Papers

We establish weighted mixed-norm estimates for fractional parabolic equations \begin{equation*} \partial_t^\alpha u=Lu-\lambda u+f \text{ in } (0,T)\times\mathbb{R}^d, \end{equation*} with nonlocal operators in both time and space. Here, $\partial_t^\alpha$ is the Caputo derivative of order $\alpha\in(0,1)$, and $L$ is a spatially nonlocal operator of order $\sigma\in(0,2)$ whose kernel is merely measurable in time. We also obtain the corresponding estimates in the odd mixed-norm spaces, where the inner integration is taken in time and the outer one in space. The estimates are robust in the limit $\alpha\to1$ and $\sigma\to2$. We also establish unique solvability when either $T<\infty$ or $\lambda>0$. The proof is based on a direction-by-direction extension argument.

We present a samplet-based framework for the efficient numerical solution of saddle-point systems arising from conditionally positive definite (CPD) kernel approximation in general and universal Kriging in particular. The vanishing moment property of samplets as well as the particular structure of the associated scaling distributions, which correspond to discrete orthogonal polynomials, allow for a numerically favorable representation of these saddle-point systems. Concretely, they enable a natural null-space reduction of the indefinite saddle-point system to a (large) linear system for the detail coefficients and a small triangular system for the polynomial coefficients. We derive error bounds for the approximation by polyharmonic splines in Beppo-Levi spaces and show that the detail coefficients span precisely the subspace on which the CPD kernel is positive definite, rendering the reduced block symmetric positive definite. In view of the quasi-sparsity of the samplet-transformed kernel matrix for asymptotically smooth kernels, the resulting method achieves O(N log N) cost for the assembly and the storage of the saddle-point system. The reduced system can efficiently be solved by a sparse Cholesky factorization. We illustrate the framework with three applications, namely Gaussian process regression with generalized covariances, landmark-based image registration via samplet-compressed thin plate splines, and three-dimensional mesh deformation.

We say that a finitely generated group equipped with a finite generating set is of type $\mathsf{R}$ if the Rips complex is contractible for sufficiently large scales. We show that the class of groups of type $\mathsf{R}$ is closed under taking graphs of groups with finite edge groups and, in particular, under taking free products. We prove that all two-dimensional right-angled Artin groups with the standard generating set are of type $\mathsf{R}$. Furthermore, we observe that products of finitely generated free abelian groups and finite groups are of type $\mathsf{R}$.

A $d$-dimensional $0$-$1$ matrix $M$ of size $n_1\times n_2\times \dots \times n_d$ can be considered as a Boolean function $M: B(n_1\times n_2\times \dots \times n_d) \to \{ 0,1\}$, where $B$ is the $d$-dimensional box of lattice points $(x_1, \dots, , x_d)\in Z^d$ with $0\leq x_i \leq n_i-1$, $1\leq i\leq d$. The $0$-$1$ matrix $M$ can also be described as a subset $P:=P(M)$ of $B$ such that $x\in P$ if and only if $M(x)=1$. A $k$-star with center $p$ in $M$ corresponds to a $(k+1)$-element subset $\{ p, p_1, \dots , p_k\} \subset B$ such that $p$ and $p_i$ differ only in one coordinate (for all $1\leq i\leq k$) and these $k$ coordinates are distinct. Here we consider the problem of determining the maximum number of $1$-entries of a $0$-$1$ matrix $M$ of dimension $d$ and size $n\times n \times \dots \times n$ that avoids all $k$-stars. Our main results are the asymptotical solution of the problem for every $d$ and $k$ (as $n\to \infty$), very close bounds for $k=d$, and the exact solution of the $d=k=3$ case. This problem has connections to several other areas of discrete mathematics, including $k$-partite hypergraphs, independent set problems, dominating set problems and covering codes. One of our tools (concerning maximal packings of induced copies of a given hypergraph) might have independent interest.

