Papers

We study inference on the strength of Gaussian multiplicative chaos from one geometric graph with unobserved vertex positions. In the planar model, the edge-count statistic has a piecewise deterministic limit with a transition at $\gamma=1$, whereas degree quantiles consistently estimate $\nu=\gamma^2/2$ throughout the subcritical range. Fractional moments give uniform finite-sample risk bounds. At finite resolution, common graph envelopes calibrate tests over continuous parameter boxes with unknown Poisson intensity. They give simultaneous coverage under adaptive refinement for a fixed statistic, and conditional coverage for an independently piloted size–degree residual on a fixed partition. For the exact periodic FFT–pixel approximation, we prove explicit total-variation rates for the annealed spatial Cox law and the observed graph law, uniformly on every compact subcritical range $0\le\gamma\le\Gamma<2$. The rates allow intensity to grow with resolution and imply uniform asymptotic coverage after recalibration at each resolution. Numerical studies examine parameter refinement, residual calibration, and multiresolution stability.

The center of distances $C(Y)$ of a metric space $(Y,\rho)$ is the set of all $t\geq 0$ for which the equation $\rho(y,p)=t $ has a solution for each point $p\in Y$. We say that a simple graph $CG_Y$ is the central graph of a metric space $(Y,\rho)$ if $Y$ is the vertex set of $CG_Y$ and distinct vertices $x,y\in Y$ are adjacent if and only if $\rho(x,y)\in C(Y)$. We prove the inequality $|E(CG_Y)| \geq \left\lceil \frac{1}{2}|Y| \right\rceil $ for all finite metric spaces $(Y,\rho)$ with $|Y|\geq 2$ and non-empty $CG_Y.$ It is also proved that the inequality $|E(CG_X)| \geq |X|-1 $ holds for each finite ultrametric space $(X,d)$. The central graphs of metric spaces $(Y,\rho)$ and ultrametric spaces $(X,d)$ satisfying $|E(CG_Y)|=\left\lceil \frac{1}{2}|Y| \right\rceil $ and, respectively, $|E(CG_X)|=|X|-1 $ are described up to graph isomorphism.

In this paper, we establish a quantitative finite-time approximation of scattering fields for nonlinear Alfvén waves in three-dimensional ideal incompressible magnetohydrodynamics near a strong constant magnetic background. Using the global weighted estimates, we prove that truncating the scattering profiles at time $T$ introduces an error in both the $L^\infty$ norm and the $H^{N_*+1}$ norm of order $$ O\bigl(\varepsilon^2(R+T)^{-\delta}\bigr), $$ where $\varepsilon\in (0,1)$ measures the size of the initial perturbation, $R\ge100$ is the scale parameter in the weighted energy, $\delta\in(0,2/3)$ is the exponent related to weights, and $N_*+1$ is the Sobolev regularity order assumed for the initial data. The pointwise estimate follows from the decay of the nonlocal pressure along nonlinear characteristics, while the Sobolev estimate combines higher-order weighted pressure bounds with estimates for derivatives of the characteristic flows. Numerical experiments measure pointwise and Sobolev pressure-integral tails to verify the quadratic dependence of the initial perturbation amplitude.

We study several fundamental properties of the category $\mathbf{Kur}$ introduced in \cite{Kim1}. By defining fibrations and weak equivalences, we show that morphisms with manifold targets admit homotopy fiber products; this is then applied to obtain a notion of generalized intersections of two submanifolds. A definition of the Weinstein category in the context of Kuranishi spaces is also presented. Finally, we establish that $\mathbf{Kur}$ forms a category of fibrant objects in the sense of Brown \cite{Brown}, and discuss its homotopy category $\operatorname{Ho}(\mathbf{Kur})$.

