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We investigate a model of $R^2$-gravity with a non-minimally coupled scalar field that gives rise to two-stage inflation with a break, that is, with an intermediate stage where inflation momentarily halts. We find that the power spectrum of the primordial curvature perturbation is significantly enhanced at the break scale, which can account for the primordial black hole (PBH) formation, without affecting the CMB constraint on large scales. The behavior of the curvature perturbation is carefully analyzed and we find a few notable new features in the spectrum. In particular, we find that the $k^3$ growth of the spectrum of toward the end of the first stage of inflation. We argue that this is a universal feature common to all two-stage models where the field dominating the second stage is heavy during the first stage. By appropriately tuning the model parameters, we find that our model can realize the scenario of PBHs as the cold dark matter of the Universe. We also find that we can choose the parameters so that the spectrum of the induced gravitational waves from the enhanced curvature perturbation fits the NANOGrav-15yr data of pulsar timing array observation., Comment: 53 pages, 14 figures, 1 table

It has been recently established that a nonlinear valley current could be generated through traveling a surface acoustic wave (SAW) in two-dimensional Dirac materials. So far, the SAW-driven valley currents have been attributed to warping Fermi surface or Berry phase effect. Here, we demonstrate that tilt mechanism can also lead to a nonlinear valley Hall current (VHC) when propagating SAW in materials with the tilted Dirac cone placed on a piezoelectric substrate. It's found that the nonlinear VHC exhibits a $\sin\theta$ dependence on the orientation of tilt with respect to SAW. In addition, this tilt-induced nonlinear acoustic VHC shows independence on the relaxation time, distinguishing from the contributions from the Berry phase or trigonal warping. Remarkably, the magnitude of the nonlinear acoustic VHC from tilt mechanism in the uniaxially strained graphene is two orders larger than those reported in MoS$_2$ stemmed from the Berry phase effect and the warping effect., Comment: 6 pages, 2 figures, Accepted by PRB letter

In this paper, we propose an empirical model for the VIX index. Our findings indicate that the VIX has a long-term empirical distribution. To model its dynamics, we utilize a continuous-time Markov process with a uniform distribution as its invariant distribution and a suitable function $h$. We determined that $h$ is the inverse function of the VIX data's empirical distribution. Additionally, we use the method of variables of separation to get the exact solution to the pricing problem for VIX futures and call options.

In this paper, we study precise deviations including precise large deviations and moderate deviations for discrete Hawkes processes for large time asymptotics by using mod-$\phi$ convergence theory.

A new nonlinear magnetothermal effect, namely unidirectional Seebeck effect(USE), has been recently reported in magnetic/nonmagnetic topological insulator (TI) heterostructure and is ascribed to the asymmetry magnon scattering. Here, we show a new mechanism to generate USE in the Rashba two dimensional electron gas (2DEG), in which the magnetism or magnetic order is completely absent. It's found that the USE has a quantum origin from the spin-momentum locking generated from Rashba spin-orbit coupling. We find that the USE exhibits a sine dependence on the orientation of the magnetic field with respect to the direction of the temperature gradient and is dominant from the states above the Lifshitz point. The USE in the semiconductor Ge(111) subsurface states has been theoretically and systematically investigated., Comment: 9 pages, 4 figures

As a candidate bound for the Binary Black Hole (BBH) merger events detected by LIGO/Virgo, Primordial Black Holes (PBHs) provide a useful tool to investigate the primordial curvature perturbations on small scales. Using the GWTC-1 to GWTC-3 catalogs, under the scenario that PBHs originate from large primordial curvature perturbations on small scales during inflationary epoch, we for the first time reconstruct the power spectrum of primordial curvature perturbations on small scales. It is found that the value of the amplitude of the primordial power spectrum is enhanced to $\mathcal{O}(10^{-2})$ on scales $\mathcal{O}(1)$ pc. This may imply the validity of PBH as a possible BBH merger candidate., Comment: 7 pages,5 figures

