‎In this paper‎, ‎we introduce the structure of a groupoid associated to a vector field‎ ‎on a smooth manifold‎. ‎We show that in the case of the $1$-dimensional manifolds‎, ‎our‎ ‎groupoid has a‎ ‎smooth structure such that makes it into a Lie groupoid‎. ‎Using this approach‎, ‎we associated to‎ ‎every vector field an equivalence‎ ‎relation on the Lie algebra of all vector fields on the smooth...

Abstract A natural approach to the construction of nearly $$G_2$$ G 2 manifolds lies in resolving spaces with isolated conical singularities by gluing asymptotically modelled on same cone. If such a resolution exits, one expects there be family manifolds, whose endpoint...

in this paper, we discuss the geometric properties of solution and lower bound estimate of ∆um−1 of the cauchy problem for a degenerate parabolic equation with periodic source term ut =∆um+ upsint. our objective is to show that: (1)with continuous variation of time t, the surface ϕ = [u(x,t)]mδq is a complete riemannian manifold floating in space rn+1and is tangent to the space rn at ∂h0(t); (2...

We gave a series of four lectures, the first by the second author and the rest by the first author. The latter was followed by a 2 hour question and answer session. The problem in question is nearly 50 years old and began with Kervaire’s paper [Ker60] of 1960 in which he defined a Z/2-valued invariant φ(M) on certain manifolds M of dimension 4m + 2. He showed that for m = 2 and M aclosed smooth...

We remark on two free-boundary problems for holomorphic curves in nearly-Kähler 6-manifolds. First, we observe that a curve geodesic ball B of the round 6-sphere meets $$\partial B$$ orthogonally must be totally geodesic. Consequently, obtain rigidity results reflection-invariant $$\mathbb {S}^6$$ and associative cones {R}^7$$ . Second, consider with boundary Lagrangian submanifold strict 6-man...

designing and optimization of the intake manifold is one of the important steps in the development of the internal combustion engines. in this paper, the flow inside the intake manifold of an internal combustion engine is investigated by numerical modeling. the convective flux models are adopted to simulate the unsteady compressible flow behavior. to solve the governing equations, for calculati...