Papers

We introduce the concept of rough additive functions, which extends rough paths theory to line integrals of distributional 1-forms. In the context of gauge theory, we use this notion to define controlled gauge transformations and holonomies via RDEs. One of our main results is a rough version of Uhlenbeck compactness for distributional connections based on rough additive functions. The main ingredient is a singular elliptic PDE to obtain a Coulomb gauge. We solve this PDE using regularity structures and a method inspired by the implicit function theorem. Surprisingly, the model needed to solve the equation is determined entirely from rough additive functions, which is a simpler and geometrically more natural object. We apply our results to the Yang-Mills measure on the unit square, showing that it has a gauge-fixed representation with optimal regularity. Although we focus on the unit square in this article, we expect our results to apply to more general surfaces.

In addition to reflection and refraction, another form of wave deviation is defined as diffraction. Notably, when incident waves strike a corner, diffracted waves emanate from the corner tip and propagate omnidirectionally. This paper proposes a novel framework for detecting rigid cornered obstacles using measured diffracted wave data. The framework first transforms the underlying initial-boundary value problems into initial value problems on conic manifolds via the method of images. Subsequently, the retrieval of obstacle information is achieved through Cheeger–Taylor functional calculus and microlocal analysis on conic manifolds. Specifically, we prove that for a given pulse, measurements of the resulting diffracted waves captured by a curve receiver uniquely determine both the location and shape of the visible portion of a polygonal obstacle. The proof is constructive, explicitly formulating the corresponding recovery scheme. This methodology offers two key advantages: first, the size and placement of the receiver can be arbitrary; second, the inversion only requires measurements of diffracted waves and obviates the need to solve wave equations within the cornered domain as in conventional methods.

We discuss an interesting distinction between the incompressible Navier-Stokes equations (the velocity equations) and its vorticity form in whole space. We show that if the initial vorticity has a Gaussian bound then the bound is inherited up to the maximal lifespan of the strong solution. However, it turns out that the velocity equations don not share the same property. In fact, $L^p$-strong solutions to the velocity equations arising from “well-localized" initial value generally behave at infinity like derivatives (of order $\geq3$) of the fundamental solution of Laplacian. To show this, a clean expansion up to maximal lifespan is derived : \begin{align} u(x,t)=-\nabla\sum_{|\alpha|=0}^{d-1}\frac{(-1)^{|\alpha|}}{\alpha !}\partial^\alpha\partial_{i,j}^2\Gamma(x)\int_0^tM_{\alpha}^{i,j}(s)ds+O\big(|x|^{-2d-1}\big)\nonumber \end{align} where ${\rm M}_{\alpha}^{i,j}(t):=\int_{\mathbb R^d}y^\alpha u^i(y,t)u^j(y,t){\rm d}y$ and $\Gamma$ is the fundamental solution of Laplacian. This improves the first order expansion given by L. Brandolese and F. Vigneron \cite{BV07}.

We establish the existence of a weak solution to the Voltage-Conductance kinetic equation for any conductivity parameter and any reasonable initial datum, and in particular without smallness condition as it was the case in [9]. The proof is based on some a priori estimates mainly drawn from [15], some trace results adapted from [14] and a DiPerna-Lions weak stability argument.

In this paper, we continue investigate the structure of the weak $\sigma$-attractor $\mathcal{A}$ for the 2D Boussinesq equations with viscosity and without heat diffusion. Specifically, we show that the projection of the attractor $\mathcal{A}$ onto the temperature component is surjective, i.e., $P_{\theta}\mathcal{A}=L^2$; and, for any $r>0$, the projection of $\mathcal{A}_r$ onto the velocity component contains a infinite-dimensional Hilbert ellipsoid, which implies $P_u\mathcal{A}_r$ is infinite dimensional immediately.

