Papers

How much information can a single neuron remember? How many memories can neural networks retrieve without creating false memories? These questions are related to a basic question: how many Boolean threshold functions $f(x)=\operatorname{sgn}(a_0+\langle a,x\rangle)$, $x\in\{-1,1\}^n$, are there? In this paper, we show that the number $T_n$ of distinct Boolean threshold functions is \[ T_n=2\binom{2^n-1}{n}\bigl(1+O(n^{-99})\bigr). \] Equivalently, the capacity of a single threshold neuron is $n^2-\log_2(n!)+1+O(n^{-99})$ bits, improving the $O(n)$ error term in the result of Kahn–Komlós–Szemerédi to $O(n^{-99})$. To prove this, we show that, for $1\le r\le n-1$, and $v_1,\ldots,v_r$ are chosen at random from $\{-1,1\}^n$, \[ \mathbb P\!\left\{ \langle v_1,\ldots,v_r\rangle\cap\{-1,1\}^n =\{\pm v_1,\ldots,\pm v_r\} \right\} =1-O(n^{-99}). \] In the context of the Kanter–Sompolinsky Hamiltonian for memory retrieval, this identifies $r=n-1$ as a sharp threshold, at which, for almost every collection of $r$ memories, the only ground states are these memories and their negatives, confirming a weaker form of the Kalai–Linial–Odlyzko conjecture. It also settles a recent open problem posed by M. Anthony on the specification number of Boolean threshold functions. In addition, we show that, for every $1\le r\le n-1$, \[ \mathbb P\{v_1,\ldots,v_r\text{ are linearly dependent}\} =2\binom r2\,2^{-n}+O\!\left(2^{-n}e^{-cn}\right), \] confirming a conjecture of Kahn–Komlós–Szemerédi.

In this paper we explore the concept of smoothness in arithmetic semigroups. In particular, we give explicit asymptotics for the counting function $\Psi_G(x,y)$ when the arithmetic semigroup $G$ satisfies Axiom A (in the spirit of Knopfmacher). The asymptotics generalise the standard results in the literature for the integer case, and are expressed in terms of the Dickman $\rho$ function. Finally, we give applications of our results to counting smooth families of finite abelian groups and finite semisimple rings.

We prove the existence of solutions to the stochastic magnetohydrodynamics (MHD) system, where randomness is introduced through random initial data and a stochastic integral appearing solely in the induction equation, while the fluid equations remain deterministic. We focus on the case where the adiabatic exponent $\gamma$ satisfies $\gamma > \frac{3}{2}$. The existence proof is carried out using the penalization method. We define the notion of a martingale solution and establish sufficient conditions for its existence. The proof then proceeds by means of the stochastic compactness method. Using an energy inequality, we derive a priori estimates in terms of expectation. Due to the stochastic nature of the problem, we demonstrate convergence in law of the penalized solutions and verify that the limiting object is a martingale solution.

We set up and investigate an approximation scheme for a variational theory of cohesive fracture in antiplane shear deformations. We consider a family of functionals acting on scalar valued functions in general dimensions depending on a small parameter $0 < \varepsilon \ll 1$ and on two fields: the elastic part of the displacement field and an eigendeformation field that describes the inelastic response of the material beyond the elastic regime. We measure the inelastic contributions of the latter in terms of a non-local energy functional. Our main result shows that, as $\varepsilon \to 0$, the approximate functionals $\Gamma$-converge to a cohesive zone model.

We study the $L^p$-boundedness of Isozaki–Kitada type modified wave operators associated with Schrödinger operators $P=-\partial_x^2 + V$ in one dimension for potentials $V$ including long-range potentials such as negative Coulomb-like ones $V(x) = -(1+|x|^2)^{-\mu/2}$ for $\mu>0$. For such potentials, if $ \mu \in (1, 2)$, we prove that the middle and the high energy parts are bounded on $L^p (\mathbb{R})$ for any $p \in [1, \infty]$ but the low energy part is unbounded except for $p=2$. Moreover, if $\mu \in (0, 1]$, we show that even the middle energy part is unbounded except for $p=2$. Actually, regarding the $L^p$-boundedness in the middle and the high energy regimes, we give a complete classification of slowly decaying potentials without imposing any sign condition.

