Papers

We study a half-interval distribution problem for polynomial residues modulo an odd prime $p$: how often the fractional part of $\varphi(x)/p$ lies in the upper half of the unit interval as $x$ ranges over $1\leq x< p/2$. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula $\#\left\{1\leq x< p/2:\left\{{\varphi(x)}/{p}\right\}>\frac12\right\} =\frac{p}{4}+O_\varphi(\sqrt p\log^2 p). $ We then show that the error term can be improved to $O_\varphi(\sqrt p\log p)$ for arbitrary quadratic polynomials and for polynomials satisfying suitable reflection symmetries. For even monomials $\varphi(x)=x^m$, we further obtain the bound $O_m(\sqrt p\log\log p)$ under the Generalized Riemann Hypothesis. Finally, in the case $m=2$, we prove an unconditional matching lower bound, showing that the factor $\log\log p$ is best possible in this setting.

We introduce a new generalized Hessian, called the Goldstein second-order $\delta$-subdifferential, and an associated notion of $(\epsilon_1,\epsilon_2,\delta)$-second-order stationary point for continuously differentiable functions with locally Lipschitz gradients. We propose a zeroth-order algorithm based on cubic regularization and Gaussian smoothing with homotopy to find such approximate second-order stationary points for Lipschitz differentiable functions, and derive the iteration complexity under a mild coercivity-type assumption on the objective function.

The main goal of this note is to relate two different Welschinger-type rules of signs that were used for definition of real enumerative invariants of toric surfaces (the invariants considered being relative to the toric boundary). The relation between them intertwines Mikhalkin's quantum index and geometry of real and complex point sets of counted real curves. As a by-product, we suggest a new definition of quantum index, which can be used for refined invariant enumeration of real algebraic curves on surfaces in a non-toric setup.

In this paper, we construct new families of imaginary and real quadratic fields that are $p$-rational. In the imaginary case, we prove that for any positive integer $k$ and any integer $m$, the imaginary quadratic field $\mathbb{Q}\left(\sqrt{-(kp+m)}\right)$ is $p$-rational for sufficiently large primes $p$. The proof relies on Louboutin's bound on the class numbers of imaginary quadratic fields. As a corollary, we recover the $p$-rationality of consecutive quadratic fields, a result due to Chattopadhyay, Laxmi and Saikia \cite{CLS}. In the real case, we give an explicit proof of the $p$-rationality of the real quadratic field $\mathbb{Q}\left(\sqrt{p(p+1)}\right)$ for any odd prime $p$, and obtain new pairs of real quadratic fields $\left(\mathbb{Q}\left(\sqrt{p(p-2)}\right),\mathbb{Q}\left(\sqrt{p(p-1)}\right)\right)$ and $\left(\mathbb{Q}\left(\sqrt{p(p+1)}\right),\mathbb{Q}\left(\sqrt{p(p+2)}\right)\right)$ for any prime $p>3$. We also construct new examples of $p$-rational triquadratic fields.

Physical limits to chemical sensing are traditionally expressed as Berg-Purcell bounds on estimation accuracy. Whether these bounds also limit the total amount of information a molecular receptor can transmit has remained unclear, despite the fact that cellular signaling performance is naturally quantified in bits rather than precision alone. Here we derive an explicit link between Berg-Purcell-type sensing limits and the information capacity of a single two-state receptor, yielding a compact expression that separates contributions from concentration range, receptor copy number, and averaging time. We show that, in the ideal fixed-time occupancy model, diffusion-limited sampling alone does not define a finite global information bound: with a perfectly specified integration window and unbounded input range, information capacity grows without bound with dynamic range, albeit slowly. A finite saturation arises when time integration is treated as an explicit physical resource. Finite timing precision yields a clock-limited bound on information capacity, and in the high-occupancy regime information transmission crosses over from diffusion-limited to clock-limited behavior. Together, these results establish a receptor-level information bound in bits that is finite once constraints on timing precision are taken into account.

