A Frequency Localized Maximum Principle Applied to the 2D Quasi-Geostrophic Equation

In this paper, we prove a maximum principle for a frequency localized transport-diffusion equation. As an application, we prove the local well-posedness of the supercritical quasi-geostrophic equation in the critical Besov spaces \mathringB^1-a_￥,q{\mathring{B}^{1-\alpha}_{\infty,q}}, and global well-posedness of the critical quasi-geostrophic equation in \mathringB⁰_￥,q{\mathring{B}^{0}_{\infty,q}} for all 1 ≤ q < ∞. Here \mathringB^s_￥,q {\mathring{B}^{s}_{\infty,q} } is the closure of the Schwartz functions in the norm of B^s_￥,q{B^{s}_{\infty,q}}.     

Analysis of the <Emphasis Type="Italic">Y</Emphasis>(4140) and related molecular states with QCD sum rules

In this article, we assume that there exist scalar D^*[`(D)]<sup>*</sup>{D}^{\ast}{\bar {D}}^{\ast}, D<sub>s</sub><sup>*</sup>[`(D)]_s^*{D}_{s}^{\ast}{\bar{D}}_{s}^{\ast}, B^*[`(B)]<sup>*</sup>{B}^{\ast}{\bar {B}}^{\ast} and B<sub>s</sub><sup>*</sup>[`(B)]_s^*{B}_{s}^{\ast}{\bar{B}}_{s}^{\ast} molecular states, and study their masses using the QCD sum rules. The numerical results indicate that the masses are about (250–500) MeV above the corresponding D ^*–[`(D)]<sup>*</sup>{\bar{D}}^{\ast}, D <sub>s</sub> <sup>*</sup>–[`(D)]_s^*{\bar {D}}_{s}^{\ast}, B ^*–[`(B)]<sup>*</sup>{\bar{B}}^{\ast} and B <sub>s</sub> <sup>*</sup>–[`(B)]_s^*{\bar {B}}_{s}^{\ast} thresholds, the Y(4140) is unlikely a scalar D_s^*[`(D)]<sub>s</sub><sup>*</sup>{D}_{s}^{\ast}{\bar{D}}_{s}^{\ast} molecular state. The scalar D<sup>*</sup>[`(D)]^*D^{\ast}{\bar{D}}^{\ast}, D_s^*[`(D)]<sub>s</sub><sup>*</sup>D_{s}^{\ast}{\bar{D}}_{s}^{\ast}, B<sup>*</sup>[`(B)]^*B^{\ast}{\bar{B}}^{\ast} and B_s^*[`(B)]<sub>s</sub><sup>*</sup>B_{s}^{\ast}{\bar{B}}_{s}^{\ast} molecular states maybe not exist, while the scalar D￠<sup>*</sup>[`(D)]^￠*{D'}^{\ast}{\bar{D}}^{\prime\ast}, D_s^￠*[`(D)]<sub>s</sub><sup>￠*</sup>{D}_{s}^{\prime\ast}{\bar{D}}_{s}^{\prime\ast}, B<sup>￠*</sup>[`(B)]^￠*{B}^{\prime\ast}{\bar{B}}^{\prime\ast} and B_s^￠*[`(B)]_s^￠*{B}_{s}^{\prime\ast}{\bar{B}}_{s}^{\prime\ast} molecular states maybe exist.     

Adaptive Sub-sampling for Parametric Estimation of Gaussian Diffusions

We consider a Gaussian diffusion X _t (Ornstein-Uhlenbeck process) with drift coefficient γ and diffusion coefficient σ ², and an approximating process Y^e_tY^{\varepsilon}_{t} converging to X _t in L ₂ as ε→0. We study estimators [^(g)]_e\hat{\gamma}_{\varepsilon}, [^(s)]²_e\hat{\sigma}^{2}_{\varepsilon} which are asymptotically equivalent to the Maximum likelihood estimators of γ and σ ², respectively. We assume that the estimators are based on the available N=N(ε) observations extracted by sub-sampling only from the approximating process Y^e_tY^{\varepsilon}_{t} with time step Δ=Δ(ε). We characterize all such adaptive sub-sampling schemes for which [^(g)]_e\hat{\gamma}_{\varepsilon}, [^(s)]²_e\hat{\sigma}^{2}_{\varepsilon} are consistent and asymptotically efficient estimators of γ and σ ² as ε→0. The favorable adaptive sub-sampling schemes are identified by the conditions ε→0, Δ→0, (Δ/ε)→∞, and NΔ→∞, which implies that we sample from the process Y^e_tY^{\varepsilon}_{t} with a vanishing but coarse time step Δ(ε)≫ε. This study highlights the necessity to sub-sample at adequate rates when the observations are not generated by the underlying stochastic model whose parameters are being estimated. The adequate sub-sampling rates we identify seem to retain their validity in much wider contexts such as the additive triad application we briefly outline.     

