28,133 research outputs found

### On the free boundary min-max geodesics

Given a Riemannian manifold and a closed submanifold, we find a geodesic
segment with free boundary on the given submanifold. This is a corollary of the
min-max theory which we develop in this article for the free boundary
variational problem. In particular, we develop a modified Birkhoff curve
shortening process to achieve a strong "Colding-Minicozzi" type min-max
approximation result.Comment: 16 page

### Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric_{g}>0$ and $2\leq n\leq 6$

In this paper, we study the shape of the min-max minimal hypersurface
produced by Almgren-Pitts in \cite{A2}\cite{P} corresponding to the fundamental
class of a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature with
$2\leq n\leq 6$. We characterize the Morse index, area and multiplicity of this
min-max hypersurface. In particular, we show that the min-max hypersurface is
either orientable and of index one, or is a double cover of a non-orientable
minimal hypersurface with least area among all closed embedded minimal
hypersurfaces.Comment: 31 pages, Section 6 reformulated and strengthened with an error
correcte

### Effective Non-vanishing of Asymptotic Adjoint Syzygies

The purpose of this paper is to establish an effective non-vanishing theorem
for the syzygies of an adjoint-type line bundle on a smooth variety, as the
positivity of the embedding increases. Our purpose here is to show that for an
adjoint type divisor $B = K_X+ bA$ with $b \geq n+1$, one can obtain an
effective statement for arbitrary $X$ which specializes to the statement for
Veronese syzygies in the paper "Asymptotic Syzygies of Algebraic Varieties" by
Ein and Lazarsfeld. We also give an answer to Problem 7.9 in that paper in this
setting

### On the existence of min-max minimal torus

In this paper, we will study the existence problem of minmax minimal torus.
We use classical conformal invariant geometric variational methods. We prove a
theorem about the existence of minmax minimal torus in Theorem 5.1. Firstly we
prove a strong uniformization result(Proposition 3.1) using method of [1]. Then
we use this proposition to choose good parametrization for our minmax
sequences. We prove a compactification result(Lemma 4.1) similar to that of
Colding and Minicozzi [2], and then give bubbling convergence results similar
to that of Ding, Li and Liu [7]. In fact, we get an approximating result
similar to the classical deformation lemma(Theorem 1.1).Comment: 31 page

### Construction and Refinement of Coarse-Grained Models

A general scheme, which includes constructions of coarse-grained (CG) models,
weighted ensemble dynamics (WED) simulations and cluster analyses (CA) of
stable states, is presented to detect dynamical and thermodynamical properties
in complex systems. In the scheme, CG models are efficiently and accurately
optimized based on a directed distance from original to CG systems, which is
estimated from ensemble means of lots of independent observable in two systems.
Furthermore, WED independently generates multiple short molecular dynamics
trajectories in original systems. The initial conformations of the trajectories
are constructed from equilibrium conformations in CG models, and the weights of
the trajectories can be estimated from the trajectories themselves in
generating complete equilibrium samples in the original systems. CA calculates
the directed distances among the trajectories and groups their initial
conformations into some clusters, which correspond to stable states in the
original systems, so that transition dynamics can be detected without requiring
a priori knowledge of the states.Comment: 4 pages, no figure

### Min-max hypersurface in manifold of positive Ricci curvature

In this paper, we study the shape of the min-max minimal hypersurface
produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a
Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all
dimensions. The min-max hypersurface has a singular set of Hausdorff
codimension $7$. We characterize the Morse index, area and multiplicity of this
singular min-max hypersurface. In particular, we show that the min-max
hypersurface is either orientable and has Morse index one, or is a double cover
of a non-orientable stable minimal hypersurface.
As an essential technical tool, we prove a stronger version of the
discretization theorem. The discretization theorem, first developed by
Marques-Neves in their proof of the Willmore conjecture \cite{MN12}, is a
bridge to connect sweepouts appearing naturally in geometry to sweepouts used
in the min-max theory. Our result removes a critical assumption of \cite{MN12},
called the no mass concentration condition, and hence confirms a conjecture by
Marques-Neves in \cite{MN12}.Comment: minor revision, 48 pages, 2 figure