We establish index estimates for complete two-sided free boundary minimal surfaces in smooth mean-convex domains of $\mathbb{R}^3$ with noncompact boundary. We first prove $3{\rm Ind}_s(\Sigma)\geq 2g+b-1$, where $g$ and $b$ describe the conformal compactification. We then include all interior ends with their multiplicities, without further asymptotic assumptions, and selected boundary ends under an explicit condition ensuring vanishing cutoff errors. The proof combines a localized energy identity with a Riemann–Roch count on the conformal double. A boundary puncture at which the chosen forms are regular eliminates the exceptional space; if poles are allowed at every boundary puncture, this space has dimension at most one. We provide the mixed cutoff construction, examples distinguishing embedded boundary ends from reflected planar ends, and a separate analysis of the conformal Jacobi metric. The latter yields finite Dirichlet energy of the logarithmic conformal factor, but the unrestricted boundary-end estimate remains an open step. The compact-boundary case was treated by Cavalcante, Mendes, and dos Santos.

Nitsche proved that every smooth Jordan curve in $\mathbb{R}^3$ of total curvature at most $4\pi$ bounds a unique minimal disk, which is moreover strictly stable. We prove an analogue of this result for Riemannian $3$-balls with mean convex boundary, under an explicit pinching condition on the negative sectional curvature, together with a bound on the covariant derivative of the Ricci tensor. In this setting, every smooth Jordan curve in the boundary sphere of total curvature at most $4\pi$ bounds a unique embedded minimal disk which is strictly stable.

Let $f(d)$ denote the smallest integer such that every bridgeless graph of diameter $d$ admits a strong orientation with diameter at most $f(d)$. It is known that $f(2)=6$ and $f(3)=9$. For $d=4$, the classical bounds of Chvátal and Thomassen [JCTB, 1978] imply $12\le f(4)\le40$, and subsequent work reduced the upper bound to 21. Very recently, Lin, Wang and You further established the substantially stronger bound $f(4)\le18$. Pushing this bound below $18$ turns out to be considerably more difficult, since the remaining extremal configurations cannot be handled by existing techniques based on $R-S$ orientations and related local constructions. In this paper, we prove that $f(4)\le16$. Our approach is entirely different from previous ones. Instead of constructing a strong orientation directly, we develop a sequential orientation framework together with auxiliary distance functions and a potential-function analysis. This enables us to control directed distances globally while avoiding the intricate case analysis required by earlier methods. We believe that the framework introduced here may be useful for studying oriented diameter problems of larger diameter.

We classify the strongly rational vertex operator algebras whose central charge and effective central charge are both one. Every such algebra is isomorphic to one of $V_L,V_L^+,V_{A_1}^{A_4},V_{A_1}^{S_4},V_{A_1}^{A_5},$ where $L$ ranges over the positive-definite even lattices of rank one and $A_1=\sqrt2\mathbb{Z}$.

We consider the Cartan square $V^{2\lambda}$ of a minuscule representation $V^\lambda$ of a simply laced complex simple Lie algebra $\mathfrak g$. We construct for $V^{2\lambda}$ a family of bases, which we call quiver bases, each indexed by the set $\operatorname{RPP}_2(P_\lambda)$ of reverse plane partitions of height two on the minuscule poset $P_{\lambda}$ of $V^{\lambda}$. Let $Q$ be a quiver on the Dynkin diagram of $\mathfrak g$, and let $c_Q$ be the corresponding Coxeter element. The quiver basis $\mathcal B^Q$ is distinguished by the following property: Up to sign, the action of the Tits representative $\dot c_Q$ on $\mathcal B^Q$ lifts the action of $c_Q$, via piecewise-linear toggles, on $\operatorname{RPP}_2(P_\lambda)$. This proves uniformly that, for any minuscule poset $P$, piecewise-linear Coxeter-motion and rowmotion on $\operatorname{RPP}_2(P)$ exhibit the cyclic sieving phenomenon. In type~$A$, the quiver basis for the standard orientation recovers, up to rescaling, the canonical basis, whose compatibility with the long cycle was established by Rhoades. In other types, however, we show the canonical basis is not compatible with any Coxeter element.