We propose a model order reduction framework for nonlinear parametrized variational inequalities arising in computational mechanics. The high-dimensional model is written in mixed primal-dual form with projection-based complementarity conditions, leading to nonlinear nonsmooth algebraic systems solved by a semi-smooth Newton method. On this basis, reduced models are constructed by proper orthogonal decomposition (POD) of both primal and dual solution snapshots. To address cases where low-dimensional linear spaces provide limited approximation efficiency, we introduce a neural-network-augmented reduced model. Two feedforward networks learn corrections in the truncated POD coordinates of the primal and dual variables from solutions of enlarged Galerkin ROMs. The learned reconstructions enter the coupled equilibrium and constraint equations directly, defining a nonlinear reduced system solved by approximate semi-smooth Newton iterations. The online cost associated with high-dimensional residual evaluations is reduced through hyper-reduction, using a sparse cubature approach based on greedy nonnegative least squares. The proposed methodology is assessed on two nonlinear variational inequalities with distinct sources of nonlinearity: a two-dimensional obstacle problem with a cubic nonlinearity in the state equation, and a three-dimensional frictional contact problem in which the Coulomb law induces a nonlinear projection in the constraint equation. Numerical results compare the high-dimensional model, the linear reduced model, the neural-network-augmented reduced model, and their hyper-reduced variants, demonstrating accurate approximations with substantial reductions in online computational cost. The performance study also evaluates feasibility over the complete constraint set and measures how errors in the reconstructed obstacle and contact forces affect the primal response.

We study orthosymplectically invariant supersphere integration at the exceptional superdimensions $M=-2u$, where the harmonic Fischer structure becomes nonsemisimple and the Pizzetti pairing degenerates. For the meromorphically continued homogeneous inverse kernels we obtain the generating function $$ \mathscr G_\mu(\rho;x,y) =\frac{\Gamma(\mu/2)}{2\pi^{\mu/2}} \bigl(1+\rho\{x,y\}+\rho^2x^2y^2\bigr)^{-\mu/2}. $$ At $\mu=-2u$, its Laurent expansion has a polynomial residue and a logarithmic finite part. We prove that these coefficients recover the complete degreewise duality structure on a fixed superspace with nonzero bosonic dimension. In degrees $k\le u$, the residue inverts a canonical renormalized pairing on $\mathcal P_k$. In the collision range $u<k\le2u$, the ordinary pairing has radical $(x^2)^{k-u}\mathcal P_{2u-k}$; the finite part reproduces the quotient, while the residue reproduces the radical after transport from the reflected degree. For $k\ge2u+1$, the finite part is the ordinary inverse kernel. We also establish the nondegenerate head–socle pairing on the generalized harmonic modules. As an application, we derive covariant right–left radial $q$-monogenic zonal symbols and identify precise degree-one obstructions to transferring scalar Pizzetti reproduction through a one-sided $q$-Fischer projection.

We study binding numbers of tight contact structures on the lens spaces $L(n,1)$. Using the $d_3-$invariant, together with restrictions on planar monodromy factorizations and the Durst-Kegel algorithm for computing $d_3$ from open books, we obtain lower bounds for the number of binding components of planar open books supporting these contact structures. As a consequence, we compute the binding number of the universally tight contact structures on $L(n,1)$, showing that it is equal to $n$.

We study generic properties of planar symplectic billiards, a symplectic analogue of classical Birkhoff billiards introduced by P. Albers and S. Tabachnikov. We prove that, for a residual set of $C^\infty$ strongly convex domains, periodic symplectic billiard trajectories are in general position and non-degenerate. We also prove a Franks' lemma for symplectic billiards and deduce that stably hyperbolic periodic points form a hyperbolic set given by the closure of the set of hyperbolic periodic points. We further show that, for a residual set of $C^\infty$ strongly convex domains, elliptic periodic points are stable and hyperbolic periodic points have transverse homoclinic points. Based on these results, we conclude that –generically– $C^\infty$ strongly convex symplectic billiard has positive topological entropy.

Delcourt and Postle reduced the Linear Hadwiger Conjecture to coloring $K_t$-minor-free graphs on $O(t\log^4 t)$ vertices. An important theorem in their proof process asserts that every sufficiently dense $K_t$-minor-free graph contains a small, highly connected subgraph. In this paper, we show that such a subgraph can be chosen to be smaller. More precisely, there exists an integer constant $C\geq 1$ such that, for all integers $t\geq 3$ and $k\geq t$, every $K_t$-minor-free graph $G$ with $d(G)\geq Ck$ contains a nonempty $k$-connected subgraph $H$ satisfying $v(H)\leq C^2t\log t$. Thus the structural bound improves from $O(t\log^3 t)$ to $O(t\log t)$. We also give a probabilistic construction showing that the $t\log t$ bound on $v(H)$ is best possible up to a constant factor. Consequently, the graphs occurring in the reduction have order $O(t\log^2 t)$ rather than $O(t\log^4 t)$.

