Papers

This paper presents the first numerical realization of Suzuki's Weil-Quadratic-Form operator, a candidate for the Hilbert–Pólya program linking spectral positivity to the Riemann Hypothesis (RH). Suzuki's 2026 construction was purely theoretical; here, the operator is instantiated via P1 finite-element discretization and Richardson extrapolation. Key results include: (R1) In the prime-free regime, the spectrum follows a closed Archimedean law $A_k(a) = \log(1/a) + \log(k-2) + B_0 + O(a)$, with $B_0 = \log q - 2\log 2$, confirmed to 30-digit precision. (R2) A Mellin double-pole argument proves the head coefficient $B(\nu)$ and shows $B_0$ depends only on the conductor $q$, independent of the Archimedean parameter. (R2b) The degree $d$ of an L-function appears directly as the logarithmic slope of the spectrum. (R3) Total spectral intensity follows the prime number theorem, $S(a) \sim (2a)^3/6$. (R4) Nontrivial zeros are not eigenvalues but occur in the explicit-formula error term of the prime symbol. (R5) The best-match line $\sigma^*(a)$ descends toward the critical line. (R6) Weil's positivity criterion is realized in operator form: bounded residual growth corresponds to all zeros on the line, while an injected off-line zero causes exponential blow-up. (R7) The lowest eigenvalue $\lambda_1(a)$ is strictly positive, decays superexponentially, and passes smoothly through the first prime threshold. (R8) The characteristic function $W(a,0;z)$ is computed for the first time, with all zeros confirmed real. (R9) Indirect traces of GUE statistics appear in the moment structure, even where direct detection is blocked. The authors emphasize that this work does not prove RH. All results are Archimedean and universal, with significance lying in the faithful numerical realization of classical identities rather than new arithmetic.

We study a family of symmetric orthogonal polynomials on the unit triangle associated with the weight \[ w_{\alpha,\gamma,\kappa}(x,y) = (xy)^\alpha(1-x-y)^\gamma |x-y|^{2\kappa+1}, \quad \alpha,\gamma,\kappa>-1,\quad \alpha+\kappa>-\frac32. \] We construct the corresponding monic symmetric orthogonal basis on the simplex chamber and prove that its elements are eigenfunctions of a formally self-adjoint second-order differential operator \(\mathcal D_1^{\alpha,\gamma,\kappa}\). A pair of adjoint ladder operators yields a second operator of order two \(\mathcal D_2^{\alpha,\kappa}\) with the same eigenfunctions. We also obtain an explicit representation of the basis in terms of one-variable Jacobi polynomials and compute its squared norms. After passing to the elementary symmetric variables, we determine the full algebra of linear partial differential operators with real polynomial coefficients having the transformed polynomials as eigenfunctions. Every such operator can be written uniquely as a polynomial in \(\mathcal D_1^{\alpha,\gamma,\kappa}\) and \(\mathcal D_2^{\alpha,\kappa}\); consequently, this algebra is isomorphic to the real polynomial ring in two variables.

We study nonlinear Hölder–Brascamp–Lieb inequalities over $\mathbb{Z}^n$. Our method embeds the integers into a suitable compactification, over which local Hölder–Brascamp–Lieb theory may be applied (e.g heat flow). Our main result is a bound for multilinear Hölder–Brascamp–Lieb functionals over suitable sublattices in $\mathbb{Z}^n$.

The global-in-time well-posedness and uniform large deviation principles (LDPs) are investigated for a wide class of Mckean-Vlasov stochastic non-local fractional $(\alpha,p)$-Laplacian equations with $\alpha \in (0,1)$ and $p>2$ driven by superlinear multiplicative noise defined on the whole space $\mathbb{R}^d$, where the non-local nonlinear fractional $(\alpha,p)$-Laplace operator is defined by a singular, symmetrical and translation invariant kernel function, the distribution-dependent drift terms have arbitrary polynomial growth and the distribution-dependent diffusion terms have superlinear growth. The global-in-time well-posedness is established under these conditions by using the monotone method and a domain expansion argument. Under additional conditions on the growth of diffusion terms, we establish the Freidlin-Wentzell and Dembo-Zeitouni uniform LDPs by using the generalized weak convergence method developed by Salins (Probab. Surv., 16:99-142, 2019). The idea of uniform tail-ends estimates, the pseudo monotone technique and the Arzelà-Ascoli theorem are combined to prove the weak-to-strong continuity of solution operators of the controlled equations in order to overcome many difficulties caused by the noncompactness of Sobolev embeddings on $\mathbb{R}^d$ and the nonlinearity of the fractional $(\alpha,p)$-Laplace operator. The superlinearly growing diffusion term is carefully controlled by using the dissipative drift terms and several algebraic inequalities.

