Papers

The present paper is devoted to a systematic study of the $p$-Brownian convergence introduced in \cite{boudabra2026stability} (in press) to study the stability of the planar Skorokhod embedding problem \cite{gross2019,Boudabra2020}. The first part is an illustration of some geometric aspects of the $p$-Brownian convergence. The second part turns this notion into a metric between domains. More precisely, we place it within the framework of optimal transport theory. Several results are obtained, namely asymptotic behavior in case of homothetic domains. Numerical illustrations are provided as well.

We investigate the spectral properties of non-Hermitian real random matrices whose entries exhibit long-range correlations decaying as~$|r-r'|^{-\alpha}$. We find a progressive breakdown of the circular law, controlled by the decrease of~$\alpha$. In all cases, the radial eigenvalue density decreases away from the origin. At~$\alpha>1$, an effective radius, reminiscent of the circular law, is retrieved, while instead, for~$\alpha<1$, the eigenvalue distribution broadens with matrix size and its spectral radius grows like a power law, with exponents numerically close to the exponents controlling the magnitude of fluctuations in the extended central limit theorem. The case~$\alpha=1$ appears as a case with self-similar eigenvalue density, and slowly growing spectral radius. Long-range correlations also enhance clustering of real eigenvalues and slow the resorption of the Saturn effect. These results reveal a correlation-driven transition and suggest the emergence of a new universality class for correlated non-Hermitian random matrices.

A graph $G=(V,E)$ is said to be word-representable if there exists a word $w$ over the alphabet $V$ such that two distinct letters $x,y\in V$ alternate in $w$ if and only if $xy \in E$. Word-representable graphs form a well-studied graph class with connections to graph orientations, combinatorics on words, and graph coloring. A near-triangulation is a planar graph in which every face except the outer face is a triangle. Several subclasses of near-triangulations have previously been investigated in the context of word-representability, including polyomino triangulations, triangulations of rectangular polyominoes with a single domino tile, $K_4$-free near-triangulations, face subdivisions of triangular grid graphs, triangulations of grid-covered cylinder graphs, and chordal near-triangulations. In this paper, we obtain a complete characterization of word-representable near-triangulations in terms of forbidden induced subgraphs. Our result unifies and extends the previously known characterizations for the above subclasses, while also correcting inaccuracies appearing in earlier works.

The Bruhat order on permutation matrices extends to alternating sign matrices via corner-sum matrices, where the order is given by entrywise domination. A classical result of Lascoux and Schützenberger states that alternating sign matrices form the Dedekind-MacNeille completion of the Bruhat order on permutations. Brualdi and Dahl introduced alternating sign hypermatrices as a three-dimensional analogue of alternating sign matrices and used them to generalise Latin squares, which may be viewed as three-dimensional analogues of permutation matrices. In this paper, in analogy with the two-dimensional case, we define and study a Bruhat order $\preceq_B$ on Latin squares and alternating sign hypermatrices. We introduce the corresponding corner-sum hypermatrices $\mathcal C_n$ and prove that entrywise domination on $\mathcal C_n$ encodes this order. We show that $\mathcal C_n$ is a distributive lattice, but that, unlike in dimension two, it is not the Dedekind-MacNeille completion of the poset of Latin squares. We further characterise the covering relations for $\mathcal C_n$ and prove rank formulae generalising the classical case of alternating sign matrices. Finally, we define monotone hypertriangles, prove that they are in bijection with $\mathcal C_n$, and show that they also encode the order by entrywise domination.

