Papers

In this paper, we establish the well-posedness and exponential stability of a class of semilinear neutral abstract evolution equations with constant time delay. Two damping mechanisms are considered: frictional damping of the form $CC^*u_t$ and viscoelastic damping with memory. Under suitable assumptions, we prove global well-posedness and exponential decay of solutions for sufficiently small initial data. This result is of considerable importance, as it provides a unified framework for the analysis of a wide class of neutral systems arising in structural mechanics, seismic isolation, and control theory. The abstract results are illustrated by some concrete examples. The analysis relies on a combination of semigroup theory, energy estimates, Duhamel's formula, and Gronwall-type arguments.

We show that singularities of the three-dimensional cubic complex Ginzburg-Landau equation developing from smooth initial data can induce non-uniqueness of solutions after the time of blowup. The solutions we study start as backward self-similar solutions, and the source of the non-uniqueness is an unstable eigenvalue of the linearization around the forward self-similar profile generated by the singularity. We establish the existence of this eigenvalue, and hence the non-uniqueness, by a computer-assisted proof for specific values of the parameters in the equation. The work is in part motivated by similarities to conjectured behavior for the Navier-Stokes equation, with which the complex Ginzburg-Landau equation shares several important properties, such as an energy inequality and the scaling symmetry.

We prove that, for any given sufficiently regular one-particle density, there exists a unique external potential for which the solution to the many-body Schrödinger equation has this prescribed density. This implies in particular that one can exactly reproduce the time-dependent density of an interacting system using non-interacting electrons, which is at the heart of time-dependent density functional theory. Our result covers the Coulomb interaction and densities arising from extended nuclei.

We provide the first fully rigorous justification of time-dependent density functional theory in the continuum: Given a time-dependent density, we prove that there exists an external potential, unique up to a time-dependent constant, such that the corresponding Schrödinger equation reproduces the given density. This potential can be obtained with an iteration scheme. Our argument covers Coulomb interactions and does not need any Taylor expansion in time. It instead requires analyticity in space for the density and potential, which is compatible with extended nuclei.

Ding-Song-Sun proved the remarkable correlation inequality that the truncated two-point correlation function of ferromagnetic Ising models with arbitrary mixed-sign external field is largest when the field vanishes. This inequality has various important consequences such as log-Sobolev inequalities for Ising and $\varphi^4$ models up to and at the critical point. We extend the DSS inequality to the class of single spin measures satisfying the GHS condition. This is an important ingredient in our proof of the log-Sobolev inequality for the massless sinh-Gordon model on $\mathbb{R}^2$ in a forthcoming article.

We construct new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the conifold $\mathcal{Z} = \{z_1^2 + z_2^2 + z_3^2 + z_4^2 = 0\} \subset \mathbb{C}^4$. These metrics have tangent cone $\mathbb{C} \times (\mathbb{C}^2 / \mathbb{Z}_2)$ at infinity and are parametrised by their Kähler class. As the Kähler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a Calabi-Yau metric on $\mathcal{Z}$ with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point and tangent cone at infinity $\mathbb{C} \times (\mathbb{C}^2/\mathbb{Z}_2)$, thereby providing a new metric realisation of the Atiyah flop.

For the one-sided Hardy–Littlewood maximal operator $M_d^+$ on $\mathbb R^d$, the natural two-weight Muckenhoupt condition was shown by Sawyer in 1986 to characterize the weak $(p,p)$ inequality in dimension one, and by Forzani, Martín-Reyes and Ombrosi in 2011 in dimension two. At the endpoint $p=1$ in dimension three, Ombrosi and Nazarov have recently given negative answers to both the Fefferman–Stein-type question and the related weighted weak-type $(1,1)$ question \cite{OmbrosiNazarov} (personal communication). In this paper, we prove that the two-weight characterization fails for every $d\geq 3$ and $p>1$. More precisely, for every $1<p<\infty$ and every $d\geq 3$, there exist weights $w$ and $v$ such that $$ A_{p,d}^+(w,v)<\infty, \qquad \|M_d^+\|_{L^p(v)\to L^{p,\infty}(w)}=\infty. $$ The proof uses a finite two-dimensional Hardy operator whose weak operator norm is bounded below by $c_p(\log N)^{1/p'}$. This operator is embedded into the three-dimensional lattice maximal operator and then transferred to the continuous setting. A tensor extension yields the same failure in every dimension $d>3$.