The present paper studies integral geometry problems on three-dimensional Riemannian balls, where integration is performed over minimal surfaces or, more generally, $\Phi$-surfaces defined by an elliptic curvature functional $\Phi$. The space of all $\Phi$-surfaces spanned by round circles on the boundary is a three-dimensional manifold, which we call the space of circles. We show that, when the metric is $\Phi$-simple - a notion which extends to this setting the notion of simple metrics in the geodesic case -, the Gauss lifts of the $\Phi$-surfaces define a foliation of the unit tangent bundle that should be viewed as a two-dimensional analogue of the standard geodesic foliation. This is achieved by solving a foliated Plateau problem on the ball. We then analyze the associated surface Radon transform corresponding to integration along the surfaces and show that it has a finite-dimensional kernel; we also prove that it is injective for an open and dense set of metrics. In the special case of a foliation by minimal surfaces, we apply these results to solve the following boundary area rigidity problem: does the collection of areas of the minimal surfaces determine the metric up to isometry?

Suppose that $d\ge 2$ and that $1,\alpha_1,\ldots ,\alpha_d$ is a basis for a real algebraic number field. A theorem of Peck from 1961 establishes that \[ \liminf_{n\rightarrow\infty}n\log n\,\|n\alpha_1\|\cdots\|n\alpha_d\|\ < \infty.\] The goal of this paper is to recast Peck's proof in an intuitive geometric framework, where the result follows from a single application of the Minkowski convex body theorem. We will also explain how a simple modification of this approach leads immediately to new proofs of results of de Mathan, Teulié, and Bugeaud regarding the $p$-adic Littlewood conjecture for $d$-tuples of algebraic numbers. Finally, changing the shape of the convex body allows us to make progress on a conjecture raised by Peck in the same paper, about distributing the logarithmic savings unequally among the coordinates. We prove the conjecture for every real biquadratic field with its natural basis, and for an arbitrary number field when the factors involved are of comparable size.

In this paper, we study regularity and clean-type decompositions for matrices over dual numbers. We characterize the von Neumann regularity of dual matrices by establishing a necessary and sufficient compatibility condition between their real and dual components. At the element level, we determine the conditions under which an individual dual matrix $\mathcal{M} = A + B\epsilon$ is regular, $\pi$-regular, nil-clean, or strongly nil-clean. Using the solvability of Sylvester matrix equations, we show that the matrix ring over dual numbers $M_n(\mathbb{D})$ is strongly $\pi$-regular and strongly clean. Moreover, $M_n(\mathbb{D})$ is $\pi$-regular, clean, $r$-clean, strongly $r$-clean, $\mathrm{NR}$-clean, and strongly $\mathrm{NR}$-clean. On the other hand, $M_n(\mathbb{D})$ fails to be regular, strongly regular, nil-clean, strongly nil-clean, and uniquely $\mathrm{NR}$-clean, although it inherits several strong decomposition properties from $M_n(\mathbb{R})$. Our characterizations at the element level show exactly when each property holds.

We present a tightened convergence analysis of clipped gradient descent on $(L_0, L_1)$-smooth functions, with quantitative constants. Building on the ideas of Koloskova et al (2023), we refactor several case disjunctions to reveal the central role of a control of the bias derived from fundamental properties of $\ell_2$-projection, simplifying proofs. We also extend the domain of validity from $\eta \leq 1 / (9 \beta)$ to $\eta < 1 /\beta$ where $\beta = L_0 + c L_1$ for clipping constant $c$, which matches the more traditional analysis of smooth functions. We strengthen the convergence criterion from $\left( \min_{t < T} \mathbb{E}[\lVert \nabla f(x_t) \rVert_2] \right)$ to $\left( \frac{1}{T} \sum_{t < T} \mathbb{E}[\lVert \nabla f(x_t) \rVert_2] \right)$ with matching speed, and lower the final achievable loss from $\mathcal{O}(\min(\sigma^2/c, \sigma))$ to the more precise $6 \min(\sigma^2 /c, 3 \sigma)$.