Most current affect scales and sentiment analysis on written text focus on quantifying valence (sentiment) -- the most primary dimension of emotion. However, emotions are broader and more complex than valence. Distinguishing negative emotions of similar valence could be important in contexts such as mental health. This project proposes a semi-supervised machine learning model (DASentimental) to extract depression, anxiety and stress from written text. First, we trained the model to spot how sequences of recalled emotion words by $N=200$ individuals correlated with their responses to the Depression Anxiety Stress Scale (DASS-21). Within the framework of cognitive network science, we model every list of recalled emotions as a walk over a networked mental representation of semantic memory, with emotions connected according to free associations in people's memory. Among several tested machine learning approaches, we find that a multilayer perceptron neural network trained on word sequences and semantic network distances can achieve state-of-art, cross-validated predictions for depression ($R = 0.7$), anxiety ($R = 0.44$) and stress ($R = 0.52$). Though limited by sample size, this first-of-its-kind approach enables quantitative explorations of key semantic dimensions behind DAS levels. We find that semantic distances between recalled emotions and the dyad "sad-happy" are crucial features for estimating depression levels but are less important for anxiety and stress. We also find that semantic distance of recalls from "fear" can boost the prediction of anxiety but it becomes redundant when the "sad-happy" dyad is considered. Adopting DASentimental as a semi-supervised learning tool to estimate DAS in text, we apply it to a dataset of 142 suicide notes. We conclude by discussing key directions for future research enabled by artificial intelligence detecting stress, anxiety and depression., Comment: 28 pages, 2 figures and 2 tables

Primordial black holes (PBHs) from the early Universe constitute an attractive dark matter candidate. First detections of black hole-neutron star (BH-NS) candidate gravitational wave events by the LIGO/Virgo collaboration, GW200105 and GW200115, already prompted speculations about non-astrophysical origin. We analyze, for the first time, the total volumetric merger rates of PBH-NS binaries formed via two-body gravitational scattering, finding them to be subdominant to the astrophysical BH-NS rates. In contrast to binary black holes, a significant fraction of which can be of primordial origin, either formed in dark matter halos or in the early Universe, PBH-NS rates cannot be significantly enhanced by contributions preceding star formation. Our findings imply that the identified BH-NS events are of astrophysical origin, even when PBH-PBH events significantly contribute to the GW observations., Comment: 7 pages, 2 figures, published in APJ

In the present paper, the Karhunen-Lo{\`e}ve eigenvalues for a sub-fractional Brownian motion are considered in the case of $H>\frac12$. Rigorous large $n$ asymptotics for those eigenvalues are shown, based on functional analysis method. By virtue of these asymptotics, along with some standard large deviations results, asymptotically estimates for the closely related problem of small $L^2$-ball probabilities for a sub-fractional Brownian motion are derived. By the way, asymptotic analysis on the Karhunen-Lo{\`e}ve eigenvalues for the corresponding "derivative" process is also established.

Vehicle color recognition plays an important role in intelligent traffic management and criminal investigation assistance. However, the current vehicle color recognition research involves at most 13 types of colors and the recognition accuracy is low, which is difficult to meet practical applications. To this end, this paper has built a benchmark dataset (Vehicle Color-24) that includes 24 types of vehicle colors, including 10091 vehicle pictures taken from 100 hours of urban road surveillance videos. In addition, in order to solve the problem of long tail distribution in Vehicle Color-24 dataset and low recognition rate of existing methods, this paper proposes a Smooth Modulated Neural Network with Multi-layer Feature Representation (SMNN-MFR) is used for 24 types of vehicle color recognition. SMNN-MFR includes four parts: feature extraction, multi-scale feature fusion, suggestion frame generation and smooth modulation. The model is trained and verified on the Vehicle Color-24 benchmark dataset. Comprehensive experiments show that the average recognition accuracy of the algorithm in the 24 categories of color benchmark databases is 94.96%, which is 33.47% higher than the Faster RCNN network. In addition, the average accuracy rate of the model when recognizing 8 types of colors is 97.25%, and the detection accuracy of algorithms in similar databases is improved. At the same time, visualization and ablation experiments also proved the rationality of our network settings and the effectiveness of each module. The code and database are published at: https://github.com/mendy-2013., Comment: in Chinese language

The properties of primordial curvature perturbations on small scales are still unknown while those on large scales have been well probed by the observations of the cosmic microwave background anisotropies and the large scale structure. In this paper, we propose the reconstruction method of primordial curvature perturbations on small scales through the merger rate of binary primordial black holes, which could form from large primordial curvature perturbation on small scales., Comment: 13 pages, 2 figures

