Papers

We study stochastic simple bilevel optimization with smooth, possibly nonconvex upper- and lower-level objectives accessed only through stochastic gradient oracles. A key challenge is that the dual multiplier induced by the lower-level constraint may become unbounded near lower-level stationary points, invalidating bounded-dual analyses and destabilizing stochastic gradient estimates. To address this, we propose Stochastic Dynamic Barrier Perturbed Gradient (SDBPG), a single-loop method that adaptively perturbs the dual formulation to regularize this degeneracy. The perturbation stabilizes the multiplier and yields controlled bias and variance even near the lower-level stationarity region. Under a rare-visit assumption governed by a parameter $\delta \in (0, \tfrac{1}{2}]$, SDBPG finds an $(\epsilon, \epsilon)$-stationary point in $\mathcal{O}(\epsilon^{-1/\delta})$ iterations, with sample gradient complexities $\mathcal{O}(\epsilon^{-2/\delta})$ and $\mathcal{O}(\epsilon^{-3/\delta})$ for the upper- and lower-level objectives, where larger $\delta$ corresponds to rarer visits to the bad region describing the negative alignment between the two objectives when the lower-level gradient is small. We further develop PR-SDBPG, a penalty-regularized variant that eliminates the rare-visit assumption, and VR-PR-SDBPG, which improves the resulting sample complexities entirely through variance reduction. To our knowledge, these are the first explicit $(\epsilon_f,\epsilon_g)$-stationarity guarantees for stochastic nonconvex-nonconvex simple bilevel optimization.

We give a quiver description of the space of based algebraic maps from $\mathbb{P}^{1}$ to the Lagrangian Grassmannian (and its orthogonal counterpart). We show our descriptions lead to an algebro-geometric refinement of some of the homotopy equivalences in real Bott periodicity. In particular, we get an isomorphism in Larson and Vakil's “naive algebro-geometric homotopy category” whose topological realization (after specializing to $\mathbb{C}$) recovers the classical homotopy equivalences $\Omega^{2}(Sp/U) \simeq BO\times \mathbb{Z}$ and $\Omega^{2}(O/U) \simeq BSp\times \mathbb{Z}$.

We study the problem of releasing counting-query outputs through a stochastic mechanism that is both consistent and \((\epsilon,\delta)\)-differentially private. Consistency requires the released value to lie within the feasible range of the query, while utility is measured by the worst-case probability of error. We first derive a closed-form expression for the minimum achievable error probability and obtain an explicit optimal mechanism. By exploiting the active differential privacy constraints satisfied by this mechanism, we then characterize the entire class of optimal mechanisms via a propagation argument, identifying the structural properties shared by all optimizers. We next extend the framework to the setting in which the privacy mechanism is cascaded with an arbitrary fixed stochastic transformation representing a predetermined portion of the communication medium between the source and the destination. We first establish necessary and sufficient conditions under which this partial fixation of the medium incurs no loss in utility. We then derive upper and lower bounds on the optimal achievable performance based on convex mixing and spectral perturbation. Finally, we apply the theory to (M)-ary phase-shift keying (PSK) transmission over an additive white Gaussian noise (AWGN) channel and show that uncoded transmission is effectively optimal in the high-privacy regime.

