On Statistical Significance of Signal

A definition for the statistical significance of a signal in an experiment is proposed by establishing a correlation between the observed p-value and the normal distribution integral probability, which is suitable for both counting experiment and continuous test statistics. The explicit expressions to calculate the statistical significance for both cases are given.Comment: 3 page

Revealing the phase transition behaviors of k-core percolation in random networks

The $k$-core percolation is a fundamental structural transition in complex networks. Through the analysis of the size jump behaviors of $k$-core in the evolution process of networks, we confirm that $k$-core percolation is continuous phase transition when $k=1,2$ while it is a hybrid first-order-second-order phase transition when $k\ge 3$. $2$-core percolation belongs to different universality class from that of $1$-core (giant component) percolation. The discontinuity of $k$-core percolation with $k\ge 3$ can be concluded from largest size jump of $k$-core which will not disappear in the thermodynamic limit while its continuous characteristic is reflected by second largest size jump which converges to zero in power law as $N\to \infty$. Furthermore, along with the previously known exponent $\beta=0.5$, we obtain a set of exponents which are independent of $k$ when $k\ge 3$ and also different from those critical exponents of $1$-core and $2$-core percolation

Finite size scaling theory for percolation with multiple giant clusters

A approach of finite size scaling theory for discontinous percolation with multiple giant clusters is developed in this paper. The percolation in generalized Bohman-Frieze-Wormald (BFW) model has already been proved to be discontinuous phase transition. In the evolution process, the size of largest cluster $s_1$ increases in a stairscase way and its fluctuation shows a series of peaks corresponding to the jumps of $s_1$ from one stair to another. Several largest jumps of the size of largest cluster from single edge are studied by extensive Monte Carlo simulation. $\overline{\Delta}_k(N)$ which is the mean of the $k$th largest jump of largest cluster, $\overline{r}_k(N)$ which is the corresponding averaged edge density, $\sigma_{\Delta,k}(N)$ which is the standard deviation of $\Delta_k$ and $\sigma_{r,k}(N)$ which is the standard deviation of $r_k$ are analyzed. Rich power law behaviours are found for $\overline{r}_k(N)$, $\sigma_{\Delta,k}(N)$ and $\sigma_{r,k}(N)$ with critical exponents denoted as $1/\nu_1$, $(\beta/\nu)_2$ and $1/\nu_2$. Unlike continuous percolation where the exact critical thresholds and critical exponent $1/\nu_1$ are used for finite size scaling, the size-dependent pseudo critical thresholds $\overline{r}_k(N)$ and $1/\nu_2$ works for the data collapse of the curves of largest cluster and its fluctuation in discontinuous percolation in BFW model. Further, data collapse can be obtained part by part. That is, $s_1(r,N)$ can be collapsed for each jump from one stair to another and its fluctuation can be collapsed around each peak with the corresponding $\overline{r}_k(N)$ and $1/\nu_2$

Finite size scaling theory for percolation phase transition

The finite-size scaling theory for continuous phase transition plays an important role in determining critical point and critical exponents from the size-dependent behaviors of quantities in the thermodynamic limit. For percolation phase transition, the finite-size scaling form for the reduced size of largest cluster has been extended to cluster ranked $R$. However, this is invalid for explosive percolation as our results show. Besides, the behaviors of largest increase of largest cluster induced by adding single link or node have also been used to investigate the critical properties of percolation and several new exponents $\beta_1$, $\beta_2$, $1/\nu_1$ and $1/\nu_2$ are defined while their relation with $\beta/\nu$ and $1/\nu$ is unknown. Through the analysis of asymptotic properties of size jump behaviors, we obtain correct critical exponents and develop a new approach to finite size scaling theory where sizes of ranked clusters are averaged at same distances from the sample-dependent pseudo-critical point in each realization rather than averaging at same value of control parameter

On the reducibility of some special quadrinomials

Let $n>m>k$ be positive integers and let $a,b,c$ be nonzero rational numbers. We consider the reducibility of some special quadrinomials $x^n+ax^m+bx^k+c$ with $n=4$ and 5, which related to the study of rational points on certain elliptic curves or hyperelliptic curves.Comment: 16 pages. Any comments are welcome

Embedding property of JJ-holomorphic curves in Calabi-Yau manifolds for generic JJ