In this work, we investigate the algorithmic aspects of two natural extensions of hereditary classes: the edge-apex class and the edge-add class, recently introduced by Singh and Sivaraman. These are defined as the graph classes obtained by at most one edge deletion or one non-edge addition, respectively, from a hereditary class $\mathcal{G}$. Building on earlier results showing that both classes remain hereditary and admit finite forbidden induced subgraph characterizations whenever $\mathcal{G}$ does, we focus on the Weighted Maximum Clique Problem (WMCP) and the Weighted Maximum Independent Set Problem (WMISP). We first present algorithms for WMCP and WMISP on both the edge-apex and edge-add classes of hereditary graph classes. Extending this framework, we introduce the notion of the $\mathcal{G}$-edge distance of a graph $G$, denoted by $\xi_{\mathcal{G}}(G)$, which quantifies how far $G$ is from the class $\mathcal{G}$ in terms of the minimum number of edge deletions or non-edge additions needed to transform it into a member of $\mathcal{G}$. By parameterizing with respect to this distance, we show that both WMCP and WMISP can be solved in $O^*(2^k)$ time on graphs whose $\mathcal{G}$-edge distance is $k$, provided these problems admit polynomial-time algorithms within the class $\mathcal{G}$. This result extends earlier algorithmic characterizations of the single edge-apex and edge-add classes to the more general setting of $k$-edge-distant graphs. By combining our general results with known properties of transitive graphs, we show that WMCP and WMISP can be solved in $O^*(2^k)$ time for graphs with transitive-edge distance $k$.

The stability and deformation theory of Einstein metrics traditionally relies on the classical Berger-Ebin transverse-traceless gauge, which structurally decouples the scalar trace from the divergence-free component of metric perturbations. In the present paper, we introduce a new spectral-geometric framework based on the Chen-Nagano gauge condition. This condition naturally arises from the harmonicity of the identity map and is intrinsically satisfied by the Ricci tensor itself via the contracted second Bianchi identity. Unlike the classical transverse-traceless framework, the Chen-Nagano gauge preserves a nontrivial interaction between the trace and trace-free sectors of a deformation. We establish a first-order differential relation proving that the divergence of the trace-free part is completely governed by the gradient of the scalar trace. Utilizing commutation formulas on Einstein manifolds, we derive a second-order spectral coupling relation that links the Lichnerowicz Laplacian to a shifted scalar operator. As a primary geometric consequence, we prove that under suitable spectral pinching assumptions, the Chen-Nagano gauge collapses to the classical transverse-traceless gauge. Specifically, we show that on compact connected negatively curved Einstein manifolds, any volume-preserving Chen-Nagano harmonic deformation whose trace-free component lies below a specific spectral threshold determined by the Einstein constant is necessarily transverse-traceless. Furthermore, we connect this rigidity to the curvature operator of the second kind, establishing explicit lower spectral bounds. Finally, we provide a dynamical interpretation within the Ricci flow framework, demonstrating that the linearized Ricci flow under the Chen-Nagano gauge reduces to a strictly parabolic equation governed by the Lichnerowicz Laplacian, ensuring exponential decay of admissible perturbations.

In this paper, we develop the foundations of the representation theory of quiver Hecke–Clifford superalgebras. We further construct a Schur–Weyl duality between quantum affine analogues of the queer Lie superalgebra and the quiver Hecke–Clifford superalgebra of type A, based on the construction due to Kwon–Lee.

In this paper, we study the convergence rate between reflected backward stochastic differential equations with quadratic generators and their penalized BSDEs. Using techniques of BMO martingales, we prove the convergence rate is at order $\frac{1}{2}$ as a function of the penalty parameter. Finally, the result is applied to study numerical approximation of reflected BSDEs with sub-quadratic generators by the Euler's polygonal line method.

We study weighted Helmholtz–Hodge decompositions of drift vector fields associated with second-order diffusion operators on $\mathbb{R}^d$, $d\ge 2$. Given a decomposition of the form \[ \mathbf{G}=A\nabla\Phi+\mathbf{B}, \] we relate the weighted divergence-free condition $\mathrm{div}_{\mu}(\mathbf{B})=0$, where $\mu=e^{2\Phi}dx$, to infinitesimal invariance of $\mu$ for the operator \[ \frac12 \mathrm{trace}(A\nabla^2)+\langle \mathbf{G},\nabla\cdot\rangle. \] We compare weighted, orthogonal, and strictly orthogonal Helmholtz–Hodge decompositions and show that uniqueness of the infinitesimally invariant measure yields uniqueness of the corresponding weighted decomposition, and hence a canonical potential. For linear vector fields, we characterize Gaussian infinitesimally invariant measures by an algebraic Riccati equation together with a trace condition. In the Ornstein–Uhlenbeck case, this gives a structural proof of the classical criterion that a finite invariant measure exists if and only if the drift matrix is Hurwitz, and it identifies the associated strictly orthogonal decomposition. Finally, we treat nonlinear polynomial perturbations that preserve a given potential and obtain explicit classes of drifts for which the invariant measure and the weighted decomposition remain unique. The results clarify the relation between Lyapunov-type potentials, non-reversible perturbations, and invariant measures for diffusion semigroups.

