The critical 2D Stochastic Heat Flow (SHF) is a universal measure-valued process that provides a notion of solution to the ill-defined 2D stochastic heat equation. We investigate the SHF in the large-time and strong-disorder regimes, proving a sharp form of local extinction: we identify the rate at which the distribution collapses to zero. We also identify the spatial scale governing the transition from vanishing mass to diverging mass, and from extinction to an averaged behavior. Corresponding results are established for the partition functions of 2D directed polymers, yielding precise free-energy estimates. Our proof provides a unified framework of change of measure and coarse-graining arguments. These results offer new insights into the 2D stochastic heat equation regularized via space-time discretization: for any regime of supercritical disorder strength $β$, including the case where $β> 0$ is kept fixed, the solution exhibits fluctuations on a superdiffusive scale.

We study the doubled edge-stage lift \[ \HL'_2(G)=L(G\otimes K_2), \] the line graph of the canonical bipartite double cover of a graph \(G\). The natural involution \((u,v)\leftrightarrow(v,u)\) has quotient isomorphic to \(L(G)\), and induces a sector decomposition \[ \Spec(\HL'_2(G))=\Spec(L(G))\cup\Spec(\mathcal A(G)), \] where \(\mathcal A(G)\) is a canonical signed refinement of the line graph. Thus the construction retains substantial edge-space information through its quotient and antisymmetric sector. For every input graph, \(\HL'_2(G)\) is perfect, claw-free, and box-perfect. In the regular case we give an explicit spectral formula, together with quantitative control of the second eigenvalue and spectral gap for non-bipartite input. Explicit families, including the complete-graph lifts and the Paley lifts, illustrate the theory; in particular, the Paley lifts furnish an explicit family of regular perfect graphs with controlled adjacency spectrum and spectral gap. The construction may be viewed both intrinsically, via ordered-edge adjacency by one-coordinate agreement, and extrinsically, as the line graph of the canonical double cover. The first viewpoint emphasizes the edge-stage nature of the lift, while the second supplies the structural proofs used here.

We introduce a nonequilibrium phenomenon, reminiscent of Anderson's orthogonality catastrophe (OC), that arises in the transient dynamics following an interaction quench between a quantum system and a localized defect. Even if the system comprises only a single particle, the overlap between the asymptotic and initial superposition states vanishes according to a power-law scaling with the number of energy eigenstates entering the initial state and an exponent that depends on the interaction strength. The presence of quantum coherence in the initial state is reflected onto the discrete counterpart of an infinite discontinuity in the quasiprobability distribution of work due to the quench transformation, and onto the subsequent power-law decay of the work distribution. The positivity loss of the work distribution is directly linked with a reduction of the minimal time imposed by quantum mechanics for the state to orthogonalize, thus leading to a quantum coherence-enhanced state-orthogonalization. We propose an experimental test of coherence-enhanced orthogonalization dynamics based on Ramsey interferometry of a trapped cold-atom system.

In this work, we use path integral techniques to predict the switching rate in a single-mode bistable open quantum system. While analytical expressions are well-known to be accessible for systems subject to Gaussian noise obeying classical detailed balance, we generalize this approach to a class of quantum systems, those which satisfy the recently-introduced hidden time-reversal symmetry [1]. In particular, in the context of quantum computing, we deliver precise estimates of bit-flip error rates in cat-qubit architectures, circumventing the need for costly numerical simulations. Our results open new avenues for exploring switching phenomena in multistable single- and many-body open quantum systems.

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. There have been recent efforts to generalize SRG analysis to Multiple-Input Multiple-Output (MIMO) systems. However, these attempts yielded only results for square systems, due to the inherent Hilbert space structure of the SRG. In this paper, we develop an SRG analysis method that accommodates non-square operators. The key element is the embedding of operators to a space of operators acting on a common Hilbert space, while restricting the input space to the original input dimension, to avoid conservatism. We generalize SRG interconnection rules to restricted input spaces and develop stability theorems to guarantee causality, well-posedness and (incremental) $L_2$-gain bounds for the overall interconnection. We show utilization of the proposed theoretical concepts on the analysis of nonlinear systems in a Linear Fractional Representation (LFR) form, which is a rather general class of systems, and the LFR is directly utilizable for control. Moreover, we provide formulas for the computation of MIMO SRGs of stable LTI operators and diagonal and non-square static nonlinear operators. Finally, we demonstrate the advantages of our embedding approach on several examples.