The present paper addresses a problem that may be of considerable interest to a broad audience since the systems considered here operate according to a switching protocol involving two distinct dynamical regimes. Starting from an initial condition, the evolution follows a first vector field until a selected state variable $y$ reaches a lower threshold $C_1$. At this moment, the dynamics switches to a second vector field. The second regime remains active until the same variable attains an upper threshold $C_2>C_1$, when the first vector field is restored. This alternating procedure is then repeated indefinitely giving rise to a piecewise smooth vector field. A complete characterization of the $\omega$-limit sets is obtained for every admissible combination of parameters and all initial condition. The analysis is carried out by combining explicit solutions of the vector fields with geometric arguments and the first return map. Beyond the classification of limit sets, the paper describes the global bifurcation structure of the family. As the parameters vary, the system undergoes qualitative transitions between distinct asymptotic regimes, including the birth and disappearance of periodic orbits, changes in their stability, the occurrence of continuum of periodic trajectories in degenerate situations, and the replacement of bounded dynamics by monotone zig-zag motions or unbounded trajectories. The corresponding bifurcation diagrams provide a complete qualitative description of the asymptotic dynamics of the model.

We prove a growth gap for the gradient of entire anisotropic minimal graphs. For each $n\geq2$ and each smooth uniformly elliptic parametric integrand $\Phi$ on $\mathbb{R}^{n+1}$, there is an exponent $\alpha=\alpha(n,\Phi)>0$ such that every smooth nonaffine entire $\Phi$-minimal graph $u:\mathbb{R}^n\to\mathbb{R}$ satisfies $\sup_{B_R^n}|Du|\geq cR^\alpha$ for all $R\geq R_0$, for some constants $c>0$ and $R_0<\infty$ depending on the solution. In particular, gradient growth $o(R^\alpha)$ forces flatness, resolving a conjecture of Mooney and the author. The result holds in every dimension, without any assumption that $\Phi$ is close to the Euclidean area integrand.

We study a general class of nonlinear reaction-diffusion equations that model pattern-forming systems. The class includes the Gierer-Meinhardt PDE, Brusselator model, and van der Pol PDE, as well as the Gray-Scott, Klausmeier, Lengyel-Epstein, Schnakenberg PDEs and others of activator-inhibitor type. In the limit in which the activator diffusivity is much smaller than that of the inhibitor, these PDEs exhibit singular Turing bifurcations, which have only recently begun to receive attention. We analytically establish that the spatially-periodic solutions that emerge in both the sub-critical and super-critical cases of singular Turing bifurcations are spatially-periodic canard solutions. These canard patterns are new types of spatially-periodic solutions that –just beyond the Turing point– have fast-slow structure in space, rather than the classical sinusoidal profile. In addition, their amplitude grows more rapidly than the classical square root growth for Turing patterns. Indeed, even for parameter values that differ by one part in a hundred from the Turing point, they can have $\mathcal{O}(1)$ amplitude. We also establish the existence of general spatially-dependent canard solutions with fast-slow structure. Our analysis focuses on the spatial ODEs that govern the time-independent solutions. We show that these ODEs have a reversible folded saddle-node singularity of type II asymptotically close to the singular Turing point, that reversible folded saddles occur for parameters away from it, and that the true and faux canards of these folded singularities are the mechanisms responsible for creating the spatial canard solutions in the general class of PDEs.

We study $p$-spread probability measures on the Boolean lattice. We show that if a family of sets $A$ is large under the product Bernoulli-$p$ measure, then no $p$-spread measure can be supported on sets that are not covered by the union of two members of $A$, answering the fractional version of Talagrand's discrete convexity problem. Consequently, we establish a coupling theorem between $p$-spread and product Bernoulli-$p$ measures.

We study graph systems generated by $\Delta$-Y and Y-$\Delta$ moves and prove that, for initial graphs of minimum degree at least $7$, $\Delta$-Y moves alone generate every reachable graph. As a consequence, for $n\ge 8$, the $\Delta$-Y system generated from the complete graph of $n$ vertices is canonically isomorphic as graded directed graphs to the space of partial Steiner systems on $n$ points.

