Papers

This manuscript addresses a connection between operations in coding theory and graph theory, as well as some operations in both areas. The concept of the Hasse diagram of a binary linear code, $\Gamma(C)$, is recall; this concept visually represents the cosets of a binary linear code with a partial ordering. The main objetive is to give a characterization of the graphs corresponding to: extended codes, punctured codes, and the direct sum of two codes, using the graph of the given binary linear code as a starting point. Relationships are established between the graphs of code operations and particular operations of graphs.

We construct a finite skew brace with soluble additive group and insoluble multiplicative group. The multiplicative group has a quotient isomorphic to $\PSL_2(7)$, and hence the example disproves Byott's conjecture.

It is known that the energy measures of any two nonconstant harmonic functions on the standard Sierpiński gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence for $L^p$-integrability of the corresponding Radon–Nikodym densities in the range \[ 1<p<\frac{\log 15}{\log 9}. \] For arbitrary ordered pairs of nonconstant harmonic functions, we prove uniform boundedness of the associated density-ratio power sums, and hence $L^p$-integrability, in the subinterval \[ 1<p<\frac{\log(35/3)}{\log 9}. \] When the denominator harmonic direction is represented by the boundary values $(0,-1,1)$, we prove boundedness throughout the full conjectured interval.

The class AC of integrands satisfying the atomic condition was introduced by De Philippis, De Rosa, and Ghiraldin in 2018. These integrands give rise to Almgren elliptic geometric functionals as proven by the second author and De Rosa in 2020. Explicit examples in intermediate dimensions include small class 2 perturbations of the area integrand and the recent anisotropies constructed by De Rosa, Firester, and Tsiamis, among them the $\ell^q$ norms in Plücker coordinates for $q$ close to $2$. We reinterpret the atomic condition in terms of convex geometry. We also propose variants of AC, related to the SAC and USAC defined by De Rosa and Tione, which we call the exposed condition EC, the strong exposed condition SEC, and the quadratic exposed condition QEC. For integrands of class 2, QEC is equivalent to validity of a Caccioppoli-type inequality and is stable under class 2 perturbations; every such QEC integrand admits Lipschitzian normals. Combining these results with a result of De Rosa, Firester, and Tsiamis, we prove that a Lipschitz graph whose anisotropic mean curvature for a QEC integrand of class 2 is $p$-integrable, with $k < p < \infty$, is of class $\mathscr{C}^{1,\alpha}$ on an open set of full measure. Finally, for all $k \ge 2$ and $n - k \ge 3$ we construct strictly polyconvex integrands - associated to inner-product norms on $\bigwedge_k \mathbf{R}^n$ - which fail the atomic condition, showing that strict polyconvexity is necessary but not sufficient for AC.

Let \(P_\eps\) be the fence poset associated with an orientation \(\eps\in\{+,-\}^{n-1}\) of a path. We define a greedy right-to-left record statistic \(\rec_\eps\) on \(S_n\) and prove \[ \sum_{\pi\in S_n}t^{\rec_\eps(\pi)}=n!\Omega(P_\eps;t), \] by a Bernstein-basis transfer between a continuous threshold recurrence and endpoint-refined order-preserving maps. Under reflection, this statistic agrees pointwise with Kahane's independently obtained greedy block statistic; the alternating specialization gives the zig-zag case posed by Ferroni, Morales, and Panova. A finite form of the transfer yields a direct recursive bijection for \(m\le n\), extended algorithmically to arbitrary alphabets. Refining by record set, direction, and terminal value identifies fixed fibers with decorated endpoint paths and pointed linear extensions of posets whose cover graphs are caterpillars. We also define cyclic records for every nonconstant orientation \(\eta\) of a cycle and prove \[ \sum_{\pi\in S_n}t^{\operatorname{crec}_\eta(\pi)} =n!\Omega(C_\eta;t). \] When the Hasse diagram is a cycle, reflection identifies these records with Kahane's circular blocks and establishes his circular-fence conjecture.