We give a survey of analytic and geometric results on `fibred cusp spaces', a large class of non-compact Riemannian manifolds which include the regular parts of singular spaces with incomplete cusp singularities as well as complete spaces with asymptotically hyperbolic cusp or asymptotically Euclidean structures at infinity. These results cover topics in spectral geometry, in particular analytic torsion and index theory, and boundary value problems. The underlying tools include a careful microlocal analysis of the resolvent and the heat kernel. We include an exposition of the geometric and analytic foundations and sketch the ideas of the proofs of the main theorems. Special emphasis is put on the common features of and the differences between the incomplete and various kinds of complete settings.

This paper explores the geometry of the space of codimension-2 submanifolds. We implicitly represent these submanifolds by a class of complex-valued functions. We show that the space of all these implicit representations admits a prequantum bundle structure over the space of submanifolds, equipped with the well-known Marsden-Weinstein symplectic structure. This bundle allows a new geometric interpretation of the Marsden-Weinstein structure as the curvature of a connection form, which measures the average of volumes swept by the deformation of the S^1-family of hypersurfaces, defined as the phase level sets of the complex function implicitly representing a submanifold.

Memory safety is traditionally characterized in terms of bad things that cannot happen. This approach is currently embraced in the literature on formal methods for memory safety. However, a general semantic principle for memory safety, that implies the negative items, remains elusive. This paper focuses on the allocator-specific aspects of memory safety, such as null-pointer dereference, use after free, double free, and heap overflow. To that extent, we propose a notion of gradual allocator independence that accurately captures the allocator-dependent aspects of memory safety. Our approach is inspired by the previously suggested connection between memory safety and noninterference, but extends that connection in a fundamentally important direction towards downgrading. We consider a low-level language with access to an allocator that provides malloc and free primitives in a flat memory model. Pointers are just integers, and as such it is trivial to write memory-unsafe programs. The basic intuition of gradual allocator independence is that of noninterference, namely that allocators must not influence program execution. This intuition is refined in two important ways that account for the allocators running out-of-memory and for programs to have pointer-to-integer casts. The key insight of the definition is to treat these extensions as forms of downgrading and give them satisfactory technical treatment using the state-of-the-art information flow machinery.

Detecting outliers in Functional Data Analysis is challenging because curves can stray from the majority in many different ways. The Modified Epigraph Index (MEI) and Modified Hypograph Index (MHI) rank functions by the fraction of the domain on which one curve lies above or below another. While effective for spotting shape anomalies, their construction limits their ability to flag magnitude outliers. This paper introduces two new metrics, the Area-Based Epigraph Index (ABEI) and Area-Based Hypograph Index (ABHI) that quantify the area between curves, enabling simultaneous sensitivity to both magnitude and shape deviations. Building on these indices, we present EHyOut, a robust procedure that recasts functional outlier detection as a multivariate problem: for every curve, and for its first and second derivatives, we compute ABEI and ABHI and then apply multivariate outlier-detection techniques to the resulting feature vectors. Extensive simulations show that EHyOut remains stable across a wide range of contamination settings and often outperforms established benchmark methods. Moreover, applications to Spanish weather data and United Nations world population data further illustrate the practical utility and meaningfulness of this methodology.

NVIDIA GPU Confidential Computing (GPU-CC) aims to provide secure execution for AI workloads. For end users, enabling GPU-CC is seamless and requires no modifications to existing applications. However, this ease of adoption relies on a proprietary and highly complex system that is difficult to inspect, creating challenges for researchers seeking to understand its architecture and security landscape. In this work, we provide a security look at GPU-CC by reconstructing a coherent view of the system. We first examine the system's blueprint, focusing on the specialized architectural engines that support its security mechanisms. We then analyze the bootstrap process, which coordinates hardware and software components to establish these protections. Finally, we conduct targeted experiments to assess whether, under the GPU-CC threat model, data transfers along different paths remain protected across the bridge between trusted CPU and GPU domains. We responsibly disclosed all security findings presented in this paper to the NVIDIA Product Security Incident Response Team (PSIRT).