In pioneering works of Meyer and of McMullen in the early 1970s, the set of Minkowski summands of a polytope was shown to be a polyhedral cone called the type cone. Explicit computations of type cones are in general intractable. Nevertheless, we show that the type cone of the product of simplices is the cone over a simplex. This remarkably simple result derives from insights about rainbow point configurations and the work of McMullen., Comment: 12 pages, 4 figures. Incorporated referee suggestions, minor edits throughout. This version focuses more on type cones of products of simplices than the previous version. Introduction has been partially rewritten to reflect this change. To appear in the Bulletin of the London Mathematical Society

In this paper, we study the affine Springer fiber $\mathcal{F} l_N$ in type $A$ for rectangular type semisimple nil-element $N$ and calculate the relative position between irreducible components. In particular, we use the affine matrix ball construction to show the relative position map is compatible with the Kazhdan-Lusztig cell structure, generalizing the work of Steinberg and van Leeuwen.

The interplay between topology and crystalline symmetries in materials can lead to a variety of topological crystalline insulator (TCI) states. Despite significant effort towards their experimental realization, so far only Pb$_{1-x}$Sn$_x$Te has been confirmed as a mirror-symmetry protected TCI. Here, based on first-principles calculations combined with a symmetry analysis, we identify a rotational-symmetry protected TCI state in the transition-metal dipnictide RX$_2$ family, where R = Ta or Nb and X = P, As, or Sb. Taking TaAs$_2$ as an exemplar system, we show that its low-energy band structure consists of two types of bulk nodal lines in the absence of spin-orbit coupling (SOC) effects. Turning on the SOC opens a continuous bandgap in the energy spectrum and drives the system into a $C_2T$-symmetry-protected TCI state. On the (010) surface, we show the presence of rotational-symmetry-protected nontrivial Dirac cone states within a local bulk energy gap of $\sim$ 300 meV. Interestingly, the Dirac cones have tilted energy dispersion, realizing a type-II Dirac fermion state in a topological crystalline insulator. Our results thus indicate that the TaAs$_2$ materials family provides an ideal setting for exploring the unique physics associated with type-II Dirac fermions in rotational-symmetry-protected TCIs., Comment: 7 pages, 7 figures, Accepted for publication in Physical Review B (2019)

Two Schubert problems on possibly different Grassmannians may be composed to obtain a Schubert problem on a larger Grassmannian whose number of solutions is the product of the numbers of the original problems. This generalizes a construction discovered while classifying Schubert problems on the Grassmannian of 4-planes in C^9 with imprimitive Galois groups. We give an algebraic proof of the product formula. In a number of cases, we show that the Galois group of the composed Schubert problem is a subgroup of a wreath product of the Galois groups of the original problems, and is therefore imprimitive. We also present evidence for a conjecture that all composed Schubert problems have imprimitive Galois groups., Comment: 24 pages

We study the primordial tensor perturbation produced from the double inflationary scenario with an intermediate break stage. Because of the transitions, the power spectrum deviates from the vacuum one and there will appear oscillatory behavior. In the case of a scalar-type curvature perturbation, it is known that the amplitude of these oscillations may be enhanced to result in the power spectrum larger than the one for the vacuum case. One might expect the similar enhancement for the tensor perturbation as well. Unfortunately, it is found that when the equation of state (EOS) parameter $w=p/\rho$ of the break stage is a constant with $w>-1/3$, the amplitude of oscillations is never large enough to enhance the power spectrum. On the contrary, the power spectrum is found to be suppressed even on those scales that leave the horizon at the first inflationary stage and remain superhorizon throughout the entire stage. We identify the cause of this suppression with the correction terms in additional to the leading order constant solution on superhorizon scales. We argue that our result is general in the sense that any intermediate break stage during inflation cannot yield an enhancement of the tensor spectrum as long as the Hubble expansion rate is non-increasing in time., Comment: 20 pages, 4 figures, Latex (version 2: one paragraph added in the conclusion section and two references added)

