# Collection prompts

Copy one complete prompt into each fresh conversation. Keep the researcher checklist separate. These blocks contain source text and the shared instruction only. No explanations have been generated.

## T01

BEGIN PROMPT

Explain the target theorem in plain English for a mathematically interested reader who is not a specialist in this field. Use the supplied context to interpret notation. You may retain a formula where it helps.

Source context:
```latex
\section{Introduction}\label{sec:introduction}

For complex functions $F,G\in L^3(\R^2)$ and
$0<\varepsilon<R<\infty$, define the annular triangular Hilbert transform by
\begin{equation}\label{eq:bilinear-truncation}
 B_{\varepsilon,R}(F,G)(x,y)
 =\int_{\varepsilon<|t|<R}F(x+t,y)G(x,y+t)\,\frac{\dd t}{t}.
\end{equation}
The two inputs are translated along different coordinate directions.
Every finite integral is absolutely convergent outside one common
null set; we use its usual representative there and set all quantities
to zero on the exceptional set. Lemma~\ref{lem:finite-truncations}
justifies this convention and continuity in the endpoints.

For $r>2$, its annular variation is
\begin{equation}\label{eq:variation-definition}
 V_r(F,G)(x,y)
 =\sup_{\substack{J\ge1\,,\ 0<t_0<\cdots<t_J\\t_j\in\mathbb Q}}
    \left(\sum_{j=1}^J
          |B_{t_{j-1},t_j}(F,G)(x,y)|^r\right)^{1/r}.
\end{equation}
The supremum is pointwise: the partition may depend on $(x,y)$, and
there is no restriction on the number of endpoints within a dyadic
scale interval. The use of rational endpoints makes measurability
immediate and, by endpoint continuity, does not change the supremum.


```

Target theorem:
```latex
\begin{theorem}[Full annular variation]\label{thm:variation}
For every real $r>2$ there is a finite constant $C_r$ such that
\[
 \|V_r(F,G)\|_{L^{3/2}(\R^2)}
 \le C_r\|F\|_{L^3(\R^2)}\|G\|_{L^3(\R^2)}
\]
for all complex $F,G\in L^3(\R^2)$.
\end{theorem}
```

END PROMPT

---

## T02

BEGIN PROMPT

Explain the target theorem in plain English for a mathematically interested reader who is not a specialist in this field. Use the supplied context to interpret notation. You may retain a formula where it helps.

Source context:
```latex
Let $G=(V,E)$ be a finite simple undirected graph. A perfect matching is a
set of edges incident to each vertex exactly once. Write
\[
 Z(G)=\#\{M\subseteq E:M\text{ is a perfect matching of }G\}.
\]
A fully polynomial randomized approximation scheme approximates $Z(G)$
within a prescribed relative error, with a prescribed success probability,
in time polynomial in the input length and the reciprocal error.
The confidence parameter enters through its logarithm. We prove the
following theorem, including a bound on the time of every execution.


```

Target theorem:
```latex
\begin{theorem}\label{thm:main}
There is a uniform classical randomized algorithm which, given $G$ and
rational parameters $0<\eps<1$ and $0<\delta<1/2$, returns a nonnegative
rational number $\widehat Z$ such that
\[
 \Prb\bigl[(1-\eps)Z(G)\le \widehat Z\le(1+\eps)Z(G)\bigr]
 \ge 1-\delta.
\]
If $Z(G)=0$, the output is zero with certainty. The worst-case bit running
time is polynomial in the input encoding length, $\eps^{-1}$, and
$\log\delta^{-1}$.
\end{theorem}
```

END PROMPT

---

## T03

BEGIN PROMPT

Explain the target theorem in plain English for a mathematically interested reader who is not a specialist in this field. Use the supplied context to interpret notation. You may retain a formula where it helps.