We propose a model of the circadian clock in a population of cells based on a network of oscillators, derived from the Kuramoto model. The coupling between oscillators is described by a global interaction term but we introduce a phase-dependent coupling mechanism, such that oscillators interact only within a specific interval of the cycle corresponding to a specific stage of the circadian cycle. We analytically demonstrate that this modified system achieves complete asymptotic phase synchronisation, provided specific conditions on the initial phase distribution and the coupling window length are met. To bridge this theoretical framework with experimental observations, we introduce a signal processing procedure based on wavelet decomposition to extract quantitative oscillatory features from Per2::luciferase reporter traces. We then calibrate the model against datasets from both wild type and Cry2 knockout hepatocyte spheroids using a two-step quasi-Monte Carlo filtering algorithm. The comparative analysis reveals significant phenotypic divergence, showing that the Cry2KO condition alters the identifiability landscape of the model's parameters and introduces new compensatory mechanisms that confound the initial population heterogeneity with long-term macroscopic signal decay.

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.

Building upon a recent result by H. Cornean, H. Garde, and A. Jensen concerning continuum limits of discrete Dirac operators, we extend the analysis to a wide class of block operator matrices. This class includes, among others, the bilayer graphene Hamiltonian. Our main goal is to find norm estimates for the difference between the resolvents of continuous operators and their discrete counterparts embedded in the continuum in a specific way. While some discretization schemes lead directly to convergence in the generalized norm resolvent sense as the mesh parameter tends to zero, others require the addition of a suitable correction term to ensure the convergence.

We introduce the notion of mixed Hodge formality, which refines classical formality and takes into account the mixed Hodge structures present on the cohomology of complex algebraic varieties. We develop an obstruction theory for mixed Hodge formality, witnessing the non-triviality of extensions of mixed Hodge structures. This allows us to understand the non-formality of certain compact Kähler manifolds in the mixed Hodge sense.

The surjectivity of various noncommutative polynomials has been studied extensively on Lie algebras over fields of different characteristics. In this article, we study the surjectivity of Engel Maps over trace zero matrices in $\mathrm{M}_2(\mathcal{O})$, where $\mathcal{O}$ is a local principal ideal ring complete with respect to its maximal ideal and has a residue field $k$ of characteristic $\neq 2$. We show that the image of $(m+1)$-th Engel map induced by the Engel polynomial $e_{m+1}(x, y) = [\cdots[[x, \underbrace{y], y], \dots, y]}_{m+1 \text{ times}}$ over $\mathrm{M}_2(\mathcal{O})$ can be determined by the image of the corresponding Engel map over $\mathrm{M}_2(k)$, for $m\geq 1$. Moreover, we prove that the $(m+1)$-th Engel map on $\mathfrak{sl}_2^{\circ}(\mathcal{O})=\left\{\ A\in \mathrm{M}_2(\mathcal{O})\ \mid\ \mathrm{tr}(A)=0\right\}$ is surjective if and only if the corresponding Engel map on $\mathfrak{sl}_2(k)$ is surjective. Under some mild condition on the residue field $k$, our results shows that every $g\in \mathfrak{sl}_2^{\circ}(\mathcal{O})$ can be expressed as $g = e_{m+1}(h_1, h_2)$ for $m\geq 1$, where both $h_1, h_2$ are in $\mathfrak{sl}_2^{\circ}(\mathcal{O})$.

In this paper, we present a characterization of strong subdifferentiability of the norm of bounded linear operators on $\ell_p$ spaces, $1\leq p<\infty$. Furthermore, we prove that the set of all bounded linear operators in ${B}(\ell_p, \ell_q)$ for which the norm of ${B}(\ell_p, \ell_q)$ is strongly subdifferentiable is dense in ${B}(\ell_p, \ell_q)$. Additionally, we present a characterization of Frechet differentiability of the norm of bounded linear operators from $\ell_p$ to $\ell_q$, where $1 < p, q < \infty$. Applying this result, we will show that the Frechet differentiability and the Gateaux differentiability of the norm of bounded linear operators on $\ell_p$ spaces coincide, extending a known theorem regarding the operator norm on Hilbert spaces.