We study generic $d$-dimensional rigidity in sparse random graphs. Our main result is that for every $d\ge 2$, the Erdős–Rényi random graph $G\sim G(n,c/n)$ undergoes a $d$-rigidity phase transition at the known, explicit, $d$-orientability threshold $c_d$: If $c<c_d$, then $G$ is asymptotically almost surely (a.a.s.) independent in the generic $d$-rigidity matroid. Moreover, in this regime $G$ has no linear-size rigidity components: it contains no induced $d$-rigid subgraphs with more than $3$ vertices, and the largest clique in its $d$-rigidity closure has size at most $o(\sqrt n)$. If $c>c_d$, then $G$ is a.a.s. not independent in the generic $d$-rigidity matroid, and we give a sharp asymptotic estimate for its rank. In addition, the $d$-rigidity closure of $G$ has a giant clique of linear size, which contains all but at most $o(n)$ vertices of the $((d+1)+d)$-core of the graph. More generally, we compute, up to a $1+o(1)$ factor, the generic $d$-rigidity rank of random graphs with a given degree distribution. For example, we show that the uniform $n$-vertex $k$-regular graph a.a.s. has rank $\min(k/2,d)n+o(n).$ Our approach is to estimate the rigidity rank of a random graph from its Galton–Watson local weak limit, using a parameter that we call local flexibility.

The 2023 paper “On Team Decision Problems With Nonclassical Information Structures” [1] presented information states and dynamic programming (DP) equations for delayed sharing information patterns, based on the concept of person-by-person (PbP) optimality of static team theory. In particular, [Section IV, 1], Theorem 5 presents recursions of the information states and Theorem 7, eqn(57), eqn(58), presents DP equations, of each team member. In this note we show that the proof of Theorem 5 and the DP eqn(57), eqn(58) of Theorem 7 of [1] are incorrect. Consequently, Lemma 8, Theorem 6, and the full statement of Theorem 7 in [1] are invalid, because their proofs rely on erroneous information states and the optimality of separated strategies (i.e., functions of the information states). We further provide the correct DP equations for PbP optimality, thereby highlighting the subtleties and challenges inherent in the analysis of delayed sharing information patterns.

We study volume-refined achievability and converse bounds for noisy permutation channels generated by strictly positive DMCs, allowing the reachable output polytope to have arbitrary affine dimension $d\ge 1$. The reachable output polytope may be lower-dimensional than the output simplex, whereas existing refined achievability analyses and fixed-error converses are not adapted to this intrinsic affine geometry. On the achievability side, we develop an affine-coordinate simplex-lattice construction adapted to the reachable output polytope, together with a nearest-neighbor decoder and a geometric error-reduction argument in the same coordinate space. This yields a Gaussian achievability approximation with an $o(1)$ remainder. On the converse side, we first use a meta-converse combined with a KL covering and a local testing estimate to obtain a fixed-error converse with a bounded remainder, which implies the logarithmic $\epsilon$-capacity $d/2$. We then apply the meta-converse with a stratified Jeffreys-mixture auxiliary output distribution. Using a local Laplace approximation and a local likelihood-ratio approximation, this choice identifies the Fisher-volume term and an explicit Gaussian testing constant, yielding a constant-order converse approximation with an $o(1)$ remainder. The achievability and converse constants arise from different constructions and are not claimed to match in general.

An edge cut C of a graph G is tight if |C \M| = 1 for every perfect matching M of G. Barrier-cuts and 2-separation cuts, also referred to as ELP-cuts, are two important types of tight cuts in matching covered graphs. Edmonds, Lovasz and Pulleyblank [Brick decompositions and the matching rank of graphs, Combinatorica 2(3) (1982) 247-274] proved that if a matching covered graph has a non-trivial tight cut, then it also has a non-trivial ELP-cut. In confirmation of a conjecture proposed by Carvalho, Lucchesi and Murty, Chen et. al. [Laminar tight cuts in matching covered graphs, J. Comb. Theory, Ser. B, 150 (2021) 177-194] showed that if C is a non-trivial tight cut of a matching covered graph G, then G has at least one C-sheltered non-trivial barrier or a 2-separation cut that is laminar with C. In this paper, we present a complete characterization of non-trivial tight cuts in matching covered graphs, from which the result of Chen et. al. can be derived directly. Moreover, we show that the lower bound of the number of C-sheltered non-trivial barrier or a 2-separation cut that is laminar with C in the result of Chen et. al. is sharp.