We study the energy-subcritical and energy-critical complex Ginzburg-Landau equations on three-dimensional Riemannian manifolds. We establish the asymptotic behavior of solutions via a purely energy-based approach and subsequently prove a fast/slow dichotomy of the solutions. Moreover, for the slow solution, we show that the solution converges to an explicit asymptotic profile. In particular, we obtain a precise description of both the amplitude and the phase of the leading-order dynamics. As a by-product, the complex Ginzburg-Landau equation approximation yields global weak solutions to the energy-subcritical and energy-critical defocusing nonlinear Schrödinger equations on compact three-dimensional Riemannian manifolds. We also investigate the long time behavior of the solutions to the energy-subcritical and energy-critical complex Ginzburg-Landau equations on a class of complete noncompact three-dimensional Riemannian manifolds and prove decay of global solutions under suitable geometric assumptions.

For $n\ge1$, we study representations of a simplicial group $G$ in the full $\infty$-subcategory of simplicial vector spaces whose normalized homology is concentrated in degrees $0,\ldots,n-1$, retaining its full mapping spaces. The resulting functor category depends only on the $n$-truncated homotopy type of $\mathbf{B}G$ and admits projectively bifibrant strict diagram and simplicial group-algebra module models. If restriction along a specified map $f:G\to H$ is an equivalence, then $\pi_0(f)$ is a group isomorphism; the proof uses ordinary extension and restriction of scalars. Yet for every $n\ge2$ and prime $q$ invertible in $k$, the nontrivial source $K(\mathbb Z/q,n-1)$ has the same $n$-representation category with vertex evaluation as the trivial group. The group-algebra augmentation is a weak equivalence, so the constant functor is also an equivalence for untruncated representations. Thus $n$-representations depend only on the $n$-truncation of $\mathbf{B}G$, but need not determine that truncation, even together with vertex evaluation.

Linearized permutation polynomials over finite fields have important applications in coding theory, cryptography, and combinatorics. In this paper, we give an explicit description of the cycle structure of a general linearized permutation polynomial in terms of the Jordan blocks of its Dickson matrix, thereby providing a Dickson-matrix formulation of a question raised by Mullen and Vaughan in 1988. Building on the cycle structure, we establish a matrix-theoretic framework to characterize their parity. For finite fields of characteristic 2, we determine all odd linearized permutation polynomials explicitly. For fields of odd characteristic, we derive parity criteria in terms of the eigenvalue structure of the associated Dickson matrix. Based on the parity criteria, we then give an algorithm for deciding the parity of a given linearized permutation polynomial.

We construct nets of curves in a rectangle $R$ with special properties. The net is formed by two families of curves, "horizontal" and "vertical", each family foliates $R$, and every horizontal curve meets every vertical curve orthogonally. Moreover, every horizontal curve started at the left edge of $R$ arrives to the right edge of $R$ at the same height, and every vertical curve started at the bottom edge of $R$ arrives to the top edge of $R$ at a prescribed in advance longitude. The latter longitude function as a map from bottom to top can be chosen in advance as any small perturbation of the identity. Our motivation for this was to design some elliptic divergence form operators, for instance on one of the domains bounded by a Reifenberg flat curve in the plane, with the scalar coefficient and the corresponding elliptic measure being absolutely continuous with respect to the natural measure on the boundary (an object of a separate paper). The construction of our net of curves in a rectangle can also be interpreted as control problem, has an interest on its own, and, likely, has other applications.

Specialization maps are natural maps from equivariant vanishing cycle cohomology to the equivariant cohomology of the ambient space. We study their injectivity problem for moduli stacks of representations of quivers with potentials, including those associated with generalized conifolds and finite subgroups of $SO(3,\mathbb{C})$. Applications to the irreducibility of modules of Reshetikhin type shifted Yangians are also given.

In this article, we investigate the boundary controllability of finite point-vortex systems in bounded multiply connected planar Lipschitz domains. The controls are applied locally on an arbitrary nonempty relatively open subset of the boundary. Zero incoming vorticity is assumed at the inflow part of the boundary. For time-dependent controls, we prove exact trajectory controllability along arbitrary prescribed collision-free vortex paths remaining away from the boundary. For stationary controls, we prove exact endpoint controllability associated with mutually disjoint real-analytic arcs. The proofs rely on harmonic interpolation, localized Runge approximation, and an implicit-function argument. The optimal control minimizing the length of trajectories is also studied. We prove the nonexistence of minimizers within classes of slightly smooth controls, and establish existence when the control time and the size of the control are penalized.