I develop a Hamiltonian framework for ${\mathbb Z}_k$ $p$-form gauge fields on arbitrary oriented cell complexes in arbitrary dimensions. Gauge qudits are defined by $p$-cells, charged boundary qudits by $(p-1)$-cells, Gauss-law generators by boundary map $\partial_p$, and magnetic checks by $\partial_{p+1}$. The same cellular structure produces local dressed Wilson operators, and at $k=2$ a Calderbank-Shor-Steane check complex relevant to quantum error correction. I then specialize to $p=2, k=2$, where the magnetic 3-cell term is absent and the one-form Gauss-law can be solved exactly. The physical Hilbert space is parameterized by plaquette electric-flux variables, while the link configuration is reconstructed as the dynamical boundary of the evolving flux domains. The reduced Hamiltonian is an Ising-type plaquette model, where its local transverse-field term is the physical image of the boundary-dressed Wilson operator $\sigma_p^z\prod_{\ell\in\partial p}\tau_\ell^z$. A tube-cap quench compares two initial flux fillings with the same initial boundary loops. Exact diagonalization on $4\times4$, $6\times4$, and $5\times5$ tori finds that the cap loses $20$-$37\%$ of its occupied-flux area, while the tube remains nearly pinned. A finite-size scaling locates a dynamical crossover of tension-to-density ratio near $(m/\varepsilon_E)_c\simeq1.89$. The unreduced plaquette-plus-link encoding provides local Gauss-law checks and a direct digital implementation, while the reduced plaquette-only Hamiltonian supplies the exact benchmark. The result places the specific top-form discharge and the cosmological constant neutralization calculation inside a general higher-form Hamiltonian and coding framework.

Let $G$ be a simple graph with maximum degree $\Delta(G)$ and chromatic index $\chi'(G)$. A graph $G$ is called edge-chromatic $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)< \chi'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>\Delta(G)\lfloor |V(G)|/2\rfloor$. In 1986, Chetwynd and Hilton proposed the influential Overfull Conjecture: If $G$ is a simple graph with $\Delta(G)>\frac{|V(G)|}{3}$, then $G$ is a Class $2$ graph if and only if $G$ contains an overfull subgraph $H$ with $\Delta(H)=\Delta(G)$. Motivated by the structural analysis for $4$-critical graphs (SIAM J. Discrete Math. 2019), we show more properties in this paper, especially four new forbidden configurations in any $4$-critical graph, and provide a new structural proof of Overfull Conjecture for graphs with maximum degree $4$.

Monadic dependence is a proposed structural dividing line for fixed-parameter tractability of first-order model checking on hereditary graph classes. A graph class is monadically dependent if the class of all graphs cannot be interpreted in its vertex-colored members using a fixed first-order formula. We prove two structural consequences of monadic dependence. First, every monadically dependent class has almost linear neighborhood complexity: for every graph $G$ in the class and every set $A\subseteq V(G)$, the family $\{N_G(v)\cap A : v\in V(G)\}$ has size $|A|^{1+o(1)}$. Second, every $n$-vertex graph in a monadically dependent class has radius-1 merge-width $n^{o(1)}$. Here, merge-width is the decomposition parameter of Dreier and Toruńczyk based on construction sequences; its radius-$r$ version measures local reachability among parts through already resolved pairs. This settles the radius-1 case of the conjectured connection between monadic dependence and almost bounded merge-width and provides the first decomposition-based structural description of monadically dependent graph classes. Our proof is algorithmic: we give an $\mathcal{O}(n^5)$-time algorithm that, given an $n$-vertex graph $G$ such that $|\{N_G(v)\cap A : v\in V(G)\}|\le O(|A|^d)$ for every $A\subseteq V(G)$, computes a construction sequence witnessing radius-1 merge-width $\mathcal{O}(n^{1-1/d}\log n)$.

Motivated by the one-dimensional and radial theory of general monotone functions, we introduce a natural multiparameter general-monotonicity class for product Fourier inequalities. The monotonicity condition is replaced by an intrinsic mixed-tail condition: a function on the positive orthant is the upper-tail potential of a complex Radon measure, and the integrated total variation of this measure is controlled by a local integral of the function. The representing measure is recovered as the full mixed distributional derivative, so the condition is intrinsic rather than coordinatewise. For coordinatewise even extensions we prove the anisotropic mixed-norm estimate \[ \|\Pi_{\alpha}\widehat f_{A}\|_{L_{\vec q}(\mathbb R^d)} \le C\|\Pi_{\beta}f\|_{L_{\vec p}(\mathbb R^d)}, \qquad \beta_k=1-\frac1{p_k}-\frac1{q_k}-\alpha_k, \] where $1\le p_k\le q_k<\infty$ and $\alpha_k>-1/q_k$. Here $\widehat f_A$ is an Abel-summed Fourier transform and agrees almost everywhere with the ordinary Fourier transform for $f\in L^1$. The exponent relation is forced by independent coordinate dilations. Thus the theorem has the same coordinatewise dilation balance as the anisotropic monotone theory, while the hypotheses are formulated in the spirit of general monotonicity: variation is controlled locally, and no product or coordinatewise monotonicity structure is imposed. The class contains all sectorial mixed-tail potentials, an infinite-dimensional cone generated by arbitrary non-product positive measures satisfying the mixed-moment condition, and real nonseparable functions which are not monotone in the coordinate variables. For compactly supported examples which are bounded below by a positive constant in a neighborhood of the origin, the range $-1/q_k<\alpha_k<1-1/q_k$ is exactly the range in which both weighted norms are finite.

