10 citations · 36 across the 15 of their papers we have counts for
15 papers
A generalized Kervaire Problem in stable homotopy groups of spheres
Petr M. Akhmet'ev
We will give an elementary self-contained description of the Mahowald element in the stable homotopy group of spheres , . Using this construction, we prove that a…
Geometric approach to stable homotopy groups of spheres II; Arf-Kervaire Invariants
Petr M. Akhmet'ev
The Kervaire Invariant 1 Problem until recently was an open problem in algebraic topology. Hill-Hopkins-Ravenel theorem clams a negative solution of the problem for all dimensions…
Minimum Quadratic Helicity States
Petr M. Akhmet'ev, Simon Candelaresi, Alexandr Yu Smirnov
Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. Those are eigenfunctions of the curl operato…
Calculations for the Practical Applications of Quadratic Helicity in MHD
Petr M. Akhmet'ev, Simon Candelaresi, Alexandr Yu Smirnov
For the quadratic helicity we present a generalization of the Arnol'd inequality which relates the magnetic energy to the quadratic helicity, which poses a lower bound. W…
Kairvaire Problems in Stable Homotopy Theory
Petr Akhmet'ev
The Kervaire Problem is an unsolved problem in Stable Homotopy Theory. The first interesting example is in dimension . There exists a closed stably-parallelizable manifold $\ti…
A remark on the Hopf invariant for spherical 4-braids
Petr Mikhailovich Akhmet'ev
An approach by J.Wu describes homotopy groups of the standard 2-sphere as isotopy classes of spherical --strand Brunnian braids is investigated in the case …
A remark on the space of 7-gons with a fixed total length in
Petr Mikhailovich Akhmet'ev
Based on the model of the space of polygons in with limited number of vertex, which was proposed by Jean-Claude Hausmann and Allen Knutson, and developed by severa…
Quadratic helicities and the energy of magnetic fields
Petr M. Akhmet'ev
Two non-local asymptotic invariants of magnetic fields for the ideal magnetohydrodynamics are introduced. The velocity of variation of the invariants for a non-ideal magnetohydrody…
On asymptotic higher analogs of the magnetic helicity invariant in MHD
Petr M. Akhmet'ev
A solution of a problem by V.I.Arnol'd about higher analog of the asymptotic Hopf invariant of divergence-free vector fields is presented. A higher invariant of magnetic fields, wh…
Geometric approach to stable homotopy groups of spheres II. The Kervaire invariant
Petr M. Akhmet'ev
A solution to the Kervaire invariant problem is presented. We introduce the concepts of abelian structure on skew-framed immersions, bicyclic structure on --framed imme…
Geometric approach to stable homotopy groups of spheres, III. Abelian, cyclic and quaternionic structure for mappings with singularities
Petr M. Akhmet'ev
Collection of (equivariant) -mappings admitting a relative abelian, cyclic, quaternionic, bicyclic, and quaternionic-cyclic structures are constructed.
Geometric approach to stable homotopy groups of spheres. I. The Hopf invariant
Petr M. Akhmet'ev
A geometric approach to the stable homotopy groups of spheres is developed in this paper, based on the Pontryagin-Thom construction. The task of this approach is to obtain an alter…
Geometric approach towards stable homotopy groups of spheres. The Kervaire invariant II
Petr M. Akhmet'ev
The notion of the geometrical --control of self-intersection of a skew-framed immersion and the notion of the -structure (the cyclic structure)…
Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I
Petr M. Akhmet'ev
In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant…
Geometric approach towards stable homotopy groups of spheres. The Hopf invariant
Petr M. Akhmet'ev
We develop a geometric approach to stable homotopy groups of spheres in the spirit of the work of Pontrjagin and Rokhlin. A new proof of the Hopf Invariant One Theorem by J.F.Adams…