We study new higher-dimensional analogs of graph chordality and review the existing ones. Our main results for simplicial complexes are: (1) $\Delta$ skeleton-E-chordal $\Rightarrow$ $\Delta^\vee$ vertex-decomposable $\Rightarrow$ $\Delta$ skeleton-clique-chordal. Moreover, for subflag complexes, $\Delta$ skeleton-E-chordal $\Longleftrightarrow$ $\Delta^\vee$ vertex-decomposable. (For $d=1$ this boils down to “$G$ chordal $\Longleftrightarrow$ $G^\vee$ vertex-decomposable”, a result closely related to Fröberg's theorem.) (2) For subflag complexes, $\Delta$ is skeleton-E-chordal $\Longleftrightarrow$ it splits as $\Delta = \Delta_1 \cup \Delta_2$, with each $\Delta_i$ a skeleton-E-chordal induced subcomplex of $\Delta$, and with $\Delta_1 \cap \Delta_2$ a complex whose $1$-skeleton is a clique. (This generalizes “$G$ chordal $\Longleftrightarrow$ $G$ splits as a union of chordal graphs that intersect in a common clique”). (3) $\Delta$ skeleton-E-chordal $\Longleftrightarrow$ every nonempty induced subcomplex of $\Delta$ has a skeleton-E-simplicial vertex. (Generalizes “$G$ chordal $\Leftrightarrow$ every nonempty induced subgraph has a simplicial vertex”.) (4) $\Delta$ underclosed $\Rightarrow$ $\Delta$ skeleton-weakly-chordal and weakly-closed. (Generalizes “$G$ interval $\Rightarrow$ $G$ chordal and co-comparability”.) (5) All pure E-chordal complexes are vertex-chordal; all pure mid-chordal complexes are weakly-vertex-chordal; all pure very-weakly-chordal complexes are weakly-ridge-chordal. (This expands Bigdeli, Yazdan-Pour and Zaare-Nahandi's work on ridge-chordality.)

In this communication we propose a discrete Cosserat rod formulation in which a slender elastic rod is represented as a chain of rigid bodies (nodes) coupled by compliant elastic forces and moments acting between adjacent node pairs. Discrete dilatation, shear, torsion and curvature strain measures are evaluated from the relative kinematics of each node pair, while the constitutive behavior is expressed in compliance form through independent stress degrees of freedom. We show that the resulting model arises rigorously from a mixed Petrov–Galerkin Cosserat rod finite element formulation (FEM) at linear kinematic interpolation order when the internal virtual work is integrated by the midpoint rule and the external and inertial contributions by the trapezoidal rule. The proposed formulation inherits the robustness and the absence of locking from the underlying mixed FEM while simultaneously exposing a two-node coupling structure that mirrors discrete rod models from the computer graphics community. This is in sharp contrast to the dense coupling of strain-parameterized reduced-order models often used in soft robotic applications. Three numerical examples involving piecewise-varying cross sections, tendon-driven actuation under different spatial discretizations, and coupled longitudinal-torsional dynamics confirm the accuracy, robustness, and convergence behavior of the presented approach.

We establish a log-majorization inequality comparing the eigenvalues of the interlaced product $Y^t X^*Y^{1-t}X$ with those of $X^*YX$, valid for every positive semi-definite $Y$ and every normal $X$, with the inequality reversing for $t \notin[0,1]$ when $Y$ is positive definite. This extends known Hermitian results to the strictly larger class of normal matrices, where normality is shown to be the exact structural hypothesis, not a technical convenience. A counterexample proves the result can fail without it. As applications, we settle a normal-matrix extension of a determinantal conjecture of Lin, proving $$\det(A^*A + |BA|^p) \le \det(AA^* + |A^*B^*|^p)$$ for arbitrary $A$, normal $B$ and $p \ge 0$, and we give a complete eigenvalue picture for products of weighted geometric means, sharpening and complementing a theorem of Hiai and Lin.

We prove that a symplectic $T^{n-1}$ action on a closed connected $2n$-dimensional symplectic manifold is Hamiltonian if and only if its orbits are contractible. This generalizes a result of Lalonde–McDuff–Polterovich on four-manifolds and theorems of McDuff and Kim on existence of fixed points. When the orbits are isotropic, we prove a stronger variant of this result, which implies non-extendability of certain symplectic circle actions on 6-manifolds to symplectic $T^2$ actions. Moreover, we prove that a symplectic $T^{n-1}$ action with isotropic orbits always splits into a maximal Hamiltonian action and a locally-free action. We end by posing several open questions on the topology of symplectic torus actions.