A graph class is $k$-WQO if its $k$-labeled graphs are well-quasi-ordered under label-preserving induced subgraph embeddings. We show that every hereditary graph class that is $2$-WQO has bounded clique-width. Combined with the recent result of Dumas and Lopez, this confirms a long-standing conjecture of Pouzet: A hereditary graph class is $2$-WQO if and only if it is $k$-WQO for all $k\geq 2$, if and only if it is $\forall$-WQO, that is, its labeled graphs are well-quasi-ordered for every possible well-quasi-ordered label set. Our proof builds on a recent structure/non-structure dichotomy for the model theoretic notion of monadic dependence by Dreier, Mählmann, and Toruńczyk. Through the non-structure characterization by forbidden induced subgraphs, we show that every hereditary $2$-WQO graph class is monadically dependent. Leveraging the Ramsey-theoretic structural properties provided by monadic dependence, we then establish bounded clique-width by ruling out the existence of large well-linked sets, which are the canonical obstructions for clique-width.

Scott analysis provides two fundamental tools for studying countable structures: Scott sentences, which characterize structures up to isomorphism, and back-and-forth relations, which measure structural similarity. A recurring phenomenon in computable structure theory is that many notions naturally associated with level $\alpha$ of Scott analysis have effective complexity at approximately $2\alpha$ jumps. This discrepancy appears both in the complexity of the back-and-forth relations and in the passage from arbitrary infinitary formulas to computable infinitary formulas. We develop two new coding tools, the Belligerent Pairs Theorem and Belligerent Jump Inversion Theorem, which allow information at complexity level $2\alpha$ to be reflected in computable structures whose distinguishing features already appear at level $\alpha$. These results extend Harrison-Trainor's finite unfriendly jump inversion uniformly throughout the computable ordinals. As applications, we determine the optimal interaction between syntactic complexity and oracle complexity for computable Scott sentences and for formulas distinguishing computable structures. For every computable infinite ordinal $\alpha$, we determine the oracle needed to compute a $\Pi_\alpha$ Scott sentence for a computable structure which has a $\Pi_\alpha$ Scott sentence. Any computable structure with a $\Pi_\alpha$ Scott sentence has a computable $\Pi_{2\alpha}$ Scott sentence. We show that both of these bounds are sharp. We prove analogous optimal results for formulas witnessing failure of the $\alpha$-back-and-forth relation. We also obtain further applications, including a resolution of a question of Chen, Gonzalez, and Harrison-Trainor concerning the complexity of back-and-forth classes.

We develop a variational formulation of Kyle's model of informed trading that accommodates stochastic liquidity and multiple traded assets. The main equilibrium result is stated first: under a martingale dual condition, a matrix-valued martingale depth process generates a linear-Gaussian equilibrium with stochastic matrix-valued price impact. We derive this martingale from a primal-dual problem, inspired by causal optimal transport, that characterizes the endogenous speed at which the insider injects private information into prices; in general, this problem admits only local martingale optimizers, and the martingale dual condition is the hypothesis that the optimizer is a true martingale. We interpret informed trading as the optimal liquidation of private information and verify the construction in the scalar and common-eigenbasis cases. The fully general matrix-valued case reduces to a coupled matrix FBSDE, which we isolate as the remaining obstruction. Along the way, we establish an independently interesting Doob-Meyer decomposition for general (not necessarily symmetric) matrix-valued submartingales.