Synchronization errors, including insertions, deletions, and substitutions, may occur in bursts in communication systems such as DNA data storage, file synchronization, and magnetic recording. In this paper, we study an error model con?sisting of one burst of t-deletions and one burst of t-substitutions. By reformulating the original sequence into a matrix form, we propose an explicit construction of error-correcting codes capable of correcting one burst of t-deletions and one burst of t-substitutions with O(log n) redundancy.

We study the (canonical) 1st derivatives of proper submersions represented as height functions and belonging to a certain class: a proper map means a map the preimage of a compact set by which is always compact. We investigate their Reeb spaces. They are the quotient spaces defined by the equivalence relations on the manifolds of the domains where we identify two points in a same connected component of a same level set of them. They have been important since the establishment of theory of Morse functions, in the 20th century, They are in certain tame situations $0$- or $1$-dimensional and graphs naturally. Related facts have been shown by Gelbukh and Saeki in the 2020s for certain proper smooth real-valued functions. In non-proper cases, even related explicit theory has been difficult, except some previously given case of the author. Our study is on a new related case.

Mathematical elasticity has a long history and a huge body of literature. Surprisingly, at its heart elasticity exhibits a rich and transparent structure that, to our knowledge, is rarely presented in an explicit and unified form. Using standard tools from functional analysis, this article reveals four orthogonal decompositions of Hilbert spaces that characterize the static equilibrium of an elastic body at small deformations, universally for arbitrary spatial dimension, arbitrary boundary conditions, and classical as well as data-driven formulations. Regarding data-driven continuum mechanics, it highlights the intrinsic structure that remains unchanged when constitutive laws are replaced by material data sets.

We study boundary free energies and half-space responses in nearest-neighbor disordered Ising models. First, we prove the existence of fixed-depth tangential pressure and identify its derivative almost everywhere with the limiting boundary response. Second, under quenched exponential boundary-response mixing, we prove Gibbs-state selection independence and exponential convergence of finite-depth pressures. We further show that uniform one-spin mixing yields DLR uniqueness and Gaussian normal localization, and verify the required mixing conditions whenever $(2d-1)\mathbb E\tanh(\beta\vert{}J\vert{})<1$. Finally, we establish an all-face surface limit in the bounded Dobrushin regime and provide counterexamples that delimit general surface and stiffness claims.

Let $(X,\Delta)\to C$ be a family of pairs over a smooth curve in positive characteristic $p>0$. We study the discrepancies of divisors over the base-change $(X,\Delta)\to C$ along Artin–Schreier and Frobenius covers of $C$, through ramification data for Artin–Schreier and degree $p$ purely inseparable extensions of DVRs. As an application, we recover a result of Hu and Zong about permanence of local stability for such base-change.

We study an $N$-species competition–diffusion system with a common competition term, for which the associated ODE system admits infinitely many equilibria. For $N=2$, Alfaro and Xiao \cite{AlfaroXiao2023} studied the long-time behavior of solutions with compactly supported initial data by constructing precise super- and sub-solutions. For $N \geq 3$, the absence of a comparison principle for the entire system presents a major difficulty. Using a weighted entropy method, we prove that every bounded nonnegative traveling profile connecting nonzero equilibria is constant, without imposing any monotonicity assumption. For nontrivial compactly supported initial populations, we further show that the species with the strictly largest Fisher-KPP speed excludes all slower species and spreads at its scalar Fisher-KPP speed. Finally, we establish heat-type lower bounds for the slower species, revealing a bump phenomenon despite their uniform extinction.

