170 problems

Let $u:\mathbb{R}^n\to\mathbb{R}$ be a bounded solution with … uniformly with respect to $x'\in\mathbb{R}^{n-1}$. A solution is one-dimensional when it depends on only one spatial…

Let $K\subset \mathbb{R}^3$ be a convex body. A plane section means the intersection of $K$ with an affine plane, and an axis of symmetry is a line in that plane whose reflection p…

Let $f\in C^1(\mathbb R)$ satisfy the positivity and Keller–Osserman conditions, and let $u$ be a solution of the boundary blow-up problem studied in the source on the unit ball…

Witten string field theory realization conjecture. The proposed $SL(K+3,\mathbb{C})$ symmetry can be obtained from Witten SFT.

Symmetry-count sufficiency conjecture. For all such groups $\Gamma$, the counts in Corollary are sufficient for $(G,\chi,p)$ to be $\Gamma$-symmetric isostatic.

Grad's conjecture. If there exists an MHD equilibrium $(X,p)$ on $M$ whose pressure has toroidally nested level sets, then the equilibrium is axisymmetric.

Lin's axial symmetry conjecture. If $\gamma>0$ and $\beta\leq 16\pi$, then every solution is axially symmetric with respect to $\vec n$.

Let $f:\mathbb{R}\to\mathbb{R}$ be Lipschitz, and let $\Omega\subset\mathbb{R}^n$ be a smooth (in fact, Lipschitz) connected domain such that $\mathbb{R}^n\setminus\Omega$ is conne…

Let $\Omega$ be a ball, and let $u\in H_0(\operatorname{curl};\Omega)\cap L^p(\Omega,\mathbb{R}^3)$ be a ground state solution of … where $2<p<6$. Here, symmetry means the property…

In relativistic particle physics, approximate symmetries may be observed even when the underlying theory has an exact symmetry. Nambu's conjecture. Some of the approximate symmetri…

Symmetry conjecture under the Main Assumption and Condition S. Any smooth compact connected embedded hypersurface $M$ in $\mathbb R^{n+1}$ satisfying the Main Assumption and Condit…

Unique-centre conjecture. If a simplex has a unique simplex centre, then it is equifacetal.

Let ${\cal C}$ be a cone, and let the adjacency decomposition method produce a set of facet orbits that is conjectured to be complete. The quotient graph has the orbits $O_i$ of fa…

Let $P_n(x)$ be the generating function for alternating sign matrices invariant under flips in the vertical and horizontal axes, with the weight defined in the source. Such matrice…

Let $Q_n(x,y)$ be the generating function for quarter-turn-symmetric alternating sign matrices. Quarter-turn polynomial factorization conjecture. There exists a sequence of polynom…

Let $Q_n(x,y)$ be the generating function for quarter-turn-symmetric $n$ by $n$ alternating sign matrices, and let $H_n$ and $A_n$ denote the half-turn-symmetric and unrestricted a…

Let $H_n(x,y)$ be the generating function for half-turn-symmetric alternating sign matrices, and let $R_n(x,\mu)$, $T_n(x,\mu)$, and the polynomials $S_{4n+1}(x),S_{4n-1}(x)$ be as…

Let $H_n(x,y)$ be the weight-generating function for half-turn-invariant $n$ by $n$ alternating sign matrices, and let $Z_n(x,y,0)$ and $Z_{n-1}(x,y,1)$ be the corresponding plane-…

Let an unbounded straight particle beam have no external fields and finite electrical current through its cross-section. Cylindrical symmetry conjecture. The beam necessarily posse…

Let $CY_n$ denote a Calabi–Yau space with $SU(n)$ holonomy, and let ternary and n-ary symmetries be higher-arity generalizations of binary Cartan–Killing–Lie symmetries. Dubrovski–…

Let a space-time be represented by a collection of primordial point objects (PPOs), viewed as vertices, with a permutation symmetry expressing their democracy. Permutation-symmetry…

A perturbed family of symmetric interval exchange maps may have periodic orbits that are not symmetric under the system's time-reversal symmetry. Hidden-symmetry conjecture. These…

Symmetric capacity conjecture. The symmetric capacity satisfies

Consider real matrix ensembles with additional linear constraints, including real symmetric or other structured real matrices. Let $\rho(\lambda)\,dx\,dy$ denote an absolutely cont…

Let an effective physical model be a model featuring an isometry, and let all fields be invariant under this isometry. Suppose the model is mapped by a reference system $S$ satisfy…