Extreme first-passage events are broadly relevant to biological, chemical, and physical processes in which the first successful arrival determines the outcome. Existing theories are largely confined to noninteracting searchers. Interacting extreme-statistics problems are notoriously difficult because correlations destroy probability factorization. We establish a general framework for interacting extreme search. A no-go theorem shows that broad classes of short-range interactions cannot beat the $1/\ln N$ extreme timescale of $N$ independent Brownian searchers, and complementary upper bounds prove that this scale is exact for repulsive interactions. We then identify three sharp mechanisms beyond the logarithmic class and derive a unified interaction-driven acceleration limit. In particular, deterministic pairwise interaction can at most reduce the extreme search time to order $1/N$, while stochastic pairwise forcing attains $1/(N\ln N)$. Our results separate acceleration due to statistical redundancy from that generated by coherent many-body transport or amplified fluctuations, deepening our understanding of interacting many-body systems.

We give counterexamples to Rota's 1970 conjecture that the sequence counting flats of varying rank in a matroid is unimodal. More specifically, inspired by Larson's recent disproof of the stronger log-concavity conjecture of Mason, we explain a mechanism which turns failures of log-concavity for flats into failures of unimodality under suitable conditions.

Nonlinear optimization problems with complicated, nonconvex, yet geometrically structured constraints can be tackled by projected-gradient methods: under weak regularity assumptions, these approaches were recently proved to possess convergence properties to the strongest stationarity conditions. In this work, we show how momentum terms, commonly used in nonlinear optimization to speed up the convergence process, can be integrated within this algorithmic framework without harming convergence guarantees. Preliminarily, we highlight an intrinsic issue induced by the direct replacement of the negative gradient with a general descent direction within the projected approach. Then, we present suitable backtracking mechanisms for the pre-projection step, allowing us to integrate momentum terms in the direction. By this technique, we can specifically ensure, without any smoothness assumptions, convergence to Mordukhovich stationarity as long as the base directions asymptotically revert to the negative gradient for small stepsizes; moreover, if the base search direction reverts exactly to the negative gradient for the smallest steps, the algorithm is proved to converge to Bouligand and Proximally stationary points, with and without (local) smoothness assumptions respectively. Finally, the proposed procedure is numerically tested on some classes of problems, namely, sparsity and bounded-rank constrained problems; the results indicate that the proposed method is computationally effective, taking advantage of the additional information provided by the momentum term.

Future wireless systems will require ever larger antenna arrays and heavier signal processing, making conventional digital multiple-input multiple-output (MIMO) architectures difficult to scale. In this paper, we show that a possible solution is to offload part of the processing from the digital to the analog domain. This can be done through linear microwave networks designed to compute directly using the communication signals at radio frequency (RF). These networks, denoted as microwave linear analog computers (MiLACs), can perform useful matrix operations instantly through wave propagation. Remarkably, although MiLACs are linear, the output signals can depend nonlinearly on the tunable parameters of the network, enabling the computation of operations beyond simple linear transforms. In particular, MiLACs can realize matrix inversion and pseudo-inversion with complexity scaling quadratically with matrix size, rather than cubically, which is essential in zero-forcing beamforming. We then review how MiLAC-aided MIMO architectures can reduce the number of RF chains, relax the resolution requirements on digital-to-analog converters (DACs) and analog-to-digital converters (ADCs), and decrease the beamforming complexity. We finally discuss the main challenges related to MiLAC and promising directions for future research.

Using the framework of critical point theory for multivalued functionals developed in \cite{Fri}, we establish some linking theorems for such functionals, which generalizes \cite[Theorem 2.11]{Wi}, \cite[Theorem 2.12]{Wi} and \cite[Theorem 2.12]{Fri}. The proofs of our abstract results rely on a general minimax principle for multivalued mappings. As an application, we obtain a nontrivial solution of a semilinear Dirichlet boundary value problem for partial differential inclusions.

