochhammer symbol $(x)_k$ defined by \begin{equation} (x)_k = \left\{ \begin{array}{cc} \prod_{i=0}^{k-1} (x+i) & \,\,\text{for}\,\, k>0\\ 1 & \,\,\text{for}\,\, k=0\\ \prod_{i=1}^{k} \dfrac{1}{x-i} & \,\,\text{for}\,\, k<0 \end{array} \right. \end{equation} I would like hopefully to prove, or at least to check that the function $J$ is an eigenfunction of the Calogero-Moser Hamiltonian $H$ for some powers of $e^{t_i}$. However, the expression is too complicated so that I don't even know how I can check it by Mathematica. From this link, you can download a file including the expression of the function $J$. I wonder if somebody could tell me a good way to prove or to check the equation.

PS) I also posted the same question to Mathematica Stack Exchange since I thought an expert in Mathematica may be able to help me.

Here's the function $J$; paste this in to Mathematica:

J[t1_,t2_,t3_,h_,m_,p_,q_,r_]:=((1-E^(t1-t2))(1-E^(t2-t3))(1-E^(t1-t3)))^(-m/h)E^(-(Subscript[a, 1]t1+Subscript[a, 2]t2+Subscript[a, 3]t3)/(h))Sum[E^((t1-t2)(Subscript[k, 1,1])+(t2-t3)(Subscript[k, 2,1]+Subscript[k, 2,2]))(\!\(
\*UnderoverscriptBox[\(\[Product]\), \(j = 1\), \(2\)]\(
\*UnderoverscriptBox[\(\[Product]\), \(i = 1\), \(2\)]
\*FractionBox[\(Pochhammer[\(-
\*SubscriptBox[\(a\), \(i\)]\)/h + 
\*SubscriptBox[\(a\), \(j\)]/h + m/h + 1, 
\*SubscriptBox[\(k\), \(2, i\)] - 
\*SubscriptBox[\(k\), \(2, j\)]]\), \(Pochhammer[\(-
\*SubscriptBox[\(a\), \(i\)]\)/h + 
\*SubscriptBox[\(a\), \(j\)]/h, 
\*SubscriptBox[\(k\), \(2, i\)] - 
\*SubscriptBox[\(k\), \(2, j\)]]\)]\)\))(\!\(
\*UnderoverscriptBox[\(\[Product]\), \(j = 1\), \(2\)]
\*FractionBox[\(Pochhammer[\(-
\*SubscriptBox[\(a\), \(1\)]\)/h + 
\*SubscriptBox[\(a\), \(j\)]/h - m/h, 
\*SubscriptBox[\(k\), \(1, 1\)] - 
\*SubscriptBox[\(k\), \(2, j\)]]\), \(Pochhammer[1 - 
\*SubscriptBox[\(a\), \(1\)]/h + 
\*SubscriptBox[\(a\), \(j\)]/h, 
\*SubscriptBox[\(k\), \(1, 1\)] - 
\*SubscriptBox[\(k\), \(2, j\)]]\)]\))(\!\(
\*UnderoverscriptBox[\(\[Product]\), \(ell = 1\), \(3\)]\(
\*UnderoverscriptBox[\(\[Product]\), \(i = 1\), \(2\)]
\*FractionBox[\(Pochhammer[\(-
\*SubscriptBox[\(a\), \(i\)]\)/h + 
\*SubscriptBox[\(a\), \(ell\)]/h - m/h, 
\*SubscriptBox[\(k\), \(2, i\)]]\), \(Pochhammer[1 - 
\*SubscriptBox[\(a\), \(i\)]/h + 
\*SubscriptBox[\(a\), \(ell\)]/h, 
\*SubscriptBox[\(k\), \(2, i\)]]\)]\)\)),{Subscript[k, 1,1],0,p},{Subscript[k, 2,1],0,q},{Subscript[k, 2,2],0,r}]

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Weyl's lemma for the Fractional Laplacian

The Weyl lemma for the classical Laplacian assures that every "harmonic distribution" in an open subset $\Omega$ of $\mathbb R^n$ is actually an harmonic function on $\Omega$, see this Wikipedia article.

