Open problems

1131 registered problems.

OP-00351 Open

Decomposing complete graphs into squares of Hamilton cycles

CO · AIM Open Applied Problems · combinatorial designs · graph decompositions · Hamilton cycles

OP-00352 Open

Excluding three correlation levels in quadratic field towers

NT · AIM Open Applied Problems · finite fields · m-sequences · Weil sums

OP-00353 Open

A zero in the nontrivial spectrum of a power permutation

NT · AIM Open Applied Problems · m-sequences · power permutations · Weil sums

OP-00354 Solution claimed

Sharp barrier-parameter growth for hyperbolic balls

OC · AIM Open Applied Problems · Hadamard manifolds · interior-point methods · self-concordant barriers

OP-00355 Open

Completeness of modes for slowly growing imaginary Schrödinger potentials

SP · AIM Open Applied Problems · completeness of eigenfunctions · non-self-adjoint operators · Schrödinger operators

OP-00356 Open

Jakobson–Naud’s essential spectral-gap conjecture

SP · AIM Open Applied Problems · hyperbolic surfaces · scattering resonances · spectral gaps

OP-00357 Open

The log-rank conjecture for deterministic communication

CO · AIM Open Applied Problems · Boolean matrices · communication complexity · matrix rank

OP-00358 Partially resolved

Low-degree convergence of relaxed minimal-deformation surface elements

NA · AIM Open Applied Problems · evolving surface finite elements · surface evolution · tangential motions

OP-00359 Open

Nonempty spectrum for a magnetic complex Airy operator at every field strength

SP · AIM Open Applied Problems · magnetic Schrödinger operators · non-self-adjoint operators · superconductivity

OP-00360 Open

The disk maximizes the first magnetic Neumann eigenvalue

SP · AIM Open Applied Problems · isoperimetric inequalities · magnetic Laplacians · Neumann eigenvalues

OP-00361 Open

Strict angle monotonicity of magnetic wedge ground energy

SP · AIM Open Applied Problems · corner domains · magnetic Laplacians

OP-00362 Solution claimed

A universal bound on excess molecular charge

SP · AIM Open Applied Problems · Coulomb systems · negative ions

OP-00363 Open

Infinitely many obstacle resonances in a fixed strip under trapping

AP · AIM Open Applied Problems · billiards · obstacle scattering · scattering resonances

OP-00364 Open

A sharp Galerkin projection bound on nonobtuse triangles

NA · AIM Open Applied Problems · Galerkin projections · hybrid high-order methods · nonobtuse triangles

OP-00365 Open

Existence of NPT bound entanglement

MP · AIM Open Applied Problems · bound entanglement · entanglement distillation · Werner states

OP-00366 Open

Is the optimal magnetic rectangle always a square?

SP · AIM Open Applied Problems · Dirichlet eigenvalues · magnetic Laplacians

OP-00367 Solution claimed

Smallest orders for periodic multidimensional Costas arrays

CO · AIM Open Applied Problems · Costas arrays · permutation arrays

OP-00368 Open

Absolute continuity for smooth periodic scalar elliptic operators

SP · AIM Open Applied Problems · periodic elliptic operators · spectral measures

OP-00369 Open

Removable cycles in cycle permutation graphs

CO · AIM Open Applied Problems · cycle permutation graphs · removable cycles · snarks

OP-00370 Open

Permutation snarks without five-cycles

CO · AIM Open Applied Problems · cycle permutation graphs · edge colourings · snarks

OP-00371 Open

Permutation snarks with order congruent to six modulo eight

CO · AIM Open Applied Problems · cubic graphs · edge colorings · snarks

OP-00372 Open

The sharp asymptotic facial length in girth-saturated plane graphs

CO · AIM Open Applied Problems · girth · graph saturation · plane graphs

OP-00373 Open

Polynomial-size nonobtuse tetrahedral meshes

MG · AIM Open Applied Problems · mesh generation · nonobtuse tetrahedra · triangulations

OP-00374 Open

Faber–Krahn inequality for graphene quantum dots

SP · AIM Open Applied Problems · Dirac operators · Faber–Krahn inequalities

OP-00375 Open

Are the standard quantum entropy inequalities complete?

