Open problems

1131 registered problems.

OP-00501 Open

The optimal asymptotic rate of binary error-correcting codes

IT · AIM Open Applied Problems · error-correcting codes · Gilbert–Varshamov bound · linear programming bounds

OP-00502 Open

Exact capacity of the binary deletion channel

IT · AIM Open Applied Problems · channel capacity · deletion channels · synchronization errors

OP-00503 Open

Symmetric capacity of the binary multiplying two-way channel

IT · AIM Open Applied Problems · channel capacity · interactive communication · two-way channels

OP-00504 Open

Capacity of the binary skew-symmetric broadcast channel

IT · AIM Open Applied Problems · broadcast channels · channel capacity · Marton's inner bound

OP-00505 Open

A complete boundary-control criterion for coupled heat systems in higher dimensions

OC · AIM Open Applied Problems · boundary controllability · coupled parabolic systems

OP-00506 Open

Continuous velocity of every sub-Riemannian length minimizer

DG · abnormal minimizers · AIM Open Applied Problems · sub-Riemannian geodesics

OP-00507 Open

The Černý synchronizing-automaton conjecture

CO · AIM Open Applied Problems · Černý conjecture · reset words · synchronizing automata

OP-00508 Open

An SRB measure for the classical Hénon parameters

DS · AIM Open Applied Problems · Hénon maps · SRB measures · strange attractors

OP-00509 Open

A sharp surface-area bound for the convex hull of a closed curve

MG · AIM Open Applied Problems · convex hulls · isoperimetric inequalities · space curves

OP-00510 Open

Maximum convex-hull volume for a closed curve of fixed length

MG · AIM Open Applied Problems · convex hulls · isoperimetric problems · space curves

OP-00511 Open

Minimum volume of a three-dimensional body of constant width

MG · AIM Open Applied Problems · Blaschke–Lebesgue problem · bodies of constant width · Meissner bodies

OP-00512 Open

A strictly convex function on every simple manifold

DG · AIM Open Applied Problems · boundary rigidity · simple manifolds · strictly convex functions

OP-00513 Open

Exactly n or infinitely many periodic orbits on a convex energy surface

SG · AIM Open Applied Problems · closed characteristics · convex hypersurfaces · Reeb flows

OP-00514 Open

The least-area convex cover of every unit-length planar curve

MG · AIM Open Applied Problems · Moser's worm problem · universal covers

OP-00515 Open

Asymptotic density of reflected wavefronts in a disk

DS · AIM Open Applied Problems · billiards · wave fronts

OP-00516 Open

An overlap-free edge unfolding for every convex polyhedron

MG · AIM Open Applied Problems · convex polyhedra · edge unfoldings · nets

OP-00517 Open

Injectivity with attenuation of finite angular degree

DG · AIM Open Applied Problems · attenuated ray transforms · geometric inverse problems · simple surfaces

OP-00518 Open

Irrational ellipticity when a convex energy surface has finitely many orbits

DS · AIM Open Applied Problems · closed characteristics · convex hypersurfaces · linear stability

OP-00519 Solution claimed

Finite-jet determination of small-time local controllability

OC · AIM Open Applied Problems · control-affine systems · nonlinear control · small-time local controllability

OP-00520 Partially resolved

Finiteness of planar central configurations for fixed masses

DS · AIM Open Applied Problems · central configurations · n-body problems · relative equilibria

OP-00521 Open

Recover an arbitrary sound speed from one incident direction

AP · AIM Open Applied Problems · fixed angle scattering · inverse scattering · sound speeds

OP-00522 Open

Capacity of the two-receiver Gaussian broadcast channel with feedback

IT · AIM Open Applied Problems · broadcast channels · channel capacity · Gaussian channels

OP-00523 Open

Capacity of the Gaussian relay channel with independent receiver noises

IT · AIM Open Applied Problems · channel capacity · Gaussian channels · relay channels

OP-00524 Open

Global attraction for Wright-type delayed feedback

DS · AIM Open Applied Problems · delay differential equations · Schwarzian derivatives · Wright's equation

OP-00525 Open

Generic stabilization by static output feedback

OC · AIM Open Applied Problems · linear control systems · pole placement · static output feedback

