This paper considers an important inverse problem in topological data analysis (TDA): How many different barcodes produce the same Betti curve? Equivalently, given a function $β\colon [n]=\{1<\cdots< n\} \to \mathbb{Z}_{\geq 0}$, how many different ways can we write $β$ as a sum of indicator functions supported on intervals in $[n]$? Our answer to this question is to connect persistent homology with the study of the Kostant partition function and the enumerative combinatorics for so-called "magic" juggling sequences studied by Ronald Graham and others. Specifically, we prove an equivalence between our inverse problem and corresponding statements in these other two settings. From an applications and statistics point of view, our work provides a quantification of how lossy the TDA pipeline is when moving from persistent homology to persistent Betti numbers.

We develop a theory of partially defined complete positivity preservers, extending Schoenberg's classical characterization to functions defined only on discrete subsets or constrained domains. We frame the extension problem through the theory of completely positive maps on operator systems -- we characterize general partially defined completely positive definite functions on general homogeneous spaces. We apply our interpolation to constrained packing problems and Delsarte theory, where one uses positive definite functions on homogeneous spaces to obtain bounds on various packing problems. We prove the specific positive definite function witnesses that a code is sharp for constrained angle codes must be from polynomials.

This paper systematically studies the subset of continuous linear functionals on the projective tensor product of Banach spaces whose norms are bounded by Grothendieck's constant $K_G$. We term such functionals Grothendieck functional integrals. The integral is defined as a linear functional on the projective tensor product space that satisfies the boundedness condition $|μ(x)| \leq K_G \|x\|_π$, where $K_G$ denotes Grothendieck's constant. We prove that such integrals admit a Hilbert space representation theorem and establish the corresponding abstract Fubini theorem to demonstrate that the order of integration may be interchanged. Furthermore, we extend this theory to the setting of multiple tensor products and provide integral representations in concrete function spaces. Our work offers a unified framework for bilinear and multilinear analysis, with a universal constant serving as the fundamental bound.

Kolmogorov-Arnold Networks (KANs) require significantly smaller architectures compared to multilayer perceptron (MLP)-based approaches, while retaining expressive power through spline-based activations. Moving boundary problems are ubiquitous in physical systems, whose numerical solutions are quite complex. We propose a shallow KAN framework combined with a Level-set formulation that directly approximates the temperature distribution $T(\mathbf{x},t)$ and the moving interface $Γ(t)$, enforcing the governing PDEs, phase equilibrium, and Stefan condition through physics-informed residuals. Numerical experiments in one and two dimensions show that the framework achieves accurate reconstructions of both temperature fields and interface dynamics, highlighting the potential of KANs as a compact and efficient alternative for moving boundary PDEs. First, we validate the model with semi-infinite analytical solutions. Subsequently, the model is extended to 2D using a level-set based formulation for interface propagation, which is solved within the KAN framework. This work demonstrates that KANs are capable of solving complex moving boundary problems without the need for measurement data.

Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in. We prove Gromov's conjecture if the manifolds are spin with a constant that also depends on a lower bound on the mean curvature $H$ (which is allowed to take negative values). If the boundary is a (not necessarily convex) hypersurface in a space form of non-negative curvature, then the constant can be made explicit in terms of the mean curvature of this model embedding. If the boundary has constant sectional curvature $κ>0$ and is a projective space of dimension $n\equiv 3 \mod 4$ or a sphere, then the constant can be expressed in terms of $κ$. If the boundary is a flat torus, then the constant can be expressed in terms of lattice data.

Given a number field $k$ with the ring of integers $\mathcal{O}_k$ and a matrix $M\in \mathrm{M}_{n}(\mathcal{O}_k)$. We prove that if $\mathcal{O}_k$ is a principal ideal domain, the local-global principle for triangularizability and diagonalizability of $M$ holds. To explain the possible failures of the local-global principle, we prove that the stratified Brauer--Manin obstruction is the only obstruction to the local-global principle for triangularizability and diagonalizability of $M$ in some special cases.