Let $K\subset\R^m$ be the closure of a bounded domain whose boundary is a compact embedded hypersurface, and let \( \overline M=\R^m\setminus\operatorname{int}K \) be the allowed target. We prove that if $\partial K$ is of class $C^{1,1}$, then every $\overline M$-valued local minimizer of the Dirichlet energy $E(u;B):=\int_B|Du|^2\dd x$ is locally $W^{2,\infty}$, and hence locally $C^{1,1}$, on its continuity set. For every $0<\alpha<1$, we also construct a convex body with $C^{1,\alpha}$ boundary and an everywhere continuous global minimizer that is not $C^{1,1}$. Thus $C^{1,1}$ is the sharp target regularity threshold in the Hölder scale.

We consider an inverse problem for a semilinear wave equation with time-independent damping, linear potential, and nonlinear potential coefficients in a bounded domain of $\mathbb{R}^n$ for $n\geq 2$. The main objective is to establish stability estimates for the simultaneous recovery of these coefficients from the associated Dirichlet-to-Neumann map. Our approach combines second-order linearization with suitably constructed geometric optics and asymptotic solutions. We establish Hölder-type stability estimates for the recovery of each of the three coefficients appearing in the semilinear wave equation under suitable a priori bounds on these coefficients. To the best of our knowledge, this is the first stability result for simultaneous determination of time-independent damping, linear and nonlinear potentials in a semiliner wave equation.

We prove sharp bounds for determining the spectral-type decomposition of Schrödinger operators in the spirit of Smale's program on the foundations of computation. For explicit one-dimensional self-adjoint Schrödinger operators $H=-{\mathrm d^2}/{\mathrm dx^2}+V$ on $L^2(\mathbb R)$, where $V\in C^\infty(\mathbb R;\mathbb R)$ is given by a finite description of all derivatives and derivative bounds, the pure point and absolutely continuous spectral sets cannot, in general, be recovered by any single limiting procedure. The singular continuous spectral set is strictly harder: it cannot, in general, be recovered by two nested limiting procedures. Analytic constructions of dichotomies realize the lower bounds: Gordon-type repetitions for pure point spectrum, high barriers for absolutely continuous spectrum, and an inverse spectral construction for singular continuous spectrum based on Riesz products, moment-killing perturbations, and a computational Gelfand–Levitan scheme. The finite-description framework also implies corresponding limitations on what can be certified in fixed formal systems (e.g., when used in computer-assisted proofs). Conversely, using wavelet-based certified computation, we prove matching upper bounds for broad classes of self-adjoint differential operators on $\mathbb R^d$ with coefficients of locally bounded variation and quantitative local variation control: two limits suffice for the pure point and absolutely continuous parts, and three for the singular continuous part. This provides a sharp hierarchy for spectral types.

Active subspaces identify low-dimensional linear structure in high-dimensional parameter-to-output maps by estimating the dominant eigenspace of a gradient covariance operator. In practice this covariance is replaced by a Monte Carlo estimator built from a limited number of gradient evaluations. Classical analyses based on controlling the covariance error in operator norm lead to sample-complexity estimates that can be substantially more pessimistic than the sampling rules commonly used in computations. This paper studies the empirical active subspace method directly in the projection-error metric relevant for ridge approximation. We derive non-asymptotic quasi-optimality bounds governed by a regularized inverse Christoffel function associated with the gradient field. Under a bounded-gradient assumption, the resulting estimates already improve the sample-complexity estimates obtained from operator-norm covariance bounds. We then show that additional smoothness of the gradient map, expressed through membership in a reproducing kernel Hilbert space, yields sharper coherence estimates and motivates tractable importance sampling from kernel diagonal measures. Furthermore, the same smoothness assumption yields a priori decay bounds for the population active subspace tail energy, which can be combined with our finite-sample estimate to prescribe rank, regularization scale, and sample size, allowing to fully characterize the a priori sample complexity. The abstract assumptions are verified for lognormal Gaussian and affine uniform parametric elliptic PDEs using weighted summability of Hermite and Legendre series expansions.