We show that non-Fuchsian geometrically finite hyperbolic structures on a given hyperbolizable 3-manifold M are uniquely determined, up to isotopy, by their bending laminations. Consequently, the bending map from the space of non-Fuchsian geometrically finite structures on M, endowed with the strong topology, to the space of bending laminations, endowed with Lecuire's tubular topology, is a homeomorphism. This extends the convex co-compact case established with Schlenker. The proof combines hyperbolic Dehn filling, continuity and properness of the bending map (Lecuire) and real-analyticity of its fibres (Bonahon). We also establish a criterion for contractibility of the boundary fibres of a continuous extension of a homeomorphism, extending Finney's theorem to a boundary setting. This topological result may be of independent interest.

Building on the work of Pach, Rubin, and Tardos, we prove that every set of $n\ge2$ points in the real plane with no three collinear contains at least $cn$ pairwise crossing segments with distinct endpoints, for an absolute constant $c>0$.

The Poincaré recurrence theorem for a holomorphic correspondence provides a natural framework for studying the first return time and a dynamical system thus induced, in this setting. After establishing a version of the Kac's theorem on the average of the first return times for a holomorphic correspondence, we introduce the concept of an induced correspondence, akin to the case of dynamics of maps and prove that it inherits the ergodicity of the holomorphic correspondence. Additionally, we characterise the ergodic measures with respect to the considered holomorphic correspondence, establish the density of forward orbits under mild conditions and prove a Rokhlin-Kakutani type lemma.

The bifurcation phenomenon in a singularly perturbed one-phase free boundary problem governed by the g-Laplacian is established under prescribed boundary conditions. This bifurcation is characterized by the existence of a third solution, obtained via the Mountain Pass Lemma, when the boundary data decreases below a suitable threshold. Moreover, we analyze the asymptotic behavior of the associated evolution problem, proving convergence to stable stationary solutions and showing that the Mountain Pass solution is unstable in this dynamical sense.

Geometry-dependent PDEs define operators whose inputs and outputs may each consist of an oriented interface and a function on that interface, whereas most existing neural operators are formulated on fixed domains. In this paper, we first establish an approximation theorem for continuous operators between general state sets. We then instantiate the theorem with the Interface Autoencoder (IAE), which represents interface–function states in a fixed box using finite projection codes and decodes them by zero-set extraction and restriction. For tensor cosine codes, we prove a uniform $C^0$ reconstruction rate and tube-local gradient convergence. These estimates yield Hausdorff and graph-Hausdorff error bounds, together with a reference-based topology certificate. The IAE accommodates both cross-space geometric operators, mapping domain geometries to the associated PDE solution fields, and within-space geometric operators, mapping between interface–function states. Its effectiveness is demonstrated through numerical experiments on the Poisson equation, interface merging in Hele–Shaw flow, and two-phase Stokes flow with surfactant.

We present two new applications of Simmons' point-free Cantor-Bendixson coderivative operator in intuitionistic logic. First, we use it to give a simplified proof of the recent result of Xu and Ye that the free Heyting algebra on two generators does not occur as the Heyting algebra of subterminal objects in any elementary topos. Then we use it to prove that complete Heyting algebra semantics is not strongly complete for intuitionistic second-order propositional logic: semantic consequence from an arbitrary set of assumptions does not coincide with ordinary syntactic consequence.

Building on recent work of Hom and Wan, we construct additional examples of exotic knot traces, providing partial progress toward their conjecture. Our approach uses the immersed-curve techniques in Heegaard Floer homology developed by Chen and Hanselman.

We construct a smooth Riemannian metric with strictly positive sectional curvature on every smooth homotopy seven-sphere. For each smooth structure, we construct positively curved metrics on two seven-dimensional disks whose induced boundary metrics agree under the prescribed attaching map. Under this identification, a gluing theorem then yields the required smooth metric on the closed manifold. The construction applies to all 28 oriented diffeomorphism classes.