Generative models often represent signals as dense grids of amplitudes, blurring sharp transients that are crucial for the correctness of physical signals. We introduce Singularity Space, a generative framework that represents signals through complex-plane singularities, rooted in the classical pole-residue representation of meromorphic functions. We learn a latent space of physically constrained, per-signal singularity configurations to solve an inverse problem from degraded or partial observations. The framework has three key properties: interpretability, in which each generated singularity configuration corresponds to a set of physical parameters; structural stability, which mitigates Gibbs artifacts at discontinuities; and resolution-free output reconstruction on arbitrary grids without retraining or interpolation. Our framework employs a transformer-based diffusion model that directly predicts samples at complex-plane singularity coordinates, subject to geometric constraints during sampling. As a controlled test case for sharp-feature recovery, we evaluate our framework on 1D Burgers shocks, where each shock is represented by 32 predicted singularities (an $8\times$ reduction versus a 1024-point grid signal). Our framework preserves signal structure ($\text{TV ratio} \approx 1$) under unseen test-time observation noise, achieves a $4.2\times$ lower reconstruction error in zero-shot sub-resolution generalization than a grid-based baseline, and recovers physical parameters to $10^{-4}$ absolute error in-distribution. These results suggest that singularity-based representations may provide a practical foundation for other transient-dominated signals such as speech and biomedical signals, with potential extension to higher-dimensional domains.

We prove the existence of global weak entropy solutions to the $p$-system with the piecewise affine flux function. The construction is based on a wave-front tracking scheme for which we identify a necessary and sufficient condition for the occurrence of infinitely many fronts. For smooth fluxes, the theory is well established. However, the lower Lipschitz regularity of the flux requires a new method to obtain the interaction estimates. We also prove the stability of the solution in the sense of \cite{bia-col-02}, showing that solutions are Lipschitz continuous with respect to the $L^{\infty}$ norm of the difference of the flux derivatives. We obtain the convergence rate of the solutions from the piecewise affine flux to smooth flux by using the stability estimate.

We investigate the combinatorial structure of the tropical characteristic polynomial of symmetric matrices using the tropical permanents of their principal submatrices. We establish new inequalities for the leading coefficients of the tropical characteristic polynomial, revealing concavity properties of the coefficient sequence and yielding necessary conditions for a sequence to arise as the coefficient sequence of the tropical characteristic polynomial of a symmetric matrix. These results provide a deeper understanding of the structure of tropical characteristic polynomials associated with symmetric matrices.

This study is devoted to anti-periodic solutions of quasilinear impulsive systems with piecewise constant argument of generalized type. We rigorously prove the existence and uniqueness of an anti-periodic solution making use of the Banach fixed point theorem. The presence of a term with piecewise constant argument is the main novelty. Appropriate examples which support the theoretical findings are provided.