For decades, the bootstrap has been a default tool for statistical inference because of its broad applicability and minimal analytic requirements. Although its validity is well understood for smooth parametric estimators, its theoretical properties for many modern semiparametric and machine-learning estimators remain largely unstudied. Nevertheless, bootstrap procedures are often used routinely in such settings, even when their validity is unknown and their computational cost is substantial. We develop the $V$-fold jackknife as a computationally efficient and theoretically justified alternative for semiparametric inference. It requires only $V$ leave-fold-out refits and uses the empirical dispersion of jackknife pseudo-values to quantify uncertainty, without deriving or evaluating an influence function. For regular asymptotically linear estimators of pathwise differentiable parameters, we show that, for fixed $V$, the Studentized $V$-fold jackknife statistic converges to a $t$-distribution with $V-1$ degrees of freedom, giving valid confidence intervals even though the jackknife variance estimator does not converge in probability. When $V\to\infty$, we establish consistency of the variance estimator at rate $V^{-1/2}$, allowing $V$ to diverge slowly, for example at rate $\log n$. We also develop simultaneous confidence bands based on the correct componentwise-Studentized limiting distribution. Finally, we extend the theory to generalized asymptotically linear estimators with diverging influence-function variance and slower-than-$\sqrt n$ convergence; scale invariance of Studentization eliminates the need to know the effective convergence rate. Simulations on the average treatment effect, Kaplan–Meier survival curve, and highly adaptive lasso dose-response curves confirm reliable inference, including where influence-function-based standard errors are anti-conservative or unstable.

We study the steady self-propelled motion of a rigid body immersed in an incompressible viscous fluid occupying an exterior domain in $\mathbb{R}^3$. In a body-fixed reference frame, the problem is described by a coupled fluid-rigid body system posed in a fixed exterior domain, where the Navier–Stokes equations are coupled with the unknown translational and angular velocities of the rigid body. The self-propulsion is generated by prescribing a boundary velocity on the body surface. Our main result asserts the existence of weak solutions under the sole assumption that the prescribed boundary flux is sufficiently small. The crucial step is the construction of a divergence-free extension of the boundary data that does not rely on either the zero-flux condition or the smallness of the boundary data. As a result, we generalize the result of Galdi [Theorem 5.1 in On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Rational Mech. Anal., 148 (1999), 53-88], where both assumptions were required.

Strong semiclassical convergence from the Hartree equation to the Vlasov equation is established in three dimensions near Penrose-stable homogeneous steady states at positive density. For sufficiently regular integrable interaction kernels, an $O(\hbar^2)$ convergence rate in weighted Sobolev spaces is proved on finite time intervals. Combining this estimate with earlier uniform-in-$\hbar$ phase-mixing and scattering bounds for the Hartree equation yields global existence and scattering for the corresponding Vlasov solution through the semiclassical limit. If the quantum and classical initial perturbations are aligned, the semiclassical convergence is uniform for all times, and the Wigner transforms of the quantum scattering profiles converge to the classical scattering profile at the explicit rate $O((\ln\hbar^{-1})^{-1/3})$.

Singular value soft-thresholding can be computed via a reduction to the matrix polar decomposition, which allows one to exploit GPU-friendly algorithms for computing the polar decomposition. Empirically, there is a significant speed-up on GPUs compared to the standard approach using the SVD. We leave the investigation of robustness to future work, but note that due to the discontinuous nature of the sign function, the reduction to the polar decomposition is likely only suitable for low-accuracy applications.

We revisit the concepts of $n$-absolutely pure and $n$-flat modules, and their relation with $n$-coherent rings in the sense of Lee. We provide new proofs for some known claims about these modules and rings, and show other new properties. We also give some corrections to the existing literature.

The spherical random geometric graph $G(n,d,p)$ is obtained by sampling $n$ independent points uniformly on the unit sphere $\mathbb{S}^{d-1}\subseteq\mathbb{R}^d$ and joining pairs of points which are sufficiently close, where the threshold is chosen so that the edge probability is $p$. The central question related to this model, and to a broad class of other models, is the following: when does the underlying geometry affect the resulting graph in a way which makes it distinguishable from the Erdős–Rényi random graph $G(n,p)$, as measured in total variation distance? The precise answer to this question was conjectured by Bubeck, Ding, Eldan, and Rácz, who predicted that $G(n,d,p)$ and $G(n,p)$ are indistinguishable precisely when $d \gg n^3p^3(\log p^{-1})^3$, and provided a test for distinguishing these models in the low-dimensional regime. Although this conjecture attracted considerable attention from researchers in probability, theoretical computer science, and high-dimensional statistics, it was previously fully proved only in the constant-density case. In this paper, we resolve the distinguishability conjecture in the broad range $1/3 \geq p \geq n^{-1/5} \text{polylog}(n)$. The key ingredient of our proof is a stronger statement which gives a precise asymptotic formula for the probability that $G(n,d,p)$ realizes a prescribed graph $H$: above the conjectured threshold, this probability is at most $(1+o(1))$ times the corresponding probability for $G(n,p)$, with the signed triangle count of $H$ appearing as the leading correction term.