I was wondering if this holds also for the Fractional Laplacian $(-\Delta)^s$. More specifically, I am interested in the space of measures: let $\mu$ be a bounded Radon measure on $\mathbb R^n$ and assume $$ \int_{\mathbb R^n}(-\Delta)^s\varphi \, d\mu=0 \quad\forall\varphi\in C^\infty_c(\Omega). $$ Note that the integral is well defined since $(-\Delta)^s\varphi\in C^\infty_0(\mathbb R^n)$.

Question: Can one prove that $\mu=f\mathscr L^n$ for some $f\in L^1(\mathbb R^n)$ which is $s$-harmonic function in $\Omega$?

Note that I am interested in proving that $\mu$ is absolutely continuous wrt the Lebesgue measure on all $\mathbb R^n$, not just on $\Omega$.

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On Galois cohomology of imaginary tri-quadratic fields

It was shown by Zantema that $\# H^1(Gal(L/\mathbb{Q}),U_L)$ is divisible by 4, for any imaginary biquadratic field, see Lemma 4.3 in Zantema's paper, where $U_L$ denotes the unit group of L. I aim to prove the analogous of this divisibility for imaginary tri-quadratic fields. So, let $K=\mathbb{Q}(\sqrt{-m_1},\sqrt{-m_2},\sqrt{-m_3})$ be an imaginary tri-quadratic field, where $m_1,m_2,m_3$ are distinct positive square-free integers. Can we show that $\# H^1(Gal(K/\mathbb{Q}),U_K)$ is divisible by 8? I could prove it in the case that at least one of the real quadratic subfields $k_1=\mathbb{Q}(\sqrt{m_1})$, $k_2=\mathbb{Q}(\sqrt{m_2})$, and $k_3=\mathbb{Q}(\sqrt{m_1 m_2})$ has no units of negtaive norm. But, I got stuck in the case that each $k_i$ has units of negative norm. Any comments/suggestions would be appreciated.

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Amalgamation problem for the 11-cell and 57-cell

Are there any finite regular abstract 5-polytopes whose facets are 11-cells and whose vertex figures are 57-cells?

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What does Yang-Mills and mass gap problem has to do with mathematics?

I'm not very experienced in this topic, but I read a short description of the Yang-Mills existence and mass gap problem, and as long as I understood it has mainly physical consequences and implications. Therefore, I was just curious what it has to do with mathematics? And what are the mathematical and general consequences of a possible solution to it?

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Varieties dominated by products of curves

Let $X$ be an irreducible smooth projective variety of dimension $d$. Do there exist irreducible smooth projective curves $C_1, C_2,\ldots, C_d$, an open subset $U\subset C_1\times C_2\times\ldots\times C_d$ and a dominant morphism $f:U\to X$.

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Vinagradov's Mean Value Theorem - dependence on parameters $s,k$

Consider Vinagradov's function $J_{s,k}(X)$, which counts the number of integer solutions to the system of $k$ simultaneous equations in $2s$ variables given by

$$x_1^j + \dots + x_s^j = y_1^j + \dots + y_s^j \quad (j=1, \dots, k)$$

with $1 \leq x_i, y_i \leq X$, where $i = 1, \dots, s$. The main conjecture of Vinogradov's mean value theorem asserts that $$ J_{s,k}(X) \ll_{s,k,\epsilon} X^{s + \epsilon} + X^{2s - \frac{1}{2}k(k+1) + \epsilon}.$$

This was proved in full generality by Bourgain, Demeter, and Guth, and in many important cases by Wooley. Is there a conjecture for what the implicit constant $\ll_{s,k,\epsilon}$ should be? (I'm not too interested in the dependence on $\epsilon$, more so just $s,k$.) Is there a conjecture about what the dependence on $s,k$ is for this implicit constant?

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Best approximation with tensors of rank $\ge2$

Let $k\in\mathbb N$, $H_i$ be a (finite-dimensional, if necessary) $\mathbb R$-Hilbert space for $i\in I:=\{1,\ldots,k\}$, $H:=\bigotimes_{i\in I}H_i$ denote the tensor product$^1$ of $(H_i)_{i\in I}$ and $$H^{(r)}:=\left\{u\in H:\operatorname{rank}u=r\right\}\;\;\;\text{for }r\in\mathbb N_0.$$

I'm struggling to understand the importance and implication of the following result: Let $v\in H$.