Robust chaos synchronization based on adaptive fuzzy delayed feedback ${\bf{\mathcal{H}}}_{\bf{\infty}}$ control

In this paper, we propose a new adaptive H_￥\mathcal H_\infty synchronization strategy, called an adaptive fuzzy delayed feedback H_￥\mathcal H_\infty synchronization (AFDFHS) strategy, for chaotic systems with uncertain parameters and external disturbances. Based on Lyapunov–Krasovskii theory, Takagi–Sugeno (T–S) fuzzy model and adaptive delayed feedback H_￥\mathcal H_\infty control scheme, the AFDFHS controller is presented such that the synchronization error system is asymptotically stable with a guaranteed H_￥\mathcal{H}_{\infty } performance. It is shown that the design of the AFDFHS controller with adaptive law can be achieved by solving a linear matrix inequality (LMI), which can be easily facilitated by using some standard numerical packages. An illustrative example is given to demonstrate the effectiveness of the proposed AFDFHS approach.     

Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions

If X = X(t, ξ) is the solution to the stochastic porous media equation in O ì R^d, 1 ￡ d ￡ 3,{\mathcal{O}\subset \mathbf{R}^d, 1\le d\le 3,} modelling the self-organized criticality (Barbu et al. in Commun Math Phys 285:901–923, 2009) and X _c is the critical state, then it is proved that ò^￥₀m(O\O^t₀)dt < ￥,\mathbbP-a.s.{\int^{\infty}_0m(\mathcal{O}{\setminus}\mathcal{O}^t_0)dt<{\infty},\mathbb{P}\hbox{-a.s.}} and lim_t?￥ ò_O|X(t)-X_c|dx = l < ￥, \mathbbP-a.s.{\lim_{t\to{\infty}} \int_\mathcal{O}|X(t)-X_c|d\xi=\ell<{\infty},\ \mathbb{P}\hbox{-a.s.}} Here, m is the Lebesgue measure and O^t_c{\mathcal{O}^t_c} is the critical region {x ? O; X(t,x)=X_c(x)}{\{\xi\in\mathcal{O}; X(t,\xi)=X_c(\xi)\}} and X _c (ξ) ≤ X(0, ξ) a.e. x ? O{\xi\in\mathcal{O}}. If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), lim_{t ? ￥} ò_K|X(t)-X_c|dx = 0{\lim_{t \to {\infty}} \int_K|X(t)-X_c|d\xi=0} exponentially fast for all compact K ì O{K\subset\mathcal{O}} with probability one, if the noise is sufficiently strong. We also recover that in the deterministic case ℓ = 0.     

Analysis of the <Emphasis Type="Italic">Y</Emphasis>(4140) with QCD sum rules

In this article, we assume that there exists a scalar D_s^*[`(D)]<sub>s</sub><sup>*</sup>D_{s}^{\ast}{\bar{D}}_{s}^{\ast} molecular state in the J/ψ φ invariant mass distribution, and we study its mass using the QCD sum rules. The predictions depend heavily on the two criteria (pole dominance and convergence of the operator product expansion) of the QCD sum rules. The value of the mass is about M<sub>Ds<sup>*</sup>[`(D)]s^*</sub>=(4.43±0.16)M_{D_{s}^{\ast}{\bar{D}}_{s}^{\ast}}=(4.43\pm0.16)  GeV, which is inconsistent with the experimental data. The D_s^*[`(D)]_s^*D_{s}^{\ast}{\bar{D}}_{s}^{\ast} is probably a virtual state and is not related to the meson Y(4140). Another possibility, such as a hybrid charmonium, is not excluded.     

