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This is a generalization of the procedure presented in [3] to construct semisimple bi-flat $F$-manifolds $(M,\nabla^{(1)},\nabla^{(2)},\circ,*,e,E)$ starting from homogeneous solutions of degree -1 of Darboux-Egorov-system. The Lam\'e…
Analogous to the famous Euler angle parametrization in three-dimensional Euclidean space, a reflection-free Lorentz transformation in (2+1)-dimensional Minkowski space can be decomposed into three simple parts. Applying this decomposition…
This paper concerns frames and equiangular lines over finite fields. We find a necessary and sufficient condition for systems of equiangular lines over finite fields to be equiangular tight frames (ETFs). As is the case over subfields of…
A class of semi-bounded solutions of the two-dimensional incompressible Euler equations satisfying either periodic or Dirichlet boundary conditions is examined. For smooth initial data, new blowup criteria in terms of the initial concavity…
We prove that given any closed $n$-manifold $M^n$, $n\geq 4$, there is an A-flow $f^t$ on $M^n$ such that the non-wandering set $NW(f^t)$ consists of 2-dimensional expanding attractor (the both, orientable and non-orientable) and trivial…
On the category of bounded complexes of finitely generated free squarefree modules over the polynomial ring S, there is the standard duality functor D = Hom_S(-, omega_S) and the Alexander duality functor A. The composition AD is an…
Three-dimensional Einstein-Maxwell theory with non trivial asymptotics at null infinity is solved. The symmetry algebra is a Virasoro-Kac-Moody type algebra that extends the bms3 algebra of the purely gravitational case. Solution space…
We classify families of free rational curves on all smooth Fano threefolds over the complex numbers. In particular, we prove the family of very free rational curves representing any fixed numerical curve class is either irreducible or…
Solutions with degenerate metric ($det(g_{\mu\nu})=0$ or $g_{\mu\nu}=0$) in the first order formalism (FOF) are physically acceptable: they may describe topology changes (Horowitz) and reduction of "metrical dimension" (Tseytlin) of…
In the Thermodynamic Bethe Ansatz approach to 2D integrable, ADE-related quantum field theories one derives a set of algebraic functional equations (a Y-system) which play a prominent role. This set of equations is mapped into the problem…
In 1984, Anatole Katok conjectured that the only closed orientable manifolds that support cohomology-free vector fields are tori and these vector fields are smoothly conjugated to Diophantine (constant) ones. In this work we present a proof…
We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We construct unique local-in-time solutions in the Lagrangian setting for $u_0 \in H^{2.5+\delta }$ such that the Rayleigh-Taylor…
We consider smooth threefolds $Y$ defined over $\Bbb{C}$ with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$. Let $X$ be a smooth projective threefold containing $Y$ and $D$ be the boundary divisor with support $X-Y$. We are interested in…
For a strongly pseudo-convex complex Finsler manifold M, a bundle U of adapted unitary frames is canonically defined. A non-linear Hermitian connection on U, invariant under local biholomorphic isometries, is given and it proved to be…
In the present paper, non-singular Morse-Smale flows on closed orientable 3-manifolds under the assumption that among the periodic orbits of the flow there is only one saddle one and it is twisted are considered. An exhaustive description…
We discuss the rigidity (or lack thereof) imposed by different notions of having an abundance of zero curvature planes on a complete Riemannian 3-manifold. We prove a rank rigidity theorem for complete 3-manifolds, showing that having…
We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle of the threefold satisfies certain…
The normality equations for the Newtonian dynamical systems on an arbitrary Riemannian manifold of the dimension $n \geq 3$ are considered. Locally the solution of such equations reduces to three possible cases: in two of them the solution…
Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) almost normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following…
We prove Grothendieck's Conjecture on Resolution of Singulari-ties for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X = SpecA, A a reduced complete Noetherian local ring of…