The seamless integration of non-terrestrial and terrestrial infrastructures is a key enabler for ubiquitous connectivity in next-generation (NG) wireless networks. We investigate a hybrid satellite-cell-free Massive MIMO system, where multiple low-Earth-orbit (LEO) satellites jointly serve users in unison with terrestrial access points (APs) under realistic imperfect channel state information and practical user association constraints. We first derive closed-form expressions of the uplink ergodic throughput by exploiting maximum ratio combining (MRC) for transmission over spatially correlated Rician fading channels. Our analysis reveals the characteristic impact of both user-satellite and user-AP association patterns on both the spectral efficiency and rate-fairness achieved. We then formulate an energy efficiency optimization problem under joint user association and power control. Since the problems are inherently NP-hard due to the binary nature of the user-association variables, we develop an improved Differential Evolution (IDE) framework that efficiently explores the feasible solutions in polynomial time. Numerical results validate our analysis and show that the proposed hybrid scheme substantially improves energy efficiency and network throughput. For large-scale scenarios, the DE framework provides practical user-satellite-AP association guidelines, enabling scalable performance gains.

This is a thorough study of quotients of hyperstructures that generalize hypergroups, namely mosaics and semimosaics. The quotients in these categories generalize those studied previously in the literature on hypergroups. We describe the effective congruences in these categories by characterizing them in terms of their underlying equivalence relation. This characterization is applied to provide new methods of constructing quotient objects modulo the action of endomorphisms, as well as to study explicit quotient mosaics of some small groups. We also show that the category of mosaics has a natural proto-exact structure.

It has been known that, for quadratic functions, quasiconvexity equals rank-one convexity but need not equal polyconvexity. While there is an explicit characterization of polyconvexity for quadratic energies, a similar characterization for quasiconvexity is not known in dimensions $n,N \geq 3.$ This paper is concerned with the search of extremal quasiconvex quadratic forms regarded as a linear elastic energy in dimensions $N=n=3$ in various symmetry classes of the elasticity tensor. In particular, we prove that for tensors with orthotropic symmetry, quasiconvexity implies polyconvexity, and thus the orthotropic class contains no nontrivial extremals. We show that this equivalence fails first at trigonal symmetry, where among some statemets, we provide a one-parameter family of non-polyconvex trigonal extreme rays. We also provide some general criteria for proving rank-one convexity, polyconvexity, and extremality, improving a result in [\ref{bib:Har.Hov.}].

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

We study the $2$-cops-move variant of the game of cops and robber, in which at most two cops may move in each turn. Let $c_2(G)$ denote the corresponding cop number. We prove that every graph embeddable on the torus, Klein bottle, or projective plane satisfies $c_2(G)\le 3$, extending a recent result of González Hermosillo de la Maza and Mohar for planar graphs.

We find a system of differential equations on an initial data surface whose solutions are in bijection with unit vector fields on the Einstein $\Lambda$-vacuum development that are proportional to a Killing vector. We refer to these conditions as the unit Killing initial data (uKID) equations, analogous to the classical Killing initial data (KID) equations. The uKID equations can be useful in a setting where only the unit-normalized part of the Killing vector is geometrically distinguished. We eliminate the scaling degree of freedom of a general Killing vector to obtain the space-time equations characterizing unit normalized Killing vector fields and also the uKID equations. These equations are also prolonged to canonical connection form, showing their finite type character. Finally, we obtain an independent derivation of the uKID equations by revisiting the propagation identity method, which has previously been used to characterize the initial data of other geometric equations.

