6 problems

For a smooth manifold $M$, let … be the set of almost invertible smooth functions. Since $S$ contains $1$, is closed under multiplication, and contains no zero divisors, define the…

For a smooth manifold $M$, let $C^\infty_d(M)$ be the $\mathbb{R}$-algebra of densely defined smooth functions $f:(D_f\subseteq M)\to\mathbb{R}$, where each domain $D_f$ is an embe…

Let $f$ be an $L$-smooth function, let $f^\star$ be its minimum value, let $x^1$ be an initial point, and write $\Delta=f(x^1)-f^\star$. Consider gradient-method iterates $x^k$ wit…

Let $f:\mathbb{R}^d\to\mathbb{R}$ be an $L$-smooth function, let $f^\star$ be its minimum value, let $x^1$ be an initial point, and write $\Delta=f(x^1)-f^\star$. For gradient-meth…

Let $K$ be a compact subset of $\mathbb{R}^d$, and let … Here, $\mathrm{(}\Omega\mathrm{)}$ denotes the property referred to in the source for vanishing ideals. Vanishing-ideal con…

Let $f$ be a nonanalytic, possibly periodic function, and let $\hat{f}(t)$ denote the associated series. Piecewise-constant series conjecture. There exist such functions $f$ for wh…