Intrinsic Ultracontractivity of Non-local Dirichlet forms on Unbounded Open Sets

In this paper we consider a large class of symmetric Markov processes $X=(X_t)_{t\ge0}$ on $\R^d$ generated by non-local Dirichlet forms, which include jump processes with small jumps of $\alpha$-stable-like type and with large jumps of super-exponential decay. Let $D\subset \R^d$ be an open (not necessarily bounded and connected) set, and $X^D=(X_t^D)_{t\ge0}$ be the killed process of $X$ on exiting $D$. We obtain explicit criterion for the compactness and the intrinsic ultracontractivity of the Dirichlet Markov semigroup $(P^{D}_t)_{t\ge0}$ of $X^D$. When $D$ is a horn-shaped region, we further obtain two-sided estimates of ground state in terms of jumping kernel of $X$ and the reference function of the horn-shaped region $D$.Comment: 47 page

Impossibility of Full Decentralization in Permissionless Blockchains

Bitcoin uses blockchain technology and proof-of-work (PoW) mechanism where nodes spend computing resources and earn rewards in return for spending these resources. This incentive system has caused power to be significantly biased towards a few nodes, called mining pools. In fact, poor decentralization appears not only in PoW-based coins but also in coins adopting other mechanisms such as proof-of-stake (PoS) and delegated proof-of-stake (DPoS). In this paper, we target this centralization issue. To this end, we first define (m, \varepsilon, \delta)-decentralization as a state that satisfies 1) there are at least m participants running a node and 2) the ratio between the total resource power of nodes run by the richest and \delta-th percentile participants is less than or equal to 1+\varepsilon. To see if it is possible to achieve good decentralization, we introduce sufficient conditions for the incentive system of a blockchain to reach (m, \varepsilon, \delta)-decentralization. When satisfying the conditions, a blockchain system can reach full decentralization with probability 1. However, to achieve this, the blockchain system should be able to assign a positive Sybil cost, where the Sybil cost is defined as the difference between the cost for one participant running multiple nodes and the total cost for multiple participants each running one node. On the other hand, we prove that when there is no Sybil cost, the probability of reaching (m, \varepsilon, \delta)-decentralization is upper bounded by a value close to 0, considering a large rich-poor gap. To determine the conditions that each system cannot satisfy, we also analyze protocols of all PoW, PoS, and DPoS coins in the top 100 coins according to our conditions. Finally, we conduct data analysis of these coins to validate our theory.Comment: This paper is accepted to ACM AFT 201

Interfacial microscopic mechanism of free energy minimization in Omega precipitate formation

Precipitate strengthening of light metals underpins a large segment of industry.Yet, quantitative understanding of physics involved in precipitate formation is often lacking, especially, about interfacial contribution to the energetics of precipitate formation.Here, we report an intricate strain accommodation and free energy minimization mechanism in the formation of Omega precipitates (Al2Cu)in the Al_Cu_Mg_Ag alloy. We show that the affinity between Ag and Mg at the interface provides the driving force for lowering the heat of formation, while substitution between Mg, Al and Cu of different atomic radii at interfacial atomic sites alters interfacial thickness and adjust precipitate misfit strain. The results here highlight the importance of interfacial structure in precipitate formation, and the potential of combining the power of atomic resolution imaging with first-principles theory for unraveling the mystery of physics at nanoscale interfaces.Comment: letter with 5 figures, submitte

Simultaneous Registration and Clustering for Multi-dimensional Functional Data

The clustering for functional data with misaligned problems has drawn much attention in the last decade. Most methods do the clustering after those functional data being registered and there has been little research using both functional and scalar variables. In this paper, we propose a simultaneous registration and clustering (SRC) model via two-level models, allowing the use of both types of variables and also allowing simultaneous registration and clustering. For the data collected from subjects in different unknown groups, a Gaussian process functional regression model with time warping is used as the first level model; an allocation model depending on scalar variables is used as the second level model providing further information over the groups. The former carries out registration and modeling for the multi-dimensional functional data (2D or 3D curves) at the same time. This methodology is implemented using an EM algorithm, and is examined on both simulated data and real data.Comment: 36 pages, 13 figure

Heat kernel estimates for time fractional equations

In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time fractional equations in metric measure spaces.Comment: 34 page

