Papers

We prove that a representation of a group embeddable or right LCM cancellative semigroup may be dilated to a representation of its reduced boundary quotient $C^\star$-algebra if and only if it extends to a completely contractive representation of the reduced semigroup operator algebra. We show that the latter property is satisfied not only by all constructible representations of amenable semigroups, but also a very large class of other representations, which encompasses several classical dilation theorems and the corresponding matricial von Neumann inequalities. In fact, our criterion of completely contractive extension to the operator algebra turns out to be an appropriate generalisation of the matricial von Neumann inequality to semigroups more general than $\mathbb{N}^k$, and dilation to the boundary quotient turns out to be an apt generalisation of unitary dilations. Thus, our theorem is in spirit and in practice a generalisation of Sz.-Nagy and Ando's dilation theorems for general semigroups. In addition, this also demonstrates that any completely contractive representation of the operator algebra dilates to a representation with additional relations among its generators, the new relations coming from the boundary quotient. In particular, for Ore semigroups, this completely characterises which representations admit unitary dilations.

We prove an operator-level convergence theorem for the mixed paracontrolled blocks \[ T^{i;j,k}_{\Lambda}(w) =I_i\bigl(w<\Psi_{j,\Lambda}\bigr)\circ\Psi_{k,\Lambda} \] on \(\mathbb T^d\), under Fourier-diagonal Gaussian covariance. At finite Galerkin cutoff, the Wick decomposition is \[ T^\tau_\Lambda=D^\tau_\Lambda+B^\tau_\Lambda, \qquad \tau=(i;j,k), \] where \(D^\tau_\Lambda\) is a deterministic Volterra multiplier on the external diagonal \(n=q\), while \(B^\tau_\Lambda\) is a centered second homogeneous Gaussian chaos with incidence \(n=q+\ell+r\). The centered coefficient has two Gaussian frequency legs and one input-output pair. Write \(\Delta_T=\{(t,s):0\le s\le t\le T\}\). An iterated rectangular non-commutative Khintchine argument, followed by the four associated oriented flattenings, yields \[ \bigl\|B^\tau_{\Lambda,N,Q,M}\bigr\|_ {L^p\bigl(\Omega;C(\Delta_T;\mathcal L(\ell_q^2,\ell_n^2))\bigr)} \lesssim_{p,\varepsilon} N^{d/2-\Gamma_\tau+\varepsilon} \bigl(M^{d/2}+Q^{d/2}\bigr), \qquad \Gamma_\tau=\lambda_i+\alpha_j+\alpha_k. \] Under the resulting strict Sobolev–Besov summability conditions, the centered cutoffs converge in \(L^p(\Omega)\) and almost surely in operator norm. The covariance contraction is kept separate from the tensor estimate: retained raw branches, or prescribed finite combinations of branches, enter through a scalar Volterra criterion imposed after the relevant cancellation has been formed. We give a frequency-envelope variant and verify the diagonal criterion for distinct-speed wave and Klein–Gordon contractions.

We study non-minuscule boundary 't Hooft lines in four-dimensional $\mathcal N=4$ $U(N)$ super Yang–Mills theory with the regular Nahm-pole boundary condition. For one-row charges we construct the exact spherical-DAHA operator $\mathbf e_+h_r(Y)\mathbf e_+$, determine its finite shift expansion and discrete Macdonald spectrum, and evaluate the S-dual Neumann Wilson two-line index as a finite-$N$ Macdonald pairing. For the first non-minuscule $U(2)$ coefficient, a local two-minuscule convolution model with Higgs branch $T^*\mathbb P^1$ has two Jeffrey–Kirwan residues, while subtracting the determinant intersection-cohomology summand supplies the additional $-\mathfrak t$ term. The same transverse $A_1$ calculation, dressed by the spectator-root factor, reproduces the first lower $U(3)$ coefficient. The local convolution calculation and its spectator lift provide low-rank checks of the complete DAHA coefficients independent of the inverse recursion, while their boundary-localization interpretation remains conjectural. Assuming an isometric oriented boundary Fourier intertwiner, the DAHA construction is compatible with the expected boundary S-duality relation.