We prove that for every finite connected simple graph $G$ on $n$ vertices with minimum degree $k\geq 2$, there is a coupling of its uniform spanning tree and its random $1$-out subgraph under which the expected number of differing edges is $O(n\log (k)/\sqrt{k})$. More generally, we establish analogous couplings between a wide class of determinantal processes and their corresponding $1$-out processes, including Kalai's determinantal hypertrees, with expected symmetric difference negligible compared to the size of the determinantal set. While the graphical case admits a surprisingly short and elegant proof, the more general result requires establishing an asymptotic equipartition property for the determinantal measures in this class, from which we also derive exponential growth rates for the homology torsion of determinantal hyperforests in higher-dimensional regular high-degree complexes.

Let $k$ be a complete, non-archimedean. We study the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over $k$ and over $\mathbb{C}$ equipped with the hybrid norm. In both settings, we show an analogue of Mané–Sad–Sullivan's theorem about the continuity of the Julia set (in the Hausdorff topology). The non-archimedean case relies on the maximum modulus principle, which we show for analytic functions over analytic spaces without boundary.

In this article, we establish uniqueness in the inverse problem of time-harmonic elastic scattering from a bounded, traction-free obstacle in two dimensions. For a single incident plane wave, it is proved that a connected polygonal obstacle is uniquely determined by the measured far-field pattern. The analysis is based on a reflection principle for the Navier equation under traction-free boundary conditions and unique continuation of elastic fields.

This paper develops a unified framework for establishing original energy dissipation of equilibrium-preserving exponential time differencing Runge–Kutta (ETDRK) methods for $L^2$ gradient flows. The analysis is based on an auxiliary continuous evolution representation of a general ETDRK method, which allows the energy variation to be controlled without imposing positive-definiteness conditions on the Butcher coefficients or introducing stabilization terms. For a broad class of equilibrium-preserving ETDRK schemes, we derive an explicitly computable time-step restriction. Explicit restrictions are obtained for representative schemes of orders two through five, and the framework also applies to arbitrarily high-order interpolation-based ETDRK methods. For the latter class, the resulting admissible step sizes are substantially larger than those obtained from the previous interpolation-specific analysis. Numerical experiments for the Allen–Cahn equation confirm the theoretical dissipation restrictions.

In the present paper, we consider a $d$-dimensional time-homogeneous stochastic differential equation with diffusion matrix $a$ and no drift coefficient. We check that the convexity in the starting position $x$ of the distribution of its solution on the path space when the codomain is endowed with the convex order is equivalent to the convexity of $\R^d\ni x\mapsto \sqrt{\zeta^*a(x)\zeta}$ for each $\zeta\in\R^d$. To do so, we introduce the seemingly stronger notion of coupling-convexity in the starting position and check that it is also equivalent to the convexity of the above square-rooted functions.

We introduce multiple $s$ values and prove that every alternating multiple zeta value is a $\mathbb{Q}$-linear combination of multiple $s$ values $s(k_1, \dots, k_r)$ with $k_i \in \{1, 2\}$. We also study when certain multiple $s$ values can be expressed as $\mathbb{Q}$-linear combinations of multiple zeta values.

The Bermond–Thomassen Conjecture asserts that every digraph with minimum out-degree at least $2k-1$ contains $k$ disjoint directed cycles; it remains open in general. We asymptotically resolve it: there exist an absolute constant $K$ and a function $g(k)=o(k)$, independent of $n$, such that every $n$-vertex finite simple loopless digraph with minimum out-degree at least $2k+g(k)$ contains $k$ disjoint directed cycles for all integers $k\ge K$ and $n\ge1$. In fact we obtain the explicit error term $g(k)=O(k^{3/4}\sqrt{\log k})$.