Source context:
```latex
\section{Introduction}\label{sec:introduction}

For nonnegative integer vectors $r=(r_1,\ldots,r_m)$ and
$c=(c_1,\ldots,c_n)$ with common total $N$, write
\[
 \Omega(r,c)=\left\{X\in\Z_{\ge0}^{m\times n}:
       \sum_jX_{ij}=r_i,\quad \sum_iX_{ij}=c_j\right\}.
\]
The uniform distribution on this finite set gives every table equal mass.
We study sampling when both dimensions vary and the margins are encoded
in binary. In particular, a polynomial bound in the numeric total $N$
would not, in general, give a polynomial bound in the input length.
There are no cell bounds or forbidden positions: every nonnegative integer
matrix with the prescribed margins is allowed. For probability laws on a
finite set, we use the convention
$\TV(\mu,\nu)=\tfrac12\sum_x|\mu(x)-\nu(x)|$.


```

Target theorem:
```latex
\begin{theorem}\label{thm:main}
For arbitrary nonnegative integer margins of equal total, the following
algorithms exist in the model of unbiased random bits and bit operations.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item Given an integer $k\ge1$, a bounded-time algorithm outputs a table
 in $\Omega(r,c)$ whose law is within $2^{-k}$ in total variation of the
 uniform law. Its running time is polynomial in
 $m,n,\log(N+1),k$.
 \item An algorithm outputs an exactly uniform table in $\Omega(r,c)$,
 terminates almost surely, and has expected running time polynomial in
 $m,n,\log(N+1)$.
\end{enumerate}
The polynomials are uniform over all margin vectors. No positivity,
sparsity, balance, or fixed-dimension hypothesis is imposed.
\end{theorem}
```

Additional source context:
```latex


For a rational tolerance $0<\eps<1$ encoded by a binary numerator and
denominator, integer comparisons compute
$k=\lceil\log_2(1/\eps)\rceil$ in time polynomial in that encoding length.
Thereafter part~\textup{(i)} has accuracy dependence polynomial in
$\log(1/\eps)$.
The algorithms below are explicit, although their exponents are deliberately
large. Part~\textup{(ii)} concerns expected time: an exceptionally rare
branch performs exhaustive computation.


```

END PROMPT

---

## T04

BEGIN PROMPT

Explain the target theorem in plain English for a mathematically interested reader who is not a specialist in this field. Use the supplied context to interpret notation. You may retain a formula where it helps.

Source context:
```latex
\section{Introduction}\label{sec:introduction}

A polynomial threshold function assigns a sign to each vertex of the
Boolean cube by evaluating a real polynomial. On the cube, the identities
$x_i^2=1$ allow every polynomial to be replaced by a multilinear polynomial
of no larger degree without changing its values. Average sensitivity
measures the expected number of coordinate changes that reverse that sign.
For $x\in\{-1,1\}^n$, let $x^{\oplus i}$ be obtained by reversing
coordinate $i$. For $f:\{-1,1\}^n\to\{-1,1\}$, define
\begin{equation}\label{eq:influence-definition}
 I(f)=\sum_{i=1}^n\Prb\{f(X)\ne f(X^{\oplus i})\},
 \qquad X\text{ uniform on }\{-1,1\}^n.
\end{equation}
This quantity is also called the total influence of $f$. Throughout,
\[
 \sgn(t)=\begin{cases}1,&t\ge0,\\-1,&t<0.\end{cases}
\]
We prove the following estimate.


```

Target theorem:
```latex
\begin{theorem}\label{thm:main}
Let $n\ge1$ and $1\le d\le n$ be integers. Let $p$ be a real multilinear
polynomial of degree at most $d$, and define
$f:\{-1,1\}^n\to\{-1,1\}$ by $f(x)=\sgn(p(x))$, with $\sgn(0)=1$.
Then, for average sensitivity under the uniform law,
\[
 I(f)\le 8d\sqrt n.
\]
\end{theorem}
```

Additional source context:
```latex


The constant is independent of both $d$ and $n$. In particular, the degree
may grow with the dimension. The polynomial may vanish on the cube, and
no regularity condition is imposed on $p$.
```

END PROMPT

---

## T05

BEGIN PROMPT

Explain the target theorem in plain English for a mathematically interested reader who is not a specialist in this field. Use the supplied context to interpret notation. You may retain a formula where it helps.