For weighted manifolds with Bakry-Émery Ricci curvature bounded from below, comparison geometric results have been established. The Bakry-Émery Ricci curvature depends on the effective dimension $N$. In this paper, the case $N = 1$ is further studied in the setting of centro-affine differential geometry. Several examples, such as rotationally symmetric ellipsoids and hyperboloids, arise in rigidity phenomena of comparison theorems.

The Collaborative Research Centre TRR 361/F90 CREATOR (2022-2030) aims at establishing a new paradigm for the simulation-driven design of electric machines. Increasing demands on efficiency, power density and sustainability require the integration of multiphysical effects, advanced materials and complex geometries into the design process. Traditional sequential workflows are no longer sufficient to address these challenges. CREATOR therefore combines expertise from electrical engineering, applied mathematics, fluid dynamics and materials science to establish integrated modelling, simulation and optimisation methodologies in a single large-scale project funded by the German and Austrian national funding agencies. This article provides an overview of the research vision, key achievements from the first funding period (2022-2026) and current developments towards a computational electric machine laboratory.

This paper explores the deformed combinatorial structures arising from the generalized Heisenberg algebra GHA which is characterized by an analytic function of the Hamiltonian f(H) and governs systems with non-linear spectra. Moving beyond the classical Heisenberg-Weyl framework, we investigate the normal ordering of the generalized number operator $N_f^n = (\fa)^n$, which naturally introduces the generalized Stirling operators of the second kind. Using the vacuum eigenvalue the Hamiltonian, we define quantum operator factorials and a generalized quantum exponential function. We explicitly construct the generalized coherent states and derive several operator identities. Notably, we prove that the powers of quantum operators expand into generalized falling factorials, and that the coherent state expectation values are explicitly given by the generalized Bell polynomial operators.

With the rapid advancement of unmanned aerial vehicle (UAV), ensuring the security of communication links among UAVs has become crucial. In this paper, we propose a novel physical layer key generation scheme based on channel codeword reconstruction. In UAV communications, the high mobility of aerial nodes leads to short channel coherence time, which together with noise causes inevitable channel estimation errors. These errors significantly degrades the performance of wireless channel-based key generation. Therefore, we propose a codeword construction algorithm that achieves a polarization characteristic, which effectively segregates reliable keys from unreliable ones. Compared to the existing quantization-based key generation scheme, our approach maximize the utilization of raw channel information and employ soft-decision decoding to generate key. Simulation results demonstrate that the proposed scheme reduces the key disagreement rate for legitimate users and increases the number of consistently generated keys. Furthermore, our method ensures a lower key consistency rate for eavesdropper, which guarantees system security.

We study the kernel-based multilevel method for approximating or learning a multivariate function from scattered data, motivated in part by recent applications in sparse grid methods. For nested families of point sets, we derive a nodal representation of the multilevel interpolant that depends explicitly on the values of the target function. This representation allows us to characterize the range of the associated interpolation operator and to construct a cardinal basis for this space. We prove that the resulting basis functions exhibit exponential decay, analogous to the localization properties known for certain kernel-based Lagrange functions. We further analyze the computational cost of the resulting formulation. For non-nested families of point sets, we derive a generalized nodal representation.

We study the tropical chromatic number of $\mathbb{R}^n$, $\chi_{\mathrm{tr}}(\mathbb{R}^n)$, the minimal number of colors needed to color $\mathbb{R}^n$, so that no two points at tropical unit distance have the same color. It is the tropical analogue of the well-known Hadwiger-Nelson problem in $\mathbb{R}^2$. We have $\displaystyle \binom{n+1}{\lfloor (n+1)/2 \rfloor} \leq \chi_{\mathrm{tr}}(\mathbb{R}^n) \leq 2^n$ for every $n$, where the lower bound comes from Sperner's antichain bound on a maximal equilateral set, as shown by Swanepoel. It is conjectured that $\chi_{\mathrm{tr}}(\mathbb{R}^n) = 2^n$, which is known to be the case for the measurable chromatic number. By constructing a graph with 62 vertices and 577 edges we demonstrate that $\chi_{\mathrm{tr}}(\mathbb{R}^3)=8$. We also construct a graph in $\mathbb{Z}^4$ with 37 vertices and 386 edges that is 11-colorable but not 10-colorable, which is above Sperner's lower bound of 10.