Stembridge introduced the notion of stability for Kronecker triples which generalize Murnaghan's classical stability result for Kronecker coefficients. Sam and Snowden proved a conjecture of Stembridge concerning stable Kronecker triple, and they also showed an analogous result for Littlewood--Richardson coefficients. Heisenberg coefficients are Schur structure constants of the Heisenberg product which generalize both Littlewood--Richardson coefficients and Kronecker coefficients. We show that any stable triple for Kronecker coefficients or Littlewood--Richardson coefficients also stabilizes Heisenberg coefficients, and we classify the triples stabilizing Heisenberg coefficients. We also follow Vallejo's idea of using matrix additivity to generate Heisenberg stable triples., Comment: 13 pages

In generalized Proca theories with vector-field derivative couplings, a bunch of hairy black hole solutions have been derived on a static and spherically symmetric background. In this paper, we formulate the odd-parity black hole perturbations in generalized Proca theories by expanding the corresponding action up to second order and investigate whether or not black holes with vector hair suffer ghost or Laplacian instabilities. We show that the models with cubic couplings $G_3(X)$, where $X=-A_{\mu}A^{\mu}/2$ with a vector field $A_{\mu}$, do not provide any additional stability condition as in General Relativity. On the other hand, the exact charged stealth Schwarzschild solution with a nonvanishing longitudinal vector component $A_1$, which originates from the coupling to the Einstein tensor $G^{\mu\nu}A_\mu A_\nu$ equivalent to the quartic coupling $G_4(X)$ containing a linear function of $X$, is unstable in the vicinity of the event horizon. The same instability problem also persists for hairy black holes arising from general quartic power-law couplings $G_4(X) \supset \beta_4 X^n$ with the nonvanishing $A_1$, while the other branch with $A_1=0$ can be consistent with conditions for the absence of ghost and Laplacian instabilities. We also discuss the case of other exact and numerical black hole solutions associated with intrinsic vector-field derivative couplings and show that there exists a wide range of parameter spaces in which the solutions suffer neither ghost nor Laplacian instabilities against odd-parity perturbations., Comment: 26 pages, 4 figures, published version

We study a model of inflation in which a scalar field $\chi$ is non-minimally coupled to Starobinsky's $R^2$ gravity. After transforming it to the Einstein frame, a new scalar field, the scalaron $\phi$, will appear and couple to $\chi$ with a nontrivial field metric, while $\chi$ acquires a positive mass via the non-minimal coupling. Initially inflation occurs along the $\phi$ direction with $\chi$ trapped near its origin by this induced mass. After $\phi$ crosses a critical value, it starts rolling down rapidly and proceeds todamped oscillations around an effective local minimum determined by the value of $\chi$, while inflation still continues, driven by the $\chi$ field at this second stage where the effect of the non-minimal coupling becomes negligible. The presence of the damped oscillations during the transition from the first to second stage of inflation causes enhancement and oscillation features in the power spectrum of the curvature perturbation. Assuming that the oscillations may be treated perturbatively, we calculate these features by using the $\delta N$ formalism, and discuss its observational implications to large scale CMB anomalies or primordial black hole formation, depending on the scale of the features., Comment: 25 pages, 6 figures, published version

The Heisenberg product is an associative product defined on symmetric functions which interpolates between the usual product and the Kronecker product. In 1938, Murnaghan discovered that the Kronecker product of two Schur functions stabilizes. We prove an analogous result for the Heisenberg product of Schur functions., Comment: 16 pages

Many models of inflation driven by vector fields alone have been known to be plagued by pathological behaviors, namely ghost and/or gradient instabilities. In this work, we seek a new class of vector-driven inflationary models that evade all of the mentioned instabilities. We build our analysis on the Generalized Proca Theory with an extension to three vector fields to realize isotropic expansion. We obtain the conditions required for quasi de-Sitter solutions to be an attractor analogous to the standard slow-roll one and those for their stability at the level of linearized perturbations. Identifying the remedy to the existing unstable models, we provide a simple example and explicitly show its stability. This significantly broadens our knowledge on vector inflationary scenarios, reviving potential phenomenological interests for this class of models., Comment: 16 pages plus references; v2 matches JCAP published version, 17 pages plus references, Section IV.E added on attractor condition against background anisotropy