Source context:
```latex
\section{Introduction}\label{sec:introduction}

Let $\Lambda$ be a finite nonempty alphabet, and let
\[
 \Phi=(\varphi_i)_{i\in\Lambda},\qquad
 \varphi_i(x)=r_i x+t_i,\qquad t_i\in\R,\quad 0<|r_i|<1.
\]
The family is indexed: different symbols may specify the same map.
Given a probability vector $p=(p_i)_{i\in\Lambda}$ with $p_i>0$, its
self-similar measure is the unique Borel probability measure satisfying
\[
 \mu=\sum_{i\in\Lambda}p_i(\varphi_i)_*\mu.
\]
Equivalently, $\mu$ is the law of
$\lim_{n\to\infty}\varphi_{I_1}\circ\cdots\circ\varphi_{I_n}(0)$,
where the symbols $I_j$ are independent with law $p$.
The limit exists uniformly in the address: if
$r_{\max}=\max_i|r_i|$ and $T_0=\max_i|t_i|$, all coding limits
have absolute value at most $T_0/(1-r_{\max})$.

For a word $w=i_1\cdots i_n$, write
$\varphi_w=\varphi_{i_1}\circ\cdots\circ\varphi_{i_n}$ and
$p_w=p_{i_1}\cdots p_{i_n}$.  An \emph{exact overlap} is an equality
$\varphi_u=\varphi_v$ for distinct words of the same positive length.
Equality here means equality of the complete affine maps, including
both the signed slope and the translation.  We allow all such overlaps.

The relevant entropy counts maps rather than addresses.  Set
\[
 G_n=\varphi_{I_1}\circ\cdots\circ\varphi_{I_n},\qquad
 \P(G_n=g)=\sum_{w\in\Lambda^n:\,\varphi_w=g}p_w.
\]
All logarithms below have base two.  For a finite-valued random variable
$Z$, its Shannon entropy is $H(Z)=-\sum_z\P(Z=z)\log\P(Z=z)$,
with $0\log0=0$.  Define the random-walk entropy rate and the Lyapunov
exponent by
\begin{equation}\label{eq:rate-and-lyapunov}
 h=h_{\mathrm{RW}}(\Phi,p)
   :=\lim_{n\to\infty}\frac{H(G_n)}n
    =\inf_{n\ge1}\frac{H(G_n)}n,
 \qquad
 \chi=-\sum_i p_i\log|r_i|>0.
\end{equation}
Indeed, a length-$(n+m)$ map is a function of two independent maps
with laws $G_n,G_m$, so $H(G_{n+m})\le H(G_n)+H(G_m)$; subadditivity
gives the displayed limit and infimum.  Also
$0\le h\le H(p):=-\sum_i p_i\log p_i$.  The ratio $h/\chi$ is
independent of the common logarithm base.

We use the lower Hausdorff dimension of a measure:
\[
 \dim_H\nu=\inf\{\dim_H E:E\subset\R\text{ Borel},\ \nu(E)>0\}.
\]
Feng--Hu's exact-dimensionality theorem
\cite[Theorem~2.8]{FengHu2009} applies to finite self-similar systems
without separation.  Indeed, the maps preserve a sufficiently large
compact interval, extend to contracting smooth diffeomorphisms, and have
derivative norm and least singular value both equal to $|r_i|$.
The system is therefore conformal, and its Bernoulli coding law is
ergodic.  These observations also cover negative ratios and repeated
indexed maps.  Thus, for the measure
$\mu$ above, there is a constant $d$ such that
\begin{equation}\label{eq:exact-dimensionality}
 \lim_{s\downarrow0}\frac{\log\mu(B(x,s))}{\log s}=d
 \quad\text{for $\mu$-almost every $x$}.
\end{equation}
For an exact-dimensional measure this constant equals both $\dim_H\mu$
and $\inf\{\dim_H E:\mu(E)=1\}$.
To see the equivalence, restrict to countably many sets on which the
local bounds $s^{d+\varepsilon}\le\mu(B(x,s))\le s^{d-\varepsilon}$
hold uniformly for all sufficiently small $s$.  The lower mass bound
gives a full-measure union of sets of dimension at most $d+\varepsilon$
by a covering argument; the upper mass bound gives dimension at least
$d-\varepsilon$ for every positive-mass set.  Let
$\varepsilon\downarrow0$.  Consequently either Hausdorff
measure-dimension convention gives the same statement below.