This paper shows that the complex projective plane $\mathbb{P}^2$ can be realized as the underlying space for a closed hyperbolic $4$-orbifold. This is the first example of a closed hyperbolic $4$-orbifold whose underlying space is symplectic, which is related to the open question as to whether or not closed hyperbolic $4$-manifolds can admit symplectic structures.

This paper introduces a quaternionic analogue of toric geometry by developing the theory of local $Q^n := Sp(1)^n$-actions on 4n-dimensional manifolds, modeled on the regular representation. We identify obstructions that measure the failure of local properties to globalize and define two invariants: a combinatorial invariant called the characteristic pair and a cohomological invariant called the Euler class, which together classify local quaternionic torus actions up to homeomorphism. We also study tetraplectic structures in quaternionic toric geometry by introducing locally generalized Lagrangian-type toric fibrations and show that such fibrations are locally modeled on $\mathbb{R}^n\times Q^n$ using a quaternionic version of the Arnold-Liouville theorem. In the last part, we show that orbit spaces of these actions acquire the structure of quaternionic integral affine manifolds with corners and Lagrangian overlaps, and we classify such spaces by establishing a quaternionic Delzant-type theorem.

Localized surface plasmons can confine light within a deep-subwavelength volume comparable to the scale of atoms and molecules, enabling ultrasensitive responses to near-field variations. On the other hand, this extreme localization also inevitably amplifies the unwanted noise from the response of local morphological imperfections, leading to complex spectral variations and reduced consistency across the plasmonic nanostructures. Seeking uniform optical responses has therefore long been a sought-after goal in nanoplasmonics. However, conventional probing techniques by dark-field (DF) confocal microscopy, such as image analysis or spectral measurements, can be inaccurate and time-consuming, respectively. Here, we introduce SPARX, a deep-learning-powered paradigm that surpasses conventional imaging and spectroscopic capabilities. In particular, SPARX can batch-predict broadband DF spectra (e.g., 500-1000 nm) of numerous nanoparticles simultaneously from an information-limited RGB image (i.e., below 700 nm). It achieves this extrapolative inference beyond the camera's capture capabilities by learning the underlying physical relationships among multiple orders of optical resonances. The spectral predictions only take milliseconds, achieving a speedup of three to four orders of magnitude compared to traditional spectral acquisition, which may take from hours to days. As a proof-of-principle demonstration for screening identical resonances, the selection accuracy achieved by SPARX is comparable to that of conventional spectroscopy techniques. This breakthrough paves the way for consistent plasmonic applications and next-generation microscopies.

The discrepancy between quantum distinguishability in Hilbert space and classical distinguishability in probability space is expressed by the gap between the quantum and classical Fisher information matrices (QFIM and CFIM, respectively). This intrinsic quantum obstruction is generally not saturable and plays a central role in both fundamental insights and practical applications in modern quantum physics. Here, we develop a geometrical framework for this gap by introducing the notion of semi-classical geometric tensor (SCGT). We relate this quantity to the quantum geometric tensor (QGT), whose real part equals the QFIM. We prove the matrix inequality between QGT and SCGT, which sharpens the standard inequality between QFIM and CFIM and provides novel multiparameter information bounds: the real part of the SCGT reproduces the CFIM plus an additional nonnegative contribution capturing quantum obstruction. This further motivates a natural extension of the Berry phase to the semi-classical setting.

This paper establishes a CLT for linear statistics of the form $\langle \mathbf{q},\boldsymbolσ \rangle$ with quantitative Berry-Esseen bounds, where $\boldsymbolσ$ is an observation from an exponential family with a quadratic form as its sufficient statistic, in the \enquote{high-temperature} regime. We apply our general result to random field Ising models with both discrete and continuous spins. To demonstrate the generality of our techniques, we apply our results to derive both quenched and annealed CLTs in various examples, which include Ising models on some graph ensembles of common interest (Erdős-Rényi, regular, dense bipartite), and the Hopfield spin glass model. Our proofs rely on a combination of Stein's method of exchangeable pairs and Chevet type concentration inequalities.