We consider the box dimension of the graphs of the generalized Riemann-type functions $G_\delta(x)=\sum_{n=1}^{\infty}g(n^{2}x)n^{-1-\delta}$ with 1-periodic real-valued continuous functions $g$ and $0<\delta\le 1$. Firstly, we establish a rational-level non-vanishing criterion for the lower bound of lower box dimension of the graph of $G_\delta$. More precisely, We prove that the lower bound $\dim_B(\mathrm{graph}\,G_\delta)\ge\frac74-\frac\delta2$ under a mild decay condition of the Fourier coefficients of $g$ and non-vanishing of the square-class chirp functional $S_{d}(a;q)$ at a single rational $a/q$. A resolution theorem then asserts that for any nonconstant real trigonometric polynomial $g$, the chirp functional $S(a;q)$ cannot vanish at every rational simultaneously; consequently, $\dim_B(\mathrm{graph}\,G_\delta)=\frac74-\frac\delta2$ for all such $g$ with $0<\delta\le1$ which gives a negative answer to \cite[problem 2]{Wu-Zhan2026}. Finally, two guiding examples distinguish structural vanishing from genuinely arithmetic vanishing related to modular elliptic curve and governed by the Prime Number Theorem.

For $X$ a conical affine symplectic singularity with $\mathbb{T}=T \times \mathbb{G}_m$-action, the fixed scheme $X^T$ and the map $X^T \rightarrow X$ carry much information about the geometry of $X$. In general, $X^T \rightarrow X$ fails to be a complete intersection. Thus, we study a derived intersection whose classical locus is the $T$-fixed subscheme $X^T$. We show that the structure of the symplectic singularity on $X$ produces a duality theorem for the structure sheaf of the derived intersection. The duality theorem allows us to study the structure of such derived intersections; in particular we describe their cohomological amplitude. An important source of symplectic singularities with $\mathbb{T}$-action are affine Grassmannian slices $\overline{W}^{\lambda}_{\mu}$. We pay particular attention to these slices when $G=\mathrm{SL}_{n+1}$, and we use the previously developed theory to characterize when $(\overline{W}^{\lambda}_{\mu})^T \rightarrow \overline{W}^{\lambda}_{\mu}$ is a complete intersection.

We develop a general theory for the dynamics of the geodesic flow of locally CAT(k) branched coverings and we show that it does not depend on the specific covering but only on the base space, the branching set and the degree of the covering. We find an explicit formula for the entropy: it equals the topological pressure of the associated natural dynamical system on the space of broken geodesics with a natural geometric potential. We find the exact asymptotic of the entropy as the number of sheets of the branched covering goes to infinity, as well as asymptotic properties of the measures of maximal entropy.

We prove an optimal Hardy inequality for every aperiodic, symmetric random walk in $\mathbb{Z}^2$ with finite variance. In particular, we verify null-criticality, and thus, optimality of the underlying Hardy weight. Under suitable moment conditions, we use fine asymptotics of the potential kernel due to Fukai and Uchiyama in order to derive the asymptotics of the weight. For the standard Laplacian, we recover the expected first order term in the asymptotics but also show that next order term is negative. Thus, our result shows that the constant in the Hardy inequality proven by Kapitanski and Laptev cannot be larger than $1/4$, which is the optimal constant in the continuum. The proof of null-criticality rests on a new criterion for general graphs beyond the locally finite case. We also recover the situation of $\mathbb{Z}^d$ with $d \geq 3$ which can also be also treated by our new method.

Compressive sensing predicts that sparse vectors (signals) can be recovered from a small number of linear measurements via efficient algorithms. This finding, which dates back two decades, has triggered a paradigm shift in signal processing and initiated many developments both in practical signal-processing applications, such as medical imaging, radar, and astronomy, and on the theoretical side. More recently, seminal connections to the field of deep learning have led to further advances in the field, such as the use of unrolled neural networks for sparse recovery and the discovery that common training algorithms (variants of gradient descent) favor sparsity in overparameterized scenarios – the so-called implicit bias phenomenon. This article gives an introduction to compressive sensing and outlines connections to deep learning. In particular, we will discuss generalization for neural networks generated by unrolling sparse recovery algorithms and implicit regularization for gradient descent applied to learning simplified linear neural networks.