Modern wireless systems require not only position estimates, but also quantified uncertainty to support planning, control, and radio resource management. We formulate localization as posterior inference of an unknown transmitter location from receiver measurements. We propose Monte Carlo Candidate-Likelihood Estimation (MC-CLE), which trains a neural scoring network using Monte Carlo sampling to compare true and candidate transmitter locations. We show that in line-of-sight simulations with a multi-antenna receiver, MC-CLE learns critical properties including angular ambiguity and front-to-back antenna patterns. MC-CLE also achieves lower cross-entropy loss relative to a uniform baseline and Gaussian posteriors. alternatives under a uniform-loss metric.

The Chernoff bound is one of the most widely used tools in theoretical computer science. It's rare to find a randomized algorithm that doesn't employ a Chernoff bound in its analysis. The standard proofs of Chernoff bounds are beautiful but in some ways not very intuitive. In this paper, I'll show you a different proof that has four features: (1) the proof offers a strong intuition for why Chernoff bounds look the way that they do; (2) the proof is user-friendly and (almost) algebra-free; (3) the proof comes with matching lower bounds, up to constant factors in the exponent; and (4) the proof extends to establish generalizations of Chernoff bounds in other settings. The ultimate goal is that, once you know this proof (and with a bit of practice), you should be able to confidently reason about Chernoff-style bounds in your head, extending them to other settings, and convincing yourself that the bounds you're obtaining are tight (up to constant factors in the exponent).

Matrix-based optimizers have attracted growing interest for improving LLM training efficiency, with significant progress centered on orthogonalization/whitening based methods. While yielding substantial performance gains, a fundamental question arises: can we develop new paradigms beyond orthogonalization, pushing the efficiency frontier further? We present \textbf{Adaptively Rotated Optimization (ARO}, a new matrix optimization framework that treats gradient rotation as a first class design principle. ARO accelerates LLM training by performing normed steepest descent in a rotated coordinate system, where the rotation is determined by a novel norm-informed policy. This perspective yields update rules that go beyond existing orthogonalization and whitening optimizers, improving sample efficiency in practice. To make comparisons reliable, we propose a rigorously controlled benchmarking protocol that reduces confounding and bias. Under this protocol, ARO consistently outperforms AdamW (by 1.3 $\sim$1.35$\times$) and orthogonalization methods (by 1.1$\sim$1.15$\times$) in LLM pretraining at up to 8B activated parameters, and up to $8\times$ overtrain budget, without evidence of diminishing returns. Finally, we discuss how ARO can be reformulated as a symmetry-aware optimizer grounded in rotational symmetries of residual streams, motivating advanced designs that enable computationally efficient exploitation of cross-layer/cross module couplings.

Motivated by earlier numerical evidence for a percolation-like transition in space-time jamming, we present an analytic description of the transient dynamics of the deterministic traffic model elementary cellular automaton rule 184 (ECA184). By exploiting the deterministic structure of the dynamics, we reformulate the problem in terms of a height function constructed directly from the initial condition, and obtain an equilibrium statistical mechanics-like description over the lattice configurations. This formulation allows macroscopic observables in space-time, such as the total jam delay and jam relaxation time, as well as microscopic jam statistics, to be expressed in terms of geometric properties of the height function. We thereby derive the associated scaling forms and recover the critical exponents previously observed in numerical studies. We discuss the physical implications of this space-time geometric approach.

We consider various notions of completeness in symplectic topology and ask two related questions. Does a complete open symplectic manifold remain complete after excising a subset? Can two sets be made arbitrarily far apart by adjusting the almost complex structure within an appropriate class of complete almost complex structures? We find rigidity phenomena when the excised set is a symplectic hypersurface. These arise from certain open Gromov-Witten invariants. We contrast this with flexibility that often occurs when the excised set is coisotropic. We also briefly touch on the opposite question of obstructions to existence of a complete symplectic structure compatible with a given complex structure. For the notion of completeness we first consider the traditional notion of geometric boundedness. We then introduce a broader notion of normalized completeness, related to the notion of intermittent boundedness of \cite{GromanFloerOpen}, which depends on $C^0$ properties and is a contractible condition. Finally we speculate about the relation to a Fukaya-categorical notion of completeness.