### On the Black Hole Masses In Ultra-luminous X-ray Sources

Ultra-luminous X-ray sources (ULXs) are off-nuclear X-ray sources in nearby
galaxies with X-ray luminosities $\geq$ 10$^{39}$ erg s$^{-1}$. The measurement
of the black hole (BH) masses of ULXs is a long-standing problem. Here we
estimate BH masses in a sample of ULXs with XMM-Newton observations using two
different mass indicators, the X-ray photon index and X-ray variability
amplitude based on the correlations established for active galactic nuclei
(AGNs). The BH masses estimated from the two methods are compared and
discussed. We find that some extreme high-luminosity ($L_{\rm X}
>5\times10^{40}$ erg s$^{-1}$) ULXs contain the BH of 10$^{4}$-10$^{5}$
$M_\odot$. The results from X-ray variability amplitude are in conflict with
those from X-ray photon indices for ULXs with lower luminosities. This suggests
that these ULXs generally accrete at rates different from those of X-ray
luminous AGNs, or they have different power spectral densities of X-ray
variability. We conclude that most of ULXs accrete at super-Eddington rate,
thus harbor stellar-mass BH.Comment: 2 figures, 2 tables, accepted by New Astronom

### Every finite group has a normal bi-Cayley graph

A graph \G with a group $H$ of automorphisms acting semiregularly on the
vertices with two orbits is called a {\em bi-Cayley graph} over $H$. When $H$
is a normal subgroup of \Aut(\G), we say that \G is {\em normal} with
respect to $H$. In this paper, we show that every finite group has a connected
normal bi-Cayley graph. This improves Theorem~5 of [M. Arezoomand, B. Taeri,
Normality of 2-Cayley digraphs, Discrete Math. 338 (2015) 41--47], and provides
a positive answer to the Question of the above paper

### Transport plans with domain constraints

This paper focuses on martingale optimal transport problems when the
martingales are assumed to have bounded quadratic variation. First, we give a
result that characterizes the existence of a probability measure satisfying
some convex transport constraints in addition to having given initial and
terminal marginals. Several applications are provided: martingale measures with
volatility uncertainty, optimal transport with capacity constraints, and
Skorokhod embedding with bounded times. Next, we extend this result to
multi-marginal constraints. Finally, we consider an optimal transport problem
with constraints and obtain its Kantorovich duality. A corollary of this result
is a monotonicity principle which gives a geometric way of identifying the
optimizer.Comment: To appear in Applied Mathematics and Optimization.
Keywords:Strassen's Theorem, Kellerer's Theorem, Martingale optimal
transport, domain constraints, bounded volatility/quadratic variation,
$G$-expectations, Kantorovich duality, monotonicity principl

### Blow-up problems for the heat equation with a local nonlinear Neumann boundary condition

This paper estimates the blow-up time for the heat equation $u_t=\Delta u$
with a local nonlinear Neumann boundary condition: The normal derivative
$\partial u/\partial n=u^{q}$ on $\Gamma_{1}$, one piece of the boundary, while
on the rest part of the boundary, $\partial u/\partial n=0$. The motivation of
the study is the partial damage to the insulation on the surface of space
shuttles caused by high speed flying subjects. We prove the solution blows up
in finite time and estimate both upper and lower bounds of the blow-up time in
terms of the area of $\Gamma_1$. In many other work, they need the convexity of
the domain $\Omega$ and only consider the problem with
$\Gamma_1=\partial\Omega$. In this paper, we remove the convexity condition and
only require $\partial\Omega$ to be $C^{2}$. In addition, we deal with the
local nonlinearity, namely $\Gamma_1$ can be just part of $\partial\Omega$.Comment: 42 page

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