A Lefschetz fibration is a smooth 4-manifold admitting a surface bundle structure over a surface except at finitely many singular fibers, whose singularities are only of nodal type. From the structure of the singular fibers, the monodromy of each singular fiber is given by a right-handed Dehn twist along a curve, called a vanishing cycle, in the fiber. Collecting all monodromy data from the singular fibers, we obtain a monodromy factorization into right-handed Dehn twists, which completely determines the Lefschetz fibration. In this paper, we construct a fibration with one singular fiber whose monodromy is periodic (that is, the monodromy homeomorphism is isotopic to a periodic map). Following the idea of Matsumoto, we give a splitting of the singular fiber into Lefschetz fibers and determine their vanishing cycles for a collection of periodic monodromies. We describe the construction of the splitting singular fibers and the procedure for reading vanishing cycles using two branched covering structures of the fibers. We also give splittings of singular fibers into Lefschetz fibers related to a family of periodic actions and determine their vanishing cycles.

With the rise of scientific machine learning, physics-informed neural networks (PINNs) have been extensively applied to a wide range of problems. Nevertheless, most theoretical analyses of PINNs remain confined to linear equations, and a substantial gap persists between PINNs and classical numerical analysis for nonlinear problems. To address this issue, we propose a robust training framework for PINNs solving nonlinear partial differential equations, together with a rigorous error analysis. Specifically, for the $p$-Laplace equation, we introduce a novel loss formulation that combines a dual residual loss measured in $W^{-1,p'}$ with a boundary loss measured in a fractional Sobolev norm $W^{1-\frac{1}{p},p}$. This formulation is designed to accommodate the limited regularity of weak solutions and enables us to establish rigorous a priori and a posteriori error estimates. Moreover, the proposed framework and its analysis are extended to a parametric setting in which the exponent, the source term, and the boundary condition may all vary with the parameters. Finally, we present numerical experiments that substantiate our theoretical findings.

We develop eigenvalue bounds for symmetric, block-tridiagonal multiple saddle-point linear systems, preconditioned with block-triangular matrices, based on approximate Schur complements. Irrespective on the number of blocks, we prove that all complex eigenvalues, with nontrivial imaginary part, are strictly contained in a circle within the complex plane, with center 1. The real and positive eigenvalues are bounded in terms of the extremal roots of a sequences of parametric polynomials. Numerical results reveal that the bounds describe very well the eigenvalue distribution of the preconditioned matrix.

A graph $G$ is \emph{$H$-induced-saturated} if $G$ has no induced subgraph isomorphic to $H$ but changing the adjacency of an arbitrary pair of vertices in $G$ creates an induced copy of $H$. The existence problem for $H$-induced-saturated graphs had previously been settled when $H$ is a complete graph, a path, an odd cycle, or an even cycle of length at most $10$. In this paper, for every integer $q\ge3$, we construct a $C_{2q+2}$-induced-saturated graph. Hence, induced-saturated graphs exist for all cycles, except for the cycle of length 3.

We construct a plane Lipschitz domain for which the range of the Laplace operator from $H^{3/2}\cap H^1_0$ to $H^{-1/2}$ is not closed. We show that rapid oscillations of the boundary create a boundary layer that leads to unbounded $H^{3/2}$ norm of the solution of the Dirichlet problem. Our proof is elementary in the sense that it is directly based on homogeneity arguments and does not use deep results of harmonic analysis such as estimates of area integrals or harmonic measures of Dahlberg or Jerison and Kenig.

This paper develops a fully discrete finite element method for three-dimensional incompressible magnetohydrodynamic (MHD) flows with nonlinear $p$-Laplace viscosity. The scheme combines spatial finite elements with a semi-implicit Euler time discretisation. Convergence to a weak solution is proved using a time-translation compactness argument together with Minty's monotonicity method. Under additional regularity assumptions, we derive unconditional error estimates for both the velocity and magnetic field, with no coupling restriction between the time step and mesh size. The framework extends finite element analysis of incompressible MHD systems to non-Newtonian shear-thickening fluids ($p>2$), while recovering the classical Newtonian case ($p=2$). Finally, numerical simulations are provided to validate the theoretical convergence rates and demonstrate the robustness of the proposed method.