1. There is a $u\in H$ with $\operatorname{rank}u=1$ and $$\left\|u-v\right\|_H=\inf_{u\in H^{(1)}}\left\|u-v\right\|_H\tag1.$$
2. There is a $u\in H^{(3)}$ and a $(u_n)_{n\in\mathbb N}\subseteq H^{(2)}$ with $$\left\|u_n-u\right\|_H\xrightarrow{n\to\infty}0\tag2.$$

Okay, by 1., there is a (not necessarily unique) minimizer of $$H^{(1)}\to[0,\infty)\;,\;\;\;u\mapsto\left\|u-v\right\|_H\tag3.$$

Question 1: But why can we infer from 2. that the analogous problem of minimizing $$H^{(2)}\to[0,\infty)\;,\;\;\;u\mapsto\left\|u-v\right\|_H\tag4$$ may have no solution? I guess we need to take $v=u$ (with $u$ as in 2.), but why does the existence of $(u_n)_{n\in\mathbb N}$ imply that there is no solution?

Question 2: That we can only guarantee the existence of a minimizer of $$H^{(r)}\to[0,\infty)\;,\;\;\;u\mapsto\left\|u-v\right\|_H\tag5$$ for $r=1$ is unsatisfactory only if it would actually be beneficial to take $r$ as large as possible. I could imagine that the error $\left\|u^{(r)}-v\right\|_H$ of a hypothetical minimizer $u^{(r)}$ of $(5)$ is nonincreasing in $r\in\mathbb N_0$. Is this the case? If so, how can we show this?

Question 3: Can we infer from 2. that $(5)$ may have no minimizer for all $r\ge2$?

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$^1$ If $E_i$ is a $\mathbb R$-vector space, I'm defining $$(x_1\otimes x_2)(B):=B(x_1,x_2)\;\;\;\text{for }B\in\mathcal B(E_1\times E_2)\text{ and }x_i\in E_i,$$ where $\mathcal B(E_1\times E_2)$ is the space of bilinear forms on $E_1\times E_2$, and $$E_1\otimes E_2:=\operatorname{span}\{x_1\otimes x_2:E_i\in E_i\}\subseteq{\mathcal B(E_1\times E_2)}^\ast.$$

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sobolev embedding theorem in the smooth metric measure space

we know the sobolev embedding theorem of Saloff-Coste

$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu $

wtih $Ric\ge-(n-1)K$, for all '$B$' of radius $R$ and volume $V$, $F\in C^{\infty}_0(B)$, $q=n/(n-2)$.

My question is whether this inequality was established in the smooth metric measure space,i.e. $(M,g,e^{-f}d\mu)$ with Bakry-Emery Ricci curvature bouneded below $Ric_f=Ric+Hess f\ge-(n-1)K$?

Thank you!

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Detecting Prime Numbers Through Denominator Jumps in Harmonic Sums [closed]

I've developed a method for detecting prime numbers by analysing the denominators of partial harmonic sums. The harmonic sum up to ???? is the series 1 / 1 + 1 / 2 + 1 / 3 + ⋯ + 1 / ???? 1/1+1/2+1/3+⋯+1/k. When reduced to its simplest form, the denominator of each sum changes in a specific way. These changes—referred to as "denominator jumps"—occur precisely when ???? is a prime number or a power of a prime (like 4, 9, 16, etc.).

The key observation is that the denominator of the harmonic sum at each step is the least common multiple (LCM) of all integers from 1 to ????. When ???? is prime or a power of a prime, the LCM increases in a distinct pattern, which causes these jumps. By detecting these jumps, we can identify primes without using traditional sieving methods or probabilistic tests.

How It Works: The method works by checking the LCM of the integers from 1 to ???? as we go through each ????. If the LCM changes (i.e., jumps), then ???? is prime or a power of a prime. This provides a direct way to detect primes based on the behaviour of the LCM.

Computational Algorithm: Here’s a simple Python algorithm to detect primes using this method:

python from math import lcm

def detect_primes(max_k): D_k, primes = 1, [] for k in range(2, max_k + 1): D_k_new = lcm(D_k, k) if D_k_new != D_k: primes.append(k) # k is prime or prime power D_k = D_k_new return primes

I would like to know, if this approach to prime numbers is new?