It remains puzzling why, despite their similar nature, Jupiter and Saturn possess a prograde equatorial jet, whereas Uranus and Neptune have a retrograde one. To understand this discrepancy, we use a two-dimensional quasi-geostrophic model to explore how the jet penetration depth, regulated by Ohmic dissipation, influences the structure and organization of jet patterns. When jets penetrate deeply into the planetary interior, the effective planetary vorticity gradient $β$ becomes negative near the equator and decreases equatorward due to spherical geometry. This $β$ profile favors dynamical modes that transport eastward momentum toward the equator, producing a prograde equatorial jet, as observed on Jupiter and Saturn. In contrast, relatively shallow systems favor a retrograde equatorial jet. In our simulations, the equatorial jet direction is primarily controlled by the gradient of $β$, as predicted by Stochastic Structural Stability Theory, rather than by its sign, as suggested by Potential Vorticity mixing. If this mechanism applies to Uranus and Neptune, the observed jet structure may suggest the presence of a stratified or Ohmic dissipation layer near their surfaces. At mid-latitudes, jet widths are constrained by the Rhines scale, yielding a scaling that explains the presence of multiple jets on Jupiter and Saturn and a single jet per hemisphere on Uranus and Neptune. Lastly, we examine how planetary parameters influence the partitioning of energy between jets and eddies. Stronger energy input, faster rotation, smaller planetary radius, or weaker large-scale damping lead to a larger fraction of the total energy in zonal jets, resulting in smoother jet structures.

Carrollian conformal field theory offers an alternative description of massless scattering amplitudes, that is holographic in nature. In an effort to build a framework that is both predictive and constraining, we construct operator product expansions (OPE) that are compatible with carrollian symmetries. In this way, we unify and extend preliminary works on the subject, and demonstrate that the carrollian OPEs indeed control the short-distance expansion of carrollian correlators and amplitudes. In the process, we extend the representation theory of carrollian conformal fields such as to account for composite operators like the carrollian stress tensor or those creating multiparticle states. In addition we classify 2- and 3-point carrollian correlators and amplitudes with complex kinematics, and give the general form of the 4-point function allowed by symmetry.

We present a homogeneous catalog of global asteroseismic parameters and derived stellar parameters for 765 Kepler main-sequence and subgiant stars. The catalog was produced by re-analyzing all available Kepler DR25 short-cadence data using pySYD, an automated pipeline to extract global asteroseismic parameters. We find 50 new detections, seven of which are also planet candidate host stars. We find excellent agreement between our $ν_{\text{max}}$ and $Δν$ measurements and literature values, with an average offset of $0.2 \pm 0.4\%$ ($σ=5\%$) and $0.2 \pm 0.7\%$ ($σ=2\%$), respectively. In addition, we derive stellar radii and masses with an average precision of $2.7\%$ and $10.4\%$, respectively, and find a mean offset of $0.8 \pm 0.2\%$ ($σ=6\%$) between our radii derived with asteroseismology and those from Gaia parallaxes. Using spectroscopic $\log{R'_{\text{HK}}}$ activity measurements from Keck/HIRES, we derive an amplitude scaling relation with an activity term for main-sequence and subgiant stars, which successfully predicts amplitudes with a precision of $\approx 8-9\%$. Our work is the largest and most homogeneous asteroseismic catalog of Kepler main-sequence and subgiant stars to date, including a total of 101 stars hosting planet candidates and 451 stars with measured rotation periods.

The extent to which the environment affects galaxy evolution has been under scrutiny by researchers for decades. With the first data from Euclid, we can begin to systematically study a wide range of environments and their effects as a function of redshift, using 63 deg2 of space-based data. In this paper, we present results from Euclid Q1, where we measured the passive-density and morphology-density relations in the redshift range z=0.25--1. We determined if a galaxy is passive using the specific star formation rate, and we classified the morphologies of galaxies using the Sersic index n and the u-r colours. We measured the local environmental density of each galaxy using the Nth-nearest neighbour method. We find that at a fixed stellar mass, the quenched fraction (the fraction of galaxies that have ceased star formation) increases with increasing local environmental density up to $z=1$. This result is indicative of the separability of the effects from the stellar mass and the environment. Similarly, at all redshifts in this work, the early-type galaxy fraction increases with increasing density at fixed stellar mass, meaning the environment also transforms the morphology of the galaxy independently of stellar mass, up to M_* < 10^10.8 Msol$. For M* > 10^10.8 Msol, almost all galaxies are early-types, with minimal impact from the environment. At z>0.75, the morphology depends mostly on stellar mass, with only low-mass galaxies being affected by the environment. Given that the morphology classifications use u-r colours, these are correlated to the star formation rate, and as such our morphology results should be taken with caution, yet future morphology classifications should help verify these results. To summarise, we successfully identify the passive-density and morphology-density relations at 0.25<z<1. Future Euclid data releases are key to confirm these trends at higher redshifts.