```

Target theorem:
```latex
\begin{theorem}\label{thm:main}
For every finite nonempty indexed family
$\varphi_i(x)=r_i x+t_i$ on $\R$ with $0<|r_i|<1$, and every
strictly positive probability vector $p$, its self-similar measure satisfies
\[
 \dim_H\mu_{\Phi,p}
 =\min\left\{1,\frac{h_{\mathrm{RW}}(\Phi,p)}{\chi(\Phi,p)}\right\}.
\]
No separation assumption is required; exact overlaps and repeated
generators are allowed.
\end{theorem}
```

END PROMPT

---

## T06

BEGIN PROMPT

Explain the target theorem in plain English for a mathematically interested reader who is not a specialist in this field. Use the supplied context to interpret notation. You may retain a formula where it helps.

Source context:
```latex
\section{Introduction}
\label{sec:introduction}

Let $(X,\mathcal F,\mu)$ be a probability space and let $T:X\to X$
be invertible, bimeasurable, and measure preserving. We say that $T$
is \emph{mixing} if
\begin{equation}\label{eq:mixing}
 \mu(A\cap T^{-r}B)\longrightarrow\mu(A)\mu(B)
 \qquad (|r|\to\infty)
\end{equation}
for every $A,B\in\mathcal F$. For a fixed tuple of bounded measurable
functions, the consecutive multiple ergodic averages are
\[
 A_N(f_1,\ldots,f_n)(x)
 =\frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x).
\]
The problem is to determine their almost-everywhere behavior, not merely
their limit in norm. We prove the following result.


```

Target theorem:
```latex
\begin{theorem}\label{thm:main}
Suppose that $T$ satisfies \eqref{eq:mixing}. For every integer
$n\ge2$ and every fixed $f_1,\ldots,f_n\in L^\infty(\mu)$,
\[
 A_N(f_1,\ldots,f_n)(x)
 \longrightarrow\prod_{j=1}^n\int_X f_j\dd\mu
 \qquad\text{for $\mu$-almost every }x,
\]
as $N$ tends to infinity through all positive integers. The probability
space need not be standard, and no rate of mixing is required.
\end{theorem}
```

Additional source context:
```latex


The exceptional null set may depend on the system and the fixed tuple.
The theorem concerns the times $k,2k,\ldots,nk$ and does not require
a common exceptional set for all bounded functions.
```

END PROMPT

---

## T07

BEGIN PROMPT

Explain the target theorem in plain English for a mathematically interested reader who is not a specialist in this field. Use the supplied context to interpret notation. You may retain a formula where it helps.

Source context:
```latex
\section{Introduction}\label{sec:introduction}

Write $[n]=\{1,\ldots,n\}$ for a positive integer $n$.
For a finite set $A\subset\N=\{1,2,\ldots\}$, write
\[
 \FS(A)=\left\{\sum_{a\in B}a:\varnothing\ne B\subseteq A\right\},
 \qquad
 \FP(A)=\left\{\prod_{a\in B}a:\varnothing\ne B\subseteq A\right\}.
\]
Thus each element of $A$ may occur at most once in an individual sum
or product, and singleton subsets are included. We prove the following.