The Combined Effect of Connectivity and Dependency Links on Percolation of Networks

Percolation theory is extensively studied in statistical physics and mathematics with applications in diverse fields. However, the research is focused on systems with only one type of links, connectivity links. We review a recently developed mathematical framework for analyzing percolation properties of realistic scenarios of networks having links of two types, connectivity and dependency links. This formalism was applied to study Erdős-Rényi (ER) networks that include also dependency links. For an ER network with average degree [`(k)]\bar{k} that is composed of dependency clusters of size s, the fraction of nodes that belong to the giant component, P <sub>∞</sub>, is given by P<sub>￥</sub>=p<sup>s-1</sup>[1-exp(-[`(k)]pP_￥) ]^sP_{\infty}=p^{s-1}[1-\exp{(-\bar{k}pP_{\infty})} ]^{s} where 1−p is the initial fraction of randomly removed nodes. Here, we apply the formalism to the study of random-regular (RR) networks and find a formula for the size of the giant component in the percolation process: P _∞=p ^s−1(1−r ^k ) ^s where r is the solution of r=p ^s (r ^k−1−1)(1−r ^k )+1, and k is the degree of the nodes. These general results coincide, for s=1, with the known equations for percolation in ER and RR networks respectively without dependency links. In contrast to s=1, where the percolation transition is second order, for s>1 it is of first order. Comparing the percolation behavior of ER and RR networks we find a remarkable difference regarding their resilience. We show, analytically and numerically, that in ER networks with low connectivity degree or large dependency clusters, removal of even a finite number (zero fraction) of the infinite network nodes will trigger a cascade of failures that fragments the whole network. Specifically, for any given s there exists a critical degree value, [`(k)]<sub>min</sub>\bar{k}_{\min}, such that an ER network with [`(k)] ￡ [`(k)]_min\bar{k}\leq \bar{k}_{\min} is unstable and collapse when removing even a single node. This result is in contrast to RR networks where such cascades and full fragmentation can be triggered only by removal of a finite fraction of nodes in the network.     

Analysis of the ΛQ baryons in the nuclear matter with the QCD sum rules

In this article, we study the Λ c and Λ b baryons in the nuclear matter using the QCD sum rules, and obtain the in-medium masses M\varLambda c*=2.335 GeVM_{\varLambda _{c}}^{*}=2.335~\mathrm{GeV}, M\varLambda b*=5.678 GeVM_{\varLambda _{b}}^{*}=5.678~\mathrm{GeV}, the in-medium vector self-energies \varSigma \varLambda cv=34 MeV\varSigma ^{\varLambda _{c}}_{v}=34~\mathrm{MeV}, \varSigma \varLambda bv=32 MeV\varSigma ^{\varLambda _{b}}_{v}=32~\mathrm {MeV}, and the in-medium pole residues l\varLambda c*=0.021 GeV3\lambda_{\varLambda _{c}}^{*}=0.021~\mathrm{GeV}^{3}, l\varLambda b*=0.026 GeV3\lambda_{\varLambda _{b}}^{*}=0.026~\mathrm{GeV}^{3}. The mass-shifts are M\varLambda c*-M\varLambda c=51 MeVM_{\varLambda _{c}}^{*}-M_{\varLambda _{c}}=51~\mathrm{MeV} and M\varLambda b*-M\varLambda b=60 MeVM_{\varLambda _{b}}^{*}-M_{\varLambda _{b}}=60~\mathrm{MeV}, respectively.     

Magnetocaloric properties of as-quenched Ni50.4Mn34.9In14.7 ferromagnetic shape memory alloy ribbons