Many time series are not stable over their full observation horizon, but may contain scientifically meaningful periods during which a signal remains stable up to a prescribed tolerance. We formulate this as an existence test for stable intervals in a non-stationary regression model with dependent, locally stationary errors. For a signal $d$ derived from the mean function, including deviations from reference levels and derivatives, stability over duration $\delta$ is encoded by $d_\infty=\inf_{t\in[0,1-\delta]}\sup_{s\in[t,t+\delta]}|d(s)|$. The hypothesis $d_\infty\ge\Delta$ states that no interval of length $\delta$ remains within the tolerance $\Delta$, while rejection provides evidence for the existence of a relevant stable period. We estimate $d$ by local linear regression and construct plug-in tests for $d_\infty$. The asymptotic distribution is determined only by near-extremal windows at which the minimax functional is attained. We formalize this localization through extremal sets and derive Gaussian and extreme value approximations for kernel estimators over possibly shrinking index sets with time-varying long-run variance. The resulting tests are consistent and have asymptotic level control. The theory also extends supremum-based relevant-change tests to time-varying long-run variance and derivative-based hypotheses. Simulations and applications to physiological and engineering time series illustrate the method.

We continue the study of compactness phenomena between the set-theoretic universe and $\mathrm{HOD}$ initiated by Goldberg–Poveda \cite{GolPov}. We focus on compactness phenomena around the power-set functions of $V$ and $\mathrm{HOD}$. We prove: (1) A singular strong limit cardinal with uncountable cofinality cannot be the first place where $\mathcal{P}(\cdot )$ and $ \mathcal{P}^{\mathrm{HOD}}(\cdot)$ disagree. (2) Assuming the existence of a measurable cardinal, $\aleph_\omega$ can be the first place where $\mathcal{P}(\aleph_\omega)\neq \mathcal{P}^{\mathrm{HOD}}(\aleph_\omega)$, answering a question of Hayut. (3) If $\kappa$ is strong limit singular of uncountable cofinality, $\mathrm{HOD}$ is correct about cardinals less than or equal to $\kappa^+$ and the GCH holds in $\mathrm{HOD}$ below $\kappa^+$ then $(\mathrm{HOD}, V)$ has the $\mathrm{cf}(\kappa)^+$-cover property. We also show that the GCH assumption in (3) is necessary, which demonstrates that Magidor's classical Covering Theorem is optimal.

We consider solving an optimization instance in which the objective is quadratic and where the decision variable is a probability measure. Our class of problems are motivated by applications arising from optimal transport (with the Gromov-Wasserstein problem being a prominent example) as well as energy landscape minimization. Because the objective depends quadratically on the decision variable, our class of problems fall outside the standard modeling framework of the Generalized Moment Problems (which requires the objective to be linear). To this end, we propose a hierarchy of convex relaxations based on searching over probability measures over products of the base space. These have a natural interpretation with the moment Sum-of-squares hierarchy-a prominent framework for solving polynomial optimization instances, which we adapt to accommodate probability measures. A key conceptual contribution is to introduce a notion of positive-semidefiniteness that extends the usual notion over matrices. Under the assumption that the decision variables satisfy certain marginal constraints (as in the Kantorovich formulation of the optimal transport problem), we establish convergence of our hierarchy towards the globally optimal solution. Under the additional assumption that the objective is a polynomial, we propose a moment-SOS type hierarchy of finite dimensional semidefinite programs whose optimal solution converges to that of the original quadratic optimization over measures. We demonstrate our framework with numerical experiments. More generally, optimization over measures where the objective and/or constraint depends on the decision in a polynomial way is a fundamental problem. It is hoped that our work provides a road-map as to how the ideas of the SOS-ordinarily developed for polynomial optimization-may be applied to a broader class of non-linear problems involving measures.