Let $V$ be an $n$-dimensional vector space over a finite field. Suppose that $\mathcal{F}$ and $\mathcal{G}$ are non-empty families of $k$-subspaces and $\ell$-subspaces of $V$, respectively. They are said to be cross-$t$-intersecting if $\dim(F\cap G)\geq t$ for any $F\in\mathcal{F}$ and $G\in \mathcal{G}$, and are further called non-trivial if $\dim(\cap_{F\in\mathcal{F}}F)<t$ and $\dim(\cap_{G\in\mathcal{G}}G)<t$. In this paper, we characterize the non-trivial cross-$t$-intersecting families with the maximum sum of sizes. When $t=1$, our result serves as the $q$-analog of the theorems in [9,11].

We identify the limit free energy of all multi-species spin glasses with centered $\pm 1$ spins. The result was previously known only under a convexity assumption on the covariance function of the Hamiltonian. We also obtain a one-species reduction of the formula for balanced multi-species models.

We study the dynamics of a d-level quantum system coupled to a bosonic reservoir when the coupling constant is large. It is known that in the limit of infinite coupling strength, the system undergoes an instantaneous nonselective measurement, resulting in the immediate decoherence in the measurement basis, followed by a unitary Zeno dynamics. Here we resolve this dynamical process by introducing a fine grained scaling regime of short times proportional to the inverse coupling. We provide a rigorous derivation of the open system dynamics in this regime of ultrastrong coupling and demonstrate how decoherence unfolds continuously in the new time scale. We show that Markovian dynamics which are not given by semigroups arise naturally, in contrast to what happens in the weak coupling theory.

Motivated by the known derived set $[1,+\infty)$ of multiple zeta-star values, we study finite multiple star harmonic sums and their infinite-depth limits. Divided differences give a cube-integral representation of their holomorphic interpolation on $\mathrm{Re}(s)>0$, which agrees with the Kawashima functions after reversal of the index. The finite-depth interpolations converge locally uniformly in $s$ as the depth tends to infinity, uniformly over all positive integer index sequences on compact subsets. For real $s>1$, these limits define a bijection from the infinite positive integer indices onto $(1,s]$. For each admissible infinite index, they tend to the corresponding multiple zeta-star limit as $\mathrm{Re}(s)\to+\infty$ with bounded imaginary part. For $\mathrm{Re}(s)>1$, we prove strict separation of every ordered pair of finite indices and of every ordered pair of infinite indices: either the difference or the difference divided by $s$ has positive real part throughout this half-plane, with the choice fixed by the pair. In particular, both index maps are injective. The proof uses a summation principle for functions with positive real part, established through positive integral representations and explicit kernel estimates.

This work investigates the stability of outliers from the circular law, via the convergence of their associated diagonal overlaps between eigenvectors - also known as the squared eigenvalue condition numbers. We consider and compare two paradigmatic cases, namely: 1) the Complex Ginibre Ensemble conditioned on the existence of an outlier, and 2) the outlier induced by a rank-one Hermitian perturbation of a Complex Ginibre matrix. In both cases, we prove almost sure convergence towards a specific constant that only depends on the radius of the outlier and its status - either conditioned or induced. These results can be generalized to other complex integrable ensembles with the same techniques, and complement our understanding of eigenvalue stability in non-Hermitian ensembles.

By the Bloch-Okounkov theorem, the $q$-bracket sends shifted symmetric functions on partitions to quasimodular forms. Following a question of van Ittersum, we study whether the Hecke action on quasimodular forms lifts through this map, in the sense of exact lifts $A_{n,2k}$ with $\langle A_{n,2k}f\rangle_q = T_{n,2k}\langle f\rangle_q$. Our main theorem is that Zagier's lowering operator $B=\frac{1}{2}(D-\partial^2)$ is injective on the genuine homogeneous subspace in every weight at least $4$, although it is not injective on the formal algebra. For exact lifts compatible with lowering, this transports Hecke actions on $q$-bracket kernels between adjacent weights, with divisibility consequences for characteristic polynomials and $p$-adic constraints on kernel eigenvalues. In fixed weight, we classify all exact lifts by an action on the kernel and a kernel-valued cocycle relative to a section. Two obstruction results show that no exact lift is an algebra homomorphism, and that even strict compatibility with multiplication by $Q_2$ already fails in weight $6$. Finally, we construct exact lifts with scalar kernel action whenever the $q$-bracket image is Hecke-stable, and exact rational computations give $q$-bracket surjectivity through weight $16$ and therefore existence of Hecke lifts through weight $16$.