OP-00526 Open

Global null control of a heat equation with cubic growth

OC · AIM Open Applied Problems · blow-up · null controllability · semilinear heat equations

OP-00527 Open

The sharp number of directions illuminating a convex body

MG · AIM Open Applied Problems · convex bodies · homothetic coverings · illumination numbers

OP-00528 Open

Quadratic growth when a Hamiltonian map has infinitely many periodic orbits

SG · AIM Open Applied Problems · Hamiltonian diffeomorphisms · periodic orbits

OP-00529 Open

Exact null control of a harmonically confined kinetic equation from a wedge

OC · AIM Open Applied Problems · hypoelliptic equations · null controllability · Ornstein–Uhlenbeck operators

OP-00530 Open

Global uniqueness from inverse backscattering data

AP · AIM Open Applied Problems · backscattering data · inverse scattering · Schrödinger operators

OP-00531 Partially resolved

An irrational polygon with nonergodic billiard flow

DS · AIM Open Applied Problems · ergodicity · polygonal billiards

OP-00532 Open

The minimum-area universal cover of planar diameter-one sets

MG · AIM Open Applied Problems · sets of constant width · universal covers

OP-00533 Open

Local scalar geodesic tomography on smooth surfaces

DG · AIM Open Applied Problems · geodesic X-ray transforms · geometric inverse problems · integral geometry

OP-00534 Open

Small-time global null control of no-slip Navier–Stokes flow

OC · AIM Open Applied Problems · Navier–Stokes equations · null controllability

OP-00535 Open

Marked length rigidity in negative curvature

DG · AIM Open Applied Problems · closed geodesics · marked length spectrum · negative curvature

OP-00536 Open

Minimum-time eradication of an expanding set inside a geographic barrier

OC · AIM Open Applied Problems · eradication strategies · moving sets · sets of finite perimeter

OP-00537 Solution claimed

Logarithmic convexity for two coupled subdiffusion orders

AP · AIM Open Applied Problems · backward problems · logarithmic convexity · time-fractional diffusion equations

OP-00538 Open

The full three-dimensional Newton minimum-resistance problem

OC · AIM Open Applied Problems · concave functions · Newton's problem of minimal resistance · shape optimization

OP-00539 Open

Global conductivity uniqueness for the p-Laplacian in higher dimensions

AP · AIM Open Applied Problems · Calderón problem · Dirichlet-to-Neumann maps · p-Laplace equations

OP-00540 Open

A closed nonplanar totally integrable wire billiard in three-space

DS · AIM Open Applied Problems · integrable billiards · twist maps · wire billiards

OP-00541 Open

Lens rigidity for nontrapping surfaces with conjugate points

DG · AIM Open Applied Problems · conjugate points · lens rigidity · scattering relations

OP-00542 Open

Approximate velocity control of a viscoelastic fluid with prescribed transport

OC · AIM Open Applied Problems · approximate controllability · boundary control · viscoelastic fluids

OP-00543 Open

Are all totally integrable outer-length billiards ellipses?

DS · AIM Open Applied Problems · integrable billiards · outer length billiards

OP-00544 Open

Unique continuation for the higher-dimensional p-Laplace equation

AP · AIM Open Applied Problems · p-harmonic functions · unique continuation

OP-00545 Open

Invertibility of the Pestov–Uhlmann reconstruction operator

DG · AIM Open Applied Problems · geodesic X-ray transforms · inversion formulas · simple surfaces

OP-00546 Open

Simultaneously recover sound speed and initial pressure

AP · AIM Open Applied Problems · photoacoustic tomography · sound speeds · wave equations

OP-00547 Open

The near-linear bound for planar halving lines

CO · AIM Open Applied Problems · halving lines · k-sets · planar point sets

OP-00548 Open

Polynomial-time detection below the planted-clique square-root threshold

ST · AIM Open Applied Problems · computational lower bounds · planted cliques · random graphs

OP-00549 Open

Positive metric entropy for billiards arbitrarily close to an ellipse

DS · AIM Open Applied Problems · convex billiards · metric entropy

OP-00550 Open

Saari's constant-inertia conjecture for planar gravitation

DS · AIM Open Applied Problems · celestial mechanics · n-body problems · relative equilibria