This paper studies a class of multi-cluster aggregative games characterized by the coexistence of cooperation and competition, where each agent's cost function depends on its own strategy and the aggregate of all agents' strategies. To address the Nash equilibrium seeking problem for such games in the non-Euclidean setting, a distributed mirror descent algorithm with gradient extrapolation is proposed over time-varying intra-cluster and inter-cluster networks. The mirror descent framework employs a general Bregman divergence as the distance measure, providing greater flexibility than Euclidean-based methods, while gradient extrapolation exploits historical gradient information to improve convergence performance. Under the restricted strong monotonicity characterized by the Bregman divergence, the convergence of the proposed algorithm is established, and it achieves the $\mathcal{O}(1/k)$ convergence rate with the appropriately selected step-size and parameters. Finally, the effectiveness of the proposed algorithm is verified by an example on the demand response of energy systems.

In this paper, we develop an algebraic framework for merge-regime convertible codes that works directly with the polynomial-evaluation structure of the underlying codes. We characterize the minimal skew polynomials associated with unions of conjugacy classes, establish an evaluation-compatible product rule, and derive a special Chinese Remainder Theorem (sCRT) tailored to these evaluation sets. Based on this machinery, we propose two conversion templates for skew polynomial evaluation codes (PECs). The first applies to distinct initial PECs, whose different evaluation sets naturally provide the moduli required by sCRT. The second treats identical initial PECs, for which distinct auxiliary evaluation sets and explicit algebraic compatibility conditions are introduced to preserve unchanged symbols and to generate written symbols from designated read symbols. When instantiated with the standard basis $1,x,\ldots,x^{k-1}$, both constructions yield merge conversions for linearized Reed–Solomon codes whose final codes are equivalent to linearized Reed–Solomon codes and achieve per-symbol access-optimal cost. The framework further specializes to the commutative ring $\mathbb{F}_q[x]$, yielding corresponding conversion constructions for ordinary PECs and recovering the known polynomial-form constructions for Reed–Solomon and Tamo–Barg codes as special cases. As a further application, to the best of our knowledge, this specialization gives the first merge-regime convertible construction with Gabidulin initial codes and a final code equivalent to a Gabidulin code, while attaining per-symbol optimal access under the symbol-access model considered in this paper.

This paper presents a topology optimization approach for the form finding of cable networks based on the force density method. The proposed framework combines the force density method with topology optimization based on continuous density variables to identify the equilibrated geometry and effective connectivity of cable structures simultaneously. Design variables are assigned to the cable network members and control their force densities through a SIMP interpolation, complemented by an explicit binary promoting term in the objective function. As a result, cable members vanish or remain active throughout the optimization process. The equilibrium configuration is obtained by enforcing the force density method equations, while the objective drives the optimized network as close as possible to a prescribed reference form. Volume constraints, passive boundary regions, and a minimum connectivity requirement at loaded joints are incorporated to yield physically meaningful and numerically stable designs. Several numerical examples based on cable net ground structures defined by horizontal, vertical, and diagonal members are used to study the proposed approach. A dedicated rounding study shows that the relaxed density variables converge to nearly discrete values at the end of the optimization process, with only a minor correction required after thresholding. A study of the volume fraction constraint further shows that reducing the permitted structural volume can trigger a qualitative transition in the optimized topology, from a redundant, densely braced configuration to a minimal load path. Finally, an assessment of solver performance shows that the proposed formulation scales reasonably well with problem size.