The equilibrium equations of elastic rods can be obtained by balancing forces and moments, or by rendering a potential energy stationary. For complex filaments the energy route asks less of one's mechanical intuition. However, in the classical Hamiltonian picture, the components of the generalized momenta are postulated a priori and their physical meaning changes at the whim of the coordinate chart. Here, we draw on two ideas from mathematical physics. First, the extended tangent bundle of the configuration Lie group is framed by left-invariant vector fields. Second, we follow a one-dimensional reading of the Cartan–Lepage–Krupka theory of variational forms: in this setting the momenta are not postulated but forced. The Legendre transform becomes a linear change of frame, and a Poisson structure on the extended tangent bundle follows. For an isolated Cosserat rod, the momenta coincide, in every encoding of the rotation group, with the material force and moment familiar from rod theories. In the presence of interactions, two cases arise: interactions that depend only on the configuration, such as gravity, leave this identification intact; interactions that depend on the strains destroy it — the conjugate momenta and the internal stresses part company. Because energies add, the momenta decompose into the internal stresses and an interaction contribution. Expressing the two factors of the Poisson bivector in different frames — one adapted to the internal stresses, the other to the conjugate momenta — then exposes the Hamiltonian flow directly in the internal variables. Applied to a tendon-actuated rod, where the standard passage to the Hamiltonian picture demands a nonlinear inversion, the construction delivers explicit equilibrium equations, the required inversion collapsing to a rank-one correction.

We study the long-term behavior of the Multiplicative Weights Update (MWU) algorithm in game settings where learning dynamics frequently fail to converge to Nash equilibria and instead exhibit Li-Yorke chaos. While such chaos precludes the prediction of specific long-term strategy profiles, it does not imply a lack of statistical structure. We demonstrate that natural invariant measures - a fundamental concept from ergodic theory - provide the rigorous framework necessary to find order within this chaos. Focusing on a two-strategy congestion game, we prove that these measures allow for a comprehensive statistical characterization of the dynamics. Crucially, we show that this framework extends beyond simple strategy frequencies to general observables, enabling the precise calculation of long-term time averages for broad classes of economic metrics - including payoffs, social cost, and regret - despite chaos. Our results reveal that this simple learning algorithm captures the full spectrum of behaviors found in one-dimensional dynamical systems, from unique or multiple absolutely continuous invariant measures to complex periodic attractors as well as coexisting chaotic and stable (periodic) behaviors. By bridging game theory and dynamical systems, we show that statistical predictability is attainable even in the absence of pointwise convergence.

The mean-field Heisenberg ferromagnet is a quantum spin model on the complete graph with isotropic spin-1/2 interactions. This non-commuting Hamiltonian is permutation and $\mathsf{SU}(2)$ invariant, and its Gibbs states undergo an $\mathsf{SU}(2)$ symmetry breaking phase transition at inverse temperature $\beta=2$. We consider the associated Davies generator, a canonical model of open-system thermalization, and prove tight asymptotic estimates for its spectral gap at all noncritical temperatures. For fixed $\beta<2$, the gap as a function of number of qubits $n$ is $\Theta(1)$, while for fixed $\beta>2$ the gap is $\Theta(n^{-1})$. The matching upper bound of the spectral gap is witnessed by the total magnetization order parameter, suggesting that the low-temperature ($\beta>2$) slowdown is associated with broken continuous symmetry. Two key ingredients in our approach are a comparison argument, which introduces auxiliary generators to bound dissipation on nontrivial representations of the symmetry groups $\mathsf{SU}(2)$ and $\mathsf{S}_n$, and a decomposition of the space of observables into spherical tensor operators to reveal a form of monotonicity.

We study the geometry of the principal curvature surface associated with a Whitney umbrella. This surface is obtained by extending the Whitney umbrella in the normal direction by its bounded principal curvature and admits a natural geometric interpretation. We prove that its intersection with the normal plane coincides with the pedal curve of the curvature parabola and deduce that the focal conic is the inversion of this pedal curve. We then investigate the geometry of the principal curvature surface along the exceptional set by studying its geodesic and normal curvatures, as well as its Gaussian and mean curvatures, and determining all possible generic numbers of zeros of these curvature functions.

Understanding the dynamics of Boolean networks is central to problems such as network reduction, design, control, and reverse engineering. As Boolean network models continue to grow in size and complexity, it becomes increasingly important to decompose networks into modules in a manner that is compatible with their dynamics. In this paper, we show that endowing the space of possible dynamics with a semiring structure enables a systematic decomposition of the dynamics of any Boolean network in terms of the dynamics of its constituent modules. This algebraic framework provides a systematic way to analyze how local dynamical behaviors combine to produce global dynamics. Our results establish a concrete algebraic foundation for network modularity and introduce new mathematical tools for the study of complex Boolean networks, and opens the door to the application of algebraic methods to problems of network analysis, decomposition, and control.