This paper concerns the topological classification of continuous Hamiltonians that find applications in biased cold plasmas and photonics. Besides a magnetic bias, the Hamiltonians are parametrized by a plasma frequency and a fixed vertical wavenumber. Eight distinct phases of matter are identified as these parameters vary. When insulating gaps are shared by two such phases, asymmetric edge modes propagate along interfaces separating the two phases. Here we apply the notion of a bulk difference invariant (BDI) to this Hamiltonian, and show by numerical diagonalizations of interface Hamiltonians that after an appropriate regularization our BDI correctly predicts edge transport as described by a bulk edge correspondence. We also derive theoretical tools to compute the BDI and show the limitations of the bulk edge correspondence (BEC) when the phase transition is too singular.

This paper proposes a quasi-maximum likelihood (QML) estimator for break points in high-dimensional factor models, specifically accounting for multiple structural breaks. We begin by establishing a necessary and sufficient condition to categorize two distinct types of breaks in factor loadings: singular changes and rotational changes. The analysis of the nearly singular subsample covariance matrices of the pseudo-factors plays a key role in our approach. It allows us to demonstrate that the QML estimator precisely identifies the true breakpoint with probability tending to one for singular changes. For rotational changes, we demonstrate that the estimator exhibits stochastically bounded estimation errors, implying break fraction consistency. Furthermore, we introduce an information criterion to estimate the number of breaks, proving that it can detect the true number with probability tending to one. Monte Carlo simulations confirm the strong finite sample performance of our proposed methods. Finally, we provide an empirical example to estimate structural breakpoints in the FRED-MD dataset spanning 1959 to 2024.

Extreme value analysis for time series is often based on the block maxima method, in particular for environmental applications. In the classical univariate case, the latter is based on fitting an extreme-value distribution to the sample of (annual) block maxima. Mathematically, the target parameters of the extreme-value distribution also show up in limit results for other high order statistics, which suggests estimation based on blockwise large order statistics. It is shown that a naive approach based on maximizing an independence log-likelihood yields an estimator that is inconsistent in general. A consistent, bias-corrected estimator is proposed, and is analyzed theoretically and in finite-sample simulation studies. The new estimator is shown to be more efficient than traditional counterparts, for instance for estimating large return levels or return periods.

In heterogeneous integration, different dies may employ distinct technologies, making floorplanning across multiple dies inherently coupled with technology assignment. By assuming a fixed technology, almost all prior floorplanning studies were developed without addressing the challenge of technology assignment. This work presents the first systematic study of multi-die floorplanning that treats technology choice as a variable. To address the challenge of variable block areas, we incorporate a recent machine learning technique for rapid PPA estimation. Our methods jointly optimize area, wirelength, performance, power, and cost, thereby highlighting the importance of technology assignment. Experimental evaluations, validated with a commercial tool for both 2.5D and 3D ICs, demonstrate that our systematic optimizations significantly outperform a greedy approach.

We analyze the susceptibility of instant runoff voting (IRV) to a lesser-studied paradox known as a \emph{reinforcement paradox}, which occurs when candidate $X$ wins under IRV in two distinct elections but $X$ loses in the combined election formed by merging the ballots from the two elections. For three-candidate IRV elections we provide necessary and sufficient conditions under which there exists a partition of the ballot set into two sets of ballots such that a given losing candidate wins each of the sub-elections. Applying these conditions, we use Monte Carlo simulations to estimate the frequency with which such partitions exist under various models of voter behavior. We also analyze the frequency with which the paradox occurs in a large dataset of real-world ranked-choice elections to provide empirical probabilities. Our general finding is that IRV is highly susceptible to this paradox in three-candidate elections.