We construct finite-dimensional de Rham subcomplexes generated by fixed-neuron shallow ReLU$^k$ neural networks, a class of spaces known to provide optimal approximation rates. For neurons of the form $s_i(x)=\omega_i\cdot x+b_i$, we introduce spaces of neural differential forms: differential $p$-forms whose coefficients are the ReLU$^k$ ridge functions $\sigma_{k-p}(s_i)$. These spaces are compatible with the exterior derivative because differentiating a ReLU power lowers its order by one, and for each fixed neuron, differentiation amounts to exterior multiplication by the fixed one-form $ d s_i$. Under a linear independence assumption on the lowest-order family $\{\sigma_{k-d}(s_i)\}_{i=1}^n$, the global complex decomposes into independent neuron-wise Koszul complexes. We prove exactness in arbitrary dimension and provide a geometric sufficient condition for the required linear independence. Numerical experiments based on the resulting complex provide evidence of stable discretizations and of convergence rates consistent with the underlying approximation theory, and exhibit no spurious modes in eigenvalue problems considered.

In overparameterized linear regression, many weak spectral directions act like a ridge penalty on the signal-bearing spectrum; negative ridge is the natural correction, pushing filters above one. The stable negative-ridge endpoint, however, is structurally limited: its pole must stay below the smallest nonzero empirical eigenvalue, and it anti-shrinks smaller eigenvalues more than larger ones. Early-stopped negative-shifted gradient descent escapes this constraint. Its filter is smooth at the would-be pole and mixed-sign-capable: above-ridgeless directions form a leading prefix, with lower directions shrunk or exposure-controlled while stopping sets the crossover. In a Gaussian spike-plus-flat model we discover a Marchenko-Pastur barrier: the shift that cancels the implicit penalty lies a bulk width above the smallest empirical eigenvalue, and the stopped path improves on every admissible endpoint by a polynomial factor in risk under explicit conditions. Our main theorem permits a general high-effective-rank tail: its trace sets the implicit floor, its squared spectrum controls exposure, and the floor-critical path recovers all head scales at once, beyond positive shrinkage and, once scales separate, every uniform rescaling of ridgeless. Handling the noncontractive shifted dynamics is the central technical challenge; localized Duhamel integrals control them. A finite-grid hold-out inequality transfers the separations to the validation-selected algorithm.

A semi-bounded operator that is bounded below by one is essentially self-adjoint if the kernel of the adjoint is trivial. This is the case if the kernel of the adjoint belongs to the form domain of the Friedrichs extension. We describe a local version of this criterion, where locality is defined in terms of bounded operators approaching the identity. The result is an abstract operator theoretic framework that generalizes the method by Wienholtz and Simader developed in the context of semi-bounded elliptic partial differential operators. As applications, we obtain competitive results on essential self-adjointness of many-particle Schrödinger operators, pseudo-relativistic Hamiltonians, and the standard model of non-relativistic quantum electrodynamics.