We study an inverse obstacle problem for the fractional Schrödinger operator $(-\Delta)^s+q$, $0<s<1$. For each exterior datum, the state is constrained by a prescribed obstacle in a bounded domain and satisfies the fractional Schrödinger equation only in the associated noncontact set. This set is unknown and depends on the coefficient, so the exterior Dirichlet-to-Neumann map is nonlinear. We show that the nonlocal character of the equation gives a direct way around this moving-free-boundary difficulty. Equality of one obstacle measurement on an exterior open set forces equality of the two corresponding obstacle states in the whole space. In their common noncontact set one then obtains $(q_1-q_2)u=0$. This identity yields recovery of the potential in the exposed region by measurable unique continuation when $s\in[1/4,1)$, and by the usual unique continuation principle together with continuity when the potentials are continuous. We also prove a geometric coverage theorem for nonnegative potentials: rational positive scalings of one nontrivial nonnegative exterior datum expose the whole domain up to a null set. Consequently, under the corresponding assumptions, a countable family of nonlinear exterior obstacle measurements determines the potential globally.

This paper investigates a variational model that describes a system of N coupled sphere-valued fields through a penalization term imposing the constraint that their sum coincides with a prescribed sphere value map. We analyze the asymptotic behavior and regularity of minimizers as the penalization parameter tends to infinity. A fundamental dichotomy is established according to the number of interacting fields. For N = 2, topological obstructions may prevent the existence of exact sphere-valued decompositions, leading to the blow-up of the minimal energy in the large-penalization regime. In contrast, for N $\ge$ 3, exact decompositions always exist, yielding uniform energy bounds and allowing the identification of the limiting constrained problem via $\Gamma$-convergence. We further derive an explicit characterization of the limiting energy by decomposing admissible configurations into an average field and fluctuation components. Finally, we establish a gap phenomenon showing that, under suitable topological assumptions, every minimizer is necessarily singular for sufficiently large values of the penalization parameter. These results highlight the fundamental role of topology in the asymptotic behavior and regularity of coupled sphere-valued variational systems and extend previous results on single-field models to the multi-field setting.

We introduce a two-relaxation-times (TRT) kinetic approximation for scalar non-linear conservation laws in one space dimension, addressing the convergence of relaxation approximations to entropy solutions. The proposed TRT system, derived from a lattice Boltzmann scheme, generalizes the classical BGK (single-relaxationtime) framework. Requesting quasi-monotonicity of the relaxation operator, we establish global existence of solutions for the TRT system and prove their convergence, using a lattice Boltzmann scheme, to the entropy solution of the original conservation law as the relaxation times vanish. Moreover, we propose a qualitative analysis, clarifying the role of relaxation parameters and equilibrium coefficients in shaping solutions, via Chapman-Enskog expansion and looking at travelling wave solutions.

This article is concerned with the study of the focusing energy-critical wave equation on the three-dimensional Riemannian manifold $\mathcal{M}=\mathbb{R}^3$ equipped with a smooth time-independent Riemannian metric $g$, which coincides with the Euclidean metric outside a compact set. Our focus is on Type II blow-up solutions, namely finite-time solutions that remain bounded in the energy space, with the goal of revealing their fundamental properties. We establish in particular lower bounds for the energy norm as the solution approaches its maximal time of existence in terms of the bounds of the ground state of the Euclidean problem.

In this paper we establish the pointwise-in-time dispersive decay for solutions to the mass-critical nonlinear Schrödinger equation in spatial dimensions $d\geq3$. Our argument relies on a delicate decomposition of the nonlinearity and an improved linear estimate, which together enable us to control the nonlinear contribution. This work unifies a framework for extending the foundational results established in an earlier paper by Fan, Killip, Visan, and Zhao [Math. Z. 311(1), Paper No. 21, 16 pp (2025)], where the same problem was addressed for spatial dimensions $d=1,2,3$, to the general higher-dimensional setting $d\geq3$.

This paper focus on a class of singular or degenerate fully nonlinear elliptic equations with Hamiltonian terms under oblique boundary conditions on $C^{1}$ domains. Under quite general conditions on the singularity/degeneracy of the model, we establish the sharp $C^{1,\alpha}$ regularity up to the boundary within a unified framework.

In this paper, we establish a doubling argument to obtain Hessian estimates for convex solutions to the Hessian quotient equation $\frac{\sigma_n}{\sigma_k}(D^2u) = f(x,u,Du)$ for $k=n-1$ and $k=n-2$ under the condition that $\log f$ is convex in the $Du$ variable. In particular, our approach is pointwise and does not make use of the Legendre transform or integral-based local maximum principles. We provide a counterexample demonstrating that interior estimates can fail if no structural assumption is imposed on $f$ in the $Du$ variable. Finally, we extend our doubling argument to general Hessian quotient equations $\frac{\sigma_l}{\sigma_k}(D^2u) = f(x,u,Du)$ for $k \in \{l-1, l-2\}$, under a similar structural condition imposed on $f$ in the $Du$ variable, alongside an additional structural concavity assumption on the operator introduced by Lu-Tsai 2026, which has very recently been established in independent works.