This expository review reformulates the BCS analyses of Haag, Emch–Guenin, and Thirring–Wehrl, together with subsequent operator-algebraic mean-field results of Bóna, Raggio–Werner, and Bru–de Siqueira Pedra, in the language of quasi-local $C^{\ast}$-algebras and state decompositions. The quasi-spin model is realized on the UHF algebra of type $2^{\infty}$, with product sectors described by von Neumann's incomplete infinite tensor products. This framework distinguishes strong limits of intensive observables, domain-restricted limits of extensive observables, and sectorwise Bogoliubov–Haag limits, including their operator criterion, gap equation, and limiting dynamics. In the degenerate model, the ground-state and thermal gauge averages admit central direct-integral decompositions over the gauge circle, and the thermal decomposition converges to the ground-state decomposition at zero temperature. Adjoining the gauge phase as a central classical variable combines the phase-dependent Bogoliubov dynamics into one automorphism group on $C(\mathbb{S}^{1},\mathcal{A})$.

In this paper we prove sharp endpoint estimates for multiparameter Marcinkiewicz multiplier operators. More precisely, this result is a consequence of a more general theorem for multiparameter $\mathcal R_{2,n}$-multipliers, a class that contains all multipliers of bounded $\mathcal{V}_q(\mathbb{R}^{\otimes n})$-variation for $1\le q<2$. The class $\mathcal R_{2,n}$ is a multiparameter generalization, introduced in this paper, of the $\mathcal R_2$-multipliers of Coifman, Rubio de Francia, and Semmes. We show that $\mathcal R_{2,n}$-multiplier operators locally map $L\log^{{3(n-1)}/{2}+{1}/{2}}L$ into $L^{1,\infty}$, and that this estimate is best possible, extending the corresponding one-parameter result of Tao and Wright to arbitrarily many parameters. We also establish the sharp bound $O((p')^{{3n}/{2}})$ for the $L^p(\mathbb R^n)\to L^p(\mathbb R^n)$ operator norms of such multiplier operators as $p \to 1^+$. The proof of our $L\log^{{3(n-1)}/{2}+{1}/{2}}L$-to-$L^{1,\infty}$ result combines a vector-valued endpoint estimate for the multipliers with an implicit square function characterization of $L\log^{\sigma/2}L$, obtained via duality from the Chang-Wilson-Wolff inequality. The latter produces, at each iterative step, auxiliary proxy functions that are fed into an intermediate one-parameter vector-valued weak-$(1,1)$ estimate.

We study the preservation of complete metrizability under maps between metric spaces. Two main approaches have been developed in this area: one based on the behavior of maps on countable compact sets, and another based on abstract conditions such as stability. In earlier work the author introduced the class of H-covering maps and proved that every such map is stable. The main result of this paper is the converse: a map is H-covering if and only if it is stable. This provides a concrete and practical description of stable maps and offers a unified framework for completeness-preserving mappings.

Let $1\le m<n$ and let $u$ be an $m$-subharmonic function on a domain in $\mathbb{C}^n$. We study local exponential and polynomial integrability, with particular attention to the sharp polynomial exponent predicted by Błocki's conjecture. Explicit radial examples show that direct analogues of the Guan–Zhou strong openness theorem and Skoda's integrability criterion formulated in terms of the $m$-Lelong number fail when $m<n$. We classify a family of radial power-logarithmic singularities and determine the exact $L^p$-integrability range for each member, including endpoint behavior. We resolve two problems posed by Benali–Ghiloufi. The normalized limit of the ball maximum always equals the $m$-Lelong number; this follows by combining their spherical-mean formula with the strong uniqueness theorem for tangents. The pointwise integrability exponent is lower semicontinuous in the base point. However, even when restricted to $SH_m$, it is not lower semicontinuous with respect to the $L^1_{\loc}$ topology. We also disprove their polynomial openness conjecture using an explicit power-logarithmic endpoint example. Finally, we introduce a scale of local Hessian-capacity conditions, denoted by $C_{m,\delta}$. The volume-capacity inequality and the layer-cake formula yield $$u\in L^s_{loc}\quad\text{for every}\quad s<\frac{(m+\delta)n}{n-m}.$$ The critical condition $\mathrm C_{m,0}=\mathrm C_m$ holds for negative functions of finite total Hessian mass with relatively compact deep sublevel sets, and for radial functions. More generally, functions in the energy class $\mathcal E_{p,m}$ satisfy $\mathrm C_{m,p}$, recovering the full Åhag–Czyż Sobolev exponent. These results provide partial progress toward Błocki's conjecture, which has remained open for more than two decades.

