Mathematics Problem Archive

Let $f$ be meromorphic in $\mathbb{C}$, and let $f^{-1}$ denote any element of the inverse function that is analytic in a neighbourhood of a point $w$...

Determine the upper and lower estimates for the growth of entire and meromorphic solutions of algebraic ordinary differential equations (AODE). (This ...

Let $F$ be a polynomial in two variables, and let $y$ be a meromorphic solution of the algebraic ordinary differential equation $F(y^{(n)},y)=0$. Is i...

Let $f$ be meromorphic in $\mathbb{C}$, and suppose that the function \[F(z)=f^{(k)}(z)+\sum^{k-2}_{j=0}a_j(z)f^{(j)}(z)\] is non-constant, where $k\g...

Let $f(z)$ be an entire function. Then Boas (unpublished) proved that there exists a path $\Gamma_\infty$ such that, for every $n$, $$ \left|\frac{f(z...

If the entire function $f(z)$ has finite order $\rho$, and the maximal density of non-zero coefficients is $\Delta$, is it true that if $\rho\Delta<\f...

If $f(z)$ is an entire function, the iterates $f_n(z), n=1,2,\ldots$ are defined inductively by \[f_{n+1}(z)=f(f_n(z)),\hspace{1cm}f_1(z)=f(z).\] A po...

We can also compare $m_0(r,f)$ with the characteristic $T(r)$. We have \[\limsup_{r\to\infty}\frac{\log m_0(r,f)}{T(r)}\geq D(\lambda)\] and ask for t...

(A width conjecture) Given a power series $\sum^\infty_{k=0}a_kz^k$, suppose that there is a non-negative $\rho$ such that all of the partial sums $S_...

Let $\Gamma$ be a rectifiable curve. Suppose $f$ is a continuous function on the plane satisfying \[\int_{\sigma(\Gamma)}f(z)dz=0\hspace{1cm}\text{ fo...

Let $f$ be a rational function and $C$ be as in Problem 2.62. We say that $g$ is a limit function for $f$ if $g$ is defined in some component $G$ of $...

Let $f$ be an entire function satisfying the condition \[\log M(r,f)\leq(1+o(1))r^\rho,\hspace{1cm}\text{ as }r\to\infty.\] Suppose that there exists ...

[(a)] ; Show that, if a function $g$ in $R_d$ has the property that its Julia set $J(g) \neq \hat{\mathbb{C}}$, then $g$ does not leave invariant a no...

Let a function $f$ in $R_d$ have the property that \[f(z)=\lambda_\alpha z+O(z^2)\hspace{1cm}\text{ as }z\to0,\] where $\lambda_\alpha=e^{2\pi i\alpha...

Let $P(z)$ be a polynomial whose zeros $z_1, z_2, \ldots, z_n$ lie in $|z|\leq1$. Is it true that $P'(z)$ always has a zero in $|z-z_1|\leq1$? (Bl. Se...

Let $f(z)=z^n+a_1z^{n-1}+\ldots+a_n$ be a polynomial of degree $n$. Cartan proved that the set $|f(z)|\leq1$, which we call $E^{(n)}_f$ can always be ...

Suppose that $P$ is a non-linear polynomial with real coefficients. Show that $P^2(z)+P'(z)$ has non-real zeros. We conjecture that the lower bound fo...

Let $\mathcal{P}$ denote the set of all polynomials of the form \[p(z)=\prod^n_{\nu=1}(z-\zeta_\nu),\] where $n\geq2$ and $|\zeta_\nu|\leq1$, $\nu=1, ...

(The corona conjecture) Let $D$ be an arbitrary domain in the plane that supports non-constant bounded analytic functions. Suppose that $f_1(z),\ldots...

Suppose that $f(z)=\sum^\infty_{n=0}a_nz^n$ is circumferentially mean $p$-valent and $f(z)\neq0$ in $\mathbb{D}$. (This latter condition is a conseque...