We initiate the study of the nonlocal to local limit for linear transport equa- tions. A nonlocal version of the linear transport equation is introduced alongside an appropriate well-posedness theory for distributional solutions with measure initial data. We proceed by deriving the associated linear transport equation as a nonlocal-to-local limit. Some of the results apply in arbitrary dimension and for measure initial data.

We investigate the initial-boundary value problem for the three-dimensional isentropic compressible Navier–Stokes equations in the upper half-space with the slip boundary conditions. We prove the global existence and uniqueness of strong solutions in the presence of vacuum and large oscillations. Although scaling frameworks for compressible flows in domains with boundaries have been developed in several settings, the global well-posedness result in the half-space remains far from complete. The system with far-field vacuum admits a natural scaling structure that preserves both the half-space geometry and the slip boundary conditions. Motivated by this scaling, we identify the following scaling-invariant initial quantity: $$ \left[ \|\rho_{0}\|_{L^{\infty}}^3 \left( \frac12\|\sqrt{\rho_0} u_0\|_{L^{2}}^{2} +\frac{1}{\gamma-1}\|P(\rho_{0})\|_{L^{1}} \right) +\|\rho_{0}\|_{L^{\infty}}^{\frac{3-\gamma}{2}} \right] \left( \|\nabla u_0\|_{L^2}^2 +\|P(\rho_{0})\|_{L^2}^2 \right). $$ Under the assumption that this quantity is sufficiently small, we establish the global well-posedness of strong solutions. This result provides a half-space counterpart of the scaling-invariant global theory for the Cauchy problem established by Wen (Adv. Math. 482 (2025), Paper No. 110628) and shows that the slip boundary condition is compatible with the system.

We study overdetermined boundary problems for the graphical translating-soliton equation \[ -\operatorname{div}\!\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) =\frac{1}{\sqrt{1+|Du|^2}} \quad\text{in }\Omega, \qquad \partial_\nu u=\Gamma H+C \quad\text{on }\partial\Omega, \] where $\Gamma, C$ are constants, $H$ is the mean curvature of the boundary $\partial\Omega$ of the regular bounded domain in $R^n$ such that $H_{\partial B_R}=-1/R$. For $\Gamma\geq0$, we prove that constant Dirichlet data force a bounded domain to be a ball. We also prove a spherical-shell rigidity theorem for a doubly connected domain with two ordered boundary heights and $a<u<b$ in the interior. The argument combines linearization under reflection, Reichel's critical-plane and annular continuation principles, curvature comparison, Serrin's corner lemma, and a radial ODE that excludes the annular alternative in the one-height problem. Finally, we give explicit counterexamples showing the sharpness of the sign, ordering, connectedness, and nesting assumptions.

This paper studies a new class of nonlocal optimal control problems where the constraints involve approximations of minimizers of quasiconvex energies. These problems are parameterized by a fractional parameter $s \in (0, 1)$ and a horizon parameter $\delta > 0$, and the energy density depends on a nonlocal fractional gradient. Here, the constraint consists of finding quasi-minimizers with respect to the quasiconvex energy. Despite the fact that the energies of interest may not have unique minimizers, we may prove the existence of solutions to this class of control problems. The constraint in Cueto-Siktar 2026 was finding global minimizers of the energy, and this problem's main limitation was an inability to prove convergence of solutions for the nonlocal control problems to those of a corresponding local, PDE-constrained optimal control problem. While this issue arises from the lack of uniqueness of minimizers for the constraining energy, we get stronger convergence results with our new choice of constraints. Namely, we obtain convergence of minimizers for nonlocal optimal control problems that have a general cost functional depending on the nonlocal gradient.

It is proved for $k>n/2$ that the Dirichlet problem of the $k$-Hessian measure on a (nonsmooth and nonuniformly) $k$-convex (also known as $k$-hyperconvex) domain for any finite Borel measure as right-hand side and boundary data taken from a $k$-convex and continuous function on the closure of the domain has a unique solution. Merely continuous boundary data are sufficient for strictly $k$-convex domains (i.e., if there exist strong barriers).