We consider a scalar valued elliptic partial differential equation on a sufficiently smooth domain $Ω$, subject to a regularized Tresca friction-type boundary condition on a subset $Γ$ of $\partial Ω$. The friction threshold, a positive function appearing in this boundary condition, is assumed to be unknown and serves as the coefficient to be recovered in our inverse problem. Assuming that (i) the friction threshold lies in a finite dimensional space with known basis functions, (ii) the right hand sides of the partial differential equation are known, and (iii) the solution to the partial differential equation on some small open subset $ω\subset Ω$ is available, we develop an iterative computational method for the recovery of the friction threshold. This algorithm is simple to implement and is based on piecewise linear finite elements. We show that the proposed algorithm converges in second order to a function $a_h$ and, moreover, that $a_h$ converges in second order in the finite element's mesh size $h$ to the true (unknown) friction threshold. We highlight our theoretical results by simulations that confirm our rates numerically.

Model-based controllers can offer strong guarantees on stability and convergence by relying on physically accurate dynamic models. However, these are rarely available for high-dimensional mechanical systems such as deformable objects or soft robots. While neural architectures can learn to approximate complex dynamics, they are either limited to low-dimensional systems or provide only limited formal control guarantees due to a lack of embedded physical structure. This paper introduces a latent control framework based on learned structure-preserving reduced-order dynamics for high-dimensional Lagrangian systems. We derive a reduced tracking law for fully actuated systems and adopt a Riemannian perspective on projection-based model-order reduction to study the resulting latent and projected closed-loop dynamics. By quantifying the sources of modeling error, we derive interpretable conditions for stability and convergence. We extend the proposed controller and analysis to underactuated systems by introducing learned actuation patterns. Experimental results on simulated and real-world systems validate our theoretical investigation and the accuracy of our controllers.

The primary objective of this section is to demonstrate that the actual pseudorandom measures of our construction are significantly smaller than the theoretical upper bounds derived from the Weil theorem. Regarding the family of sequences, we note that the construction $E_{f,g,h}$ allows for a large variety of sequences by choosing different triples of polynomials. While the detailed analysis of the cross-correlation measure of such a family is a challenging problem and lies beyond the scope of the present paper, the structure of the construction suggests that sequences generated by different polynomials will remain nearly orthogonal. Indeed, since each sequence is built from distinct Legendre symbol sequences with proven low correlation, their combinations are expected to maintain the same level of independence.

For a semisimple multiring category with left duals, we prove that the unit object is simple if and only if the tensor functors by any non-zero algebra are separable (resp. faithful, resp. Maschke, resp. dual Maschke, resp. conservative). This induces a list of equivalent definitions of fusion category. As an application, we describe the connectness of a class of weak Hopf algebras by the separability of tensor functors. We also consider applications to transfer of simplicity between the unit objects, semisimple indecomposable module category and Grothendieck ring.

We study the construction of $d$-deletion-correcting binary codes by formulating the problem as a Maximum Clique Problem (MCP). In this formulation, vertices represent candidate codewords and edges connect pairs whose longest common subsequence (LCS) distance guarantees correction of up to $d$ deletions. A valid codebook corresponds to a clique in the resulting graph, and finding the largest codebook is equivalent to identifying a maximum clique. While MCP-based formulations for deletion-correcting codes have previously been explored, we demonstrate that applying Penalty-Guided Clique Search (PGCS), a lightweight stochastic clique-search heuristic inspired by Dynamic Local Search (DLS), consistently yields larger codebooks than existing graph-based heuristics, including minimum-degree and coloring methods, for block lengths $n = 8,9,\dots,14$ and deletion parameters $d = 1,2,3$. In several finite-length regimes, the resulting codebooks match known optimal sizes and outperform classical constructions such as Helberg codes. For decoding under segmented reception, where codeword boundaries are known, we propose an optimized LCS-based decoder that exploits symbol-count filtering and early termination to substantially reduce the number of LCS evaluations while preserving exact decoding guarantees. These optimizations lead to significantly lower average-case decoding complexity than the baseline $O(|C| n^2)$ approach.