We construct a branch of travelling periodic `roll wave' solutions to the free-boundary incompressible Navier–Stokes equations on an inclined plane in two dimensions. These solutions bifurcate from a parallel shear flow, under natural assumptions on the related Orr–Sommerfeld equation. Using techniques from analytic global bifurcation theory, we extend the local branch to a global curve of solutions. A key step of the proof is reformulating the problem, including the unknown free boundary, as an elliptic system in the sense of Agmon–Douglis–Nirenberg. Finally, we verify the hypotheses on the Orr–Sommerfeld equation for two regimes: small wavenumber and low Reynolds number.

We investigate a three-dimensional inverse scattering problem for tracking a rigidly moving target from far-field data generated by a single incident field. Extending our recent two-dimensional study, we develop a Bayesian optimization framework for simultaneously tracking the target's location and orientation over successive time steps, with the translational and rotational motions modeled as independent stochastic processes. We derive analytical formulas for the far-field pattern under translations and rotations and use them to design a Bayesian optimization procedure tailored to the tracking problem. We further establish posterior consistency for the underlying probabilistic model. When the target shape is unknown, its shape is identified at the initial time using a fully connected neural network trained on a precomputed dataset. Numerical experiments validate the effectiveness of the proposed framework.

This paper is devoted to the rigidity of weak solutions for anisotropic $N$-Laplacian equations with Neumann or Robin boundary conditions on smooth bounded convex domains of $\mathbb{R}^N$. The anisotropic operator is given by $$a(\xi) = H^{N-1}(\xi)\nabla H(\xi),$$ where $H$ stands for a norm on $\mathbb{R}^N$; this formulation contains the classical $N$-Laplacian as a special case. We establish a key integral inequality involving the anisotropic gradient and the second fundamental form of the domain boundary, which acts as the core technical tool in our proofs. Under natural monotonicity assumptions on the nonlinearity, we prove that all weak solutions to the Neumann boundary problem are constant, without requiring any a priori boundedness assumption on the solution. Furthermore, we extend this rigidity result to Robin boundary value problems by imposing suitable constraints on the boundary nonlinear term. Moreover, our rigidity results remain valid not only on bounded convex domains but also on suitable unbounded domains. By working under substantially weaker assumptions than those previously available, we establish rigidity results that fill the gaps in the existing literature for anisotropic $N$-Laplacian equations with nonlinear boundary conditions and substantially extend the rigidity theory of anisotropic quasilinear elliptic equations at the critical exponent $p=N$.

We give a self-contained proof of Gödel's completeness theorem entirely within the formalism of first-order Boolean doctrines (an algebraic approach to classical many-sorted first-order logic). Moreover, we show that Gödel's completeness theorem entails that the fiberwise Stone dual of a first-order Boolean doctrine is its type space functor; roughly speaking, this means that the Stone dual of the Boolean algebra of formulas in context $X$ is the Stone space of $X$-pointed models modulo elementary equivalence.

We study a system that is governed by neural dynamics. Neural dynamics are a model for deep neural networks with a large number of layers. For a differential equation where the right-hand side is given by a neural network, we analyze the exact ensemble controllability of the system. This property plays an essential role in Machine Learning. It concerns the ability to learn to perform different tasks simultaneously with a single neural dynamics: Exact ensemble controllability requires that $N$ different initial states are steered to corresponding $N$ target states with a single set of control parameters. We present a constructive solution to the problem. The construction is based on the solution of a neural interpolation problem. We show that if the right-hand side of the differential equation is given by a neural network of depth two, the interpolation problem can be reduced to the solution of a system of linear equations.

In the paper we introduce vector-valued dynamic extended graphons in order to obtain the mean-filed limit for certain class of problems with adaptive weights. We impose fairly general assumptions on the matrix of initial connections, which allows us to cover the case of sparse graphs.