We derive and analyze a binary Poisson-Nernst-Planck system with steric interactions and interspecies drag through the energetic variational approach. The steric effects are incorporated into the free energy, while the drag mechanism enters the dissipation functional; eliminating the transport velocities yields a non-diagonal, concentration-dependent Onsager mobility and an entropy-production structure that is not coercive in the standard $L^2(0,T;H^1)$ sense. For the resulting drag-modified steric PNP system, we prove the existence of global finite-energy weak solutions using an entropy-variable approximation, weighted gradient estimates, and a vacuum-compatible square-root formulation of the weighted entropy gradients. In the pure Neumann equal-mass setting, we establish a sublevel entropy-entropy production inequality, obtain exponential relaxation for approximation-generated weak solutions, and identify the sharp small-sublevel limit of the optimal entropy-production constant through an explicit linearized formula involving the drag mobility, steric Hessian, Poisson coupling, and Neumann spectrum. We further show that the same linearized constant governs the local nonlinear relaxation of sufficiently small strong perturbations of the homogeneous equilibrium. Finally, we discuss the rank-one steric limit and clarify the role of the positive definiteness of the steric matrix in the finite-energy compactness theory.

We develop an inverse theory for time–frequency localization operators, formulated as a free-boundary problem in which the localization domain is unknown and its boundary is recovered from prescribed spectral data. The guiding principle is that an eigenfunction can itself be regarded as geometric data determining a localization domain, with strong consequences for the associated variational problem. We obtain five main results. First, for every polynomial $f_0$ sufficiently close to the Gaussian and every $\lambda\in(0,1)$, we construct a real-analytic domain $U_\lambda$ such that $f_0$ is an eigenfunction of the corresponding localization operator with eigenvalue $\lambda$, providing a general inverse construction of localization domains. Second, we recover the null-set invariant Abreu–D"orfler characterization of disks as the only simply connected localization domains having a Hermite polynomial as eigenfunction. Third, we show that simple connectivity is essential: a weighted Hele–Shaw flow yields an infinite-dimensional family of non-radial, doubly connected analytic domains for which the Gaussian is an eigenfunction. Fourth, we prove the optimality of the exponent $1/2$ in the quantitative stability inequality of G'omez–Guerra–Ramos–Tilli. Finally, we prove that every local maximizer of the Gaussian Faber–Krahn problem is a disk and, using a Fock-space concentration–compactness profile decomposition, obtain a new proof of the Nicola–Tilli concentration theorem.

Let $M=\mathbb{H}^n\times\mathbb{S}^m$, where $n\geq 2$, $m\geq 1$, and $N=n+m$. Let $P_k$ be the order-$2k$ GJMS operator, with $1\leq k<N/2$, and assume that $\Lambda_0=\inf\sigma_{L^2(M)}(P_k)>0$. We study$$P_kU-\lambda U=|U|^{q-2}U,\qquad q=\frac{2N}{N-2k},\qquad 0<\lambda\leq\Lambda_0,$$and attainment of the associated critical quotient $S_{\lambda,k}(M)$. Let $S_{N,k}$ be the Euclidean best Sobolev constant. For $0<\lambda<\Lambda_0$, the inequality $S_{\lambda,k}(M)<S_{N,k}$ implies attainment and a nontrivial weak solution. Localized Euclidean extremals establish this inequality when $N\geq4k$, or when $2k+2\leq N<4k$ and $\lambda>\Lambda_{\mathrm{loc}}$, where $\Lambda_{\mathrm{loc}}$ is explicit. If $N\geq2k+2$ and $S_{\Lambda_0,k}(M)<S_{N,k}$, attainment also holds at $\lambda=\Lambda_0$ in the threshold form completion. At the threshold, $L^2$-coercivity fails precisely on the constant spherical eigenspace. We combine cocompactness for its hyperbolic coefficient with a profile decomposition relative to the critical transformations preserving $\mathcal{A}=\mathbb{R}^{n-1}\times{0}$. Under the threshold hypotheses above, the strict Euclidean inequality excludes concentration escaping $\mathcal{A}$ from normalized minimizing sequences. If $r_j^{(J)}$ denotes the remainder after the first $J$ extracted profiles, then$$\lim_{J\to\infty}\limsup_{j\to\infty}|r_j^{(J)}|_{L^q(\mathbb{R}^N)}=0,$$which yields compactness modulo the axis-preserving transformations.

