Mathematics Problem Archive

: Let S_(n) be a subset of 2^(n), interpreted as a family of truth assignments to x_(1),...,x_(n). Let S_(n)-SAT be the problem of determining satisfi...

Bixby-Flint-Miklos : Let G be a bicolored graph, and let H be the graph whose vertices are the valid pressing sequences of G and whose edges connect t...

Let $A=(a_{ij})$ be an $n\times n$ matrix with nonnegative entries and total entry sum $n$. Define $$\phi(A)=\prod_{i=1}^n\sum_{j=1}^n a_{ij}+\prod_{j...

A superpermutation on $n$ symbols is a string containing every permutation of the $n$ symbols as a contiguous substring. Determine the minimum possibl...

For positive integers $N,q,a$, let $Q(N;q,a)$ be the number of perfect squares among $a,a+q,\ldots,a+(N-1)q$, and let $Q(N)=\max_{q,a\geq1}Q(N;q,a)$. ...

For partitions $\lambda,\mu,\nu$ of $n$, the Kronecker coefficient $g_{\mu\nu}^{\lambda}$ is defined by $$V_\mu\otimes V_\nu\cong\bigoplus_\lambda g_{...

Let $M(n)$ be the number of monotone Boolean functions of $n$ variables, equivalently the number of antichains of subsets of an $n$-element set. Deter...

Let $W(r,k)$ be the least $N$ such that every coloring of $\{1,\ldots,N\}$ with $r$ colors contains a monochromatic arithmetic progression of length $...

Let $\boldsymbol{d}=(d_1,\ldots,d_n)$ be an unordered partition of $4g-4$ with $d_i\in\{-1,0,1,2,\ldots\}$, and let $\widehat\Pi_{4g-4}$ be the set of...

Let $\boldsymbol{d}=(d_1,\ldots,d_n)$ be an unordered partition of $4g-4$ with $d_i\in\{-1,0,1,2,\ldots\}$, and let $\widehat\Pi_{4g-4}$ be the set of...

Let $\boldsymbol{d}=(d_1,\ldots,d_n)$ be an unordered partition of $4g-4$ with $d_i\in\{-1,0,1,2,\ldots\}$, and let $\widehat\Pi_{4g-4}$ be the set of...

Fix singularities $a_1,\ldots,a_n$ and real exponent differences $\alpha_1,\ldots,\alpha_n$ for second-order Fuchsian equations on the Riemann sphere....

For the Heun equation $$y''+\left(\sum_{j=0}^2\frac{1-\alpha_j}{z-a_j}\right)y'+\frac{Az-\lambda}{(z-a_0)(z-a_1)(z-a_2)}y=0,$$ where $\alpha_j>0$, $A=...

Classify the entire solutions of $F(y^{(k)},y)=0$ when $F$ is irreducible and its highest-degree homogeneous part has a single distinct linear factor,...

Let $f$ be a nonlinear entire function and let $D$ be a Fatou component on which all limit functions of the iterates $f^n$ are constant. Can the set o...

Let $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be rational with Julia set $J$, and suppose that a component $D$ of $\widehat{\mathbb C}\setminus J$...

How many completely invariant components can the Fatou set of a transcendental entire function have? In particular, can there be more than one?...

Does there exist a rational function having an analytic invariant Jordan curve on which it is topologically conjugate to an irrational rotation, where...

Is the number of degenerate Herman rings of a rational function finite, and can it be bounded in terms of the degree of the rational function?...

Let $C$ be an analytic invariant curve of a rational function $f$, suppose $f:C\to C$ is not a homeomorphism, and assume $C$ contains a repelling fixe...

Let $D\subset\mathbb R^n$ have regular boundary for the Dirichlet problem. Prove or disprove that the following are equivalent: (PI) there is a nonzer...

A spherical quadrilateral is a disk with four marked boundary vertices, curvature-one metric, geodesic sides, and interior angles $\pi\alpha_j>0$. Cla...

Let $A$ be a set of vectors $a=(a_1,\ldots,a_n)\in\mathbb C^n$ such that every $n$ of them are linearly independent, and assign to every $a\in A$ a $\...

Let $A$ be a set of vectors $a=(a_1,\ldots,a_n)\in\mathbb C^n$ such that every $n$ are linearly independent, and let $K_a$ be plane convex compact set...

Classify entire functions $f,g$ for which $f\circ g$ is periodic. Prove that, up to the natural equivalences, the possibilities are exhausted by: $g$ ...

Does there exist an entire function whose zeros lie on the positive ray and whose $1$-points lie on two rays making angles $\pm\alpha$ with it, for so...