In this paper, we prove that for a generic choice of tame (or compatible) almost complex structures $J$ on a symplectic manifold $(M^{2n},\omega)$ with $n \geq 3$ and with its first Chern class $c_1(M,\omega) = 0$, all somewhere injective $J$-holomorphic maps from any closed smooth Riemann surface into $M$ are \emph{embedded}. We derive this result as a consequence of the general optimal 1-jet evaluation transversality result of $J$-holomorphic maps in general symplectic manifolds that we also prove in this paper.Comment: The exposition of the paper much improved with some clarifications in the Fredholm setting. The proof of 1-jet evaluation transversality is much clarified by correcting minor error in the Fredholm analysis and making precise the choice of Sobolev norms in the proof. A few more references are adde

Floer trajectories with immersed nodes and scale-dependent gluing

We define an enhanced compactification of Floer trajectories under Morse background using the adiabatic degeneration and the scale-dependent gluing techniques. The compactification reflects the 1-jet datum of the smooth Floer trajectories nearby the limiting nodal Floer trajectories arising from adiabatic degeneration of the background Morse function. This paper studies the gluing problem when the limiting gradient trajectories has length zero through a renomalization process. The case with limiting gradient trajectories of non-zero length will be treated elsewhere. An immediate application of our result is a proof of the isomorphism property of the PSS map : A proof of this isomorphism property was first outlined by P\"unihikin-Salamon-Schwarz \cite{PSS} in a way somewhat different from the current proof in its details. This kind of scale-dependent gluing techniques was initiated in [FOOO07] in relation to the metamorphosis of holomorphic polygons under Lagrangian surgery and is expected to appear in other gluing and compactification problem of pseudo-holomorphic curves that involves `adiabatic' parameters or rescales the targets.Comment: 122 pages, 6 figures. Submitted version. Quadratic estimates added, presentation of error estimates improve

Ultrafast modulation of near-field heat transfer with tunable metamaterials

We propose a mechanism of active near-field heat transfer modulation relying on externally tunable metamaterials. A large modulation effect is observed and can be explained by the coupling of surface modes, which is dramatically varied in the presence of controllable magnetoelectric coupling in metamaterials. We finally discuss how a practical picosecond-scale thermal modulator can be made. This modulator allows manipulating nanoscale heat flux in an ultrafast and noncontact (by optical means) manner.Comment: 11 pages, 2 figure

Thick-thin Decomposition of Floer Trajectories and Adiabatic Gluing

This is a sequel to [OZ1] in which we studied the adiabatic degeneration of Floer trajectories to "disk-flow-disk" configurations and the recovering gluing, where the gradient flow part had length 0. In the present paper we study the case when the gradient flow part has a positive length. Unlike the standard gluing problem, we study the problem of gluing 1-dimensional gradient segments and 2-dimensional (perturbed) J-holomorphic curves. The two also have different convergence rates near the ends: linear convergence for the finite gradient segments and exponential convergence for J-holomorphic maps. As an immediate application, we outline the proof that when a finite number of Hamiltonian deformations of a non-exact Lagrangian submanifold collapse simultaneously, the pearl complex moduli spaces used in [BC] are diffeomorphic to the J-holomorphic polygon moduli spaces in [FOOO] provided the dimension of the moduli spaces is sufficiently small. This is enough to prove that the A_{\infty}-structures appearing in the two pictures are isomorphic to each other. The main gluing theorem also provides another proof of the isomorphism property of PSS map which is different from that of [OZ1]: It bypasses the nodal Floer trajectories by going directly from "disk-flow-disk" configurations to resolved Floer trajectories.Comment: 82 pages, 1 figure. Submitted version. Section 14 expanded. Improved expositio

The Complete Structure of the Cohomology Ring and Associated Symmetries in D=2D=2 String Theory

We determine explicitly all structure constants of the whole chiral BRST cohomology ring in $D=2$ string theory including both the discrete states and tachyon states. This is made possible by establishing several identities for Schur polynomials with operator argument and exploring associativity. Furthermore we find that the (chiral) symmetry algebra of the charges obtained by using the descent equations can actually be read off from the cohomology ring structure by simple operation involving the ghost field $b$. We also determine the enlarged symmetry algebra which contains the charges having ghost number $-1$ and $1$. Finally the complete symmetry transformation rules are derived for closed string discrete states by carefully combining the left and right sectors. It turns out that the new states introduced recently by Witten and Zwiebach are naturally created when symmetries act on the old states.Comment: 48 pages, UU-HEP/92/6, with postscript file for one figure; with input line correcte