Mathematics Problem Archive

In the ring of bounded analytic functions on the unit ball or the polydisc in $n$ variables, is the intersection of two finitely-generated ideals agai...

For which simply-connected domains $D$ (with $0 \in D$) is it true that there is a constant $K = K(D)$ such that $$ {\int\int}_D|f|^2\,dx\,dy\leq K{\i...

Let $\mu\geq1$ be a singular measure on the boundary of the unit disc $\mathbb{D}$; and let $S_\mu$ be the corresponding singular inner function \[S_\...

Let $f_1,\ldots,f_n$ and $g$ be $H^\infty$ functions on $\mathbb{D}$. If $|g(z)| \leq |f_1(z)| + \ldots + |f_n(z)|$, are there necessarily functions $...

Let $w\geq0$ be an integrable function on the circle or on the line. Then $w$ is said to belong to the class $A_2$ if \[\sup_l\Big(\frac{1}{|l|}\int_l...

Do there exist inner functions in any strictly pseudo-convex domain of $\mathbb{C}^n$, $n\geq2$? Alexandrov (no citation) and independently L{\o}w (no...

In Pompeiu's formula, $f(z)$ and $\int_\Gamma f(\zeta)/(\zeta-z)\,d\zeta$ make sense for merely continuous functions, but \[\int\int_\Omega\frac{\part...

Let the sequence $\{M_k\}^\infty_0$ of positive numbers be such that \[M_0=1\hspace{1cm}\text{ and }\hspace{1cm}\frac{M_{k+j}}{M_kM_j}\geq\begin{pmatr...

Let $\psi:S^1\to S^1$ be a direction-reversing homeomorphism, and let $A_\psi$ denote the set of functions $f:S^1\to\mathbb{C}$ such that both $f$ and...

Let $\psi:S^1\to S^1$ be a homeomorphism. When is it true that $\text{Re }A = \text{Re }A \circ \psi$? That is, when is each function $f$ in the real ...

A sequence $\{z_n\}^\infty_1$ in $|z|< 1$ is interpolating for bounded analytic (harmonic) functions if, for each bounded sequence $\{\alpha_n\}^\inft...

Let $f(z)\in H^\infty$, and let $\{z_n\}$ be a Blaschke sequence \[\sum^\infty_{n=1}(\-|z_n|)<\infty.\] [(a)] ; Does there always exist a Blaschke pro...

[(a)] ; Suppose that $f, f_1, f_2,\ldots,f_n\in H^\infty$, and \[|f|\leq|f_1|+|f_2|+\ldots+|f_n|.\] Do there necessarily exist $h_1,h_2,\ldots,h_n\in ...

For each pair of $f,g\in H^\infty$, does there necessarily exist another pair of functions $a,b\in H^\infty$ such that $$ af+gb\neq0,\hspace{1cm}|z|<1...

Let $K_1, K_2, K_3$ be disjoint closed sets in the extended complex plane, and $C_1, C_2, C_3$ constants. Let $\rho_n(f)$ be the best rational approxi...

Let $D$ be an open subset of the extended complex plane with non-empty boundary $\partial D$, and let $F$ be a relatively-closed subset of $D$. Let $f...

Let us call a closed set $E$ in $\mathbb{C}$ a weak Arakelian set if, corresponding to each function $g(z)$ continuous on $E$ and analytic in the inte...

Let $\gamma$ be a Jordan arc in $\mathbb{C}^n$, $n\geq2$ such that the projections $\gamma_j$ on the complex coordinate planes $j=1,\ldots,n$ have are...

Does the condition $\sum 1/p_n<\infty$ for positive integers $p_n$ guarantee that the sequences of powers $\{z^{p_n}\}$ fails to span $C(\gamma)$ for ...

Let $F$ be a closed subest of $\mathbb{R}^n$, $n\geq2$. Call $F$ a set of harmonic approximation if every function continuous on $F$ and harmonic in t...

Let $D$ be a planar domain. A sequence $\{z_j\}$ of points in $D$ is said to be an interpolating sequence if whenever $\{\alpha_j\}\in\ell^\infty$ the...

Let $\Gamma\subset\mathbb{C}$ be a Jordan curve of logarithmic capacity $1$, and let $\phi$ be a conformal map from the exterior of $\Gamma$ to the ex...

Let $K$ be a compact subset of $\mathbb{R}^n$, $n\geq3$. For $\phi\in\mathcal{D}$, let $D(\phi)$ be a least-diameter disc containing $\text{spt }\phi$...