We construct an explicit infinite family of simple two-dimensional Cayley complexes over $\mathbb{F}_2^n$ whose degree is polynomial in $n$ and whose nontrivial vertex-link eigenvalues lie in $[-\lambda,\lambda]$ for every fixed $\lambda>0$. For every fixed $d\ge2$, we also obtain an explicit infinite family of weighted $d$-dimensional Cayley complexes over $\mathbb{F}_2^n$ with codimension-two local spectral norm at most $1/d$ and Cayley degree $\Theta_d(n)$. Our two-dimensional construction uses evaluation at rational points of algebraic curves to produce projective direction sets and many functions affine along these directions, which may be useful for further constructions and improvements.

We consider the weakly damped stochastic nonlinear Schrödinger (NLS) equation on the real line, driven by a noise that is white in time and smooth in space. Assuming that the noise is sufficiently non-degenerate, we prove that the equation has a unique stationary measure in the class of probability measures concentrated on $H^2$, and establish polynomial mixing in the dual-Lipschitz metric over $H^1$. We do not impose any restriction on the size of the damping. The proof is based on a coupling argument, whose key ingredient is a Foiaş-Prodi-type estimate in the $H^1$-norm, derived by means of a Lyapunov functional adapted to the linearized NLS dynamics. To compensate for the loss of compactness, we combine this estimate with a truncated Poincaré inequality and a space-time weight function quantifying the spatial decay of solutions.

Covariance-sensitive analyses of Gaussian annealing for sampling from a convex body require controlling how much covariance can grow under a radial Gaussian tilt. For an isotropic convex body $K\subseteq\mathbb{R}^n$, let $\mu_{K,t} (\mathrm{d} x) \propto e^{-t\| x \| ^2} \mathbb{1}_K(x)\,\mathrm{d} x$, and let $Q_n$ be the supremum of $\|\operatorname{Cov}(\mu_{K,t})\|_{\mathrm{op}}$ over all such $K$ and all $t>0$. We prove the sharp bound $Q_n=\Theta(n^{2/5})$, closing the gap between the known $\Omega(n^{1/3})$ lower bound and the $O(\sqrt{n\log(en)})$ upper bound. The upper bound applies not only to uniform measures on convex bodies but to every compactly supported isotropic logconcave probability measure. It combines a dimension-free variance bound for quadratic forms with a Rényi comparison at a nearby time, projected moment estimates, and relative-entropy control along the Gaussian-tilt path. For the matching lower bound, we construct an explicit unconditional convex body whose axial coordinate is coupled to the transverse quadratic energy. Moderate-deviation estimates show that an appropriate tilt creates directional variance $\Omega(n^{2/5})$.

This paper is concerned with exact high-order generalized polarization tensor (GPT) cancellation in two-dimensional noncircular multilayers with finite positive isotropic coating conductivities. We first prove that, for every admissible cyclically symmetric core–shell geometry, there exists a unique positive coating conductivity for which the structure is weakly neutral to every uniform incident field. For higher orders, we establish locally unique coating conductivities that cancel the complete \(K\)-contracted GPTs (K-CGPTs) block on a broad class of fixed noncircular multilayers with cyclic symmetry. This includes homothetic multilayers generated by strictly star-shaped symmetric domains, without assuming that the geometry is close to concentric disks. We also replace the near-background condition by a small-core condition covering finite, insulating, and perfectly conducting cores. By varying selected interface modes together with the coating conductivities near a radial GPT-vanishing structure, we further construct exact families with neither rotational nor reflection symmetry. For three coatings, the relevant nondegeneracy conditions and an explicit chiral example are rigorously certified. Finally, we show that the resulting cancellation yields enhanced near-cloaking for every entire harmonic incident field. A set of certified numerical examples fully corroborates the theoretical predictions established in this work.

For $d\geqslant3$, let $p^*=\frac{2d}{d-2}$. We prove that the Born-Jordan distribution maps $L^{2,p^*}(\mathbb R^d)\times L^{2,p^*}(\mathbb R^d)$ boundedly into $L^{p^*}(\mathbb R^{2d})$. The Lorentz index $p^*$ cannot be increased in either variable. In particular, this gives the endpoint $L^2\times L^2\to L^{p^*}$ estimate, settling the critical boundedness problem raised by Stra, Svela, and Trapasso \cite{SST26}.