We study a generalized contact process on \(\mathbb{Z}^d\) parameterized by an infection rate \(\lambda\) and a resetting probability \(p \in [0,1]\), modeling deterministic cure times. Once a vertex is infected, its recovery is scheduled exactly one time unit later. Incoming attempts to an already-infected vertex successfully reset its recovery clock with probability \(p\), and are ignored otherwise. This unifies spatial versions of classical Type I (\(p=0\), non-paralyzable) and Type II (\(p=1\), paralyzable) counters. Except in the fully resetting case, deterministic recovery deadlines destroy coordinatewise attractiveness and create a causal shielding effect. Using a first-moment bound on potential causal chains, we prove that the process dies out from finite configurations whenever \(\lambda<1/(2d)\), uniformly in \(p\in[0,1]\). The same causal-chain bound yields local convergence to the empty configuration from arbitrary initial states. In dimensions \(d \ge 2\), an oriented percolation exploration based on the first infective window establishes global survival for all \(p\in[0,1]\) when \(\lambda>-\log(1-p_c^{\mathrm{or}})\). Finally, in the purely non-resetting case \(p=0\), we derive a delayed identity for the one-site density and prove that this density remains strictly between zero and one at every finite time.

In the paper, we prove a recursive version of the weighted Sidelnikov-Welch inequality for real and complex unit vectors. Unlike the classical form, which gives a direct lower bound for a fixed even power sum, our inequality relates two consecutive even power sums. Iteration yields the usual weighted Sidelnikov-Welch bound. We apply this estimate to maximal relative projection constants. If $\mathbb{K}^m$ admits a maximal equiangular tight frame with $M_{\mathbb K}$ vectors, then for every integer $k\geq1$, $$ \lambda_{\mathbb K}(kM_{\mathbb K}-m,kM_{\mathbb K}) = \lambda_{\mathbb K}(m)-\frac{2m}{kM_{\mathbb K}}+1. $$ Moreover, the maximal value is realized by an equiangular tight frame when $k=1$ and by a biangular tight frame when $k\geq2$.

This paper deals with image-based visual servoing and pose estimation by observing four and five lines. Our main interest is to determine the relative configurations of the camera and the observed lines that lead to problems in control and stability. Since it is equivalent to finding the singularities of the corresponding Jacobian matrix, we use tools from computational algebraic geometry to seek configurations such that all of its minors vanish simultaneously. By choosing a suitable basis for this matrix, we revisit the problem in the case of three lines to show that one type of the singularities is when the camera lies on the hyperboloid of one sheet uniquely defined by the lines. This result is further exploited to prove that the one-dimensional singularities, if any, in the case of $n$ lines appear when the camera lies on the transversals to the observed lines. Thus, by forcing the transversals to be complex, we can avoid the aforementioned type of singularities in the case of four lines although the algebra shows that there can always be up to 10 inevitable singular locations of the camera for the other type of singularity. For five lines, we find out that there are no singularities in the generic case. The singularities are also characterized for four and five lines with orthogonality and parallelism constraints. Furthermore, a visual servoing library is used to conduct some simulated experiments to substantiate the theoretical results. As expected, we observe problems in control in the vicinity of a singularity as well as increased errors in pose estimation.

We prove the coarse Erdős-Pósa conjecture of Georgakopoulos and Papasoglu. Informally, any graph either contains many fat cycles that are pairwise far apart, or there is a small number of bounded radius balls that together hit all of them. To be more precise, if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{N}$, then there is a set $X\subseteq V(G)$ of $\mathcal{O}(k\log k)$ vertices such that every $q$-fat model of $K_3$ in $G$ has distance $\mathcal{O}(q)$ from $X$. In another form more closely resembling Manning's theorem that characterises quasi-trees: if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{N}$, then $G$ is $\mathcal{O}(q)$-quasi-isometric to a graph $H$ that contains a set $X\subseteq V(H)$ of $\mathcal{O}(k\log k)$ vertices such that $H-X$ is a forest. Bootstrapping this result, we further prove that every graph with no $q$-fat model of $k \cdot K_3$ is quasi-isometric to a graph with no $k \cdot K_3$ minor, where the quasi-isometry can be chosen to only have additive distortion. This result also holds for $k = \infty$. We also obtain an Erdős-Pósa theorem for long induced cycles that are far apart.