We prove local $W^{1,q}$-regularity for weak solutions to fractional $p$-Laplacian type equations with right-hand side $f\in L^r_{\mathrm{loc}}(Ω)$. Assuming $p>1$, $s\in(0,1)$, and $sp'>1$, solutions belong to $W^{1,q}_{\mathrm{loc}}(Ω)$ for the optimal exponent $q=q(n,p,s,r)$. We obtain quantitative local gradient estimates involving nonlocal tail terms. The optimality of $q$ is confirmed by a counterexample.

We show the existence of non-trivial self-expanding harmonic map flows starting from non-energy-minimizing 0-homogeneous maps to a regular ball or a closed hemisphere. In particular, given a non-minimizing but stationary 0-homogeneous harmonic map $u_0$ to a closed hemisphere, we construct infinitely many different weak solutions to harmonic map flow starting from $u_0$, all of which satisfy the parabolic monotonicity formula. This answers a question of Struwe.

We initiate the study of a multivariate graph polynomial $Φ_G(x,y,z)$ that interpolates between classical counting polynomials for matchings and for cycle structures arising in the Harary--Sachs expansion of the characteristic polynomial. We focus on analytic properties and computational consequences. Our main contribution is an explicit, degree-uniform zero-free region for $Φ_G$ on bounded-degree graphs, obtained via the Fernández--Procacci convergence criterion for abstract polymer gases.

Our main objective in this paper is to study the average rank of the $2$-Selmer group of the elliptic curve associated with the $\fracπ{3}$-congruent number problem. Following Heath-Brown's strategy, we could find an asymptotic formula for the size of the relaxed $2$-Selmer groups, which has several consequences towards the average of $2$-Selmer ranks and $\fracπ{3}$-congruent number problem. Indeed, we could find an unconditional positive density of $2$-Selmer rank being $1$ or $3$, among the positive square-free integers $n\equiv 13\pmod{24}$ having all the prime divisors congruent to $1$ modulo $4$ and an unconditional positive density of $2$-Selmer rank being $0$ or $2$, among the positive square-free integers $n\equiv 5\pmod{24}$ having all the prime divisors congruent to $1$ modulo $4$.

This paper addresses the problem of exponential and accelerated finite-time, as well as nearly fixed-time, stabilization of switched linear MIMO systems. The proposed approach relies on a generalized homogenization framework for switched linear systems and employs implicit Lyapunov functions for control design, covering both common and multiple Lyapunov function settings. Linear matrix equations and inequalities are derived to characterize the dilation generator and to synthesize the controller gains. Robustness of the resulting control laws with respect to system uncertainties and external disturbances is analyzed. The effectiveness of the proposed approach is illustrated through numerical examples.

In this note we analyse the scrollar invariants of $k:1$ covers of $\mathbb P^1$ that factor through the normalisation of a nodal curve in the $m$-th Hirzebruch surface $\mathbb F_m$. We then give an existence theorem for nodal curves in $\mathbb F_m$ having fixed class and singular locus.

The bounded mean betting procedure serves as a crucial interface between the domains of (1) sequential, anytime-valid statistical inference, and (2) online learning and portfolio selection algorithms. While recent work in both domains has established the exponential wealth growth of numerous betting strategies under any alternative distribution, the tightness of the inverted confidence sets, and the pathwise minimax regret bounds, little has been studied regarding the asymptotics of these strategies under the null hypothesis. Under the null, a strategy induces a wealth martingale converging to some random variable that can be zero (bankrupt) or non-zero (non-bankrupt, e.g. when it eventually stops betting). In this paper, we show the conceptually intuitive but technically nontrivial fact that these strategies (universal portfolio, Krichevsky-Trofimov, GRAPA, hedging, etc.) all go bankrupt with probability one, under any non-degenerate null distribution. Part of our analysis is based on the subtle almost sure divergence of various sums of $\sum O_p(n^{-1})$ type, a result of independent interest. We also demonstrate the necessity of null bankruptcy by showing that non-bankrupt strategies are all improvable in some sense. Our results significantly deepen our understanding of these betting strategies as they qualify their behavior on "almost all paths", whereas previous results are usually on "all paths" (e.g. regret bounds) or "most paths" (e.g. concentration inequalities and confidence sets).