For $\omega=e^{2\pi i/5}$, is an entire solution of $$f(\omega z)f(\omega^{-1}z)=f(z)-1$$ unique up to rotation of the variable $z$?...

Let $f$ be entire and suppose every zero of every derivative $f^{(n)}$, $n\ge0$, lies in the closed lower half-plane. Must $f$ lie in the compact-open...

For a local inverse germ $\phi_z$ of a meromorphic function $f$ at a noncritical value $w=f(z)$, Gross's theorem gives analytic continuation along alm...

Let $F$ be entire in two variables and let a holomorphic germ $\phi$ satisfy $F(z,\phi(z))=0$ near a nonsingular point. Must $\phi$ admit analytic con...

Let $\mu$ be a positive plane measure with $\mu(\{|z|\le r\})\le cr^\alpha$ for some $0<\alpha<1/2$. Can its logarithmic potential $$u(z)=\int\log\lef...

Let subharmonic functions $u_k$ on the unit square converge uniformly to $u(x,y)=x$. If $D_k=\{u_k<0\}$ and $D_k^*$ is the component containing $-1/2$...

Let $u$ be subharmonic on $1<|z|<2$, and let pairwise disjoint open sets $D_k$ be unions of components of $\{u<0\}$. Suppose that for every $r\in(1,2)...

Let positive masses $a_k$ be placed at a discrete set $x_k\in\mathbb R^n$ and suppose $\sum_k a_k/|x_k|^{n-1}<\infty$, so that $$F(x)=\sum_k\frac{a_k(...

Let $f$ be holomorphic in the unit disk with exactly one simple zero $z_0$ and two $1$-points $z_1,z_2$, counted with multiplicity. Determine the larg...

Let $f$ be holomorphic in the unit disk, with $f(0)=0$, $f'(0)\ne0$, no other zeros, and exactly two solutions $z_1,z_2$ of $f(z)=1$, counted with mul...

Solve the two-one-point extremal problem for real holomorphic $f$: determine the minimum of $\max(|z_1|,|z_2|)$ and maximum of $|f'(0)|$ when $f(0)=0$...

Let $f$ be real and holomorphic in the unit disk, with one simple zero at $0$ and two simple $1$-points at $\pm ia$. Determine the minimum possible va...

For rational functions of each fixed degree, determine the analogues of Goldberg's constant and the unit-disk zero/one-point extremal quantities, and ...

For $\phi(n)=\sup_{\deg p=n}\int_{|z|<1}|p'|/(1+|p|^2)\,dm$, let $\alpha=\limsup\log\phi(n)/\log n$. For a regular compact set $E$, define $\beta_E$ f...

Obtain better rigorous estimates for the Littlewood exponents $\alpha$, $\beta$, and for $\sup_c P_c$, where $P_c$ is the pressure for the hyperbolic ...

Are the iterates $p_c^n$ of hyperbolic quadratic polynomials extremal, or nearly extremal, for the Littlewood exponent $\alpha$ governing mean spheric...

For which parameters $c$ is the pressure $P_c$ of the hyperbolic quadratic polynomial $p_c(z)=z^2+c$, for the potential $|(p_c^n)'|^{-1}$, close to or...

For a regular connected compact plane set $E$, let $\beta_E=\limsup_{\varepsilon\to0}\log l(\varepsilon)/(-\log\varepsilon)$, where $l(\varepsilon)$ i...

If the definition of $\sup_E\beta_E$ is extended from connected regular compact plane sets to all regular compact sets, is the supremum attained on co...

For finitely many positive charges $a_k$ at points $x_k\in\mathbb R^3$, is the critical set of $u(x)=\sum_{k=1}^n a_k/|x-x_k|$ always finite?...

If the critical set of $u(x)=\sum_{k=1}^n a_k/|x-x_k|$ for positive point charges in $\mathbb R^3$ is finite, how many points can it contain? In parti...

For a monic polynomial $p$ of degree $d$, determine the maximum length of the lemniscate $E(p)=\{z:|p(z)|=1\}$. Is the extremal asymptotically $p(z)=z...

For the rectangular lattice $\Lambda=\{an+ibm:n,m\in\mathbb Z\}$ with $a^2+b^2=1$ and $a\in(0,1)$, let $f_a:\mathbb D\to\mathbb C\setminus\Lambda$ be ...

Let $u_1,u_2,u_3$ be subharmonic in the plane with $u_j(0)=0$, and let $v_1\le v_2\le v_3$ be their pointwise increasing rearrangement. Put $I(r,v)=\i...