We give a negative solution to the complemented subspace problem for Banach spaces with unconditional bases over both the real and complex fields. For every $\rho>0$, we construct a projection $P_\rho$ of norm less than $1+\rho$ on a separable superreflexive space \begin{equation*} X_\rho=\left(\bigoplus_{j=1}^{\infty}\ell_{p_j}^{N_j}\right)_2, \qquad p_j\downarrow2, \end{equation*} such that $Z_\rho=P_\rho(X_\rho)$ and its dual $Z_\rho^*$ have Schauder bases but admit no unconditional bases. Over the real field, both spaces have Gordon–Lewis local unconditional structure (GL-lust) but fail Dubinsky–Pełczyński–Rosenthal local unconditional structure (DPR-lust), disproving a conjecture of Figiel, Johnson and Tzafriri. In particular, neither is isomorphic to a Banach lattice, giving a negative solution to the separable Banach-lattice complemented subspace problem. A modification of the construction also shows that the class of separable real Banach lattices is not primary. A Lean 4 formalisation of the main results accompanies the paper.

We prove a refinement of Hirzebruch's signature formula on the individual Hochschild diagonals of a smooth projective complex variety. Natural Hermitian forms, obtained by symmetrizing a Todd-corrected pairing, have explicitly computable positive and negative indices and are preserved by derived equivalences. On diagonals of the same parity as the dimension, their signatures separate the even and odd contributions to the Hochschild-Kostant-Rosenberg decomposition; on the other diagonals, their radicals and signatures are governed by the first Chern class. We deduce the derived invariance of $h^{n-2,0}$, $h^{n-1,1}$ and $h^{n-3,1}$ in any dimension, and of $h^{2,0}$ when the canonical bundle is trivial. Combining the invariance of $h^{2,0}$ with the multiplicative structure of Hochschild cohomology and Verbitsky's orthogonal action, we give a new short proof of the theorem of Huybrechts and Nieper-Wißkirchen that a derived partner of an irreducible holomorphic symplectic variety is again irreducible holomorphic symplectic. In dimension five with trivial canonical bundle, the signature invariants and the Libgober-Wood identity together establish the derived invariance of all Hodge numbers

Motivated by a log-convexity question of Bennett and Tao, we consider Gowers-type functionals defined by $L^p$ norms of multiple autocorrelations. We prove sharp bounds in terms of $L^q$ norms and characterise the near-extremisers on Euclidean spaces as well as locally compact abelian groups. We also establish two families of degree-lowering inequalities and study their near-extremisers. As a byproduct of this broader theory, we show that the constant in the log-convexity estimate for Gowers norms is strictly less than unity, confirming the aforementioned conjecture of Bennett and Tao. Finally, we characterise the $p$ for which the Gowers-type functionals necessarily satisfy the triangle inequality on nonnegative measurable functions.

We ask how small an increasing integer model of a finite real set can be while preserving all equalities between equal-length sums of at most $q$ elements, with repetitions allowed. The increasing correspondence must preserve exactly which sums are equal; the signs of unequal comparisons may change. Let $H_m(q)$ be the least diameter sufficient for every ordered real set of size $m$. For every integer $q\ge2$ we prove $H_4(q)=q(q+1)$ and $H_5(q)=q^2(q+1)$, and we determine $H_6(2)=24$. The proofs use the linear space determined by the additive equalities and the inequalities prescribing the order of the elements. They also give bounds for larger sets, determine the first two asymptotic terms for each fixed $m\ge4$, and yield further families of sharp examples. An application bounds the integer alphabets needed to realize finite additive-square-free spectra. The supplementary data support the finite six-element classification.

The classical Melan equation modeling suspension bridges is considered. We first study the explicit expression and global properties of the analytical solution for the simplified “less stiff” model, based on which we derive an a priori estimate for the original classical Melan equation and establish, under an explicit load condition, that it has a unique solution with nonnegative integral under downward live loads, thereby showing the uniqueness of the corresponding deflection curve in the engineering setting. We also develop an efficient iterative approximation method by taking the solution of the simplified “less stiff” model as the first iterate, and prove its geometric convergence with explicit error estimates. The applicability and computational efficiency of the method are demonstrated through calculations for two actual bridges, which also quantify the influence of the nonlinear nonlocal term on the solution and clarify the relationship between the simplified and original models. Several engineering observations are verified and explained, and some related open problems are suggested.