We study the calibration of interaction-driven ordinary differential equation (ODE) models for traffic and pedestrian dynamics by means of optimal control and stochastic gradient descent. The interaction forces are either prescribed by classical physics-inspired laws, whose parameters are identified from trajectory data, or represented by feed-forward neural networks whose weights act as control variables. For the neural-network-driven particle system we prove well-posedness of the state equation, Lipschitz-continuous dependence of the trajectories on the initial data and on the network parameters, existence of an optimal parameter set, and a first-order optimality system with an explicit backward adjoint equation. The resulting adjoint representation of the gradient is combined with a projected mini-batch ADADELTA scheme, so that each gradient evaluation requires one forward and one backward ODE solve, independently of the number of network parameters. Building on this framework we propose two new second-order models, one for vehicular traffic and one for crowds, that combine goal-seeking, learned interaction, repulsion-alignment or dissipation, and external control terms. Numerical experiments compare six models (a follow-the-leader model of Lighthill-Whitham-Richards type, a neural-network traffic model, the social force model, a neural-network pedestrian model, and the two new models) in terms of cost reduction and root-mean-square trajectory deviation. The neural-network pedestrian model achieves the lowest final cost, while the richer second-order models trade a higher calibration cost for individualised, behaviourally interpretable dynamics. We conclude by discussing limitations and directions for incorporating richer behavioural and environmental effects.

In this paper, we prove the existence of stationary pulse solutions for a bulk-surface reaction-diffusion system modeling cell polarization in two-dimensional domains. Using a singular perturbation method, we construct pulse solutions whose centers lie near non-degenerate critical points of a geometry-dependent potential expressed in terms of the Neumann Green's function, as formally predicted in \cite{watanabe2026}. The main analytical difficulty arises from the half-Laplacian induced by the logarithmic singularity of the Green's function, resulting in algebraically rather than exponentially decaying inner solutions. Our analysis may also serve as a useful guide for applying singular perturbation methods to other bulk-surface type systems.

We study preservation of spatial convexity for viscosity solutions of first order Hamilton-Jacobi evolution equations with Neumann-type boundary conditions in bounded convex domains. Under a monotonicity assumption on the Hamiltonian in the outward normal direction, every viscosity supersolution of the Neumann problem is also a state-constraint supersolution. This reduces the boundary difficulty to a state-constraint problem and enables us to prove convexity preservation via the convex-envelope argument established by Alvarez, Lasry and Lions (1997). We also obtain propagation estimates for semiconvexity and strong convexity under similar structural assumptions.

We develop a semisimple obstruction theory for finite-length abelian categories. To each object \(X\) we associate a canonical obstruction subobject \(\mathcal O(X)\), consisting of the part invisible to all morphisms into semisimple objects. The quotient \(X/\mathcal O(X)\) is semisimple and universal among such morphisms, yielding a canonical semisimplification reflector and characterizing semisimplicity by the vanishing of obstruction. Iterating the construction gives an obstruction filtration which, in finite-length module categories, recovers the classical radical filtration and Loewy theory. Obstruction is also heart-dependent: under Happel–Reiten–Smalø tilting, extension data can appear as new obstruction layers, leading to obstruction shadows and visible torsion. Thus classical radical theory arises as the fixed-heart module-theoretic instance of a categorical framework for semisimple reconstruction, persistence, and variation of heart.

We present a non-perturbative formulation of quantum gravity in asymptotically flat spacetimes using Witten's cubic open string field theory (OSFT) on unstable D-branes. Given that the massless spin-2 sector of the closed string spectrum inherently encodes the degrees of freedom of quantum gravity, we establish this connection by mapping on-shell closed string excitations directly to gauge-invariant operators in OSFT, realizing a concrete flat-space analogue of the AdS/CFT correspondence. By evaluating the OSFT correlation functions of these operators perturbatively around the tachyon vacuum, we systematically reproduce the all-orders genus expansion of the closed string S-matrix for the associated states. This demonstrates that closed string theory-and thus quantum gravity-emerges dynamically from the open string tachyon vacuum, establishing Witten's OSFT as a fully calculable, non-perturbative framework for flat-space quantum gravity.