We study the algebraic compatibility between quantum homomorphic encryption and stabilizer quantum error correction in an honest-but-curious model, restricting the encryption to the transversal block-Pauli mask $(X^aZ^b)^{\otimes n}$ controlled by two secret key bits $a$ and $b$. We prove a necessary and sufficient criterion for an $[[n,1,d]]$ stabilizer code to remain compatible with this mask, namely that the transversal $X$ and $Z$ operators commute with every stabilizer generator, and we specialize it to a classical-code criterion for Calderbank–Shor–Steane codes. Combining code-space preservation with the logical action of the mask, every stabilizer code falls, with respect to each of the two transversal mask operators, into exactly one of three classes: incompatible, compatible but trivially acting, or compatible with a nontrivial logical Pauli action; the two operators need not fall into the same class. We verify this classification by exact symplectic computation for rotated and planar surface codes, toric codes, triangular and toric color codes, the $[[15,1,3]]$ code and several small codes, and find that the class is fixed by the lattice size and the boundary presentation rather than by the topological phase: the surface-code family realizes all three classes, whereas color codes realize only the two compatible ones. For encrypted computation we show that the Shor code does not implement the logical $T$ gate by a transversal $T^{\otimes 9}$, whereas suitable triorthogonal codes do so up to a Clifford correction. We formulate logical-Pauli masking as a code-independent algebraic template. Syndrome extraction and recovery are key-independent whenever the mask preserves the code space, so that the client only needs the block size and the class of the code to decrypt.

We derive confidence intervals and confidence sequences for causal effects in situations where the back-door criterion is applicable. Our tightest confidence intervals hold in the standard setting where the training data consists of IID observations over a system described by a given causal diagram. When interventions are allowed to depend on the past data, our confidence intervals become wider and involve a term coming from the law of the iterated logarithm, even where the number of observations is known in advance. In the sequential setting where the number of observations is not given, our confidence intervals, arranged into a confidence sequence for causal effects, involve more iterated logarithm terms and become even wider.

For operators defined on locally convex spaces we define the notions of boundedness and ergodicity associated to an infinite matrix. Given two matrices $ A$ and $ B$, we study when $ A$-bounded operators are $ B$-ergodic. Using this approach we obtain equivalent formulations for the classical notions of power boundedness, Cesàro boundedness and mean ergodicity.

We study the problem of multiclass PAC learning with bandit feedback in the realizable setting. In this framework, there is an unknown data distribution over an instance space $\mathcal{X}$ and a label space $\mathcal{Y}$, as in classical multiclass PAC learning, but the learner does not observe the labels of the i.i.d. training examples. Instead, in each round, it receives an unlabeled instance, predicts its label, and receives bandit feedback indicating only whether the prediction is correct. Despite this restriction, the goal remains the same as in classical PAC learning. We provide a general characterization of the optimal sample complexity of this problem, sharp for every non-trivial concept class, up to logarithmic factors. Our characterization is based on a new combinatorial dimension, termed the bandit $\mathrm{DS}$ dimension, defined via generalized combinatorial structures we call pseudo-boxes. These extend the pseudo-cubes underlying the $\mathrm{DS}$ dimension by allowing a different number of neighbors in each coordinate. In contrast to the $\mathrm{DS}$ dimension, which governs the full-information setting by counting the number of coordinates in the pseudo-cube, the bandit $\mathrm{DS}$ dimension aggregates the number of neighbors across coordinates, leading to a characterization in which the sample complexity scales with the total number of neighbors. We also propose a general learning algorithm achieving the upper bound, based on an algorithmic principle called ListCascade, which connects bandit learning to list learning and may be of independent interest.