As superconducting kinetic inductance detectors (KIDs) continue to grow in popularity for sensitive submillimeter detection and other applications, there is a drive to advance toward lower-loss devices. We present measurements of diagnostic thin-film aluminum coplanar waveguide (CPW) resonators designed to inform ongoing KID development at NASA Goddard Space Flight Center. The resonance frequencies span $f_0$ = 3.5-4 GHz and include quarter-wave and half-wave resonators with varying coupling capacitor designs. We present measurements of the device film properties and an analysis of the dominant mechanisms of loss in the resonators measured in a dark environment, demonstrating quality factors of $Q_i^{-1} \approx 3.64-8.57\ \times 10^{-8}$. We observe an enhanced level of suppression in the loss contributions from two-level systems (TLS) at intermediate-to-high read powers, and a regime at these powers and low temperatures where contributions from intrinsic processes $Q_i^{-1}$,other dominate the total loss. We also observe deviations from the standard TLS loss model at low powers and temperatures below 60 mK, and use a modified model to describe this behavior.

We carefully analyse the challenges posed by the construction of type IIB chiral global embeddings of Fibre Inflation with $\overline{ \rm D3}$ uplift to a de Sitter vacuum. We present an explicit example involving an $h^{1,1}=4$ Calabi-Yau manifold with a K3 fibration and a del Pezzo divisor supporting non-perturbative effects. The chosen orientifold involution induces O3-planes that can be placed on top of each other at the tip of the throat of a deformed conifold singularity. The D7-brane sector contains standard magnetised branes and a Whitney brane. The former induce chiral matter and generate quantum effects that stabilise the Kähler moduli, while the latter helps to increase the total D3-charge. We study in detail the constraints on the parameter space leading to an observationally viable inflationary dynamics, finding several regions where the effective field theory is under control.

How can we perform a metrological task if only limited control over a quantum system is given? Here, we present systematic methods for conducting nonlinear quantum metrology in scenarios lacking a common reference frame. Our approach involves preparing multiple copies of quantum systems and then performing local measurements with randomized observables. First, we derive the metrological precision using an error propagation formula based solely on local unitary invariants, which are independent of the chosen basis. Next, we provide analytical expressions for the precision scaling in various examples of nonlinear metrology involving two-body interactions, like the one-axis twisting Hamiltonian. Finally, we analyze our results in the context of local decoherence and discuss its influences on the observed scaling.

The INTENTAS project aims to develop an atomic sensor utilizing entangled Bose-Einstein condensates (BECs) in a microgravity environment. This key achievement is necessary to advance the capability for measurements that benefit from both entanglement-enhanced sensitivities and extended interrogation times. The project addresses significant challenges related to size, weight, and power management (SWaP) specific to the experimental platform at the Einstein-Elevator in Hannover. The design ensures a low-noise environment essential for the creation and detection of entanglement. Additionally, the apparatus features an innovative approach to the all-optical creation of BECs, providing a flexible system for various configurations and meeting the requirements for rapid turnaround times. Successful demonstration of this technology in the Einstein-Elevator will pave the way for a future deployment in space, where its potential applications will unlock high-precision quantum sensing.

We consider a class of infinite-dimensional dynamical systems driven by non-linear parabolic partial differential equations with initial condition $θ$ modelled by a Gaussian process `prior' probability measure. Given discrete samples of the state of the system evolving in space-time, one obtains updated `posterior' measures on a function space containing all possible trajectories. We give a general set of conditions under which these non-Gaussian posterior distributions are approximated, in Wasserstein distance for the supremum-norm metric, by the law of a Gaussian random function. We demonstrate the applicability of our results to periodic non-linear reaction diffusion equations \begin{align*} \frac{\partial}{\partial t} u - Δu &= f(u) \\ u(0) &= θ\end{align*} where $f$ is any smooth and compactly supported reaction function. In this case the limiting Gaussian measure can be characterised as the solution of a time-dependent Schrödinger equation with `rough' Gaussian initial conditions whose covariance operator we describe.

We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$, thus extending Anderson localization from the linear (cf. Bourgain [Geom. Funct. Anal., 17(3):682--706, 2007]) to a nonlinear setting, and the random (cf. Bourgain-Wang [J. Eur. Math. Soc., 10(1):1--45, 2008]) to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrarily dimensional phase space, and the application of Bourgain's geometric lemma in [Geom. Funct. Anal., 17(3):682--706, 2007].