Mathematics Problem Archive

Determine for which orders Williamson matrices exist; such matrices give a construction of Hadamard matrices....

For each order $n$, determine the largest possible absolute determinant of an $n\times n$ matrix whose entries are all $1$ or $-1$....

Let $V$ be a complex algebraic variety with coordinate ring $R$. If the module of derivations of $R$ is free over $R$, must $V$ be smooth?...

Let $m,p\ge 2$, put $d=m+p-1$, and let $F(x)=(1,x,\ldots,x^d)$ be the rational normal curve. For $j=1,\ldots,mp$, let $X_j$ be the $p$-plane spanned b...

Find an integer cosine rule for integer triangles in integer trigonometry....

{\bf(IKEA problem.)} Classify all $n$-tuples of LLS-sequences for the angles that form integer $n$-gons....

Classify all combinatorial possible types of faces....

Classify all empty simplices of dimension $n$ up to lattice congruence....

Which $n$-gons are realizable as faces of an $m$-dimensional continued fraction? Here are two essentially geometrically different subcases: ; {\bf Fac...

Describe all finite two-dimensional sails (and the corresponding continued fractions)....

{\bf (Multidimensional IKEA problem.)} Describe the collections of the sails of the cones for all polytopes of a given combinatorial type....

{\bf (V. Arnold.)} Does there exist an algorithm to decide whether a given type of fundamental domain is realizable by a periodic continued fraction?...

{\bf (V. Arnold.)} Torus decompositions of integer noncongruent Klein sails are distinct....

{\bf (V. Arnold.)} Describe all torus decompositions that are realized by periodic two-dimensional continued fractions....

{\bf (V. Arnold.)} Classify continued fractions that correspond to the same cubic extension of the field of rational numbers....

Prove the existence of a cone for a single non-periodic combinatorial structure ($n\ge 3$)....

Find frequencies on $n$-dimensional continued fractions with the highest relative frequencies....

For every positive integer constant $C$ there exist only finitely many pairwise integer non-congruent faces with frequencies exceeding $C$....

Is that true that sum of all relative frequencies for all possible faces is finite for higher dimensions $(n\ge 3)$?...

In case of positive answer to the above question find the generalization of the Gauss map and compare the corresponding frequencies of faces with the ...

Find a natural generalization of the Farey tessellation to higher-dimensional hyperbolic geometry....

{\bf (Jacobi's last theorem.)} Let $K$ be a totally real cubic number field. Consider arbitrary elements $y$ and $z$ of $K$ such that $0<y,z<1$ (here ...

Study geometric properties of Markov spectrum....

Generalize continued fractions to describe 3-bridge knots....

Prove the existence of energy minimizers for elastostatics for quasiconvex stored-energy functions $W$ satisfying $W(A)\to\infty$ as $\det A\to0^+$ ....

Find useful ways of verifying polyconvexity and quasiconvexity for stored-energy functions arising in anisotropic nonlinear elasticity....

Determine when the minimizer $y^*$ in Theorem 2.1 of the source is smooth....

Can the Lavrentiev phenomenon occur for elastostatics under growth conditions ensuring that all finite-energy deformations are continuous?...

Prove or disprove that, under reasonable growth conditions on $W$, energy minimizers satisfy the weak Euler-Lagrange equations....

Prove or disprove that, under reasonable growth conditions on $W$, an energy-minimizing deformation satisfies $\det Dy^*(x)\ge\varepsilon>0$....

Justify the Ciarlet-Nečas minimization problem, or an appropriate modification, in situations involving smooth self-contact....

Prove or disprove uniqueness of sufficiently smooth equilibrium solutions for pure-displacement problems in homogeneous bodies homeomorphic to a ball ...

Devise general methods for proving the existence of local but nonglobal minimizers and other weak equilibria in nonlinear elastostatics....

Develop local and global bifurcation theories for nonlinear elastostatics with mixed displacement-traction boundary conditions....

Clarify the status of models based on the fracture energy functional (2.31) in the source relative to classical fracture and nonlinear elastostatics....

Prove global existence and uniqueness for suitable initial-boundary-value problems in dynamic nonlinear elasticity....

Develop a qualitative dynamics for dynamic theories of elasticity....

Develop criteria for dynamic stability and instability of equilibria in nonlinear elasticity....

Establish the status of elasticity theory with respect to atomistic models....

For the set of energy-minimizing gradients $K(\theta)$ defined in the source, determine its quasiconvex hull $K(\theta)^{qc}$ for $\theta\le\theta_c$....

For free-energy functions $\psi(A,\theta)$ of elastic crystals, determine boundary conditions under which the minimum is attained and conditions under...

Give a rigorous derivation of models of rods, plates, and shells from three-dimensional elasticity as thickness tends to zero....

Under which additional assumptions does this principle become a rigorous theorem?...

Consider two Lagrangian fibrations $\phi: M^4 \to B$ and $\phi': {M'}^4 \to B'$. Assume that $B$ and $B'$ are affinely equivalent in the sense that th...

Do local symplectic invariants exist for diffeomorphic degenerate singularities? How many and of what kind are they? This question makes sense even in...

Describe the symplectic invariants of stable rank-one singularities described by V. V. Kalashnikov. For such singularities, one of the action variable...

Assume that we know explicit formulas for the action variables $I_1,\dots, I_n$ so that we are able to analyse their asymptotic behaviour in a neighbo...

Describe all topological types of singularities that may appear in algebraically integrable systems with a small ($\leq 3$) number of degrees of freed...

[\'{A}. Pelayo] Extend the classification of semitoric systems $F=(J,H)$ in to allow for $F$ having non-degenerate singularities with hyperbolic block...

[\'{A}. Pelayo] Consider a compact connected $2n$-dimensional symplectic manifold $M$, endowed with a Hamiltonian $(S^1)^{n-1}$-action; these are call...