Critical Lagrange multipliers are known to be responsible for the potentially slow convergence of Newton-type methods, and noncriticality of a given multiplier has been tied to an error bound which estimates the distance of a primal-dual pair associated with a perturbed problem to the primal solution and the multiplier set of the unperturbed one, and is decisive for a local convergence analysis. We investigate this relation in an abstract setting where merely a set-valued mapping assigning primal-dual pairs to a parameter is available, and characterize the error bound of interest in terms of noncriticality of the underlying multiplier and two calmness-type properties of the associated multiplier mapping, one of which is a novel weak inner calmness in the fuzzy sense. No structural assumptions on the underlying problem are needed for that, and the two calmness-type conditions are not only sufficient but also necessary. Afterwards, we specify these findings for composite optimization problems. Whenever the subdifferential of the outer function is a polyhedral mapping, both conditions hold automatically, which explains the equivalence of noncriticality and the error bound observed in the polyhedral setting. Whenever the outer function is $C^2$-decomposable, they are secured by calmness of a restricted multiplier mapping together with a closedness condition, which recovers and extends the known characterization for $C^2$-cone reducible constraint systems.

We introduce a new approach to studying correlations between arithmetic functions and linear phases with respect to highly irrational frequency over short intervals. As an application, we prove that \[ \Big|\sum_{x<n \leq x+H}\mu(n) e(n\alpha)\Big| \ll H(\log x)^{-1/3+\varepsilon } \] whenever $x^{0.58+\varepsilon }\leq H\leq x$. In the same range of $H$ we also show that \[ \Big|\sum_{x<n \leq x+H} \Lambda (n) e(n\alpha)\Big| \ll H(\log x)^{-A} \] for arbitrarily large $A>0$ unless there exists an integer $1\leq q\ll (\log x)^{O_A(1)}$ such that $\|q\alpha \| \ll x (\log x)^{O_A(1)}/H^2$. This breaks the $3/5$ barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T. Zhan, "On the representation of large odd integer as a sum of three almost equal primes," Acta Mathematica Sinica 7.3 (1991), 259-272]. Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials would lead to a corresponding improvement in the lower bound for $H$.

Let $M^n\subset S^{n+m}$ be a compact minimal submanifold of the unit sphere, and let $\lambda_1\geq\cdots\geq\lambda_m\geq0$ be the eigenvalues of its fundamental matrix. We prove that $\sum\limits_{\alpha=1}^{\min\{n,m\}}\lambda_\alpha+\lambda_2\leq n$ forces $M$ to be a totally geodesic subsphere, a generalized Clifford torus, or a Veronese manifold. We first show that equality everywhere in the pinching condition forces the second fundamental form to be parallel. Next, by computing the eigenvalues of the fundamental matrices of irreducible symmetric $R$-spaces, we obtain a complete classification of the equality cases among minimal submanifolds with parallel second fundamental form.

We study the classical problem of scheduling jobs with random processing times on $m$ identical machines to minimize the expected sum of completion times, for Bernoulli jobs: job $j$ takes time $p_j$ with probability $q_j$ and time $0$ otherwise, and its outcome is revealed when it starts. The benchmark is an optimal adaptive policy. We give a quasi-polynomial-time approximation scheme for every number of machines. Previously, quasi-polynomial time was known to give an $O(\log N)$-approximation, and approximation schemes were known only for a constant number of distinct sizes. The scheme rests on a simple observation: it suffices to round the times at which machines become free, rather than the times at which jobs start, to a grid whose width is proportional to the job size, and an optimal policy stretched by a factor close to one already respects such grids. We also show that every policy that fixes the order of the jobs in advance loses a factor $\Omega(\log N)$, already on two machines, so a constant factor requires adapting the order to the observed outcomes. Further results include a polynomial-time adaptive rule with ratio $\min\{m,1+\sum_j q_j\}$, a simpler approximation scheme for a constant number of sizes, and #P-hardness of computing the optimal expected cost on two machines.

We establish local volume doubling and a local $L^2$-Poincaré inequality for metric measure spaces tamed by a measure in the extended Kato class that satisfy the bounded interpolation property. To this end, we construct a bi-Lipschitz time change to a metric measure space satisfying a uniform lower Ricci curvature bound. As an application, we obtain pointed measured Gromov-Hausdorff precompactness for such spaces.

