265 research outputs found
Abelian Equations and Rank Problems for Planar Webs
We find an invariant characterization of planar webs of maximum rank. For
4-webs, we prove that a planar 4-web is of maximum rank three if and only if it
is linearizable and its curvature vanishes. This result leads to the direct
web-theoretical proof of the Poincar\'{e}'s theorem: a planar 4-web of maximum
rank is linearizable. We also find an invariant intrinsic characterization of
planar 4-webs of rank two and one and prove that in general such webs are not
linearizable. This solves the Blaschke problem ``to find invariant conditions
for a planar 4-web to be of rank 1 or 2 or 3''. Finally, we find invariant
characterization of planar 5-webs of maximum rank and prove than in general
such webs are not linearizable.Comment: 43 page
On a class of linearizable planar geodesic webs
We present a complete description of a class of linearizable planar geodesic
webs which contain a parallelizable 3-subweb.Comment: 7 page
Geodesic Webs on a Two-Dimensional Manifold and Euler Equations
We prove that any planar 4-web defines a unique projective structure in the
plane in such a way that the leaves of the foliations are geodesics of this
projective structure. We also find conditions for the projective structure
mentioned above to contain an affine symmetric connection, and conditions for a
planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric
surface. Similar results are obtained for planar d-webs, d > 4, provided that
additional d-4 second-order invariants vanish.Comment: 15 page
Theory of linear G-difference equations
We introduce the notion of difference equation defined on a structured set.
The symmetry group of the structure determines the set of difference operators.
All main notions in the theory of difference equations are introduced as
invariants of the symmetry group. Linear equations are modules over the skew
group algebra, solutions are morphisms relating a given equation to other
equations,symmetries of an equation are module endomorphisms and conserved
structures are invariants in the tensor algebra of the given equation. We show
that the equations and their solutions can be described through representations
of the isotropy group of the symmetry group of the underluing set. We relate
our notion of difference equations and solutions to systems of classical
difference equations and their solutions and show that our notions include
these as a special case.Comment: 34 page
Geodesic Webs of Hypersurfaces
In the present paper we study geometric structures associated with webs of
hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional
manifold there is naturally associated a unique projective structure and,
provided that one of web foliations is pointed, there is also associated a
unique affine structure. The projective structure can be chosen by the claim
that the leaves of all web foliations are totally geodesic, and the affine
structure by an additional claim that one of web functions is affine.
These structures allow us to determine differential invariants of geodesic
webs and give geometrically clear answers to some classical problems of the web
theory such as the web linearization and the Gronwall theorem.Comment: 11 pages, in Russia
Geodesic Webs and PDE Systems of Euler Equations
We find necessary and sufficient conditions for the foliation defined by
level sets of a function f(x_{1},...,x_{n}) to be totally geodesic in a
torsion-free connection and apply them to find the conditions for d-webs of
hypersurfaces to be geodesic, and in the case of flat connections, for d-webs
(d > n) of hypersurfaces to be hyperplanar webs. These conditions are systems
of generalized Euler equations, and for flat connections we give an explicit
construction of their solutions.Comment: 9 page
Influence of Lateral Incision on Inhomogeneous Deformation of a Nickel [001] - Single Cristal at Axial Compression
We used the scanning electron microscopy and the electron backscattered diffraction to investigate a deformation relief and a crystallographic disorientation of a nickel [001] single crystal with {100} faces with a lateral incision. We identified that the lateral incision can change the shear domains distribution pattern in the sample by creating additional deformation domains near the incision. The change in the patterns of the misorientation accumulation on the mutually perpendicular faces accompanies this deformation. We established that the orientation alteration occurs toward the increase of the Schmid factor for the slip systems in two of the four (previously equally loaded) slip planes. This method of shear deformation contributes to an optimal mutually consistent deformation in the adjacent areas of the single crystal
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