The temperature dependences of magnetic entropy change and refrigerant capacity have been calculated for a maximum field change of Δ H=30 kOe in as-quenched ribbons of the ferromagnetic shape memory alloy Ni_50.4Mn_34.9In_14.7 around the structural reverse martensitic transformation and magnetic transition of austenite. The ribbons crystallize into a single-phase austenite with the L2₁-type crystal structure and Curie point of 284 K. At 262 K austenite starts its transformation into a 10-layered structurally modulated monoclinic martensite. The first- and second-order character of the structural and magnetic transitions was confirmed by the Arrott plot method. Despite the superior absolute value of the maximum magnetic entropy change obtained in the temperature interval where the reverse martensitic transformation occurs (|\varDelta S_M^max|=7.2 J kg^-1 K^-1)(|\varDelta S_{\mathrm{M}}^{\max}|=7.2\mbox{ J}\,\mbox{kg}^{-1}\,\mbox{K}^{-1}) with respect to that obtained around the ferromagnetic transition of austenite (|\varDelta S_M^max|=2.6 J kg^-1 K^-1)(|\varDelta S_{\mathrm{M}}^{\max}|=2.6\mbox{ J}\,\mbox{kg}^{-1}\,\mbox{K}^{-1}), the large average hysteretic losses due to the effect of the magnetic field on the phase transformation as well as the narrow thermal dependence of the magnetic entropy change make the temperature interval around the ferromagnetic transition of austenite of a higher effective refrigerant capacity (RC^magn_eff=95J kg^-1\mathrm{RC}^{\mathrm{magn}}_{\mathrm{eff}}=95\mbox{J}\,\mbox{kg}^{-1} versus RC^struct_eff=60J kg^-1)\mathrm{RC}^{\mathrm{struct}}_{\mathrm{eff}}=60\mbox{J}\,\mbox{kg}^{-1}).     

Controllability Issues for Continuous-Spectrum Systems and Ensemble Controllability of Bloch Equations

We study the controllability of the Bloch equation, for an ensemble of non interacting half-spins, in a static magnetic field, with dispersion in the Larmor frequency. This system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum, whose controllability is not well understood. We provide several mathematical answers, with discrimination between approximate and exact controllability, and between finite time or infinite time controllability: this system is not exactly controllable in finite time T with bounded controls in L ²(0, T), but it is approximately controllable in L ^∞ in finite time with unbounded controls in L^￥_loc([0,+￥)){L^{\infty}_{loc}([0,+\infty))}. Moreover, we propose explicit controls realizing the asymptotic exact controllability to a uniform state of spin + 1/2 or −1/2.     

Asymptotics for Large Time of Global Solutions to the Generalized Kadomtsev–Petviashvili Equation

We study the large time asymptotic behavior of solutions to the generalized Kadomtsev-Petviashvili (KP) equations $ \left\{\alignedat2 &u_t + u_{xxx} + \sigma\partial_x^{-1}u_{yy}= - (u^{\rho})_x, &;&;\qquad (t,x,y) \in {\bold R}\times {\bold R}^2,\\ \vspace{.5\jot} &u(0,x,y) = u_0 (x,y),&;&; \qquad (x,y) \in{\bold R}^2, \endalignedat \right. \TAG KP $ \left\{\alignedat2 &u_t + u_{xxx} + \sigma\partial_x^{-1}u_{yy}= - (u^{\rho})_x, &;&;\qquad (t,x,y) \in {\bold R}\times {\bold R}^2,\\ \vspace{.5\jot} &u(0,x,y) = u_0 (x,y),&;&; \qquad (x,y) \in{\bold R}^2, \endalignedat \right. \TAG KP where † = 1 or † = m 1. When „ = 2 and † = m 1, (KP) is known as the KPI equation, while „ = 2, † = + 1 corresponds to the KPII equation. The KP equation models the propagation along the x-axis of nonlinear dispersive long waves on the surface of a fluid, when the variation along the y-axis proceeds slowly [10]. The case „ = 3, † = m 1 has been found in the modeling of sound waves in antiferromagnetics [15]. We prove that if „ S 3 is an integer and the initial data are sufficiently small, then the solution u of (KP) satisfies the following estimates: ||u(t)||_￥ ￡ C (1 + |t|)^-1 (log(2+|t|))^k, ||u_x(t)||_￥ ￡ C (1 + |t|)^-1 \|u(t)\|_\infty \le C (1 + |t|)^{-1} (\log (2+|t|))^{\kappa}, \|u_x(t)\|_\infty \le C (1 + |t|)^{-1} for all t ] R, where s = 1 if „ = 3 and s = 0 if „ S 4. We also find the large time asymptotics for the solution.     

