Mathematics Problem Archive

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 ...

Study geometric properties of Markov spectrum....

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....

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

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

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

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

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

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

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

[N. T. Zung] Study the topology and geometry of these singular fibers and their small neighbourhoods....

[N. T. Zung] Give a clear description of these special singular fibers....

Complete the above table: {\rm (1)} construct new examples of natural Hamiltonian systems on closed two-dimensional surfaces admitting polynomial inte...

Construct stationary axially symmetric 4-dimensional Einstein metrics admitting Killing tensors of higher order....

[Gilkey ] In the Riemannian case, is every $1$-homogeneous manifold locally homogeneous?...

{\it Construct a natural Hamiltonian system on the 2-sphere with a nonconstant potential which is superintegrable by integrals of degree $\ge 3$ and a...

Prove the algebraic version of Birkhoff conjecture for outer billiards in a non-Euclidean surface of constant curvature....

Prove Conjecture [source label: Descon] it the case that the foliation admits (i) a rational, (ii) an algebraic first integral....

Is it possible to choose the domain so that the dynamics of the corresponding billiard map are locally (near the 2-periodic orbit) conjugated to the d...

Are there plane billiards, other than ellipses, that possess rational caustics with two different values of the rotation numbers? Same question for ou...

{\rm (V. Retakh) Establish complete integrability of the noncommutative version of the leapfrog map. Define noncommutative versions of the pentagram m...

Consider the action of $ Aut (V,J)$ on $V$. Describe the partition of $V$ into $ Aut (V,J)$-orbits. More generally, describe the action of $ Aut (V,J)...

Find necessary and sufficient conditions for the bi-Lagrangian Grassmannian $LG(V,J)$ to be a smooth algebraic variety. Describe the partition of $LG(...

Do bi-integrable systems exist for each algebraic type?...

Are there any restrictions on the algebraic type of the pencils $\mathcal{A}_{x+\lambda a}$? Which algebraic types can be realised by means of an appr...

Study examples of ``quadratic $+$ linear'' Poisson pencils. Compute their algebraic types and construct complete families of polynomials in bi-involut...

Is it true that for any quadratic Poisson bracket (defined on a vector space), there exists a polynomial integrable system?...

Describe closed manifolds $M$ which admit Nijenhuis operators $L(x)$ that are algebraically regular at each point $x\in M$....

Let us fix a certain algebraic type of a linear operator, i.e., its Segre characteristic (see above). Does there exist a Nijenhuis operator $L$ in $\m...

Describe all the functions $f(x,y)$ of two variables defined in a neighbourhood of $(0,0)\in\mathbb{R}^2$ such that ; $f_y(0,0)\not\equiv 0$; ; $f_y(0...

Consider a smooth map $\Phi=(\sigma_1,\dots,\sigma_n): U(0) \to \mathbb{R}^n$, where $U(0)$ is a neighbourhood of the origin $0\in \mathbb{R}^n$. We a...

Describe/classify the collections of algebraically independent homogeneous polynomials $\sigma_1, \dots, \sigma_n$, $\deg \sigma_k = k$, in $n$ variab...

Describe the structure of singularities of $\Phi=(\sigma_1,\dots,\sigma_n): M \to \mathbb{R}^n$ in terms of the singular points of the recursion opera...

Which foliations with affine leaves can be described as the image of the moment map?...

Consider a Poisson manifold $M$ of even dimension such that it is symplectic on a dense set $U\subset M$. Assume that $M$ is endowed with a toric acti...

[S. V\ u Ng{\d o}c] Given a set of semiclassical operators that verify conditions [source label: item:self-adjoint ] and [source label: item:commute] ...

[S. V\ u Ng{\d o}c] Define (and detect) the quantum Chern class....

[\'{A}. Pelayo] Can one make progress in counting the number of fixed points by studying the spectrum of the quantisation of $\mu \colon M \to S^1$?...

What are necessary and/or sufficient conditions on a metric $g$ such that every polynomial integral of its geodesic flow is quantisable?...

[Miranda-Presas-Solha ] Modify this scheme to get finite dimensional representation spaces for focus-focus and hyperbolic singularities that still cap...

[folklore, very irritating] For any set of charges of the same sign in $\mathbb{R}^n$, the set of its points of equilibrium is finite....

[A. Gabrielov, D. Novikov, B. Sh., seems good, but no progress] Let $(x_1,y_1),(x_2,y_2),\dots, (x_N,y_N)$ be a collection of points in $\mathbb{R}^2$...

[B. Sh., looks bad, but very important] Does there exist an upper bound for the number of real roots valid for all non-trivial solutions of all equati...

[D. Khavinson, I. Itenberg, B. Sh., apparently bad] Find the maximal possible number $\#(2k, l)$ of isolated zeros for real non-negative polynomials o...

[G. Ottaviani, B. Sh., seems good] Find the maximal possible number $\widetilde\#(2k, l)$ of isolated zeros for real non-negative polynomials of degre...

[G. Ottaviani, B. Sh., seems good] For any number of variables, $\widetilde\#(2k,l)=k^l$....

[S. Fisk, seems bad, see , p. 575] Given a pair of real polynomials $(p,q),$ give restrictions on the location of the roots of $p+iq$ in terms of the ...