Population gradient flow is a common tool for reasoning about how neural networks adapt, including after pretraining. We show that it can mispredict finite-batch stochastic gradient descent (SGD) qualitatively, and we trace the discrepancy to a specific mechanism. In a two-unit ReLU regression, a source task drives the two neurons toward positive proportionality and a target task rewards separating them. After source training for time $T$, gradient flow recovers on the target in time linear in $T$. Online SGD with batch size $b$ and step size $\eta$ in both phases instead fails with high probability throughout a horizon of order $e^{c/\eta}$ once $T \gtrsim \log(b/\eta)$, uniformly on an explicit set of initializations with Gaussian probability above one percent. For each fixed $T$, small-step SGD still recovers, so the failure requires the joint limit of small steps and long pretraining. At the target clone, the population instability is carried entirely by inputs on which the two ReLU gates disagree. For units at angle $\delta$ these inputs form a wedge of probability $\delta/\pi$, and weight decay shrinks the angle exponentially during pretraining. On every other input both units receive the same random linear update, which contracts their separation in conditional expectation. Bounding the cumulative probability of sampling the wedge along the exact online recursion, without a diffusion approximation, shows that recovery with fixed probability from an identical source-gradient-flow checkpoint, within $e^{c/\eta}$ updates, requires $Nb \gtrsim e^{\lambda T}$ target samples and batch size $b \gtrsim \eta e^{\lambda T}$, where $N$ counts updates and $\lambda$ is the weight decay. In simulations, recovery is approximately a function of the disagreement budget $b\delta/\eta$ and saturates in the horizon.

We construct a positive, finitely additive, left-invariant measure on the congruence cylinder ring of a smooth $d$-dimensional algebraic group over an $n$-dimensional local field. The measure takes values in $\mathbb{R}((X_2))\cdots((X_n))$. A normalized cylinder of depth $(i_1,\ldots,i_n)$ has volume $q^{-di_1}\prod_{\ell=2}^nX_\ell^{di_\ell}$, where $q$ is the final residue-field cardinality. We also obtain coordinate-change and quotient integration formulas for this measure.

We show global existence of weak solutions to the following system of partial differential equations with nonlinear diffusion in both equations and anisotropic signal production \begin{equation} \begin{cases} u_t=\nabla\cdot((u+1)^{m_1-1}\nabla u)-\nabla\cdot(u\nabla v), \\ v_t=\nabla\cdot((v+1)^{m_2-1}\nabla v)+\nabla\cdot(u\nabla v)-v+u, \end{cases} \end{equation} in a smooth bounded domain $\Omega\subseteq\mathbb{R}^n$, $n\in\mathbb{N}$ with nonnegative initial data and homogenuous Neumann boundary conditions. Moreover, we show boundedness of $u$ in $L^\infty((0,\infty);\:L^{m_1}(\Omega))$ and $v$ in $L^\infty(\Omega\times(0,\infty))$. The strategy will be to show global existence and boundedness of classical solutions $(u_\varepsilon,v_\varepsilon)$ to the approximating system \begin{equation} \begin{cases} u_{\varepsilon t}=\nabla\cdot((u_\varepsilon+1)^{m_1-1}\nabla u_\varepsilon) -\nabla\cdot(\frac{u_\varepsilon\nabla v_\varepsilon}{1+\varepsilon\vert\nabla v_\varepsilon\vert}), \\ v_{\varepsilon t}=\nabla\cdot((v_\varepsilon+1)^{m_2-1}\nabla v_\varepsilon)+\nabla\cdot(u_\varepsilon\nabla v_\varepsilon)-v_\varepsilon+u_\varepsilon. \end{cases} \end{equation} Passing to the limit $\varepsilon\searrow 0$, we obtain convergence of weak solutions of the second to weak solutions of the first system. One of the main challenges lie in the proof of weak convergence of $\frac{\nabla v_\varepsilon}{1+\varepsilon\vert\nabla v_\varepsilon\vert}$ to $\nabla v$ in $L^2(\Omega\times(0,T))$ for $T\in(0,\infty)$.