IT · AIM Open Applied Problems · entropy cones · entropy inequalities · von Neumann entropy

OP-00376 Open

The exact real Grothendieck constant

FA · AIM Open Applied Problems · Grothendieck inequality · semidefinite relaxations · sharp constants

OP-00377 Open

Reed's bound for the chromatic number

CO · AIM Open Applied Problems · chromatic number · graph colorings

OP-00378 Open

Can bound entanglement produce a device-independent secret key?

IT · AIM Open Applied Problems · bound entanglement · quantum information · quantum key distribution

OP-00379 Open

Subpolynomial compact-set peaks of Hecke–Maass waves

NT · AIM Open Applied Problems · arithmetic quantum chaos · Maass forms · sup norms

OP-00380 Open

Global uniqueness of fine-mesh semilinear Galerkin solutions

NA · AIM Open Applied Problems · control problems · finite element methods · semilinear elliptic equations

OP-00381 Open

Simon’s multidimensional weighted square-integrability conjecture

SP · absolutely continuous spectrum · AIM Open Applied Problems · decaying potentials · Schrödinger operators

OP-00382 Open

Global convergence of the simple CRDT conformal-mapping iteration

NA · AIM Open Applied Problems · cross-ratios · numerical conformal mapping · Schwarz–Christoffel maps

OP-00383 Open

Anderson localization for the cosine skew-shift model at every coupling

SP · AIM Open Applied Problems · Anderson localization · discrete Schrödinger operators · skew-shifts

OP-00384 Open

Sharp permutation bound for triangular parts of correlation matrices

NA · AIM Open Applied Problems · correlation matrices · SOR methods · triangular truncation

OP-00385 Open

The Weinstock inequality for convex domains in a sphere

SP · AIM Open Applied Problems · convex domains · isoperimetric inequalities · Steklov eigenvalues

OP-00386 Solution claimed

Reality of the spectrum for a spherical strip with PT-symmetric boundary conditions

SP · AIM Open Applied Problems · non-self-adjoint operators · PT-symmetric operators · Robin boundary conditions

OP-00387 Open

Dynamic optimality of splay trees

CO · AIM Open Applied Problems · binary search trees · dynamic optimality · splay trees

OP-00388 Open

Optimal stable manifold widths of Besov balls

NA · AIM Open Applied Problems · Besov spaces · nonlinear approximation · nonlinear widths

OP-00389 Open

Optimal star-discrepancy lower bound for infinite sequences

NT · AIM Open Applied Problems · discrepancy · low-discrepancy sequences · quasi-Monte Carlo methods

OP-00390 Solution claimed

Unbounded first Steklov multiplicity with fixed boundary count

SP · AIM Open Applied Problems · eigenvalue multiplicities · Riemannian surfaces · Steklov eigenvalues

OP-00391 Open

Divisible point sets as unions of parallel lines

CO · AIM Open Applied Problems · direction problems · divisible codes · finite geometry

OP-00392 Open

A strongly polynomial pivot rule for the simplex method

OC · AIM Open Applied Problems · linear programming · pivot rules · simplex method

OP-00393 Open

Sunada’s no-gap conjecture for abelian covers

SP · abelian coverings · AIM Open Applied Problems · hyperbolic surfaces · spectral gaps

OP-00394 Solution claimed

A linear genus–girth bound for facial cycles

CO · AIM Open Applied Problems · facial cycles · girth · graphs on surfaces

OP-00395 Solution claimed

An optimal Weyl remainder for interior transmission eigenvalues

SP · AIM Open Applied Problems · interior transmission eigenvalues · non-self-adjoint operators · Weyl's law

OP-00396 Open

Maximum treewidth at a prescribed edge count

CO · AIM Open Applied Problems · extremal problems · tree decompositions · treewidth

OP-00397 Partially resolved

The factor-two relation between triangle deletion and packing

CO · AIM Open Applied Problems · triangle covers · triangle packings · Tuza's conjecture

OP-00398 Partially resolved

Uniformly effective algebraic multigrid for diagonally dominant systems

NA · AIM Open Applied Problems · algebraic multigrid · diagonally dominant matrices · preconditioners

OP-00399 Open

The sharp vector Riesz inequality for multidimensional potential fields

CA · AIM Open Applied Problems · Riesz transforms · sharp constants · singular integrals

OP-00400 Solution claimed

Optimal spectral signings of regular graphs

CO · AIM Open Applied Problems · graph lifts · Ramanujan graphs · signed graphs