Compact and accurate variational wave functions of three-electron atomic systems constructed from semi-exponential radial basis functions

The semi-exponential basis set of radial functions [A.M. Frolov, Phys. Lett. A 374, 2361 (2010)] is used for variational computations of bound states in three-electron atomic systems. It appears that the semi-exponential basis set has a substantially greater potential for accurate variational computations of bound states in three-electron atomic systems than was originally anticipated. In particular, the 40-term Larson’s wave function improved with the use of semi-exponential radial basis functions now produces the total energy –7.4780581457 a.u. for the ground 1²S-state in the ^￥Li^\infty{\rm Li} atom (only one spin function c₁\chi_1 = aba\alpha\beta\alpha - baa\beta\alpha\alpha was used in these calculations). This variational energy is very close to the exact ground state energy of the ^￥Li^\infty{\rm Li} atom and is substantially lower than the total energy obtained with the original Larson’s 40-term wave function (–7.477944869 a.u.).     

Thermal States in Conformal QFT. I

We analyze the set of locally normal KMS states w.r.t. the translation group for a local conformal net A{{\mathcal A}} of von Neumann algebras on \mathbb R{\mathbb R} . In this first part, we focus on the completely rational net A{{\mathcal A}} . Our main result here states that, if A{{\mathcal{A}}} is completely rational, there exists exactly one locally normal KMS state j{\varphi} . Moreover, j{\varphi} is canonically constructed by a geometric procedure. A crucial r?le is played by the analysis of the “thermal completion net” associated with a locally normal KMS state. A similar uniqueness result holds for KMS states of two-dimensional local conformal nets w.r.t. the time-translation one-parameter group.     

The ζ-Determinant and the Additivity of the η-Invariant on the Smooth, Self-Adjoint Grassmannian

In this paper we discuss the existence of the ‘-determinant of a Dirac operator \Dd\Dd on a compact manifold with boundary. We show that the determinant is well defined in the case of the operator \Dd\Dd with a domain determined by a boundary condition from the smooth, self-adjoint Grassmannian \Grass_￥^*(\Dd)\Grass_{\infty}^*(\Dd) discussed in the papers [5, 13, 29]. We prove a generalization of a pasting formula for the m-invariant (see [34]). The results of the paper are used in the recent proof of the projective equality of the ‘-determinant and Quillen determinant on \Grass_￥^*(\Dd)\Grass_{\infty}^*(\Dd) (see [30, 31]).     

The value of the cosmological constant

We make the cosmological constant, Λ, into a field and restrict the variations of the action with respect to it by causality. This creates an additional Einstein constraint equation. It restricts the solutions of the standard Einstein equations and is the requirement that the cosmological wave function possess a classical limit. When applied to the Friedmann metric it requires that the cosmological constant measured today, t _U , be L ~ t_U^-2 ~ 10^-122{\Lambda \sim t_{U}^{-2} \sim 10^{-122}} , as observed. This is the classical value of Λ that dominates the wave function of the universe. Our new field equation determines Λ in terms of other astronomically measurable quantities. Specifically, it predicts that the spatial curvature parameter of the universe is W_k0 o -k/a₀²H²=-0.0055{\Omega _{\mathrm{k0}} \equiv -k/a_{0}^{2}H^{2}=-0.0055} , which will be tested by Planck Satellite data. Our theory also creates a new picture of self-consistent quantum cosmological history.     

Global Solutions to the 3-D Incompressible Anisotropic Navier-Stokes System in the Critical Spaces

In this paper, we consider the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations with initial data in the critical Besov-Sobolev type spaces B{\mathcal{B}} and B^{-\frac12,\frac12}₄{\mathcal{B}^{-\frac12,\frac12}_4} (see Definitions 1.1 and 1.2 below). In particular, we proved that there exists a positive constant C such that (ANS _ν ) has a unique global solution with initial data u₀ = (u₀^h, u₀³){u_0 = (u_0^h, u_0^3)} which satisfies ||u₀^h||_B exp(\fracCn⁴ ||u₀³||_B⁴) ￡ c₀n{\|u_0^h\|_{\mathcal{B}} \exp\bigl(\frac{C}{\nu^4} \|u_0^3\|_{\mathcal{B}}^4\bigr) \leq c_0\nu} or ||u₀^h||_{B^{-\frac12,\frac12}4} exp(\fracCn⁴ ||u₀³||_{B^{-\frac12,\frac12}4}⁴) ￡ c₀n{\|u_0^h\|_{\mathcal{B}^{-\frac12,\frac12}_{4}} \exp \bigl(\frac{C}{\nu^4} \|u_0^3\|_{\mathcal{B}^{-\frac12,\frac12}_{4}}^4\bigr)\leq c_0\nu} for some c ₀ sufficiently small. To overcome the difficulty that Gronwall’s inequality can not be applied in the framework of Chemin-Lerner type spaces, [(L^p_t)\tilde](B){\widetilde{L^p_t}(\mathcal{B})}, we introduced here sort of weighted Chemin-Lerner type spaces, [(L²_{t, f})\tilde](B){\widetilde{L^2_{t, f}}(\mathcal{B})} for some apropriate L ¹ function f(t).     