Control barrier functions (CBFs) have become an important controller design tool for autonomous systems subject to safety constraints. Despite their popularity, recent works have shown that CBF-based controllers can destabilize the internal dynamics of the system. In this paper, we consider linear systems with affine safety constraints and design linear feedback controllers that satisfy high-order CBF (HOCBF) constraints while rendering the origin globally exponentially stable. We first characterize the exact class of all linear gain matrices that globally satisfy the HOCBF constraints, including necessary and sufficient conditions for when this class is nonempty. Then, by leveraging the recently introduced notion of CBF output dynamics and CBF internal dynamics, we provide the necessary and sufficient conditions for the existence of stabilizing gain matrices within that class. Finally, we show that Linear Quadratic Regulator (LQR) and robust control problems can be solved while being constrained within this class of safe and stabilizing gain matrices, through standard linear control techniques such as algebraic Riccati equations (AREs) and Linear Matrix Inequalities (LMIs). We illustrate our results in a simulation example.

We develop a coupled model for electroosmotic pumping and passive-solute dispersion of a solvent-free simplified Phan-Thien-Tanner fluid in a compliant slit microchannel. Pressure, wall deformation, axial field, velocity, and dispersion are evaluated self-consistently along the finite-throughput pump characteristic. Lubrication theory, Debye-Huckel electrostatics, an elastic-foundation wall law, and Taylor-Aris macrotransport yield a closed-form flux relation for combined electroosmotic and pressure-driven forcing. Because the shear rate depends cubically on the total shear stress, the two contributions cannot be superposed. The flux decreases monotonically with pressure gradient, ensuring a unique inversion at prescribed throughput. Current conservation couples the axial field to the deformed gap under constant-current and constant-voltage operation. In pressure-free flow, thinning the electric double layer produces a plug-like profile and the Newtonian Taylor coefficient decays as the inverse square of the Debye parameter. Under hydraulic loading, an adverse pressure gradient drives a sheared core counterflow that persists in the thin-double-layer limit, causing the coefficient to approach a finite plateau. At fixed nonzero throughput, partial cancellation between electroosmotic and pressure-driven shear yields a maximum plate number at finite double-layer thickness. This optimum is conditional: joint optimization over throughput and double-layer thickness shifts the overall optimum toward free-flow, thin-double-layer operation. Viscoelasticity can enhance or suppress loaded dispersion, while compliance shifts the pump characteristic and separation optimum. Brownian dynamics validates the reduced model. The resulting load-resolution relation identifies conditions that balance pressure delivery and separation performance.

In a discrete choice model, choice probabilities observed for a finite collection of choice sets may not identify a counterfactual choice probability under an unobserved choice set. We represent this counterfactual probability as a linear functional of a mixing distribution. Because the target is a functional of a distribution whose support is not restricted to a finite set, the parameter space is infinite-dimensional, while the data impose only finitely many moment restrictions. Therefore, observed choice probabilities need not point identify such a target. The identified set is defined as the set of target values compatible with observed choice probabilities. Rather than imposing conditions to ensure point identification, we characterize the identified set, and conduct inference on its lower and upper endpoints. We represent each endpoint as the value of a linear program over probability measures, and give conditions to obtain pathwise differentiability of the identification bounds. As a consequence, we are able to prove asymptotic normality of plug-in endpoint estimators. Finally, we provide an Expectation-Maximization-like algorithm for certifying membership of candidate values in the identified set and establish local convergence guarantees.

We develop a regularity theory for equivariant Allen–Cahn solutions on closed Riemannian manifolds with a Lie group acting isometrically. When the cohomogeneity of the action is between $3$ and $7$, we show that a sequence of equivariant Allen–Cahn solutions with uniformly bounded energy and equivariant index converge to embedded minimal hypersurfaces with optimal regularity, meaning that the singular set is at least codimension $7$ and lies in the union of all non-principal orbits. When the cohomogeneity is $2$ and the action has no exceptional orbits, we show the same result but the minimal hypersurfaces may be immersed. As a result, any closed Riemmanian manifold with cohomogeneity $2$ Lie group action and no exceptional orbits admits a minimal hypersurface with optimal regularity. A key tool is the regularity theory of Chodosh–Mantoulidis, building on the work of Wang–Wei. However, we adapt their arguments to a modified Allen–Cahn equation with a drift Laplacian. We also show that appropriate index bounds hold for the limiting minimal hypersurface when it is smooth. We also extend the variational constructions of solutions of the Allen–Cahn equation of Guaraco and Gaspar–Guaraco by defining an equivariant mountain pass invariant, as well as the equivariant Allen–Cahn $p$-widths. This builds on the work of Gromov and is the Allen–Cahn parallel to Wang's equivariant volume spectrum in the Almgren-Pitts setting. We show that in the limit as $\epsilon$ tends to $0$, the equivariant Allen–Cahn $p$-widths converge to the equivariant $p$-widths, as defined by Wang.