This paper considers neural channel state information (CSI) compression for time-varying massive multiple-input multiple-output (MIMO) channels in frequency division duplex (FDD) systems with limited feedback resources. The main challenge lies in obtaining a compact and efficient representation of the CSI given that it exhibits strong temporal correlation across successive snapshots. Existing memoryless compression models do not exploit this property, while simple temporal extensions often incorporate multiple observations without explicitly modeling the latent dynamics. We propose a context-aware compression framework based on a k-memory Markov variational autoencoder (k-MMVAE), which uses a finite temporal window to capture the evolution of CSI in the latent space. The model introduces Markov-structured latent dynamics with finite memory, enabling efficient use of temporal dependencies for compression. Simulation results show that the proposed approach improves target CSI reconstruction performance compared to memoryless and weakly sequential baselines, particularly at low and moderate compression rates. These results suggest that explicit latent temporal modeling can provide an effective mechanism for CSI compression under limited feedback constraints.

Neural operator models trained on simulation data often lose accuracy when applied to experimental measurements due to the sim-to-real gap. Standard fine-tuning with limited real data can reduce this gap, but it may also damage the core physics-relevant representations learned during pretraining. Although knowledge-preserving adaptation has been widely investigated in vision or language tasks, it remains unclear whether these methods are suitable for neural operators whose architectures and protected knowledge are fundamentally different. Neural operators need to preserve core-scale physical structures rather than semantic or visual features. We propose PhysGuard, a physics-preserving framework for accurate sim-to-real adaptation of neural operators. Specifically, PhysGuard uses the empirical Fisher Information Matrix computed on simulation data to identify physics-critical parameter directions, then restricts fine-tuning updates to directions that do not interfere with them. A layer-wise Gram-matrix formulation makes this efficient for models with millions of parameters, while an adaptive threshold automatically determines the protected subspace size. A spectral probe experiment shows that the dominant Fisher directions are strongly associated with low-frequency output structures. Experiments on benchmark across four neural operator architectures and different physical systems show that PhysGuard performs strongly on most evaluation metrics compared to baselines. The benefits are most evident under severe domain shift, where it reduces low-frequency error by up to 32% compared to standard fine-tuning while maintaining adaptability. Our code is available at https://github.com/Data-Driven-Computational-Geotechnics/PhysGuard.

Several useful concepts of almost frames or Riesz bases of Hilbert spaces gave us the motivation to review the theory of C*-module frames and C*-module Riesz bases for Hilbert C*-modules, searching for similar useful structures. We give a survey of the background in C*-module frame theory, summarizing and particularly extending or correcting earlier results. The concepts of near-Riesz bases, pseudo-frames, pseudo-Riesz sequences, and pseudo Riesz bases of Hilbert C*-modules are introduced and equivalently related to (semi-)C*-Fredholm properties of their synthesis operators. The notion of the excess of a C*-module frame is introduced, taking values in related C* algebras. Indicators for possible erasures of frame elements are described with the aim to preserve robustness of frames. Introducing a new notion of pseudo-Riesz-Weyl sequences in the standard Hilbert C*-module $l_2(A)$ we reverse the found correspondence by assuming the associated synthesis operator of a Bessel sequence to be (upper semi-)C*-Weyl. Furthermore, we deduce from the invariance of (upper semi-)C*-Fredholm and C*-Riesz-Weyl properties of synthesis operators with respect to ”compact” perturbations the perturbation invariance of the respective classes of module Bessel sequences.

We study semantic compression of reversible-execution evidence for rollback reasoning. A run is a finite fact base; rollback semantics are modeled by a monotone closure operator induced by function-free Horn rules. A single edit replaces one fact by another; fidelity is the Jaccard discrepancy of the resulting closures, yielding a finite-alphabet distortion for rate-distortion analysis. A deterministic deletion scan decomposes the log into an irredundant core–preserving the closure–and a redundant remainder. Under admissible reconstructions (facts entailed by the original log), redundant facts are distortion-invisible, reducing the semantic rate-distortion function to a core-only optimization scaled by the core probability mass. At zero distortion, the optimal rate is a hypergraph entropy induced by overlaps of zero-distortion reconstruction sets on the core. We introduce a rollback-task loss based on a rollback observable, deriving parallel endpoint and factorization laws. The framework is instantiated on reversible causal nets and discussed in the event-structure view, showing how reversing disciplines yield different cores and compression frontiers. Numerical evaluation uses Blahut-Arimoto to design single-letter test channels and Monte Carlo reconstruction to assess end-to-end degradation at the log level.