We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of degree eight whose ruling forms a genus one curve. We classify which curves arise, classify the face lattices of the convex hulls, and determine which are spectrahedra. We also discuss an approach to these convex hulls using projective duality., Comment: 20 pages, many color .eps pictures

We consider the finite interactions of the generalized Proca theory including the sixth-order Lagrangian and derive the full linear perturbation equations of motion on the flat Friedmann-Lema\^{i}tre-Robertson-Walker background in the presence of a matter perfect fluid. By construction, the propagating degrees of freedom (besides the matter perfect fluid) are two transverse vector perturbations, one longitudinal scalar, and two tensor polarizations. The Lagrangians associated with intrinsic vector modes neither affect the background equations of motion nor the second-order action of tensor perturbations, but they do give rise to non-trivial modifications to the no-ghost condition of vector perturbations and to the propagation speeds of vector and scalar perturbations. We derive the effective gravitational coupling $G_{\rm eff}$ with matter density perturbations under a quasi-static approximation on scales deep inside the sound horizon. We find that the existence of intrinsic vector modes allows a possibility for reducing $G_{\rm eff}$. In fact, within the parameter space, $G_{\rm eff}$ can be even smaller than the Newton gravitational constant $G$ at the late cosmological epoch, with a peculiar phantom dark energy equation of state (without ghosts). The modifications to the slip parameter $\eta$ and the evolution of growth rate $f\sigma_8$ are discussed as well. Thus, dark energy models in the framework of generalized Proca theories can be observationally distinguished from the $\Lambda$CDM model according to both cosmic growth and expansion history. Furthermore, we study the evolution of vector perturbations and show that outside the vector sound horizon the perturbations are nearly frozen and start to decay with oscillations after the horizon entry., Comment: 23 pages, 3 figures

We consider a massive vector field with derivative interactions that propagates only the 3 desired polarizations (besides two tensor polarizations from gravity) with second-order equations of motion in curved space-time. The cosmological implications of such generalized Proca theories are investigated for both the background and the linear perturbation by taking into account the Lagrangian up to quintic order. In the presence of a matter fluid with a temporal component of the vector field, we derive the background equations of motion and show the existence of de Sitter solutions relevant to the late-time cosmic acceleration. We also obtain conditions for the absence of ghosts and Laplacian instabilities of tensor, vector, and scalar perturbations in the small-scale limit. Our results are applied to concrete examples of the general functions in the theory, which encompass vector Galileons as a specific case. In such examples, we show that the de Sitter fixed point is always a stable attractor and study viable parameter spaces in which the no-ghost and stability conditions are satisfied during the cosmic expansion history., Comment: 23 pages, 7 figures, revised version

For a massive vector field with derivative self-interactions, the breaking of the gauge invariance allows the propagation of a longitudinal mode in addition to the two transverse modes. We consider generalized Proca theories with second-order equations of motion in a curved space-time and study how the longitudinal scalar mode of the vector field gravitates on a spherically symmetric background. We show explicitly that cubic-order self-interactions lead to the suppression of the longitudinal mode through the Vainshtein mechanism. Provided that the dimensionless coupling of the interaction is not negligible, this screening mechanism is sufficiently efficient to give rise to tiny corrections to gravitational potentials consistent with solar-system tests of gravity. We also study the quartic interactions with the presence of non-minimal derivative coupling with the Ricci scalar and find the existence of solutions where the longitudinal mode completely vanishes. Finally, we discuss the case in which the effect of the quartic interactions dominates over the cubic one and show that local gravity constraints can be satisfied under a mild bound on the parameters of the theory., Comment: 20 pages, 3 figures, revised version

We investigate the nonlocal gravity theory by deriving nonlocal equations of motion using the traditional variation principle in a homogeneous background. We focus on a class of models with a linear nonlocal modification term in the action. It is found that the resulting equations of motion contain the advanced Green's function, implying that there is an acausality problem. As a consequence, a divergence arises in the solutions due to contributions from the future infinity unless the Universe will go back to the radiation dominated era or become the Minkowski spacetime in the future. We also discuss the relation between the original nonlocal equations and its biscalar-tensor representation and identify the auxiliary fields with the corresponding original nonlocal terms. Finally, we show that the acusality problem cannot be avoided by any function of nonlocal terms in the action., Comment: 18 pages, no figure, comments and references added