For $Y$ a quasi-projective complex variety and $w \colon Y \to \mathbb C$ a regular function, we construct a Chern character from the Grothendieck group of the category of matrix factorizations of $w$ to the critical cohomology of $w$, and show that it factors through a certain topological $K$-theory group. We prove a Grothendieck-Riemann-Roch theorem with respect to this Chern character, and verify several functorial properties.

In this paper, we propose and study a robust variant of the smart predict-then-optimize approach that accounts for prediction shifts due to disturbance in the covariate feature space. While traditional integrated-learning-and-optimization models assume that side information is perfectly revealed, empirical data-driven features are frequently corrupted or noisy at the time of decision-making, leading to fragile operational policies. To bridge this gap, we integrate principles of robust optimization directly into the predictive-prescriptive pipeline via a smart predict-then-robustly optimize loss and establish a computationally tractable convex surrogate, designed to hedge against worst-case feature perturbations. On the theoretical front, we formalize the structural validity of this surrogate by proving its approximation error probability decays exponentially according to a sub-Gaussian concentration profile. Furthermore, we establish that under mild assumptions, the surrogate is Fisher consistent with high probability. We also prove necessary conditions under which our framework outperforms standard smart predict-then-optimize and maintain its superiority even when the standard method is equipped with regularized upstream predictions. Numerical experiments validate that our robust framework consistently yields significant performance improvements over standard methods, both in out-of-sample terms and in training stability.

We introduce the concept of component-perfect multidimensional discrete Morse (MDM) functions on finite regular CW complexes. While perfect discrete Morse functions in the classical sense require the number of critical cells of index $p$ to exactly equal the $p$-th Betti number, the multiparameter framework necessitates a topological dynamics perspective, where critical cells naturally cluster into connected components. Building upon the existing notion of strictly perfect MDM functions, we propose the more flexible definition of component-perfection. We also show how to restrict and extend MDM functions on finite regular CW complexes, laying the groundwork for more complex topological operations.

We prove quantitative stable regularity lemmas for binary real-valued functions, extending the work of Malliaris and Shelah for stable graphs. The statements of our results are modeled after non-quantitative theorems for stable functions due to Chavarria, Conant, and Pillay. One of the key tools in our quantitative proof is an "analytic symmetry lemma", which gives a function-theoretic analogue of the fact that a pair of good sets in a graph has density close to 0 or 1. We also develop a function-theoretic treatment of Malliaris and Shelah's random sampling method for refining partitions consisting of good sets into equipartitions.

We prove function-theoretic analogues of a quantitative result of Hodges on extracting the order property from a sufficiently large 2-tree coded in a binary relation. Similar analogues for functions were previously obtained by Daskalakis and Golowich and by Anderson and Benedikt. These results are from statistical learning theory, where 2-trees are captured by sequential fat-shattering dimension, and the order property is controlled by various notions of "thresholds". Our first main result (Theorem 1.11) focuses on extracting a less restrictive kind of threshold from a tree, and yields significantly better bounds compared to what can be obtained from earlier results focusing on more restrictive versions. Part of the motivation for Theorem 1.11 lies in a companion paper, where this theorem is used to obtain efficient bounds in quantitative regularity lemmas for "stable functions". Here will use Theorem 1.11 to reprove a result of Anderson and Benedikt in a stronger form and with improved bounds. We also use Theorem 1.11 to prove an at most double-exponential bound on dual sequential fat-shattering, which resolves an open problem. In our second main result (Theorem 1.14), we give a new proof of a result of Daskalakis and Golowich on extracting "tight thresholds" from large sequential fat-shattering dimension, with improved bounds. This resolves another open problem related to correcting the proof of a result claimed by Jung, Kim, and Tewari.

Given an isotropic, exchangeable, and unconditional random vector $\mathbf X$, we consider the sample covariance matrix constructed from i.i.d. copies of several tensor models of $\mathbf X$, such as the tensor power $\mathbf{X}^{\otimes d}$. Under appropriate moment conditions on $\mathbf X$, we show that almost surely, the empirical spectral distribution converges weakly to the Marchenko-Pastur law. This extends previous results which required the coordinates of $\mathbf X$ to be independent. As we demonstrate, our extension applies to many new random vectors $\mathbf X$ of interest.