We relate two recent enrichments of the Hochschild theory of a finite-dimensional algebra $\Lm$: the $\tau$-Hochschild (co)homology of Cibils, Lanzilotta, Marcos and Solotar, built from Iyama's higher Auslander–Reiten translates of the regular bimodule, and the Coxeter automorphism $\sigma_\Lm$ of the Tamarkin–Tsygan calculus. We show that the Nakayama functor of the enveloping algebra transforms Happel's minimal resolution into a complex representing $\D\Lm\Ltimes_\Lm \D\Lm$, the square of the Serre bimodule whose shift generates $\sigma_\Lm$, and that the $\tau$-translates $\tau_n\Lm$ are precisely the cycle bimodules of this complex. This produces extensions $0\to \B_n\to \tau_n\Lm\to \Tor_n^\Lm(\D\Lm,\D\Lm)\to 0$ whose outer term is dual to $\Ext^n_{\Lme}(\Lm,\Lme)$ and whose inner term is a strictly Morita-theoretic residue of the minimal model. In top degree $d=\gldim\Lm$ the residue vanishes and $\tau_d\Lm$ is the dual of the degree-one component of the $(d+1)$-preprojective algebra of Iyama–Oppermann; for $\Lm=\kk Q$ hereditary, $\tau_{\Lme}\Lm\cong \D\Pi(Q)_1$ and $\HH^1_\tau(\kk Q)$ is the degree-one part of the zeroth Hochschild homology of the preprojective algebra. For self-injective algebras, the derived part vanishes identically, which explains structurally the growth of $\tau$-cohomology for the Buchweitz–Green–Madsen–Solberg algebras. Taking Euler characteristics in the Cibils–Lanzilotta–Marcos–Solotar dimension formulas recovers Happel's trace formula $\sum_i(-1)^i\dim\HH^i(\Lm)=-\tr\sigma_\Lm$. We prove that the two refinements are transversal, propose the combined Morita invariant, exhibit derived-equivalent algebras of finite global dimension whose $\tau$-translates have identical dimension but opposite composition, and pose the problem of derived invariance of $\tau$-Hochschild theory over the smooth locus.

We develop a twofold generalization of classical Yang-Mills theory, extending it from principal bundles to the setting of possibly non-transitive and non-integrable Lie algebroids. The classical theory is recovered when one considers the Atiyah algebroid of a principal bundle. In our framework, principal bundle connections are replaced by the more general notion of (infinitesimal) multiplicative Ehresmann connections. An action functional for such connections is constructed, now including a curvature 3-form contribution, alongside the usual curvature 2-form term, and the resulting variational problem is naturally constrained by a cohomological condition. We derive the associated Euler-Lagrange equations, and define a class of self-dual solutions (instantons) in both 4 and 5 dimensions. We also show that the solution space is invariant under gauge transformations, and compute its tangent space at a solution. As an important example, we show that our framework produces a Yang-Mills theory for connections on bundle gerbes.

Self-supervised data splitting has emerged as a promising paradigm for sparse-view CT reconstruction, enabling training from incomplete measurements without fully sampled ground truth. However, the influence of key design choices, including partitioning strategy, preprocessing, and inference, remains insufficiently understood. In this work, we introduce a unified framework that decomposes splitting-based reconstruction into these three components, enabling controlled comparison of existing methods and two incremental extensions: multi-partition splitting and an alternative inference strategy. Experiments on simulated LoDoPaB-CT data under independent and correlated noise, together with validation on the real-world 2DeteCT dataset, show that the optimal partitioning strategy strongly depends on the measurement noise structure. Lattice-based splitting performs favorably under independent noise, whereas angular masking is more robust under correlated noise and real measured data. Multi-partition splitting consistently improves over pure projection-wise splitting in several settings. Complementary perceptual and structural metrics, including LPIPS and HaarPSI, reveal differences between masking strategies that are less apparent from PSNR and SSIM alone. These results provide practical guidelines for designing self-supervised sparse-view CT reconstruction methods and highlight the limitations of common independence assumptions in realistic imaging environments.

Certifying the safety of a control action in real time, from streaming partial observations of a nonlinear, interconnected system under non-stationary disturbances, is a problem no existing data-driven framework solves. Batch methods such as data-enabled predictive control require a pre-collected dataset and offer no stability certificate for nonlinear dynamics; informativity-based approaches characterize data sufficiency offline and non-recursively; neither exploits the known graph topology of networked systems as a structural prior. This paper addresses both limitations. First, we develop a streaming contraction certificate beta_cert(t) = beta_hat(t) - rho(t), where beta_hat(t) is estimated recursively via integral regression on a sliding window of partial input-output observations, and rho(t) is a data-dependent uncertainty radius mapping estimation error to a conservative bound on the true closed-loop contraction rate. The certificate issues a provably safe deployment signal the moment beta_cert(t) crosses and sustains above zero. Second, we introduce a topology-aware estimator enforcing known graph adjacency as exact zero constraints on the Jacobian, reducing the effective parameter count per row from O(N) to O(d_max) for maximum node degree d_max. On a five-node nonlinear benchmark under heavy-tailed Laplace disturbances with two observed nodes, the streaming certificate achieves certified deployment at t*=2.6s from 130 samples, 17 seconds earlier than an offline batch baseline, with 16x lower accumulated error during the unprotected window. The topology-aware estimator cuts certification time by 59% (1.62s vs 3.98s) and accumulated disturbance cost by 58%, with the advantage persisting across all window sizes below 40 samples. The framework is domain-agnostic and applies to any large-scale nonlinear networked system under streaming data and partial observations.