Nonlinear filtering with correlated noise leads to a Duncan-Mortensen-Zakai (DMZ) equation in the form of a stochastic partial differential equation (SPDE). Unlike the independent noise case, the presence of correlation prevents the classical invertible transformation that reduces the DMZ equation to a deterministic partial differential equation, requiring a direct numerical treatment of the SPDE. This paper develops a tensor train (TT) based framework for solving medium- to high-dimensional DMZ equations with correlated noise. Spatial discretization transforms the SPDE into a high-dimensional stochastic differential system, which is efficiently compressed using TT approximation. A semi-implicit Milstein scheme is employed for temporal integration to ensure stability and accuracy. Under suitable regularity assumptions, we establish a convergence analysis of the proposed method. In particular, the spatial error is controlled by both the mesh size and the prescribed TT approximation accuracy. In the temporal direction, the convergence is proved by estimating stochastic integrals involving drifted observations, without invoking a change-of-measure argument. Numerical experiments demonstrate that the proposed method achieves stable and accurate performance for cubic sensor problems. In challenging multi-modal settings, where particle filter and extended Kalman filter deteriorate, the proposed method maintains accuracy and effectively captures the posterior distribution.

We characterize the fixed points of the operator $\Gamma_p\Pi_p^\ast$ for $n\geq 3$ and $1<p<n$. More precisely, we prove that a convex body $K\in\mathcal K_o^n$ satisfies \[\Gamma_p\Pi_p^\ast K=cK\] for some $c>0$ if and only if $K$ is an origin-centered ellipsoid, thereby settling the Lutwak–Yang–Zhang fixed-point conjecture in this range. Our proof is based on a variational analysis along linear reflection shadow systems. To address the nonlinear structure of the $L_p$ setting, we introduce the $L_p$-Projection Rolodex, which provides a dimensional reduction of the volume of the polar $L_p$-projection body to weighted lower-dimensional sectional functionals. A suitable change of variables, together with Ball's harmonic Prékopa–Leindler inequality, yields the convexity needed along the shadow system. Under the fixed-point condition, a first-variation identity then forces $\operatorname{vol}_n(\Pi_p^\ast K_t)$ to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.

In this paper, we provide techniques to obtain left non-degenerate set-theoretic solutions of the Yang-Baxter equation, drawing on the class of right groups. To this end, we introduce the new algebraic structures of left $RG$-semibraces, which include left (cancellative) semibraces as a proper subclass.

Recently, for a limited class for bipartite links, the complicated Khovanov-Rozansky matrix factorization technique was reduced to an analogue of elementary Kauffman-Khovanov cycle calculus for an arbitrary $N$. In this note, we demonstrate the consistency of such reduction with the computation of the bipartite Khovanov polynomials for $N=2$. Namely, we explain how the Kauffman-Khovanov $2^2$-hypercube is shrinked to the bipartite 3-hypercube.

While motivated by structural problems in mathematical music theory, this article introduces a novel combinatorial framework that advances the classification of cyclic cubic bipartite graphs. We extend the classical study of Levi graphs by endowing their vertices with an internal algebraic anatomy – specifically, treating them not as empty geometric nodes, but as defined subsets of a cyclic base space Z_n. This internal structure allows us to formalize and classify a highly restricted class of graph isomorphisms: those strictly induced by global affine bijections f(x) = ax+b (mod n) operating directly on the underlying base set. By applying this framework to generalized tone networks (Tonnetze) unrolled via the Chinese Remainder Theorem in composite dimensions – specifically the classic 12-TET (3x4) and the decaphonic 10-TET (2x5) – we reveal absolute geometric anchors for these spaces, namely the (9,4) and (6,5) systems respectively. We completely classify the topological orbits of these structured graphs, proving a fundamental architectural dichotomy: while the isomorphic landscape of 12-TET splits into an orientation-preserving family and an orientation-reversing chiral mirror (providing a rigorous foundation for musical Negative Harmony), the 10-TET space is unconditionally orientation-preserving. Finally, we demonstrate that these abstract combinatorial properties manifest as rigidly coherent, parallel auditory universes through explicit structural voice-leading maps and acoustic physical modeling synthesis.

