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Two-dimensional (2D) solid-liquid transition (SLT)~\cite{Mermin1966Absence,Mermin1968Crystalline,Kosterlitz1972Long} concerns fundamental concepts of long-range correlations vital to magnetism, superconductivity, superfluidity, and topological matters. A long sought-after example is the melting of a Wigner Crystal (WC)~\cite{Wigner1934Interaction} of electrons. Detection efforts have targeted distinctive collective modes such as pinning by disorder, resonant-frequency absorption of Goldstone modes, and melting transition. However, only one-step second-order melting of softly-pinned modes was reported. Without the evidence of genuine pinning as exhibited in the charge density waves (CDWs)~\cite{PinningCDW}, these modes are likely intermediate phases which are only part of a complete SLT. To verify if there is a WC-intermediate phase transition will not only provide a solid proof of a WC, but will also unveil the nature of the SLT in relation to the two-stage Kosterlitz-Thouless (KT) model~\cite{Kosterlitz1972Long,Kosterlitz1973Ordering,Halperin1978Theory,Nelson1979Dislocation,hexatic_ceperley,hexatic_nelson, Young1979Melting}. %(or dislocation pairs) above a critical temperature $T_c$. Through transport studies of ultra-dilute high-purity 2D systems, this work presents evidence for not only a WC, but also for a two-stage WC-liquid SLT mediated by a first-order WC-intermediate phase transition.

In response to the growing need for theoretical tools that can be used in QCD to describe and understand the dynamics of gluons in hadrons in the Minkowski space-time, the renormalization group procedure for effective particles (RGPEP) is shown in the simplest available context of heavy quarkonia to exhibit a welcome degree of universality in the first approximation it yields once one assumes that beyond perturbation theory gluons obtain effective mass. Namely, in the second-order terms, the Coulomb potential with Breit-Fermi spin couplings in the effective quark-antiquark component of a heavy quarkonium, is corrected in one-flavor QCD by a spin-independent harmonic oscillator term that does not depend on the assumed effective gluon mass or the choice of the RGPEP generator. The new generator we use here is much simpler than the ones used before and has the advantage of being suitable for studies of the effective gluon dynamics at higher orders than the second and beyond the perturbative expansion., Comment: 8 pages, 2 figures, journal version

An instructive paradox concerning classical description of energy and momentum of extended physical systems in special relativity theory is explained using an elementary example of two point-like massive bodies rotating on a circle in their center-of-mass frame of reference, connected by an arbitrarily light and infinitesimally thin string. Namely, from the point of view of the inertial observers who move with respect to the rotating system, the sums of energies and momenta of the two bodies oscillate, instead of being constant in time. This result is understood in terms of the mechanism that binds the bodies: the string contributes to the system total energy and momentum no matter how light it is. Its contribution eliminates the unphysical oscillations from the system total four-momentum. Generality of the relativistic approach, applied here to the rotor example, suggests that in every extended physical system its binding mechanism contributes to its total energy and momentum., Comment: 33 pages, 6 figures, to be published in American Journal of Physics http://aapt.scitation.org/journal/ajp , added one reference

We test the renormalization group procedure for effective particles (RGPEP) on a model of fermion-scalar interaction based on the Yukawa theory. The model is obtained by truncating the Yukawa theory to just two Fock sectors in the Dirac front form of Hamiltonian dynamics, a fermion, and a fermion and a boson, for the purpose of simple analytic calculation that exhibits steps of the procedure., Comment: 7 pages

Schr\"odinger equation with potential $-g/r^2$ exhibits a limit cycle, described in the literature in a broad range of contexts using various regularizations of the singularity at $r=0$. Instead, we use the renormalization group transformation based on Gaussian elimination, from the Hamiltonian eigenvalue problem, of high momentum modes above a finite, floating cutoff scale. The procedure identifies a richer structure than the one we found in the literature. Namely, it directly yields an equation that determines the renormalized Hamiltonians as functions of the floating cutoff: solutions to this equation exhibit, in addition to the limit-cycle, also the asymptotic-freedom, triviality, and fixed-point behaviors, the latter in vicinity of infinitely many separate pairs of fixed points in different partial waves for different values of $g$., Comment: 6 pages, 3 figures

We provide sharp two-sided estimates on the Dirichlet heat kernel $k_1(t,x,y)$ for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel $k_1(t,x,y)$ in terms of its global two-sided sharp estimates., Comment: 14 pages, Second version changes: corrected typos