We examine combinatorial structures that generalize $(k,n)$-nets, orthogonal arrays, and mutually orthogonal Latin squares. By a reticulation we mean a point set and two collections (types) of families of lines such that two lines of different types meet in exactly one point and each family of lines partitions the point set. The number of points incident with any line depends only upon the type of the line, and every point is incident with the same number of lines of a given type. Each choice of a single line family of each type leads to an arrangement of the points into a rectangular grid. Recording the line containing a given point in the corresponding position of an array gives a generalization of sets of mutually orthogonal Latin squares, dubbed a cooperative system. A cooperative system consists of a collection of column-Latin matrices and a collection of row-Latin matrices such that each column-Latin matrix is orthogonal to each row-Latin matrix. Recording lines which contain a given point as a tuple gives a generalization of certain orthogonal arrays, dubbed semi-orthogonal arrays. Notions of parastrophy and isotopy for cooperative systems, corresponding to permuting families of lines and lines within each family, are introduced. Constructions of small reticulations and three recursive constructions of larger reticulations are given.

Growing interest in space interferometry for detecting bio-signatures of exoplanets has led to the development of several low-cost mission concepts involving Earth-orbiting formation flying techniques to perform nulling interferometry of incoming light from exoplanetary systems. Pursuing these developments, this work proposes a multi-objective dynamic programming scheme to optimize the balance between fuel expenditure and scientific outcomes of low-cost formation flying space interferometry missions by leveraging several insights from mission design. This scheme accounts for nominal concept of operations constraints, fuel expenditure, and observability conditions to produce a globally optimal Pareto front policy for exoplanetary system observation, and represents one of the first applications of multi-objective optimization to distributed space telescope astronomy.

This paper gives some technical results about cofibrant symmetric spectra of simplicial sets, and uses them to produce a model of the $K$-theory of a Waldhausen category that lands in the subcategory of (projective) cofibrant spectra.

The dual lattices $E_6^*$ and $E_7^*$ are of particular interest in source coding and data compression applications. Among all known lattices in dimensions six and seven they attain the smallest normalized second moments, i.e., the smallest average quantization error. Their use in practice requires fast closest-point (nearest-lattice-point) algorithms. The known approach, due to Conway and Sloane and completed for $E_6$ and $E_6^*$ by Takizawa, Yagi, and Kawabata (TYK), decodes these lattices as unions of cosets of root lattices $A_n$: each coset is decoded separately, and the best result is kept. This requires four coset decodings for $E_7^*$ and six for $E_6^*$, together with explicit distance computations. This paper shows that all these coset decodings can be collapsed into a single sweep. Reformulated in terms of glue vectors, the TYK decompositions state that $E_7^*$ is the union of the even glue classes of $A_7^*$, and that $E_6^*$ is a parity-matched sublattice of $A_1^*\oplus A_5^*$. The candidate chain constructed by the closest-point algorithm of McKilliam, Clarkson, and Quinn (MCQ) for $A_n^*$ visits every glue class of $A_n$ exactly once and is optimal within each class. Consequently, one sorted sweep per coordinate block yields the closest points of all glue cosets simultaneously, and $E_6^*$ and $E_7^*$ are decoded at roughly the cost of a single $A_5^*$ or $A_7^*$ quantization. Rough operation counts indicate a $4$–$6\times$ reduction for $E_6^*$ and $3$–$4\times$ for $E_7^*$ relative to coset-by-coset decoding. We also discuss further constant-factor improvements available from recent refinements of the $A_n^*$ algorithms, and an open question concerning sort-free linear-time decoding.