Whittle index policies offer a scalable method for restless multi-armed bandits, but under partial observability even determining the indifference subsidy at a single belief requires solving an infinite-horizon belief-state problem with no closed-form value function. Liu [10] addresses this difficulty by linearizing the unknown decision boundary, leading to a linear system and a closed-form approximate Whittle index. However, the resulting threshold uses only a one-step active–passive comparison and does not account for longer-horizon continuation values. We extend this framework to a \emph{$t$-step lookahead threshold policy}. For each subsidy $m$, the threshold is defined by the active-minus-passive advantage under $t$-step finite-horizon value iteration. At $t=1$, the threshold is $m$-independent and recovers the linear threshold of Liu [10]; for $t>1$, it becomes subsidy-dependent through the induced first-crossing structure and tracks the exact decision boundary more closely. The proposed algorithmic framework does not require indexability as an input and includes an indexability verification. Under the original Whittle indexability, we prove that the $t$-step approximate Whittle index converges geometrically to the exact Whittle index, \[ |\widehat W_t(\omega)-W(\omega)|=O(\beta^t). \] Numerically, all 2,715 tested three-state instances are verified with computable priority index functions according to the proposed criterion. The P95 index error decreases from $2.18\times10^{-2}$ at $t=1$ to $8.93\times10^{-4}$ at $t=8$. In an exact-comparable instance with $\beta=0.9999$, $t=2$ already recovers the exact Whittle-index ordering. Moderate-depth threshold policies also outperform the one-step baseline and remain close to the optimal dynamic-programming benchmark, while runtime grows mildly with $t$.

Let $K\subset\mathbb{R}^n$ be a convex body with support function $h_K$. For $\nu\in\mathbb{S}^{n-1}$, $p\in\mathbb{R}$, and $u\in\nu^\perp$, we introduce the slicing support function $h_\nu(u,p)$, defined as the support function of the slice $K\cap\{x\cdot\nu=p\}$ in the direction $u$. For each fixed $p$, this is precisely the support function of the corresponding translated fiber appearing in the construction of the convex fiber body of Mathis and Meroni \cite{MathisMeroni2023}. We derive an infimal representation of $h_\nu$ in terms of $h_K$, together with a corresponding minimax identity. Using the Fenchel–Moreau theorem, we prove that $h_K$, and hence $K$, can be recovered from the slicing support function without any regularity assumption on $\partial K$. We also obtain a differential recovery formula when $K$ is strictly convex and $\partial K$ is of class $C^1$. In dimension three, we establish a cylindrical Monge–Ampére-type determinant identity expressed in terms of the spherical curvature matrix of $\partial K$. When the relevant tangent directions are principal directions, this determinant reduces to a weighted ratio of the corresponding principal radii of curvature. We further characterize this principal-direction condition by showing that, for convex bodies with $C^2$-boundary and positive Gaussian curvature, the spherical coordinate directions are principal directions away from the poles if and only if, up to translation, the body is a body of revolution. Finally, we extend the construction to higher-codimensional iterated slicing support functions and derive a full-Hessian determinant identity via the Schur complement.

Dual active galactic nuclei (DAGN) mark a critical phase in the evolution of merging galaxies and the pairing of supermassive black holes, yet they remain difficult to identify in large imaging surveys because of projection effects and limited spatial resolution. Compact foreground stars and unresolved substructure can mimic dual nuclei through chance superposition, complicating automated detection. We revisit the 46,061 galaxies flagged but rejected as DAGN candidates by the GOTHIC pipeline, primarily because the two nuclei fell within the SDSS fibre aperture or exceeded its separation threshold. We train a supervised deep-learning framework based on the YOLOv11 oriented-bounding-box architecture on annotated SDSS imaging to separate genuine dual nuclei from foreground stellar contaminants and other spurious alignments. The final model attains a validation precision of 0.919, recall of 0.905, and $F_1$ of 0.912, and yields 29,605 dual-nucleus candidates after removing star-dominated and blended detections. Structured visual inspection indicates that $54.5$–$62\%$ are consistent with genuine dual nuclei, implying $\sim(1.4$–$1.8)\times10^{4}$ plausible systems. Cross-calibrating the YOLO separation against the deterministic GOTHIC centroid measurement and restricting to the compact regime ($d \le 6.87''$) gives a conservative subset of $\sim 13{,}672$ candidates, reaching calibrated separations of $\sim 0.56''$. Spectroscopy of the most compact ($\le 1$ kpc) systems shows spectra dominated by stellar absorption, with no resolved double-peaked emission, so confirmation requires higher-resolution follow-up. The catalogue is a statistically refined list of candidates, not confirmed DAGN. Nonetheless, deep-learning detection substantially reduces contamination and expands the plausible DAGN census.