The $n$-$(n+1)$-$(n+2)$ theorem, recently established by Cohen and Shusterman, says that if two groups of type $\mathrm{F}_{n+1}$ map onto a common quotient $Q$ of type $\mathrm{F}_{n+2}$ and one of the two kernels is of type $\mathrm{F}_n$, then the associated fibre product is of type $\mathrm{F}_{n+1}$. We prove a stronger, symmetric variant of this theorem when $Q$ is virtually nilpotent. More precisely, for $i\in\{1,2\}$, let $1\to N_i\to \Gamma_i\overset{\small\pi_i}{\to} Q\to 1$ be short exact sequences of groups and let $P=\{(\gamma_1,\gamma_2)\in\Gamma_1\times\Gamma_2\colon \pi_1(\gamma_1)=\pi_2(\gamma_2)\}$ be their fibre product. Let $k,l,m,n\in\mathbb N_0$, assume that $N_1$ and $N_2$ are of type $\mathrm{FP}_k$ and $\mathrm{FP}_l$, respectively, and that $\Gamma_1$ and $\Gamma_2$ are of type $\mathrm{F}_m$ and $\mathrm{F}_n$, respectively. We prove that $P$ is of type $\mathrm{F}_{\min\{k+l+1,m,n\}}$. The same formula holds with the homological finiteness properties $\mathrm{FP}_r$ in place of $\mathrm{F}_r$. This provides an alternative proof of the homotopical and homological Virtual Surjections Theorem, which gives a finiteness criterion for subgroups of direct products in terms of their embedding in the ambient product.

We study the dynamics of a many-body quantum system strongly interacting with a bosonic reservoir. The coupling is given by a potential operator for the system and it is linear in the field. We show that in the limit of infinite coupling strength the system undergoes instantaneous spatial decoherence. To resolve the decoherence process in time we consider large coupling strengths $\lambda$ and short times scaling as $t\propto \lambda^{-\alpha}$, with $\alpha\ge0$, in the limit $\lambda\rightarrow\infty$. On short time scales $\alpha>1$ the dynamics is trivial while on longer ones $0\le\alpha<1$ the decoherence is instantaneous. We show that the decoherence process is resolved exactly for $\alpha=1$, defining the fine-grained time scale $\tau=\lambda t$. We construct an approximate effective evolution map of the many-body system with controlled error estimates. Generically, the effective dynamics is markovian, but not given by a dynamical semigroup. As a physical application we give a rigorous description of the phenomenon of localization of macroscopic quantum objects in position space.

For every subsonic speed $c\in(0,\sqrt2)$, we prove the existence of a finite-energy traveling wave for the planar Gross–Pitaevskii equation. This resolves the longstanding problem of the existence of prescribed-speed traveling-wave solutions in two dimensions, explicitly stated as open by Mariş (Ann. of Math., 2013) and Bellazzini and Ruiz (Amer. J. Math., 2023). The proof relies essentially on the energy estimate \begin{equation*} E(\psi)\le C_J\bigl(I_c(\psi)+\ind(\psi)\bigr), \qquad c\in J, \end{equation*} where $E$ is the energy, $I_c$ the action at speed $c$, $\ind$ the real Morse index, $J$ is any compact interval contained in $(0,\sqrt2)$, and $C_J$ is a positive constant depending only on $J$. We also prove finite-bubble compactness, including splitting of the energy, action, potential energy, and momentum, and attainment of the action among nonconstant waves of Morse index at most one.

We prove several new structural and classification results about $d$-regular graphs with positive Lin–Lu–Yau (LLY) curvature. We show that any positively curved $d$-regular graph has diameter at most $2d-2$, which improves the previously best known diameter bound obtained from the Bonnet-Myers-type theorem for positively curved graphs. We further show that every positively curved $d$-regular graph with $d\ge 3$ is $3$-connected. We classify all positively curved $3$-regular graphs, as well as all positively curved regular planar graphs.