Let $K, L$ be convex bodies in $\mathbb{R}^n$ with $K$ centered. Assume that $|K \cap \theta^{\perp}| \le |L|\theta^{\perp}|$ for all $\theta \in S^{n-1}$. We prove that $|K| \le c\sqrt{n}|L|$, which is sharp up to the choice of the absolute constant. The result gives the sharp isomorphic order in a mixed section-projection comparison problem, complementing the isomorphic Busemann-Petty and Shephard problems. It also removes the John's position assumption from an earlier result of the author, up to an absolute constant factor.

We extend, in Isabelle/HOL, the deep-and-shallow embedding methodology of our prior work from propositional to first-order modal logic (FML) with constant-domain Kripke semantics. Three embeddings of FML into classical higher-order logic (HOL) are provided side by side: a deep embedding, a heavyweight maximal-shallow embedding, and a lightweight minimal-shallow embedding. The minimal-shallow embedding is presented as an Isabelle/HOL locale, parametrised by an accessibility relation, a world-indexed interpretation, a universe of worlds, and a variable assignment; the locale form admits a global faithfulness theorem, stating that quantifying over all minimal-shallow interpretations recovers exactly deep validity. A central technical contribution is a mechanisation, for FML under constant-domain Kripke semantics, of the (countable) downward Löwenheim-Skolem theorem, which underpins the automation of our faithfulness proof between the deep and minimal-shallow embeddings. Deploying it inside an extension of the minimal-shallow locale resolves the surjectivity problem that arises against an uncountable domain of individuals – where the locale's variable assignment, having countable domain V = nat, cannot be surjective onto the domain – and thereby yields faithfulness over the full domain. Since prior work treats only the propositional fragment, we develop here the substitution machinery (free/bound-variable predicates, the fresh-variable function, capture-avoiding substitution, alphabetic renaming, the substitutability predicate, the substitution lemma, and size-based induction principles) needed for the first-order quantifiers.

We study the population gradient flow of an infinitely wide two-layer neural network learning a misspecified single-index model in high dimension. The two layers are optimized jointly, with a perturbative parameter tuning the relative training speed between the first and second layer. This setting was considered by Berthier, Montanari and Zhou in \cite{berthier2024learning}, who conjectured a hierarchical learning scenario with explicit timescales as the second layer is trained faster than the first. In this paper, we prove that the constant and linear components of the hidden link function are indeed recovered within the predicted timescales, at sharp explicit thresholds. We then analyze the onset of learning of the quadratic component and show that the components learned at earlier stages continue to influence the dynamics in an essential way. Our proof is based on quantitative approximation results for singularly perturbed flows evolving near a manifold defined by integral constraints. At a phenomenological level, we also show that the empirical measure of the weights displays singular behaviour when reaching the quadratic component of the hidden link, with a small fraction of neurons growing significantly while the remaining ones rearrange to preserve the components already learned.

We develop a two-axis interpolation framework for the O$(n)$ universality family, treating the spatial dimension $D$ and the spin-component number $n$ as independent continuous parameters connecting exact limiting solutions. On the spatial axis, anchoring between the Onsager solution at $D=2$ and mean-field theory at $D\to\infty$ yields a closed-form prediction for the 3D Ising critical coupling that agrees well with Monte Carlo benchmarks $K_c = 0.2204$ (benchmark: $0.22165$) with no adjustable parameters. Wilson–Fisher-constrained polynomial interpolation gives $\nu=2/3$, $\beta=31/96$, and $\eta=35/864$ at $D=3$ (benchmarks: $0.6299$, $0.3265$, $0.0362$), and reproduces conformal-bootstrap results across $3 \le D < 4$. On the spin axis, we establish a necessary compatibility criterion: two-anchor interpolation succeeds only for observables that vary monotonically between the anchor values. The critical coupling $K_c(n)$ violates this criterion because the Heisenberg value falls below the spherical limit, whereas the correlation-length exponent $\nu(n)$ satisfies it. A perturbative $1/n^2$ expansion yields $\nu(3) = 0.7493$ (benchmark: $0.7112$), and propagation through exact scaling relations gives $\beta(3) = 0.3797$ (benchmark: $0.3689$) and $\gamma(3) = 1.489$ (benchmark: $1.396$), without introducing additional parameters. The framework naturally extends to non-integer spin, producing the prediction $\nu(2.5) = 0.7143$ for the O$(2.5)$ universality class. These results establish dimensional and spin interpolation as a unified and predictive approach to critical phenomena, while clarifying the structural conditions under which interpolation succeeds.