If $f$ in $S$, Baernstein has shown that \[\int^{2\pi}_0|f(re^{i\theta})|^p\,d\theta\leq\int^{2\pi}_0|k(re^{i\theta})|^p\,d\theta,\hspace{1cm}0<r<1,\h...

Let $\Omega$ be a simply-connected domain in the finite plane whose complement contains $n$ disjoint closed balls with centres on the interval $[0,1]$...

Show that each quasiconformal mapping of $\mathbb{R}^n$ onto $\mathbb{R}^n$ has a quasiconformal extension to $\mathbb{R}^{n+1}$ . This has been estab...

Suppose that $A$ and $B$ are disjoint linked Jordan curves in $\mathbb{R}^3$ which lie at a distance $1$ from each other. Show that the length of $A$ ...

Let $\mathcal{W}$ be a hyperbolic Riemann surface, $G_\omega$ the Green's function with pole $\omega\in\mathcal{W}$, and $\Gamma=\{[\gamma_n]\}$ the f...

Let $V$ be the zero set of some analytic function in a strictly pseudo-convex domain $\Omega$ in $\mathbb{C}^2$. If $V$ has finite area inside $\Omega...

Prove or disprove the following statements about analytic capacity $\gamma$. [(a)] ; If $E\subseteq\mathbb{C}$ is compact and $\phi$ is a $C^1$-diffeo...

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...

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 ...

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

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_...

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...

Part (3) of the problem: Let $s\leq d\leq m$ and $z_1,\dots,z_m\in \mathbb{R}^d$ i.i.d. random vectors with i.i.d. entries $\left( z_k\right)_j = \lef...

Give a tight upperbound to $\mathbb{E}\left\| A^{\otimes k} -\mathbb{E} A^{\otimes k} \right\|.$...

Does the Paley Equiangular tight frame satisfy the Restricted Isometry Property pass the square root bottleneck? (even by logarithmic factors?)....

Explicit deterministic constructions of codes achieving the GV bound Is the GV bound tight?...

Prove or disprove: $R_A(\alpha n,\beta n,n)=\alpha+(1-\alpha)R_A(1,\beta n,(1-\alpha)n)+o(1)$....

What are the asymptotics of $\mathcal{D}\left(n;\frac12\right)$ ? \item An interesting aspect of the Deletion Channel is that different messages may h...

Given a $d$ -regular graph on $n$ nodes $G$ . Let $MaxBis(G)$ and $MinBis(G)$ denote, respectively, the size of its largest and smallest bisection. Is...

Is the k-medians Linear Programming relaxation tight even for point clouds coming from generative models that do not have a community structure?...

Can one give conditions for integrality of the k-medians LP or the k-means SDP based on stability type properties (on the fact that the data is “well-...

Is the Semidefinite programming relaxation for the positive Principal Component Analysis problem tight with high probability for Wigner matrices?...

Is the SDP for angular synchronization tight (with high probability) for noise levels $\sigma$ essentially until the solution of angular synchronizati...

For which levels of noise is the SDP for Multireference Alignment tight?...

Is the Maximum likelihood for Multireference Alignment consistent? (after fixing the power spectrum) What is the sample complexity of the Multireferen...

Problem 1. Find a good characterization for undirected graphs having an acyclic orientation so that the in-degree of every node is even. Problem 2. Fi...

Let D be a digraph without loops and k a positive integer. For a partition $\Pi$ of V(D) into directed paths (a path partition) let $|\Pi|_k=\sum_{P \...

Let $B=\{b_1,b_2,\ldots ,b_k\}\subset\{0,1,\ldots ,n-1\}$, and let $B_i=\{b_1+i, b_2+i,\ldots, b_k+i\}$ where addition is modulo n. That is, ${\mathca...

In a soccer tournament of n teams, every pair of teams plays one match. The winner gets 3 points, the loser gets 0, while both teams receive 1 point i...