Diameters in supercritical random graphs via first passage percolation

We study the diameter of $C_1$, the largest component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ in the emerging supercritical phase, i.e., for $p = \frac{1+\epsilon}n$ where $\epsilon^3 n \to \infty$ and $\epsilon=o(1)$. This parameter was extensively studied for fixed $\epsilon > 0$, yet results for $\epsilon=o(1)$ outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter, however these did not cover the entire supercritical regime (namely, when $\epsilon^3 n\to\infty$ arbitrarily slowly). {\L}uczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of $1000/7$. We show that throughout the emerging supercritical phase, i.e. for any $\epsilon=o(1)$ with $\epsilon^3 n \to \infty$, the diameter of $C_1$ is with high probability asymptotic to $D(\epsilon,n)=(3/\epsilon)\log(\epsilon^3 n)$. This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of $C_1$ is w.h.p. asymptotic to $(2/3)D(\epsilon,n)$, and the maximal distance in $C_1$ between any pair of kernel vertices is w.h.p. asymptotic to $(5/9)D(\epsilon,n)$.Comment: 25 pages; to appear in Combinatorics, Probability and Computin

Anomalous Scaling of the Penetration Depth in Nodal Superconductors

Recent findings of anomalous super-linear scaling of low temperature ($T$) penetration depth (PD) in several nodal superconductors near putative quantum critical points suggest that the low temperature PD can be a useful probe of quantum critical fluctuations in a superconductor. On the other hand, cuprates which are poster child nodal superconductors have not shown any such anomalous scaling of PD, despite growing evidence of quantum critical points. Then it is natural to ask when and how can quantum critical fluctuations cause anomalous scaling of PD? Carrying out the renormalization group calculation for the problem of two dimensional superconductors with point nodes, we show that quantum critical fluctuations associated with point group symmetry reduction result in non-universal logarithmic corrections to the $T$-dependence of the PD. The resulting apparent power law depends on the bare velocity anisotropy ratio. We then compare our results to data sets from two distinct nodal superconductors: YBa$_2$Cu$_3$O$_{6.95}$ and CeCoIn$_5$. Considering all symmetry-lowering possibilities of the point group of interest, $C_{4v}$, we find our results to be remarkably consistent with YBa$_2$Cu$_3$O$_{6.95}$ being near vertical nematic QCP, and CeCoIn$_5$ being near diagonal nematic QCP. Our results motivate search for diagonal nematic fluctuations in CeCoIn$_5$.Comment: Published version, 12 pages, 3 figures, 1 tabl

Time Fractional Poisson Equations: Representations and Estimates

In this paper, we study existence and uniqueness of strong as well as weak solutions for general time fractional Poisson equations. We show that there is an integral representation of the solutions of time fractional Poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator ${\cal L}$ and the corresponding subordinator associated with the time fractional derivative. This integral representation has an integral kernel $q(t, x, y)$, which we call the fundamental solution for the time fractional Poisson equation, if the semigrou for ${\cal L}$ has an integral kernel. We further show that $q(t, x, y)$ can be expressed as a time fractional derivative of the fundamental solution for the homogenous time fractional equation under the assumption that the associated subordinator admits a conjugate subordinator. Moreover, when the Laplace exponent of the associated subordinator satisfies the weak scaling property and its distribution is self-decomposable, we establish two-sided estimates for the fundamental solution $q(t,x, y)$ through explicit estimates of transition density functions of subordinators

Electrical transport in small bundles of single-walled carbon nanotubes: intertube interaction and effects of tube deformation

We report a combined electronic transport and structural characterization study of small carbon nanotube bundles in field-effect transistors (FET). The atomic structures of the bundles are determined by electron diffraction using an observation window built in the FET. The single-walled nanotube bundles exhibit electrical transport characteristics sensitively dependent on the structure of individual tubes, their arrangements in the bundle, deformation due to intertube interaction, and the orientation with respect to the gate electric field. Our ab-initio simulation shows that tube deformation in the bundle induces a bandgap opening in a metallic tube. These results show the importance of intertube interaction in electrical transport of bundled nanotubes

Anatomy of a young giant component in the random graph

We provide a complete description of the giant component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ as soon as it emerges from the scaling window, i.e., for $p = (1+\epsilon)/n$ where $\epsilon^3 n \to \infty$ and $\epsilon=o(1)$. Our description is particularly simple for $\epsilon = o(n^{-1/4})$, where the giant component $C_1$ is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for $C_1$). Let $Z$ be normal with mean $\frac23 \epsilon^3 n$ and variance $\epsilon^3 n$, and let $K$ be a random 3-regular graph on $2\lfloor Z\rfloor$ vertices. Replace each edge of $K$ by a path, where the path lengths are i.i.d. geometric with mean $1/\epsilon$. Finally, attach an independent Poisson($1-\epsilon$)-Galton-Watson tree to each vertex. A similar picture is obtained for larger $\epsilon=o(1)$, in which case the random 3-regular graph is replaced by a random graph with $N_k$ vertices of degree $k$ for $k\geq 3$, where $N_k$ has mean and variance of order $\epsilon^k n$. This description enables us to determine fundamental characteristics of the supercritical random graph. Namely, we can infer the asymptotics of the diameter of the giant component for any rate of decay of $\epsilon$, as well as the mixing time of the random walk on $C_1$.Comment: 42 page