Determining the equilibrium configuration and shape of curved two-dimensional films with (generalized) liquid crystalline (LC) order is a difficult infinite dimensional problem of direct relevance to the study of generalized polymersomes, soft matter and the fascinating problem of understanding the origin and formation of shape (morphogenesis). The symmetry of the free energy of the LC film being considered and the topology of the surface to be determined often requires that the equilibrium configuration possesses singular structures in the form of topological defects such as disclinations for nematic films. The precise number and type of defect plays a fundamental role in restricting the space of possible equilibrium shapes. Flexible closed vesicles with spherical topology and nematic or smectic order, for example, inevitably possess four elementary strength $+1/2$ disclination defects positioned at the four vertices of a tetrahedral shell. Here we address the problem of determining the equilibrium shape of flexible vesicles with generalized LC order. The order parameter in these cases is an element of $S^1/Z_p$, for any positive integer $p$. We will focus on the case $p =3$, known as triatic LCs. We construct the appropriate order parameter for triatics and find the associated free energy. We then describe the structure of the elementary defects of strength $+1/3$ in flat space. Finally, we prove that sufficiently floppy triatic vesicles with the topology of the 2-sphere equilibrate to octahedral shells with strength $+1/3$ defects at each of the six vertices, independently of scale., Comment: New results and new sections added, 4 new figures and updated abstract

We show that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression then for every $N\ge 2$ there exist (unilateral or bilateral) subshifts $\Sigma$ over $N$ symbols, with entropy arbitrarily close to $\log N$, uncorrelated to $y$. In particular, for $y=\mu$ being the M\"obius function, we get that there exist subshifts as above which satisfy the assertion of Sarnak's conjecture. The existence of positive entropy systems uncorrelated to the M\"obius function is claimed in Sarnak's survey \cite{sarnak} (and attributed to Bourgain), however, to our knowledge no examples have ever been published. We fill in this gap and by the way we show that this has nothing to do with more advanced algebraic properties (for instance multiplicativity) of the considered sequence., Comment: 10 pages

Semantic mapping is the incremental process of "mapping" relevant information of the world (i.e., spatial information, temporal events, agents and actions) to a formal description supported by a reasoning engine. Current research focuses on learning the semantic of environments based on their spatial location, geometry and appearance. Many methods to tackle this problem have been proposed, but the lack of a uniform representation, as well as standard benchmarking suites, prevents their direct comparison. In this paper, we propose a standardization in the representation of semantic maps, by defining an easily extensible formalism to be used on top of metric maps of the environments. Based on this, we describe the procedure to build a dataset (based on real sensor data) for benchmarking semantic mapping techniques, also hypothesizing some possible evaluation metrics. Nevertheless, by providing a tool for the construction of a semantic map ground truth, we aim at the contribution of the scientific community in acquiring data for populating the dataset.

How the two dimensional (2D) quantum Wigner crystal (WC) transforms into the metallic liquid phase remains to be an outstanding problem in physics. In theories considering the 2D WC to liquid transition in the clean limit, it was suggested that a number of intermediate phases might exist. We have studied the transformation between the metallic fluid phase and the low magnetic field reentrant insulating phase (RIP) which was interpreted as due to WC formation [Qiu et al, PRL 108, 106404 (2012)], in a strongly correlated 2D hole system with large interaction parameter $r_s$ ($\sim~$20-30) and high mobility. Instead of a sharp transition, we found that increasing density (or lowering $r_s$) drives the RIP into a state where the incipient RIP coexists with Fermi liquid. This apparent mixture phase intermediate between Fermi liquid and WC also exhibits a non-trivial temperature dependent resistivity behavior which may be qualitatively understood by the reversed melting of WC in the mixture, in analogy to the Pomeranchuk effect in the solid-liquid mixture of Helium-3.

In this paper, we provide two-sided estimates and uniform asymptotics for the solution of $d$-dimensional critical fractal Burgers equation $u_t-\Delta^{\alpha/2}u+b\cdot \nabla\left(u|u|^q\right)=0$, $\alpha\in(1,2)$, $b\in\mathbb R^d$ for $q = (\alpha-1)/d$ and $u_0 \in L^1(\mathbb R^d)$. We consider also $q > (\alpha-1)/d$ under additional condition $u_0 \in L^\infty(\mathbb R^d)$. In both cases we assume $u_0\geq0$, which implies that the solution is non-negative. The estimates are given in the terms of the function $P_t u_0$, where $P_t$ is the stable semigroup operator., Comment: 19 pages

In this paper, we provide two-sided estimates for the source solution of $d$-dimensional critical fractal Burgers equation $u_t-\Delta^{\alpha/2}+b\cdot \nabla\left(u|u|^q\right)=0$, $q=(\alpha-1)/d$, $\alpha\in(1,2)$, $b\in\mathbb{R}^d$, by the density function of the isotropic $\alpha$-stable process., Comment: 11pages

We examine the density functions of the first exit times of the Bessel process from the intervals [0,1) and (0,1). First, we express them by means of the transition density function of the killed process. Using that relationship we provide precise estimates and asymptotics of the exit time densities. In particular, the results hold for the first exit time of the n-dimensional Brownian motion from a ball.