Proximal gradient methods are widely used in imaging, and their speed of convergence can be accelerated by incorporating variable metrics and/or extrapolation steps. Recent works have shown that preconditioning strategies can significantly enhance this acceleration, in particular, for image deblurring problems. In parallel, a multilevel framework has been introduced to speed up inertial and inexact forward-backward schemes for image restoration problems. In this paper, we combine preconditioning and multilevel strategies to design a robust and consistent acceleration framework for both standard and inexact forward-backward schemes applied to regularized convex optimization problems. Numerical experiments in image deblurring confirm that our approach yields a substantial improvement in convergence speed compared to standard methods.

On square-free moduli $q\in(X,2X]$ having at most five prime factors, we prove that each sign of the normalized Kloosterman sum $\operatorname{Kl}(1;q)$ occurs $\gg X/\log X$ times. This improves the recent unconditional result of Zhang and Zhong for moduli with at most six prime factors. Building on their analytic estimates and optimized Selberg sieve, we replace their truncated divisor penalty by a geometric half-weight. The new weight retains the $P_5$ exclusion threshold and is a positive linear combination of two standard two-parameter truncated divisor weights, so the Zhang–Zhong transference argument applies without alteration. After transference, the relevant pointwise coefficient is reduced from $5/16$ to $5/32$, which yields a positive final sieve margin.

In this paper, we establish the theoretical foundations for the generic optimization problem in a Hilbert space whose feasible set is an affine-transversal Hilbert submanifold given by the intersection of a nonlinear manifold and an affine subspace. We develop the geometric and analytical toolkit, such as the tangent-space characterization, projection operators, and implicit retraction operators, to essentially ensure the feasibility of iterates for algorithmic design. We also derive weak-form expressions for the derivatives of lifted objective functionals. In particular, we introduce a projection-induced Riemannian metric whose induced norm is uniformly equivalent to the ambient norm, under which the projection-induced gradient becomes an exact Riemannian gradient with an explicit formula. This construction replaces the implicit tangent-space Riesz representation underlying classical Riemannian optimization with directly computable operator evaluations, yielding a practical variable-metric framework for algorithmic design while preserving the geometric structure required for convergence analysis. With these theoretical foundations, it becomes possible to apply standard techniques in Euclidean spaces to design algorithms with strictly feasible iterates for the optimization problem on an affine-transversal Hilbert submanifold. We also propose the theoretical frameworks for algorithmic design on affine-transversal Hilbert submanifolds by showcasing the Riemannian line-search and trust-region algorithms with rigorous convergence analysis.

Although neural ordinary differential equations (NODEs) are a powerful framework for learning continuous-time dynamics, they generally do not preserve essential qualitative properties, such as positivity. We propose a structure-preserving Neural ODE framework based on nonstandard finite difference (NSFD) discretization. The learned dynamics are parameterized by nonnegative production and destruction rates, yielding an explicit, differentiable update that integrates seamlessly into standard automatic differentiation pipelines. We prove that the resulting scheme unconditionally preserves positivity for arbitrary time-step sizes while retaining first-order consistency. We outline an extension based on Patankar-type discretizations that preserves conservation laws exactly. Numerical experiments on an SIR epidemic model show that our approach generates physically meaningful trajectories, remains robust under coarse discretizations, and outperforms conventional NODEs in preserving the qualitative structure of the learned dynamics.