We study the \(N=2\), large–\(d\) sector of BFSS/BMN-type matrix quantum mechanics on the lattice in the Gaussian regime. We develop a radial endpoint formulation in which the bulk, gauge, and longitudinal degrees of freedom are integrated out, leaving transverse endpoint variables governed by an effective holonomy potential. We show that this planar endpoint formulation is equivalent to the angular Molien–Weyl description of the gauge-projected partition function, up to a universal spectator factor. This relation allows the low-temperature expansion of the endpoint partition function to be obtained from the Molien–Weyl result, whose quadratic coefficient \(d(d+1)/2\) counts Gaussian singlet states above the vacuum. We then analyze the continuum limit of the quadratic coefficient and show that it separates into a Gaussian contribution, a \(D\)-channel, and a \(\beta\)-channel. The naive Gaussian term becomes trivial, while the exact holonomy kernel generates finite continuum contributions through singular dependence on the endpoint Gaussian width and anisotropic coupling. We then study the geometry of the holonomy potential and show that its relevant saddle is a constrained boundary saddle on the aligned branch, rather than an unconstrained critical point. The associated transverse expansion captures the local saddle geometry, but any finite polynomial truncation has a trivial continuum limit. Finally, we introduce a non-polynomial toy model based on \(V_{\rm toy}(B)=-\log\cosh B\), which provides a completion of the transverse expansion and reproduces exactly the continuum \(D\)-channel contribution \(-2d\). This prepares the geometric interpretation of the \(D\)-channel as a Wishart–Stiefel entropy associated with an emergent four-dimensional geometry embedded \(\mathbb R^d\) in the endpoint formulation.

In this paper, we study the eigenvalue problem \[\left\{\begin{array}{cl}-\hbox{div}\left(a(x)\frac{Du}{|Du|}\right)=\Lambda\, b(x)\frac{u}{|u|} & \text{in }\Omega\\u=0 & \text{on }\partial\Omega,\end{array}\right.\] where $a(x)$ and $b(x)$ are suitable nonnegative functions. We prove that the first eigenvalue coincides with the weighted Cheeger constant. To see this identity, we analyze the behavior of the first Dirichlet eigenvalue of the weighted $p$-Laplacian operator as $p$ goes to $1$. In the case that the weight $a(x)$ is Lipschitz-continuous, we show that the limit of eigenvalues of the weighted $p$-Laplacian exists, and it is the weighted Cheeger constant. In addition, we check that the sequence of normalized $p$-eigenfunctions converges to the normalized eigenfunction of our limiting problem, which turns out to be bounded. For more general weights, we identify the first eigenvalue of the weighted 1-Laplacian operator with the weighted Cheeger constant and prove that the associated eigenfunction is bounded.

We study sharp weighted Sobolev-type inequalities of the form\[ \int_{0}^{1}|u(x)|\rho(x) \,\mathrm{d}x \leqslant \Lambda \Bigl(\int_{0}^{1}|u^{(k)}(x)|^2 \,\mathrm{d}x \Bigr)^{1/2}, \qquad u\in H_0^k(0,1), \]where $\rho$ is a non-negative weight. We characterize the minimizers and identify the optimal constant $\Lambda(k,\rho)$ by showing that every minimizer has a constant sign and therefore solves a nonlinear eigenvalue problem of polyharmonic type. This yields an explicit characterization of extremizers for a broad class of weights. Moreover, we even provide with a an explicit computation of the optimal constant in terms of the weight function. The new weighted estimates turn to be very useful and, among other applications, allow us to recover several previous sharp estimates and Hardy type inequalities on finite intervals.

We study how much a linear program (LP) or an integer program (IP) can be compressed when solved repeatedly over a fixed feasible region with varying objective coefficients. We provide a framework that compresses LP and IP formulations with guarantees on the probability of preserving optimality, using either an uncertainty set for the cost vector or historical samples. Our framework exposes a tunable tradeoff between compression and the probability of recovering an optimal solution. We characterize the minimum dimension of affine reformulations preserving the full optimizer set over a given cost region and provide sample-based learning algorithms with fast-rate guarantees, without requiring a nondegeneracy assumption for LPs. With high confidence over the training data, the learned reformulation preserves the optimizer set for an independent test cost with probability at least $1-\rho-\widetilde{O}(d^\star_\rho/\hat n)$, where $\rho$ bounds the excluded probability mass, $d^\star_\rho$ is the retained region's decision-relevant dimension, and $\hat n$ is the number retained from $n$ training samples. Existing data-driven approaches to LP and IP reduction do not, to our knowledge, learn a single low-dimensional affine reformulation with a prescribed probability of preserving the full optimizer set. Experimental results show substantial improvements over the compared baselines in accuracy, sample efficiency, and computational speed.