```

Target theorem:
```latex
\begin{theorem}\label{thm:main}
Let $r,m\ge1$ be integers, and fix real numbers $R\ge2$ and $D\ge1$.
For every coloring $\chi:\N\to[r]$,
there are distinct positive integers $a_1<\cdots<a_m$ and a color
$c\in[r]$ such that
\[
 \chi\left(\sum_{j\in J}a_j\right)
 =\chi\left(\prod_{j\in J}a_j\right)=c
 \quad\text{for every }\varnothing\ne J\subseteq[m].
\]
Equivalently, $\FS(A)\cup\FP(A)$ is monochromatic for a set
$A\subset\N$ of cardinality $m$.
The elements can additionally be chosen to satisfy
\begin{equation}\label{eq:separated-elements}
 a_1>R,\qquad
 a_d>R\left(\sum_{k<d}a_k+\prod_{k<d}a_k\right)^D
 \quad(2\le d\le m).
\end{equation}
\end{theorem}
```

END PROMPT

---

## T08

BEGIN PROMPT

Explain the target theorem in plain English for a mathematically interested reader who is not a specialist in this field. Use the supplied context to interpret notation. You may retain a formula where it helps.

Source context:
```latex
\section{The result and its context}

Foulkes' conjecture compares two ways of composing symmetric powers.
For a finite-dimensional complex vector space \(V\) and integers
\(1\le a\le b\), it asks for a \(\GL(V)\)-equivariant injection
\[
             \Sym^a(\Sym^b V)\hookrightarrow\Sym^b(\Sym^a V).
\]
Over \(\CC\), complete reducibility makes this equivalent to an
inequality between the multiplicities of every irreducible polynomial
representation. Write \(\Schur_\lambda V\) for the Schur module indexed
by a partition \(\lambda\), and \(s_\lambda\) for its character.
If \(h_r\) denotes the complete homogeneous symmetric function, then
\(h_a[h_b]\) is the stable character of \(\Sym^a(\Sym^b V)\), where
brackets denote plethysm. The conjecture says that
\(h_b[h_a]-h_a[h_b]\) is Schur-positive: all its Schur coefficients
are nonnegative integers. We prove the sixth case.


```

Target theorem:
```latex
\begin{theorem}\label{thm:main}
For every integer \(b\ge6\) and every finite-dimensional complex vector
space \(V\), there is a \(\GL(V)\)-equivariant injection
\[
                 \Sym^6(\Sym^b V)\ \hookrightarrow\ \Sym^b(\Sym^6 V).
\]
Equivalently, \(h_b[h_6]-h_6[h_b]\) is Schur-positive for every \(b\ge6\).
\end{theorem}
```

Additional source context:
```latex


Theorem~\ref{thm:main} resolves this case of Foulkes' conjecture
positively, with no restriction on \(\dim V\). The diagonal case
\(b=6\) is immediate; the content is the comparison for every larger
\(b\).
```

END PROMPT

---

## T09

BEGIN PROMPT

Explain the target theorem in plain English for a mathematically interested reader who is not a specialist in this field. Use the supplied context to interpret notation. You may retain a formula where it helps.

Source context:
```latex
\section{Introduction}
\label{sec:introduction}

For a subset $S$ of a von Neumann algebra $M$, write $W^*(S)$ for the
smallest unital von Neumann subalgebra containing $S$. Thus adjoints
are included and generation is taken in the weak operator topology.
The generator problem asks whether every von Neumann algebra with
separable predual has the form $W^*(x)$ for one bounded operator $x$.
Equivalently, it asks for two self-adjoint generators: their sum
$a+ib$ recovers $a$ and $b$ as its real and imaginary parts.

We prove this assertion for type $\mathrm{II}_1$ factors by first
establishing a relative generation theorem. For a finite factor with
normalized trace $\tau$, put $\norm{x}_2=\tau(x^*x)^{1/2}$.
An inclusion $P\subset M$ is \emph{irreducible} when
$P'\cap M=\mathbb C1$. Write $\mathcal U(M)$ for the unitary group
of $M$.