We study Gaussian fluctuations for a lower-triangular system of interacting diffusions in which particle $i$ interacts only with its predecessors. Although the empirical measure of this system converges to the same McKean–Vlasov limit as in the corresponding exchangeable mean-field system, the sequential structure remains visible at the $N^{-1/2}$ fluctuation scale. We introduce the logarithmically weighted fluctuation fields \[ Y_t^{N,n} = \frac{1}{\sqrt{N}} \sum_{i=1}^N \frac{\bigl(\log(N/i)\bigr)^n}{n!} (\delta_{X_t^i}-\bar\rho_t), \qquad n\ge 0, \] and prove joint convergence of the entire family in a countable product of weighted negative Sobolev path spaces to the unique probabilistically strong solution of a linear hierarchy in which $Y^n$ couples to $Y^{n+1}$. In particular, the limit of the empirical fluctuation field $\sqrt{N}(\mu_t^N-\bar\rho_t)=Y_t^{N,0}$ is not governed by the closed fluctuation SPDE arising in the classical exchangeable case. The proof combines conditional-measure replacement, deterministic estimates for the logarithmic weights, a martingale argument, and a weighted Volterra estimate.

The influence of nonautonomous perturbations on a system of two coupled nonlinear non-identical oscillators is studied. Both the coupling intensity and the perturbation strength are assumed to decay in time according to a power law. We focus on resonance effects associated with the commensurability of the natural frequencies of the oscillators at some energy levels. In particular, we describe the conditions on perturbations and coupling parameters under which phase-locked solutions appear with the oscillator energies remaining asymptotically close to resonant levels. By combining the averaging method with the construction of Lyapunov functions, we derive a nonautonomous model system governing the perturbed dynamics and analyse the stability of resonant solutions in the phase-locking regime. The theoretical results are illustrated for a system of non-identical Duffing oscillators with decaying coupling.

A Rotating Detonation Combustor (RDC) is a promising technology for increasing efficiency in propulsion and power generation applications. The dynamics of the RDC are governed by continuously propagating detonation waves within an annular combustion chamber. Multiple operating modes can be observed, including nonlinear interactions between counter-rotating waves and the emergence of standing wave patterns. Koopman operator theory provides a framework to globally linearize nonlinear dynamical systems by representing their evolution in the space of observables rather than states. In this work, finite-dimensional approximations of the Koopman operator are constructed using variants of Dynamic Mode Decomposition (DMD) applied to high-speed video data capturing the natural flame luminosity of the detonation waves in the RDC at the Technical University (TU) Berlin. By introducing time-delay embeddings as a dictionary of observables, this approach overcomes the limitations of standard DMD methods, particularly for accurate reconstruction of standing wave patterns and for capturing nonlinear interactions. In addition, a technique is presented to mitigate the influence of sensor noise in the luminosity measurements. Finally, it is shown that the DMD-based models provide insight into the dynamics of different operating modes by decomposing the reconstructed signal into its characteristic features.

This report explores and compares numerical stabilization methods for transport-dominated flows arising in atmospheric modeling. The Streamline-Upwind (SU) and Streamline-Upwind Petrov-Galerkin (SUPG) stabilization formulations are implemented in Julia's ClimaCore.jl package and Python's Firedrake package for a test problem with slotted-cylinder initial conditions. These methods are compared for their ability to mitigate spurious oscillations while preserving sharp features against existing hyperdiffusion and quasi-monotone limiter methods, alongside various combinations. For this benchmark, quasi-monotone limiters were most effective at minimizing un-physical extrema, at the expense of diffusing the overall structure. The SUPG method was not found to improve upon the no stabilization case, for this test case in the computed error metrics. However, it performs best at preserving sharp feature and overall structure, among tested stabilized runs. Theoretical properties of SUPG are analyzed, and an asymptotics-based SUPG algorithm is proposed as future work.

Piecewise Deterministic Monte Carlo (PDMC) algorithms utilize continuous time Markov processes to generate samples from continuous distributions, and provide a modern alternative to discrete time Markov chain Monte Carlo algorithms. In this work we provide a survey of recent results on scaling limit arguments to understand the behaviour and efficiency of two often-used Piecewise Deterministic Monte Carlo algorithms: the Bouncy Particle Sampler and the Zig-Zag Process. In particular we discuss a Functional Central Limit Theorem, scaling in the high-dimensional regime, scaling under anisotropy, and scaling in the big data regime. This work is intended as part of the proceedings of the 2024 Isaac Newton Institute Programme "Stochastic systems for anomalous diffusion".