An ab initio molecular dynamics study of ionic conductivity in hexagonal lithium lanthanum titanate oxide La0.5Li0.5TiO3

To date, the fastest lithium ion-conducting solid electrolytes known are the perovskite-type ABO₃ oxide, with A = Li, La and B = Ti, lithium lanthanum titanate (LLTO) Li_3x La_{( 2 \mathord}

Maximum Solutions of Normalized Ricci Flow on 4-Manifolds

We consider the maximum solution g(t), t ∈ [0,  + ∞), to the normalized Ricci flow. Among other things, we prove that, if (M, ω) is a smooth compact symplectic 4-manifold such that and let g(t), t ∈ [0, ∞), be a solution to (1.3) on M whose Ricci curvature satisfies that |Ric(g(t))| ≤ 3 and additionally χ(M) = 3τ (M) > 0, then there exists an , and a sequence of points {x _j,k ∈ M}, j = 1, . . . , m, satisfying that, by passing to a subsequence,

Leading-order actions of Goldstino fields

This paper starts with a self-contained discussion of the so-called Akulov–Volkov action S_AV\mathcal{S}_{\mathrm{AV}}, which is traditionally taken to be the leading-order action of the Goldstino field. Explicit expressions for S_AV\mathcal{S}_{\mathrm{AV}} and its chiral version S_AV^ch\mathcal{S}_{\mathrm{AV}}^{\mathrm{ch}} are presented. We then turn to the issue on how these actions are related to the leading-order action S_NL\mathcal{S}_{\mathrm{NL}} proposed in the newly proposed constrained superfield formalism. We show that S_NL\mathcal{S}_{\mathrm{NL}} may yield S_AV/S_AV^ch\mathcal{S}_{\mathrm {AV}}/\mathcal{S}_{\mathrm{AV}}^{\mathrm{ch}} or a totally different action S_KS\mathcal{S}_{\mathrm{KS}}, depending on how the auxiliary field in the former is integrated out. However, S_KS\mathcal{S}_{\mathrm{KS}} and S_AV/S_AV^ch\mathcal{S}_{\mathrm {AV}}/\mathcal{S}_{\mathrm{AV}}^{\mathrm{ch}} always yield the same S-matrix elements, as one would have expected from general considerations in quantum field theory.     

Boundary Concentration for Eigenvalue Problems Related to the Onset of Superconductivity

We examine the asymptotic behavior of the eigenvalue w(h) and corresponding eigenfunction associated with the variational problem m(h) o inf_{y ? H¹(W;C )} \fracò_W \abs(i?+hA)y² dx dy ò_W\absy² dx dy \mu(h)\equiv\inf_{\psi\in H^{1}(\Omega;{\bf C} )} \frac{\int_{\Omega } \abs{(i\nabla+h{\bf A})\psi}^{2}\,dx\,dy} {\int_{\Omega }\abs{\psi}^{2}\,dx\,dy} in the regime h>>1. Here A is any vector field with curl equal to 1. The problem arises within the Ginzburg-Landau model for superconductivity with the function w(h) yielding the relationship between the critical temperature vs. applied magnetic field strength in the transition from normal to superconducting state in a thin mesoscopic sample with cross-section W ì \R²\Omega\subset\R^{2}. We first carry out a rigorous analysis of the associated problem on a half-plane and then rigorously justify some of the formal arguments of [BS], obtaining an expansion for w while also proving that the first eigenfunction decays to zero somewhere along the sample boundary ?W\partial \Omega when z is not a disc. For interior decay, we demonstrate that the rate is exponential.