```

Target theorem:
```latex
\begin{theorem}\label{thm:relative-generation}
Let $P\subset M$ be an irreducible inclusion of type $\mathrm{II}_1$
factors, with $M$ of separable predual. The set
\[
 \{u\in\mathcal U(M):W^*(P,u)=M\}
\]
is a dense $G_\delta$ subset of the unitary group $\mathcal U(M)$
for the trace $2$-norm topology.
\end{theorem}
```

Additional source context:
```latex
Throughout the paper,
\emph{separable} for a finite von Neumann algebra means that its
predual is separable. This ensures separability of $L^2(M,\tau)$;
it does not require operator-norm separability of $M$.


```

END PROMPT

---

## T10

BEGIN PROMPT

Explain the target theorem in plain English for a mathematically interested reader who is not a specialist in this field. Use the supplied context to interpret notation. You may retain a formula where it helps.

Source context:
```latex
\section{Introduction}\label{sec:introduction}

Let $\Omega\subset\R^3$ be a bounded connected domain with smooth boundary.
For real functions $\lambda,\mu\in C^\infty(\overline\Omega)$ satisfying
\begin{equation}\label{eq:energy-positivity}
  \mu>0,\qquad 3\lambda+2\mu>0
  \quad\text{on }\overline\Omega,
\end{equation}
define the strain, stress, and static elasticity operator by
\begin{equation}\label{eq:physical-operator}
 e(u)=\frac{\nabla u+(\nabla u)^T}{2},\qquad
 \sigma_{\lambda,\mu}(u)=\lambda(\diverg u)I+2\mu e(u),\qquad
 L_{\lambda,\mu}u=\diverg\sigma_{\lambda,\mu}(u).
\end{equation}
Here $\mu$ is the shear modulus and $\lambda+2\mu/3$ is the bulk modulus.
The inequalities in \eqref{eq:energy-positivity} give uniformly positive
elastic energy. For each $f\in H^{1/2}(\partial\Omega;\R^3)$ there is a
unique $u_f\in H^1(\Omega;\R^3)$ with $L_{\lambda,\mu}u_f=0$ and trace $f$.
The displacement-to-traction map is defined weakly by
\begin{equation}\label{eq:physical-dn}
 \langle\Lambda_{\lambda,\mu}f,g\rangle
 =\int_\Omega\left[
 \lambda(\diverg u_f)(\diverg v_g)+2\mu e(u_f):e(v_g)
 \right]\dd x,
\end{equation}
where $v_g\in H^1(\Omega;\R^3)$ has trace $g$, and $A:B=\tr(A^TB)$.
The weak equation makes this expression independent of the extension.
For smooth boundary data it is the traction
$\sigma_{\lambda,\mu}(u_f)n$, with $n$ the outward normal.


```

Target theorem:
```latex
\begin{theorem}\label{thm:main}
Let $\Omega\subset\R^3$ be any bounded connected domain with
$C^\infty$ boundary. For $j=1,2$, let
$\lambda_j,\mu_j\in C^\infty(\overline\Omega;\R)$ satisfy
$\mu_j>0$ and $3\lambda_j+2\mu_j>0$ on $\overline\Omega$.
If
\[
 \Lambda_{\lambda_1,\mu_1}
 =\Lambda_{\lambda_2,\mu_2}
 \colon H^{1/2}(\partial\Omega;\R^3)
       \longrightarrow H^{-1/2}(\partial\Omega;\R^3),
\]
then $\lambda_1=\lambda_2$ and $\mu_1=\mu_2$ throughout $\Omega$.
\end{theorem}
```

Additional source context:
```latex


Thus Theorem~\ref{thm:main} gives a positive resolution of the smooth
three-dimensional isotropic elastic Calder\'on uniqueness problem under
\eqref{eq:energy-positivity}. It uses the full zero-frequency boundary
operator in fixed Euclidean coordinates. The coefficients need not be
analytic or close to constants, and their agreement near the boundary is
not an additional hypothesis.
```

END PROMPT