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

The simple vigorous groups form a broad class of groups of homeomorphisms of Cantor space that includes Thompson's group $V$, its various generalisations and many others such as Nekrashevych's groups of dynamical origin. Bleak, Elliott and Hyde (2024) proved that every finitely generated simple vigorous group is $2$-generated, and, in this paper, we give several strong generation results for this class of simple groups. For example, we prove that if $G$ is a finitely generated simple vigorous group, then $G$ is generated by three involutions, $G$ is generated by an element of order $m$ and an element of order $n$ for any choice of $m \geq 2$ and $n \geq 3$, $G$ has a minimal generating set of size $k$ for all $k \geq 2$, every nontrivial element of $G$ is contained in a generating pair and the direct power $G^n$ is $2$-generated for all $n$. These results are analogous to well-known results for finite simple groups, but of course the proofs in this context are quite different. One consequence of our results is that Thompson's group $V$ is $(2, 3)$-generated, which answers a question of Sapir (2017). Another consequence is that every finitely generated group quasi-isometrically embeds in a $(2, 3)$-generated simple group, strengthening theorems of Hall (1974) and Bridson (1998). All of our proofs are constructive, and we establish several new generation criteria for these groups, which we expect to be of wider interest, even just for Thompson's group $V$.

We propose an r-adaptive neural algorithm for Isogeometric Analysis (IGA) based on residual minimization. The boundary-value problem is solved using a standard conforming Galerkin formulation, while a neural network relocates the interior knots. A strong-form residual in the sense of physics-informed neural networks (PINNs) controls a norm stronger than the energy (H^1) error. We therefore weight it by classical a posteriori theory: element residuals scaled by the local mesh size, interface flux jumps, and Neumann boundary residuals yield a computable estimator of the energy error, which we minimize with respect to the knots. For coercive problems on admissible mesh families, this estimator is reliable and locally efficient up to oscillation terms; beyond that regime, the same loss remains well-defined and extends differentiable r-adaptivity to indefinite and advection-dominated problems. In the parametric setting, the network maps each parameter to a knot-density function in a single evaluation; since it outputs a density rather than a fixed-dimensional vector of knot locations, one trained network produces an admissible mesh at any refinement level. Mesh gradients are obtained by reverse-mode automatic differentiation through the discrete solution equation. Numerical experiments in one and two dimensions illustrate that the method concentrates degrees of freedom near singularities, material interfaces, and boundary layers, improving accuracy for a fixed number of degrees of freedom.

The present work introduces a new relation between closed elastica knots with the solution of the nonlinear Schrödinger equation (NLSE) through the Hasimoto transformation. An elastica knot is defined in terms of the Frenet-Serret curvature $\kappa(s,t)$ as a function of the arclength $s$ along the spatial curve ${\bf r}(s,t)$ at a fixed time $t$, which is a solution of the curvature differential equation $\partial^{2}_{s}\kappa(s,t) = -\;\kappa^{3}/2 + k_{0}^{4}\tau_{0}^{2}\;\kappa^{-3} + \lambda\,k_{0}^{2}\kappa/2$. Here, the Frenet-Serret torsion $\tau(s,t)$ satisfies the conservation law $\kappa^{2}(s,t)\,\tau(s,t) \equiv k_{0}^{2}\,\tau_{0}$, while the constant of integration $\lambda$ is associated with the constrained variation. After briefly reviewing the Hasimoto transformation from a space curve ${\bf r}(s,t)$ to the nonlinear Schrödinger equation (NLSE) $-\,iD^{-1}\partial_{t}\psi = \partial^{2}_{s}\psi + \frac{1}{2}\,|\psi|^{2}\psi$, where the constant $D$ has units of fluid circulation (m$^{2}$/sec), we show how the traveling-wave solution $\psi(s,t) = \Psi(s_{t} \equiv s - c\,t) \equiv \kappa(s_{t})\;\exp[i\theta(s_{t})]$ is mapped onto the curvature equation for an elastica knot, with $\theta^{\prime}(s_{t}) \equiv c/(2D) + k_{0}^{2}\tau_{0}/\kappa^{2}(s_{t})$ and the elastica-knot constant $k_{0}^{2}\lambda = -\frac{1}{2}\,(c/D)^{2}$ expressed in terms of the traveling-wave NLSE parameters $(c,D)$. The constraint of a closed 3D elastica knot imposes spatial periodicity conditions that introduce a unique set of knot parameters for which the NLSE traveling wave can exist. The present work shows that the NLSE traveling-wave solution on a closed elastica knot requires an extension of the classical elastica-knot parameter space.

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