We study homogeneous gravitational instantons, conventionally called the Hawking-Moss (HM) instantons, in bigravity theory. The HM instantons describe the amplitude of quantum tunneling from a false vacuum to the true vacuum. Corrections to General Relativity (GR) are found in a closed form. Using the result, we discuss the following two issues: reduction to the de Rham-Gabadadze-Tolley (dRGT) massive gravity and the possibility of preference for a large $e$-folding number in the context of the Hartle-Hawking (HH) no-boundary proposal. In particular, concerning the dRGT limit, it is found that the tunneling through the so-called self-accelerating branch is exponentially suppressed relative to the normal branch, and the probability becomes zero in the dRGT limit. As far as HM instantons are concerned, this could imply that the reduction from bigravity to the dRGT massive gravity is ill-defined., Comment: 20 pages, 2 figures, comments and references added

We consider two toy models of open inflation and investigate their ability to give a suppression of scalar power on large scales whilst also satisfying observational constraints on the spatial curvature of the universe. Qualitatively we find that both models are indeed capable of fulfilling these two requirements, but we also see that effects of the quantum tunnelling must be carefully taken into account if we wish to make quantitative predictions., Comment: 11 pages, 13 figures. Updated to match published version: figures 7 and 11 updated, references added, typos corrected

We study the Coleman-de Luccia (CDL) instanton characterizing the tunneling from a false vacuum to the true vacuum in a semi-classical way in dRGT (deRham-Gabadadze-Tolley) massive gravity theory, and evaluate the dependence of the tunneling rate on the model parameters. It is found that provided with the same physical Hubble parameters for the true vacuum $H_{\rm T}$ and the false vacuum $H_{\rm F}$ as in General Relativity (GR), the thin-wall approximation method implies the same tunneling rate as GR. However, deviations of tunneling rate from GR arise when one goes beyond the thin-wall approximation and they change monotonically until the Hawking-Moss (HM) case. Moreover, under the thin-wall approximation, the HM process may dominate over the CDL one if the value for the graviton mass is larger than the inverse of the radius of the bubble., Comment: v2, references and one comment added, 19 pages, 5 figures

Thermal decomposition behaviors of TiH2 powder under a flowing helium atmosphere and in a low vacuum condition have been studied by using in-situ EXAFS technique. By an EXAFS analysis containing the multiple scattering paths including H atoms, the changes of hydrogen stoichiometric ratio and the phase transformation sequence are obtained. The results demonstrate that the initial decomposition temperature is dependent on experimental conditions, which occurs, respectively, at about 300 and 400 degree in a low vacuum condition and under a flowing helium atmosphere. During the decomposition process of TiH2 in a low vacuum condition, the sample experiences a phase change process: {\delta}(TiH2) - {\delta}(TiHx) - {\delta}(TiHx)+{\beta}(TiHx) - {\delta}(TiHx)+{\beta}(TiHx)+{\alpha}(Ti) - {\beta}(TiHx)+{\alpha}(Ti) - {\alpha}(Ti)+{\beta}(Ti). This study offers a way to detect the structural information of hydrogen. A detailed discussion about the decomposition process of TiH2 is given in this paper., Comment: 9 pages, 7 figures

It is known that the no-boundary proposal in the traditional Einstein gravity does not prefer inflation, that is, the probability of realizing a large number of e-folds is exponentially suppressed. This situation may be changed drastically in a class of nonlinear massive gravity theories recently proposed by deRham, Gabadadze and Tolley, called dRGT massive gravity. We show that the contribution from the massive gravity sector can enhance the probability of a large number of e-folds substantially for a sufficiently large mass parameter mg comparable to the Hubble parameter during inflation, say $m_g \gtrsim 10^{12}$ GeV. We illustrate possible models to trigger such a large mass parameter in the early universe while it is negligibly small in the present universe. This opens a new window to explore the inflationary scenario in the context of quantum cosmology., Comment: 5 pages, 1 figure