Let $f$ be continuous on a compact subset $K$ of $\mathbb{C}$. If there exists a sequence $\{f_n\}^\infty_1$ of functions analytic near $K$ for which ...

Let $f_1$ and $f_2\in H^\infty(\text{unit disc}) = H^\infty$; and let the function $g\in H^\infty$ satisfy the inequality \[|g(z)|\leq|f_1(z)|+|f_2(z)...

Let $\Gamma$ be a curve of the form \[\{x+iA(x):-\infty<x<\infty\}\] with \[|A(x_1)-A(x_2)|\leq M|x_1-x_2|.\] Let $E$ be a compact subset of $\Gamma$,...

Let $K$ denote the $\frac{1}{3}$-Cantor set on $\mathbb{R}$; let $E = K\times K$, and let $\Omega=\mathbb{C}^*\setminus E$. Prove the corona theorem f...

For nonconvex subsets of $\mathbb{R}$ and $\mathbb{R}^2$, is a set spectral if and only if it tiles by translations?...

For an iteration without memory that uses $n$ evaluations of a function or its derivatives per step, is its convergence order always at most $2^{n-1}$...

Given a complex polynomial $f$ of degree $d\geq2$ and $z\in\mathbb{C}$, must there be a critical point $c$ of $f$ such that $|f(z)-f(c)|\leq |f'(z)|\,...

Characterize the domains for which there exists a nonzero function whose integral vanishes over every congruent copy of the domain....

Does the Flint Hills series $\sum_{n=1}^{\infty} 1/(n^3\sin^2 n)$ converge?...

Establish global regularity, or exhibit breakdown, for solutions of the Vlasov–Maxwell equations from appropriate smooth initial data....

Are there infinitely many Lehmer pairs of zeros in the sense used in the theory of the de Bruijn–Newman constant?...

Is the global attractor of the periodically forced two-dimensional Navier–Stokes equations conjugate to a smooth finite-dimensional dynamical system? ...

For viscous incompressible flow in $\mathbb{R}^3\setminus B$ past a fixed obstacle $B$, with velocity approaching a nonzero constant vector at infinit...

Develop a global-in-time theory of the Cauchy problem for the one-dimensional Euler–Fourier system for initial data constrained only by finite energy ...

Develop a compensated-compactness calculus for symmetric matrices when compensated compactness yields only inequalities. As a first application, prove...

For compressible Navier–Stokes equations with constant viscosities near vacuum, does the unphysical one-dimensional consequence identified by Hoff and...

Does the compressible Euler system in odd spatial dimension admit a nontrivial smooth eternal solution having finite nonzero mass and energy?...

Prove existence of a regular reflection for compressible flow against a wedge without assuming that the flow is irrotational....

Let $X$ be a centered Gaussian random vector in $\mathbb{R}^n$ with known covariance matrix, and for an orthonormal basis $B=(b_1,\ldots,b_n)$ let $N_...

For a $d\times d$ real Gaussian matrix $G_{\mathbb{R}}$ and complex Gaussian matrix $G_{\mathbb{C}}$, both normalized to entry variance $1/d$, define ...

Let ${W}$ denote a symmetric Wigner matrix with i.i.d. entries ${W_{ij}\sim \mathcal{N}(0,1)}$ . Also, given ${B\in\mathbb{R}^{n\times n}}$ symmetric,...

Is there a polynomial time algorithm that is able to find the largest clique of $G$ (with high probability) for $\omega \ll \sqrt{n}$ ? For example, f...

Does there exist $c>0$ such that it is NP-hard, given a graph $G$ and $\phi>0$, to distinguish $h_G\le\phi$ from $h_G\ge c\sqrt{\phi}$?...

Given a symmetric matrix ${M}$ with small condition number, is there a quasi-linear time (on ${n}$ and the number of non-zero entries of ${M}$ ) proce...

Let ${G=(V,E,W)}$ be a graph and ${k}$ a positive integer, is the following true? $\rho_G(k) \leq \mathrm{polylog}(k) \sqrt{\lambda_k}. \ \ \ \ \ (2)$...

Let $A_1,\dots,A_n\in \mathbb{R}^{d\times d}$ be symmetric matrices and $g_1,\dots,g_n\sim\mathcal{N}(0,1)$ i.i.d.. Does the following hold? $\mathbb{...

Let $X\in\mathbb{R}^{d\times d}$ be a symmetric matrix with (otherwise) independent gaussian entries. Prove (or disprove): $\mathbb{E} \|X\| \lesssim\...