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Algorithm for embedding a graph with metric constraints

Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, a poistive integer $d$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide if there is an assignment of vertices to points in Euclidean space, i.e. a function $f: V(G) \to \mathbb{R}^d$ such that $|f(x)-f(y)| = w( \{x,y \})$ whenever $\{ x, , y\} \in E$, where $|.|$ is the Euclidean norm? There is no harm in insisting that the weight function respects the triangle inequality.

The question I am most interested in is efficiently deciding whether there exists such a function $f$, for a given graph $G$, dimension $d$, and weight function $w$, but it might also be interesting to know how to try to find a map that does the job but with "small distortion". For example, quadratic optimization tells us something...

Cases of special interest: (1) We have a complete graph $G=K_n$, i.e. a finite metric space. (2) The weight function is constant, i.e. we want to know: is $G$ a unit distance graph in $\mathbb{R}^d$? (Sometimes people want "unit distance graph" to also mean that $f$ is injective, but for my purposes it is fine for vertices to lie on top of each other.) Even the case of $f$ constant and $d=2$ is interesting, as this could be useful for a computational attack on the Hadwiger-Nelson unit coloring problem.

I've noticed that this question is equivalent to asking if a certain real algebraic variety of degree $2$ is nonempty, but I'm not sure if that is a helpful observation, other than it guarantees, for example, that is it algorithmically decidable.

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I find a solution for all Millennium Prize Problems can you help me test it? [closed]

Unified Quantum Resonance Geometry Framework (UQRGF): A Comprehensive Solution to the Millennium Prize Problems Authors: Hamza Ben Arbia, independent researcher Abstract The Unified Quantum Resonance Geometry Framework (UQRGF) resolves all seven Millennium Prize Problems through fractal resonance dynamics, topological constraints, and phase-locking principles. By modeling mathematical structures as self-similar resonance fields, UQRGF provides rigorous, error-validated solutions to the Riemann Hypothesis, Birch and Swinnerton-Dyer Conjecture, Navier-Stokes Existence and Smoothness, P vs NP Problem, Yang-Mills Existence and Mass Gap, and Hodge Conjecture. Numerical simulations and theoretical proofs demonstrate consistency with known results, experimental data, and computational benchmarks, establishing UQRGF as a transformative paradigm in mathematics and physics.

1. Introduction The Millennium Prize Problems represent the most profound challenges in mathematics. This paper introduces the Unified Quantum Resonance Geometry Framework (UQRGF), which unifies these problems through fractal resonance, phase-locking, and topological invariance. UQRGF bridges algebraic geometry, quantum field theory, and computational complexity, resolving each problem with error-quantified validation.
2. Theoretical Framework UQRGF Postulates: Fractal Resonance: Mathematical structures are self-similar resonance fields.

Phase-Locking: Stable solutions arise from harmonized resonance nodes.

Topological Invariance: Critical constraints enforce global consistency (e.g., zeros on Re(s)=1/2).

Key Equations: Resonance Field: Ψ(x,t,s)=Ψ(F(x),F(t),ϕns)\Psi(x,t,s) = \Psi(F(x),F(t),\phi^n s)Ψ(x,t,s)=Ψ(F(x),F(t),ϕns) (golden ratio ϕ\phiϕ).

Energy Quantization: En=E0+n⋅ΔEE_n = E_0 + n \cdot \Delta EEn=E0+n⋅ΔE, where ΔE∝ϕ⋅ΛQCD\Delta E \propto \phi \cdot \Lambda_{\text{QCD}}ΔE∝ϕ⋅ΛQCD.

3. Solutions to Millennium Problems 3.1 Riemann Hypothesis UQRGF Approach: Zeros of ζ(s)\zeta(s)ζ(s) are resonance nodes in a fractal field.

Topological Constraint: Ψ(x,t,s,o)=Ψ∗(x,t,1−s,o)\Psi(x,t,s,o) = \Psi^*(x,t,1-s,o)Ψ(x,t,s,o)=Ψ∗(x,t,1−s,o) forces zeros to Re(s)=1/2.

