7,645 research outputs found

    Refined intersection products and limiting linear subspaces of hypersurfaces

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    Let XX be a hypersurface of degree dd in Pn\Bbb P^n and FXF_X be the scheme of Pr\Bbb P^r's contained in XX. If XX is generic, then FXF_X will have the expected dimension (or empty) and its class in the Chow ring of G(r+1,n+1)G(r+1,n+1) is given by the top Chern class of the vector bundle SdUβˆ—S^dU^*, where UU is the universal subbundle on the Grassmannian G(r+1,n+1)G(r+1,n+1). When we deform a generic XX into a degenerate X0X_0, the dimension of FXF_X can jump. In this case, there is a subscheme FlimF_{lim} of FX0F_{X_0} with the expected dimension which consists of limiting Pr\Bbb P^r's in X0X_0 with respect to a general deformation. The simplest example is the well-known case of 2727 lines in a generic cubic surface. If we degenerate the surface into the union of a plane and a quadric, then there are infinitely many lines in the union. Which 2727 lines are the limiting ones and how many of them are in the plane and how many of them are in the quadric? The goal of this paper is to study FlimF_{lim} in general.Comment: AmS-Tex, 26 page

    Residual Intersections and Some Applications

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    We give a new residual intersection decomposition for the refined intersection products of Fulton-MacPherson. Our formula refines the celebrated residual intersection formula of Fulton, Kleiman, Laksov, and MacPherson. The new decomposition is more likely to be compatible with the canonical decomposition of the intersection products and each term in the decomposition thus has simple geometric meaning. Our study is motivated by its applications to some geometric problems. In particular, we use the decomposition to find the distribution of limiting linear subspaces in degenerations of hypersurfaces. A family of identities for characteristic classes of vector bundles is also obtained as another consequence. This paper will appear in Duke Math. Jour.Comment: 23 pages, Ams-Tex Version 2.

    Chern classes and degenerations of hypersurfaces and their lines

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    We study limiting lines on degenerations of generic hypersurfaces in PnP^n.Comment: 18 pages, AmS-Tex V2.

    Global performance of multireference density functional theory for low-lying states in sdsd-shell nuclei

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    We present a comprehensive study of low-lying states in even-even Ne, Mg, Si, S, Ar isotopes with the multireference density functional theory (MR-DFT) based on a relativistic point-coupling energy density functional (EDF). Beyond mean-field (BMF) effects are taken into account by configuration mixing of both particle-number and angular-momentum projected axially deformed states with generator coordinate method (GCM). Global performance of the MR-DFT for the properties of both ground state and of the first 2+,4+2^+, 4^+ states is examined, in comparison with previous studies based on nonrelativistic EDFs and available data. Our results indicate that an EDF parameterized at the BMF level is demanded to achieve a quantitative description

    Mixing Time of Random Walk on Poisson Geometry Small World

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    This paper focuses on the problem of modeling for small world effect on complex networks. Let's consider the supercritical Poisson continuous percolation on dd-dimensional torus TndT^d_n with volume ndn^d. By adding "long edges (short cuts)" randomly to the largest percolation cluster, we obtain a random graph Gn\mathscr G_n. In the present paper, we first prove that the diameter of Gn\mathscr G_n grows at most polynomially fast in ln⁑n\ln n and we call it the Poisson Geometry Small World. Secondly, we prove that the random walk on Gn\mathscr G_n possesses the rapid mixing property, namely, the random walk mixes in time at most polynomially large in ln⁑n\ln n.Comment: 23 page

    Existence of Positive Solutions for a class of Quasilinear Schr\"{o}dinger Equations of Choquard type

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    In this paper, we study the following quasilinear Schr\"{o}dinger equation of Choquard type βˆ’β–³u+V(x)uβˆ’β–³(u2)u=(IΞ±βˆ—βˆ£u∣p)∣u∣pβˆ’2u,Β Β x∈RN, -\triangle u+V(x)u-\triangle (u^{2})u=(I_\alpha *|u|^p)|u|^{p-2}u, \ \ x \in \mathbb{R}^{N}, where Nβ‰₯3N\geq 3,\ 0<Ξ±<N0<\alpha<N, 2(N+Ξ±)N≀p<2(N+Ξ±)Nβˆ’2\frac{2(N+\alpha)}{N}\leq p<\frac{2(N+\alpha)}{N-2} and IΞ±I_\alpha is a Riesz potential. Under appropriate assumptions on V(x)V(x), we establish the existence of positive solutions

    On The Modified Newman-Watts Small World and Its Random Walk

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    It is well known that adding "long edges (shortcuts)" to a regularly constructed graph will make the resulted model a small world. Recently, \cite{W} indicated that, among all long edges, those edges with length proportional to the diameter of the regularly constructed graph may play the key role. In this paper, we modify the original Newman-Watts small world by adding only long special edges to the dd-dimensional lattice torus (with size ndn^d) according to \cite{W}, and show that the diameter of the modified model and the mixing time of random walk on it grow polynomially fast in ln⁑n\ln n.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1703.0825

    On a Lower Bound for the Time Constant of First-Passage Percolation

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    We consider the Bernoulli first-passage percolation on Zd(dβ‰₯2)\mathbb Z^d (d\ge 2). That is, the edge passage time is taken independently to be 1 with probability 1βˆ’p1-p and 0 otherwise. Let ΞΌ(p){\mu(p)} be the time constant. We prove in this paper that ΞΌ(p1)βˆ’ΞΌ(p2)β‰₯ΞΌ(p2)1βˆ’p2(p2βˆ’p1) \mu(p_1)-\mu({p_2})\ge \frac{\mu(p_2)}{1-p_2}(p_2-p_1) for all 0≀p1<p2<1 0\leq p_1<p_2< 1 by using Russo's formula.Comment: 7 page

    Uniqueness of the critical probability for percolation in the two dimensional Sierpinski carpet lattice

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    We prove that the critical probability for the Sierpinski carpet lattice in two dimensions is uniquely determined. The transition is sharp. This extends the Kumagai's result to the original Sierpinski carpet lattice.Comment: 22pages; typos added(erased pre-finel comments

    On The Time Constant for Last Passage Percolation on Complete Graph

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    This paper focuses on the time constant for last passage percolation on complete graph. Let Gn=([n],En)G_n=([n],E_n) be the complete graph on vertex set [n]={1,2,…,n}[n]=\{1,2,\ldots,n\}, and i.i.d. sequence {Xe:e∈En}\{X_e:e\in E_n\} be the passage times of edges. Denote by WnW_n the largest passage time among all self-avoiding paths from 1 to nn. First, it is proved that Wn/nW_n/n converges to constant ΞΌ\mu, where ΞΌ\mu is called the time constant and coincides with the essential supremum of XeX_e. Second, when ΞΌ<∞\mu<\infty, it is proved that the deviation probability P(Wn/nβ‰€ΞΌβˆ’x)P(W_n/n\leq \mu-x) decays as fast as eβˆ’Ξ˜(n2)e^{-\Theta(n^2)}, and as a corollary, an upper bound for the variance of WnW_n is obtained. Finally, when ΞΌ=∞\mu=\infty, lower and upper bounds for Wn/nW_n/n are given.Comment: 12 pages, 1 figur
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