We prove that Cannon–Thurston maps do not exist for hyperbolic normal subgroups of hierarchically hyperbolic groups, both for Morse and hierarchically hyperbolic boundaries. This result includes a large class of normal subgroups of the mapping class group of a surface of genus $g\geq 2$. As a corollary, we also prove a non-existence result for subgroups of most free by cyclic groups. Moreover, we prove that the visual boundary of a CAT(0) group with isolated flats is homeomorphic to the relatively hierarchically hyperbolic boundary of the space it acts on, thus recovering a previously known non-existence result for CAT(0) groups with isolated flats.

We prove a martingale representation theorem for cylindrical martingale-valued measures defined on a separable Banach space. The main tool for establishing the theorem, is a new theory of non-radonifying stochastic integration in reflexive Banach spaces. A second one is the study and characterization of cylindrical white noise measure processes. As consequences of our representation theorem, we prove analogous versions for Hilbert space-valued measures and for cylindrical square integrable martingales. Finally, we apply the results to characterize the solutions to the weak martingale problem for SDEs driven by cylindrical white noise measures.

We study two interacting urns with power reinforcement $W(n)=n^\alpha$, for $\alpha>1$. Using Qin's stochastic-approximation theorem as input, we reduce the phase transition problem for domination to the sign analysis of a single analytic function $\Phi_p$. This deterministic reduction identifies the sharp threshold and proves that the two natural critical parameters for domination coincide. We determine the first-order behavior of the critical interaction parameter as $\alpha\downarrow1$ and show that it is not nondecreasing in the reinforcement exponent, answering two questions posed by Qin.

As data collecting technologies advance, data structures are getting more and more complex, from single vectors to multi-dimensional tensors. This article is motivated by a variable selection problem to detect important genes from an ultrahigh dimensional pool that are associated with human facial shape variations. We propose a data-driven trimmed feature screening method based on a tensor ridge regression model (TrimTenRidge) through setting thresholds on the tensor coefficients to perform a feature screening procedure. Unlike existing approaches, the TrimTenRidge does not require any sparse structures. In addition, it not only detects important predictors but also locates specific regions/components of the tensor response that are associated with each of the selected predictors. We prove the theoretical selection consistency and also assess its empirical performance through various simulation settings. The approach copes with ultra-high dimensional predictors and tensor responses simultaneously and contributes to the literature from theoretical, methodological, and five applicational aspects. We further apply the TrimTenRidge approach to genome-wide human facial shape data, from which the entire facial shapes form a $2,342\times 7,160\times 3$ tensor, and we successfully detect several novel genetic loci and also confirm some existing findings that are associated to facial shape.

The aim of this paper is to obtain a version of the classical Szegö limit theorem, where instead of the operator of second derivative on a circle we consider the Heisenberg-Hörmander Laplacian in $L^2(\Bbb R^3)$. Besides, we derive a sharp inequality for convex functions that we call Szegö-type inequality.

Coded computing is a reliable and fault-tolerant paradigm for executing large-scale computational tasks over distributed worker nodes. Among existing coded computing frameworks, Berrut Approximated Coded Computing (BACC) enables distributed computation of arbitrary non-polynomial functions through rational interpolation. Although BACC provides provable approximation guarantees and resilience against straggling workers, its robustness against Byzantine workers remains largely unexplored. To fill this research gap, we propose Robust Berrut Approximated Coded Computing (RBACC), which establishes a coding-theoretic framework for BACC by enabling error localization and error correction in the presence of Byzantine workers. In particular, RBACC introduces a new choice of evaluation points that establishes a connection between Berrut interpolation and Discrete Cosine Transform (DCT) codes, thereby enabling error localization and error correction under finite-precision arithmetic. We derive analytical upper bounds on the approximation error of RBACC under multiple operating scenarios, including straggler-only systems and systems with Byzantine workers under finite-precision arithmetic. Building upon this analysis, we formulate several optimization problems for selecting the DCT code dimension and for assigning encoded evaluations to unreliable workers. We show that these are previously unexplored design parameters that can be systematically optimized to improve the reconstruction accuracy. Experimental results demonstrate that the proposed RBACC framework effectively mitigates stragglers and Byzantine workers while offering improved reconstruction accuracy over the baselines.