The inability to see and quantify systemic financial risk comes at an immense social cost. Systemic risk in the financial system arises to a large extent as a consequence of the interconnectedness of its institutions, which are linked through networks of different types of financial contracts, such as credit, derivatives, foreign exchange and securities. The interplay of the various exposure networks can be represented as layers in a financial multi-layer network. In this work we quantify the daily contributions to systemic risk from four layers of the Mexican banking system from 2007-2013. We show that focusing on a single layer underestimates the total systemic risk by up to 90%. By assigning systemic risk levels to individual banks we study the systemic risk profile of the Mexican banking system on all market layers. This profile can be used to quantify systemic risk on a national level in terms of nation-wide expected systemic losses. We show that market-based systemic risk indicators systematically underestimate expected systemic losses. We find that expected systemic losses are up to a factor four higher now than before the financial crisis of 2007-2008. We find that systemic risk contributions of individual transactions can be up to a factor of thousand higher than the corresponding credit risk, which creates huge risks for the public. We find an intriguing non-linear effect whereby the sum of systemic risk of all layers underestimates the total risk. The method presented here is the first objective data driven quantification of systemic risk on national scales that reveal its true levels., Comment: 15 pages, 6 figures

In this paper we study the local regularity of fractional integrals of Fourier series using several definitions of the H\"older exponent. We especially consider series coming from fractional integrals of modular forms. Our results show that in general cusp forms give rise to pure fractals (as opposed to multifractals). We include explicit examples and computer plots., Comment: 18 pages, 6 pdf figures, uses .bbl file

We study a recursively defined sequence which is constructed using the least common multiple. It has been conjectured that every term of that sequence is $1$ or a prime. In this paper we show that this claim is connected to a strong version of Linnik's Theorem, which is yet unproved. We also study a generalization on which composite numbers may appear depending on the initial term., Comment: 16 pages, 1 figure; multiple typos added, modified title, many other small corrections

The main subject of the paper, motivated by a question raised by Boshernitzan, is to give criteria for a bounded complex-valued sequence to be uncorrelated to any strictly ergodic sequence. As a tool developed to study this problem we introduce the notion of correlation between two shift-invariant measures supported by the symbolic space with complex symbols. We also prove a "lifting lemma" for generic points: given a joining $\xi$ of two shift-invariant measures $\mu$ and $\nu$, every point $x$ generic for $\mu$ lifts to a pair $(x,y)$ generic for $\xi$ (such $y$ exists in the full symbolic space). This lemma allows us to translate correlation between bounded sequences to the language of correlation of measures. Finally, to establish that the property of an invariant measure being uncorrelated to any ergodic measure is essentially weaker than the property of being disjoint from any ergodic measure, we develop and apply criteria for ergodicity of four-jump cocycles over irrational rotations. We believe that apart from the applications to studying the notion of correlation, the two developed tools: the lifting lemma and the criteria for ergodicity of four-jump cocycles, are of independent interest. This is why we announce them also in the title. In the Appendix we also introduce the notion of conditional disjointness., Comment: 21 pages

We provide sharp two-sided estimates of the Fourier-Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0,1) killed at 1 and killed or reflected at 0., Comment: 11 pages

Consider the $\lambda$-Green function and the $\lambda$-Poisson kernel of a Lipschitz domain $U\subset \mathbb H^n=\left\{x\in\mathbb R^n:x_n>0\right\}$ for hyperbolic Brownian motion with drift. We provide several relationships that facilitate studying those objects and explain somehow theirs nature. As an application, we yield uniform estimates in case of sets of the form $S_{a,b}=\{x\in\mathbb H^n:x_n>a, x_1\in(0,b)\}$, $a,b>0$, $a,b>0$, which covers and extends existing results of that kind., Comment: 16 pages

Let (B^{(1)}_t ;B^{(2)}_t ;B^{(3)}_t + \mu t) be a three-dimensional Brownian motion with drift \mu, starting at the origin. Then X_t = ||(B^{(1)}_t ;B^{(2)}_t ;B^{(3)}_t +\mu t)||, its distance from the starting point, is a diffusion with many applications. We investigate the distribution of the supremum of (X_t), give an infinite-series formula for its density and an exact estimate by elementary functions.

The work in this article is concerned with two different types of families of finite sets: separating families and splitting families (they are also called "systems"). These families have applications in combinatorial search, coding theory, cryptography, and related fields. We define and study generalizations of these two notions, which we have named $n$-separating families and $n$-splitting families. For each of these new notions, we outline their basic properties and connections with the well-studied notions. We then spend the greatest effort obtaining lower and upper bounds on the minimal size of the families. For $n$-separating families we obtain bounds which are asymptotically tight within a linear factor. For $n$-splitting families this appears to be much harder; we provide partial results and open questions.

We propose Mecke-Palm formulas for multiple integrals with respect to a Poisson random measure interlaced with its intensity measure. We apply such formulas to multiple mixed L\'evy systems of L\'evy processes and obtain moment formulas for products of interlaced multiple Poisson integrals., Comment: added example on p. 24-25

We describe a proof of the Central Limit Theorem that has been formally verified in the Isabelle proof assistant. Our formalization builds upon and extends Isabelle's libraries for analysis and measure-theoretic probability. The proof of the theorem uses characteristic functions, which are a kind of Fourier transform, to demonstrate that, under suitable hypotheses, sums of random variables converge weakly to the standard normal distribution. We also discuss the libraries and infrastructure that supported the formalization, and reflect on some of the lessons we have learned from the effort.