We study the evolution of a viscous incompressible fluid whose initial vorticity is supported on a smooth open curve. We show that, for $\nu t\ll 1$ and sufficiently small Reynolds number $\Gamma/\nu$, the vorticity is described at leading order by a Lamb–Oseen type vortex concentrated around a curve evolving according to the binormal flow predicted by the localized induction approximation. The solution is written as an explicit leading-order profile plus a lower-order perturbation, which is controlled in a Morrey $\mathcal{M}^{\infty}$ norm. Then, we apply this construction to the Hasimoto soliton. In this case, the estimates are uniform with respect to the torsion parameter, allowing us to consider a large-torsion regime in which the soliton undergoes a macroscopic displacement. We show that the corresponding Navier–Stokes solution contains a localized portion of the kinetic-energy distribution, associated with the binormal velocity, which remains concentrated inside a moving physical region and undergoes an order-one displacement during an admissible time interval.

Let $n\ge3$ and $1<p<n$. We first prove the local trace analogue of the sharp one-bubble critical-point stability theorem of Liu and Zhang~\cite{LiuZhang2025}: near a positive trace-bubble, the Euler–Lagrange residual controls the gradient distance to the normalized trace-bubble manifold with the sharp power $\max\{1,p-1\}$. Then, we establish a Struwe-type compactness theorem for the critical trace functional, which gives the trace counterpart of the Mercuri–Willem decomposition~\cite{MercuriWillem2010}. Combining Struwe-type compactness with the local stability estimate yields a sharp quantitative one-bubble critical-point stability theorem.

This paper investigates inverse two-phase Stefan-type problems for parabolic heat equations with unknown time-dependent source and reaction coefficients. Suitable transformations reduce the original free-boundary problems to equivalent equations on fixed spatial domains, and Fourier spectral expansions are used to derive reconstruction formulas for the unknown coefficients. In the first formulation, the source coefficients are identified from integral, pointwise, and nonlocal additional data, leading to Volterra integral equations. In the second formulation, an exponential transformation is applied to recover the reaction coefficients. A principal advantage of the model with two moving boundaries is that each boundary provides its own Stefan condition which sufficient to determine the two unknown time-dependent coefficients without additional overdetermination conditions. Under appropriate assumptions the existence, uniqueness, boundedness and regularity of weak and strong solutions are established. Illustrative examples confirm the applicability of the reconstruction procedure and show that the coefficients remain stable under the noisy data. The proposed approach provides a rigorous base for identifying time-dependent thermal parameters in two-phase phase-change processes.

We study a free boundary problem of minimising a functional containing a non-local term rewarding depth into the zero phase: for $u\geqslant 0$ on a bounded, open set $\Omega\subset\mathbb R^d$, we minimise $$J(u) = \int_\Omega \left(\frac12|\nabla u|^2 - fu\right) \;-\; \int_{\{u=0\}} F\big(\mathrm{dist}(x,\partial \{u=0\})\big)\;\mathrm{d}x. $$ This kind of functional arises, for example, from a two-membranes problem with an adhesive contact energy. We first address a well-definedness issue caused by the non-local, boundary-sensitive nature of the functional and prove existence of minimisers, establishing along the way a weak lower semicontinuity result for the non-local term. We then derive stationarity conditions for minimisers, including a variational (Euler–Lagrange type) inequality, a PDE on the positivity set, and, under a mild non-degeneracy assumption, a free boundary condition obtained via inner variations and a Danskin-type differentiation of the distance function.