The importance of functional data analysis has increased substantially in recent years. In machine learning, nonlinear function regression based on deep neural networks is referred to as operator learning, and many of its applications involve functional time series data. However, the theoretical understanding of nonlinear models in functional time series analysis remains limited, as most existing works focus on linear models. In this paper, we derive basic properties for analyzing adaptive learning in nonlinear functional autoregressive (NFAR) models. Specifically, we derive sufficient conditions for NFAR models to be exponentially mixing. We provide an example with a Hammerstein operator under which these conditions are satisfied. As an application of exponential mixing, we consider operator learning for NFAR models with Urysohn operators and derive convergence rates for adaptive estimators based on deep neural networks.

We extend the closed-form privacy-subsidy result of Nakamura~(2026, arXiv:2605.15746) from the single-period Kyle model to continuous-time. A committed Bayesian automated market maker observes the aggregate order flow perturbed by an independent Brownian privacy channel of diffusion intensity $\sigma_\varepsilon$. Under the Markovian linear equilibrium, the price-impact coefficient is $\lambda = \sigma_v / \sqrt{\sigma_u^2 + \sigma_\varepsilon^2}$ – constant in time – and the cumulative expected transfer from the protocol's liquidity pool to traders over $[0,1]$ is $|\Pi_M| = \sigma_v \sigma_\varepsilon^2 / \sqrt{\sigma_u^2 + \sigma_\varepsilon^2}$. We then establish a structural correspondence between this cumulative privacy subsidy and Loss-Versus-Rebalancing (Milionis et al.~2022), identifying privacy-noise welfare as the order-flow observation analog of LVR's price observation gap. The result completes the continuous-time Kyle leg of the program of quantifying break-even fees for committed-AMM exchanges under privacy-aggregated information environments.

This paper establishes a rigorous functional analytic framework for weighted Weyl-Sonine fractional operators on semi-infinite intervals. While the classical Phillips functional calculus relies strictly on completely monotonic Bernstein functions to subordinate linear semigroups, we demonstrate that this structural restriction can be entirely bypassed. By exploiting the intrinsic topological decay induced by an admissible amnesia weight ($\omega$), we prove that the abstract integral representation converges absolutely for general classes of tempered Sonine kernels, thereby unifying the treatment of both diffusive relaxation and oscillatory non-local dynamics within a single framework. Employing the theory of strongly continuous semigroups, we characterize these operators as the infinitesimal generators of a deformed translation semigroup that simultaneously incorporates subjective aging scales ($\psi$) and history-attenuating amnesia effects. This similarity-based approach not only provides a sound abstract foundation for the hypersingular Marchaud representation of these operators but also guarantees their closedness and the analytical well-posedness of the associated abstract fractional Cauchy problems in general Banach spaces. Furthermore, we introduce the associated generalized fractional integral and prove the fundamental inversion theorem, demonstrating the complete analytical symmetry of the theory. Finally, we establish pointwise decay estimates for the solutions via weighted Sobolev embeddings.

We use the fixed point method and toroidal compactifications to establish general lower bounds for the essential dimension of congruence covers $\Gamma' \backslash \mathcal{X}^0 \rightarrow \Gamma \backslash \mathcal{X}^0$ of mixed Shimura varieties. Our main result shows that $\mathrm{ed}_{\mathbb{C}}(\Gamma' \backslash \mathcal{X}^0 \rightarrow \Gamma \backslash \mathcal{X}^0; p)$ is bounded from below by the dimension of certain unipotent subgroups associated with the rational boundary components of the given mixed Shimura datum. This generalizes theorems of Brosnan and Fakhruddin to the case of an arbitrary mixed Shimura datum. As a consequence, we obtain incompressibility results for congruence covers of universal families of principally polarized abelian varieties. We also describe explicit fixed points for $(GL_2, \mathcal{H}_2)$ and $(V \rtimes GL_2, \mathcal{Y}_2)$.