We propose a geometric model predictive control framework for quantum systems subject to smoothness and state constraints. By formulating quantum state evolution intrinsically on the projective Hilbert space, we penalize covariant accelerations to generate smooth trajectories in the form of Riemannian cubics, while incorporating state-dependent constraints through potential functions. A structure-preserving variational discretization enables receding-horizon implementation, and a Lyapunov-type stability result is established for the closed-loop system. The approach is illustrated on the Bloch sphere for a two-level quantum system, providing a viable pathway toward predictive feedback control of constrained quantum dynamics.

In this note, we show that the category of Latin (resp. commutative) medial quandles is equivalent to the category of affine modules over a certain Laurent polynomial ring (resp. the dyadic rationals). As applications, we describe free objects in these categories and obtain a structure theorem for finitely generated medial commutative quandles. We also characterize racks whose duals are commutative. Collectively, this solves two open problems of Bardakov and Elhamdadi (arXiv:2601.07057v2).

We establish the global well-posedness theory of small BV weak solutions to a one-dimensional compressible Navier--Stokes model for reacting gas mixtures in dynamic combustion. The unknowns of the PDE system consist of the specific volume, velocity, temperature, and mass fraction of the reactant. For initial data that are small perturbations around the constant equilibrium state $(1, 0, 1, 0)$ in the $L^1(\mathbb{R}) \cap {\rm BV}(\mathbb{R})$-norm, we establish the local-in-time existence of weak solutions via an iterative scheme, show the stability and uniqueness of local weak solutions, and prove the global-in-time existence of solutions for initial data with small BV-norm via an analysis of the Green's function of the linearised system. The large-time behaviour of the global BV weak solutions is also characterised. This work is motivated by and extends the recent global well-posedness theory for BV weak solutions to the one-dimensional isentropic Navier--Stokes and Navier--Stokes--Fourier systems developed in [T.-P. Liu, S.-H. Yu, Commun. Pure Appl. Math. 75 (2022), 223--348] and [H. Wang, S.-H. Yu, X. Zhang, Arch. Ration. Mech. Anal. 245 (2022), 375--477].

We prove that an admissible $p$-adic Banach representation of $\text{GL}_2K$ whose locally analytic vectors have an infinitesimal character has Gelfand-Kirillov dimension $\leq[K\colon\mathbf Q_p]$, where $p>2$ and $K$ is a $p$-adic field. We also prove this for the group of units of the quaternions over $K$ replacing $\text{GL}_2K$. In the process, we make some observations in the theory of $p$-adic Banach representations that might be of independent interest.

In this paper, we introduce a new notion of algebra over a linear $\infty$-operad and a corresponding notion of coalgebra over an $\infty$-cooperad. We next extend the Koszul duality between linear $\infty$-operads and linear $\infty$-cooperads from our previous paper (arXiv:2105.11943) to their categories of algebras and coalgebras. This duality theorem specialises to the known duality in the case of algebras over classical (non-infinity) operads, but our proof is different. In fact, it is based on a much more general duality between presheaves and copresheaves on a category of trees. The latter duality is a priori independent of the (co)algebra structures, but we show that it can be lifted to (co)presheaves supporting such a structure. Based on this duality, we define the homology of an algebra over an $\infty$-operad, and prove that it can be described in terms of the homology of the same category of trees with coefficients in a presheaf.

We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.

We construct a realizability model of linear dependent type theory from a linear combinatory algebra. Our model motivates a number of additions to the type theory. In particular, we add a universe with two decoding operations: one takes codes to cartesian types and the other takes codes to linear types. The universe is impredicative in the sense that it is closed under both large cartesian dependent products and large linear dependent products. We also add a rule for injectivity of the modality turning linear terms into cartesian terms. With all of the additions, we are able to encode (linear) inductive types. As a case study, we consider the type of lists over a linear type, and demonstrate that our encoding has the relevant uniqueness principle. The construction of the realizability model is fully formalized in the proof assistant Rocq.