We analyze the training dynamics of variational physics-informed neural networks applied to linear coupled parabolic convection–diffusion–reaction systems, called heterogeneous when only a subset of the components undergoes convective transport. In the neural tangent kernel regime, the gradient flow on the space-time variational residuals reduces to a linear differential system whose operator is a Gram matrix built from the space-time symbol of the system operator and the matrix tangent kernel. The main result is a double screening theorem. Under dominant convection, the Schur complement of this matrix relative to the block of convective components converges to an expression that involves only the diffusive block of the symbol, with no coupling term, together with the Schur complement of the tangent kernel. From this we derive four consequences, namely an exact identity quantifying the screened coupling energy, a degradation law for the training rate governed by the canonical correlations of the kernel, a bound on the condition number in terms of the Péclet number, and the non-participation of temporal frequencies in the screening mechanism. The growth of the condition number slows down shared-step gradient descent, whereas the continuous flow suffers no slowdown, which makes the training difficulty attributable to the optimizer rather than to the approximation. We finally show that the Adam optimizer, through its adaptive scaling, mitigates this difficulty provided that the architecture separates the parameters associated with the convective and diffusive components, confirming the architectural prescription that follows from the second screening. The predictions are validated numerically down to machine precision, on a two-dimensional exchanger, and by the full training of finite-width networks.

We prove that given any sufficiently small and regular solution $B$ of the stationary Euler equations there exists an infinite dimensional family of solutions $(u,b)$ of the non-resistive magnetohydrodynamics equations (MHD) that relax to $(0, B)$. More precisely, $(u,b) \to (0, B)$ exponentially fast as $t \to +\infty$. This family may be viewed as lying in the stable manifold of the non-resistive MHD equations around the equilibrium state $(0, B)$. The problem whether it actually coincides with the stable manifold remains open. As a byproduct of our result, we provide a large class of global regular solutions of the non-resistive MHD equations. Another consequence is that any sufficiently small and regular solution of the stationary Euler equation is (non-trivially) topologically accessible via MHD from a large class of magnetic fields according to the definition of Moffatt and, in this scenario, the topology of the magnetic lines is (entirely) preserved in the limit $t \to + \infty$.

The paper studies a dynamical system in the Wasserstein space, where the evolution is governed by a Fokker-Planck-Kolmogorov equation with coefficients chosen from a prescribed set that depends at each point on that point and on the current measure. Under the assumption that the multivalued mapping determining the constraints is convex-valued, upper semicontinuous and satisfies certain growth conditions, we prove an existence theorem and establish the compactness of the solution set. Moreover, we study an optimal control problem for such dynamical systems.

In this paper, we consider the Navier-Stokes initial boundary value problem in exterior domains with initial data in the Lebesgue space $L^p$, $p\in (2,3)$, and show the existence of a global weak solution. For the quoted solution, we also furnish a structure theorem. In particular, we see that the weak solution becomes regular after a certain instant of time and it is also regular a. e. in time. Although a general $L^p$-theory for local strong/mild solutions is well-established in the literature, a corresponding theory for global weak solutions in exterior domains seems to be new. Of course, our results hold in the particular cases of the Cauchy problem and the initial boundary value problem in a half-space.

We study the spectrum of the Orr–Sommerfeld operator associated with the incompressible plane Poiseuille flow in the high-Reynolds-number regime. In the Tollmien–Schlichting eigenvalue region, we prove the existence and uniqueness of the lower and upper branches of the neutral curve. For each sufficiently small fixed wavenumber $\alpha$, the viscosities corresponding to the lower and upper neutral branches satisfy $\nu\sim |\alpha|^7$ and $\nu\sim |\alpha|^{11}$, respectively. Equivalently, for each sufficiently small viscosity $\nu$, the corresponding lower and upper neutral wavenumbers satisfy $\alpha^2\sim \nu^{2/7}$ and $\alpha^2\sim \nu^{2/11}$, respectively. We also establish the simplicity of the neutral eigenvalues and verify the transversal crossing condition on both neutral branches. The proof is based on a boundary-adapted version of the Rayleigh–Airy iteration scheme, together with precise expansions and refined estimates for the correction terms and their parameter derivatives. These spectral results verify the assumptions required in the classical Hopf bifurcation framework of Joseph–Sattinger \cite{JS1972} and Iooss \cite{Iooss1972}, and hence yield traveling-wave solutions bifurcating from the plane Poiseuille flow at the neutral points.

Global existence and boundedness of weak solutions for a fully parabolic degenerate chemotaxis system with indirect signal production are proved for any initial data in the subcritical case and under smallness conditions in the critical and supercritical cases. To construct weak solutions, a time discrete scheme is set up, for which the first equation has a gradient flow structure with respect to the 2-Wasserstein distance, while the other two equations feature an L\textsuperscript{2}-variational structure. The proof relies in particular on the flow interchange method and discrete maximal regularity.