Mathematics Problem Archive

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...

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 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)|\,...

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

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 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...

Prove or disprove: there exists a universal constant $C$ such that, for any choice of $n$ symmetric matrices $H_1,\dots,H_n\in\mathbb{R}^{n\times n}$ ...

Consider a $A\in\mathbb{C}^{M\times N}$ obtained by sampling random rows of a Discrete Fourier Tranform. How large does $M$ need to be in order for, w...

How many mutually unbiased bases are there in 6 dimensions?...

What is the approximation ratio (or integrality gap) for the Sum-of-Squares (SOS) relaxation of degree 4 for the Max-Cut problem? What about other con...

What is the value of the (real) Grothendieck constant?...

What is the clique number of the Paley graph? Can the the SOS degree 4 analogue of the theta number help upper bound it?...

The Betti poset of a monomial ideal $I$ determines the Stanley projective dimension of $S/I$ and $I$. More precisely, if $I\subseteq S$ and $I'\subset...

Given a graph $G(V,E)$, find a minimum cardinality set $S \subset V$ of nodes such that adding a complete graph on $S$ renders the graph $G+K_S$ globa...

Problem 1. Given an undirected graph $\displaystyle G=(V,E)$ and $\displaystyle s,t\in V,\;\; k\in N$, decide whether the graph has an acyclic orienta...

Is it true that every t-perfect graph is strongly t-perfect?...

Are there deletion-contraction formulas for the polymatroid Tutte polynomial?...

Let M be a matroid on ground set V, let H=(V,E) be a hypergraph with maximum degree $\Delta$, let c(v) be the cost of node v, and let $l(e) \leq u(e)$...

Let D=(V,A) be a digraph with arc-capacities $c : A \to \mathbb{N}$ and a root node $r_0\in V$. A k-arborescence is the arc-disjoint union of k spanni...

Let G=(A,B;E) be a bipartite graph, and $w:E \to \{1,-1\}$ a conservative weighting. Can we determine in polynomial time the maximum number of positiv...

Is every t-perfect graph 4-colourable?...

A hypergraph $H=(V,E)$ has the Kőnig-property if there is a set $\mathcal{D}\subseteq E$ of pairwise disjoint hyperedges such that there is a vertex c...

If G is an undirected graph with even degrees then call two closed Eulerian walks compatible if no pair of incident edges occurs consecutively in both...

Is there a polynomial algorithm for computing a $v_0$-reduced divisor equivalent to a given divisor of an undirected multigraph?...

Let $G$ be an undirected graph. What is the complexity of computing the rotor-router action of the sandpile group of $G$ on the spanning trees of $G$?...

Is the chip-firing reachability problem co-NP-hard for general digraphs?...

Is the chip-firing halting problem in P for Eulerian digraphs with multiple edges?...

Is it true that the chip-firing halting problem for simple digraphs is NP-complete?...

Is it true that a clutter has the MFMC property if and only if it has the packing property?...