A nonlocal gravity model which does not assume the existence of a new dimensional parameter in the action and includes a function $f(\Box^{-1} R)$, with $\Box$ the d'Alembertian operator, is studied. By specifying an exponential form for the function f and including a matter sector with a constant equation of state parameter, all available power-law solutions in the Jordan frame are obtained. New power-law solutions in the Einstein frame are also probed. Furthermore, the relationship between power-law solutions in both frames, established through conformal transformation, is substantially clarified. The correspondence between power-law solutions in these two frames is proven to be a very useful tool in order to obtain new solutions in the Einstein frame., Comment: 35 pages, uses jcappub.sty, v2: Sections 3.6 and 4 added, references added, accepted for publication in JCAP

As a first step toward understanding a lanscape of vacua in a theory of non-linear massive gravity, we consider a landscape of a single scalar field and study tunneling between a pair of adjacent vacua. We study the Hawking-Moss (HM) instanton that sits at a local maximum of the potential, and evaluate the dependence of the tunneling rate on the parameters of the theory. It is found that provided with the same physical HM Hubble parameter $H_{HM}$, depending on the values of parameters $\alpha_3$ and $\alpha_4$ in the action, the corresponding tunneling rate can be either enhanced or suppressed when compared to the one in the context of General Relativity (GR). Furthermore, we find the constraint on the ratio of the physical Hubble parameter to the fiducial one, which constrains the form of potential. This result is in sharp contrast to GR where there is no bound on the minimum value of the potential., Comment: v2, 14 pages, 3 figures, comments and references added

We consider a model of non-local gravity with a large bare cosmological constant, $\Lambda$, and study its cosmological solutions. The model is characterized by a function $f(\psi)=f_0 e^{\alpha\psi}$ where $\psi=\Box^{-1}R$ and $\alpha$ is a real dimensionless parameter. In the absence of matter, we find an expanding universe solution $a\propto t^n$ with $n<1$, that is, a universe with decelarated expansion without any fine-tuning of the parameter. Thus the effect of the cosmological constant is effectively shielded in this solution. It has been known that solutions in non-local gravity often suffer from the existence of ghost modes. In the present case we find the solution is ghost-free if $\alpha>\alpha_{cr}\approx0.17$. This is quite a weak condition. We argue that the solution is stable against the includion of matter fields. Thus our solution opens up new possibilities for solution to the cosmological constant problem., Comment: 7 pages, 1 figure, LaTeX, V2:Some clarifications and references added

We discuss a possible mechanism to screen a cosmological constant in non-local gravity. We find that in a simple model of of non-local gravity with the Lagrangian of the form, $R+f(\Box^{-1}R)-2\Lambda$ where $f(X)$ is a quadratic function of $X$, there is a flat spacetime solution despite the presence of the cosmological constant $\Lambda$. Unfortunately, however, we also find that this solution contains a ghost in general. Then we discuss the condition to avoid a ghost and find that one can avoid it only for a finite range of `time'. Nevertheless our result suggests the possibility of solving the cosmological constant problem in the context of non-local gravity., Comment: LaTeX 7 pages, no figure

We propose a two-sample test for the means of high-dimensional data when the data dimension is much larger than the sample size. Hotelling's classical $T^2$ test does not work for this "large $p$, small $n$" situation. The proposed test does not require explicit conditions in the relationship between the data dimension and sample size. This offers much flexibility in analyzing high-dimensional data. An application of the proposed test is in testing significance for sets of genes which we demonstrate in an empirical study on a leukemia data set., Comment: Published in at http://dx.doi.org/10.1214/09-AOS716 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We generalize the field theory of global monopole to Dirac-Born-Infeld(DBI) field and investigate the gravitational property of a DBI global monopole in four-dimensional spherically symmetric spacetime. The coupled equations for the metric and the DBI scalar field are solved asymptotically and numerically. It is found that, just as a canonical global monopole, the gravitational effect of DBI global monopole is equivalent to that of a deficit solid angle in the metric plus a negative mass at the origin. However, compared with a canonical global monopole, for the same false vacuum and symmetry breaking scale, a DBI global monopole has a relatively smaller core and a larger absolute value of effective mass. Thus, it can give out a larger deflect angle when the light passing by. Especially, when the scale of warp factor is small enough, the effective mass of a DBI global monopole does not depend apparently on the value of the false vacuum, which is qualitatively different from that of a canonical global monopole., Comment: 10 pages, 3 figures, accepted for publication in EPJC