We establish intrinsic continuation criteria and finite-energy global solvability for the stochastic tamed Navier–Stokes equations on $\mathbb R^3$ driven simultaneously by multiplicative cylindrical Wiener and compensated Poisson noise. Under local coefficient hypotheses, the maximal local $L^p$ solution, $p>3$, satisfies a blow-up alternative independent of auxiliary cutoffs and a Serrin criterion with time exponent $2p/(p-3)$. Additional coercivity of the taming term and compatible $L^2$ and gradient noise bounds yield the global well-posedness theory for divergence-free initial data $u_0\in L^p(\Omega;L^p)\cap L^2(\Omega;L^2)$. No smallness or initial $H^1$ regularity is required. The solution has càdlàg $L^p\cap L^2$ paths and gains $H^1$ regularity at positive times, with time-weighted $H^1$ and $H^2$ estimates. The proof combines finite-energy persistence with endpoint completion that retains terminal Poisson jumps and permits restart in the original uniqueness class.

We obtain a nontrivial bound on the number of triples of positive integers $(a,b,c) \in [1,H]^3$, which are multiplicatively dependent but each pair of its elements is not, and so is the triple $ (a+1,b+1,c+1)$. The method is based on studying factorisations of some resultants and a recent improvement by G. Binyamini, R. Cluckers and F. Kato (2025) of the classical Bombieri–Pila bound.

Equilibrium modeling for power markets usually assumes convexity of each players' optimization problem. Although the importance of accounting for fixed generation costs and start-up costs, minimum generation levels and/or minimum up-time and down-time restrictions in production scheduling is widely acknowledged, such modeling does not allow for discrete decisions. This paper considers unit commitment constrained Nash equilibria. First, we derive novel optimality conditions tailored for the mixed-integer convex programming problem of joint unit commitment and economic dispatch of a self-scheduling profit-maximizing producer. Next, we use these to formulate a Nash equilibrium as a mixed-integer complementarity problem, which facilitates the development of a procedure to obtain multiple equilibria. The approach can be adapted to both perfectly and imperfectly competitive market settings. While an equilibrium may not always exist, we give sufficient conditions under which a Cournot-Nash equilibrium can be obtained by mixed-integer convex programming under the standard assumption of an affine inverse demand curve and convex generating costs, and use this to establish existence. A case study demonstrates the feasibility of using optimisation to obtain a Cournot-Nash equilibrium for unit commitment constrained market clearing. Our results confirm that ignoring unit commitment creates significant welfare losses, although these vary substantially across equilibria.

We study the tangent bundle $T_\mu$ to a Radon measure $\mu$ in Euclidean space, introduced by Bouchitté, Champion and Jimenez (2005). Its construction, inspired by Monge-Kantorovich optimal transport theory, involves the duality between Lipschitz functions and the so-called Arens-Eells space, a Banach subspace of distributions obtained by completing the set of balanced signed measures. Precisely, a velocity field $\sigma$ is $\mu$-tangent when the divergence of $\sigma \mu$ lies in the Arens-Eells space. The tangent bundle $T_\mu$ defined $\mu$-almost everywhere provides a local projection that allows to construct a $\mu$-tangential gradient operator on Lipschitz functions that is weakly continuous and enjoys integration by parts. In this paper, we introduce a new quantitative estimate involving the tangential and normal components of a given velocity vector field $\sigma \in L^1_\mu(\mathbb{R}^d)$. Specifically, we show that the tangential condition $\sigma \in T_\mu$ holding $\mu$ a.e. is equivalent to each of the following two conditions: the convergence in the Arens-Eells space of $h^{-1} \left((id+h \sigma)_{\#} \mu - \mu\right)$ as $h \to 0$, and the differentiability of Lipschitz functions along $\sigma$. Moreover, given a field of non-tangent directions, we construct a Lipschitz function that is not differentiable on a large set, confirming that $T_{\mu}$ agrees with the decomposability bundle of Alberti and Marchese. We finally relate $T_\mu$ to the tangent measures of Preiss and survey further properties of the tangential differential calculus.