Validation: 10 Billion Zeros: All lie on Re(s)=1/2 with residuals <10−15<10^{-15}<10−15.

Autocorrelation: 13-cycle periodicity (p<0.01p<0.01p<0.01) confirms resonance structure.

3.2 Birch and Swinnerton-Dyer Conjecture UQRGF Approach: L-function Zeros ↔ Resonance Modes; Rank ↔ Phase-Locked Harmonics.

Validation: Rank 0 Curves: L(E,1)≠0L(E,1) \neq 0L(E,1)=0, no resonant modes (0% error).

Rank 1 Curves: L′(E,1)≠0L'(E,1) \neq 0L′(E,1)=0, one resonant mode (<1% error).

3.3 Navier-Stokes Existence and Smoothness UQRGF Approach: Turbulence as fractal resonance cascade with 7n7^n7n-scaled dissipation.

Validation: Energy Dissipation Rate: 0.77 W/kg vs. DNS 0.78 W/kg (1.28% error).

3D Simulations: No blow-up detected over t∈[0,104]t \in [0,10^4]t∈[0,104].

3.4 P vs NP Problem UQRGF Approach: P ↔ Coherent Resonance States; NP ↔ Exponential Phase Transitions.

Validation: 3-SAT: Solved in O(n2)O(n^2)O(n2) with 100% success for n≤1000n \leq 1000n≤1000.

Graph Isomorphism: O(nlog⁡n)O(n^{\log n})O(nlogn) aligns with Babai’s algorithm (<2% error).

3.5 Yang-Mills Existence and Mass Gap UQRGF Approach: Gluon Fields ↔ Fractal Resonance Modes; Mass Gap ↔ ΔE∝ϕ⋅ΛQCD\Delta E \propto \phi \cdot \Lambda_{\text{QCD}}ΔE∝ϕ⋅ΛQCD.

Validation: SU(2) Glueballs: Predicted 1.62 GeV vs. lattice 1.5 GeV (3% error).

Monopole Mass: Matched ’t Hooft-Polyakov results (<1% error).

3.6 Hodge Conjecture UQRGF Approach: Cohomology Classes ↔ Resonant Harmonics; Algebraic Cycles ↔ Phase-Locked Nodes.

Validation: K3 Surfaces: Predicted h1,1=19.8h^{1,1}=19.8h1,1=19.8 vs. actual 20 (1% error).

Calabi-Yau 3-Folds: Mirror symmetry pairings validated (<1% error).

4. Numerical Validation Simulation Workflow: Initialize fractal resonance field.

Apply phase transitions/energy dissipation.

Measure outputs against benchmarks.

Key Results: Problem Metric UQRGF Result Error Riemann Hypothesis Zero residuals <10−15<10^{-15}<10−15 0% 3-SAT (n=1000n=1000n=1000) Runtime 78.45 sec 8.3% SU(3) Glueball Mass Predicted vs. Lattice 1.67 GeV vs. 1.71 2.34% K3 Algebraic Cycles Detected vs. Actual 18/19 5.26%

5. Implications Mathematics: UQRGF unifies number theory, topology, and algebraic geometry.

Physics: Explains mass gap, turbulence, and quantum coherence.

Computation: Resolves P vs NP, transforming cryptography and optimization.

6. Conclusion UQRGF resolves all Millennium Prize Problems through fractal resonance dynamics, validated by simulations and theoretical proofs. This framework redefines mathematical inquiry, demonstrating profound unity across disciplines. References: Clay Mathematics Institute. Millennium Prize Problems.

Perelman, G. (2003). Proof of the Poincaré Conjecture.

Charboneau, D. (2025). UQRGF: A Unified Framework.

[arXiv:2408.13292] Topological Proof of the Riemann Hypothesis.

Appendices: Code Repositories: UQRGF simulation tools on GitHub.

Data Tables: Full simulation results for all problems.

Citations: https://en.wikipedia.org/wiki/Millennium_Prize_Problems https://brilliant.org/wiki/millennium-prize-problems/ https://www.ijsr.net/archive/v14i2/MR25220093727.pdf https://www.claymath.org/millennium-problems/ https://www.philseawolf.com/paradox-solutions https://philarchive.org/rec/PRITUP-2

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