We characterize the actions on bifoliated planes that arise as orbits spaces of transitive pseudo-Anosov flows on orientable closed 3-manifold. We use branched covers, veering triangulations, and a new compactness criterion to extend previous work that handled the special case where the plane has no odd-prong singularities and the action preserves a leafwise orientation on the foliations.

For a subset $D \subset \mathbb{N}$, denote by $S\left(D,n\right)$ the total sum of all odd parts minus the sum of all even parts of all partitions of $n$ in which parts from $D$ do not repeat. In this paper, we derive the generating function for $S\left(2\mathbb{N},n\right)$ and use it to give some congruences modulo 4. We also derive the generating function for $S\left(2\mathbb{N}-1,n\right)$ and give two congruences for these functions, one of which was previously proved by Garvan and Sarma.

Let $D$ be an unbounded domain in $\mathbb{C}^n$, and let $f$ be a bounded continuous function prescribed on the boundary of $D$. We show that, if $D$ has $r$-peak points on its boundary and is of bounded type, $f$ extends to a maximal bounded continuous function $F$ on $\overline{D}$ that is $q$-plurisubharmonic and $(n-q-1)$-plurisuperharmonic (i.e., $(dd^c F)^n=0$) on $D$, and that coincides with the Perron-Bremermann envelope created with respect to bounded $q$-plurisubharmonic functions on $\overline{D}$.

In this article, we study permutation orbifolds and simple current extensions in the framework of vertex operator (super)algebras. We extend the construction of permutation-twisted modules for tensor products of vertex operator algebras to vertex operator superalgebras with $\frac12\mathbb Z$-grading, including the effect of the canonical involution. Using tensor category methods and simple current extensions, we build an induction theory for permutation-twisted modules associated with solvable automorphism groups, arising from semidirect products of simple current automorphisms and cyclic permutations. In particular, we describe the structure and classification of irreducible twisted modules in terms of stabilizer subgroups and associated projective representations, and determine their multiplicities explicitly. As applications, we illustrate the theory with explicit examples from code vertex operator algebras, lattice-type simple current extensions, and the Moonshine vertex operator algebra.

We extend the entropy formula of Menon and Yu for the real Deep Linear Network (DLN) to its complex and quaternionic analogues, obtaining a unified formula for DLNs over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$.

Monte Carlo methods have proven highly effective for elliptic partial differential equations through algorithms such as Walk on Spheres and Walk on Stars, which evaluate solutions at individual points without volumetric meshing or global linear solves. Extending these methods to the transient regime has remained an open challenge: parabolic equations couple space and time through an anisotropic scaling, requiring joint sampling of spatial displacements and backward time steps whose distribution was not previously available in a unified, exact form. We present Walk on Heat Stars, a grid-free Monte Carlo solver that closes this gap by extending the boundary integral framework of Walk on Stars to the parabolic setting. Our method introduces a non-cylindrical boundary integral formulation that accommodates the time-varying domains induced by heat-ball sampling. The heat ball geometry is parameterized by a logarithmic time coordinate and a spatial direction, revealing that the double-layer kernel factorizes into independent Gamma and uniform components. This parameterization enables exact directional importance sampling of the recursive next walk position, the Neumann flux contribution, and the volumetric source term, yielding unbiased Monte Carlo estimators for all three components. We additionally derive a preliminary gradient estimator that expresses spatial derivatives as weighted boundary integrals of the solution, requiring no recursion on the gradient, and adapt a heteroscedastic regression-based denoiser to the space-time domain for variance reduction. We validate our method on analytical solutions across a range of geometries and spatial frequencies, confirm convergence at the expected Monte Carlo rate, and demonstrate practical applicability on heat sink and cooling scenes with mixed or pure Neumann boundary conditions.

We extend the Kauffman bracket skein module of three manifolds to a homology theory in a combinatorial way. The resulting homology depends on Heegaard splittings of three manifolds.

Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors. In this study, we investigate this limitation by examining the failure of shallow ReLU neural networks in fitting high-frequency functions. This observation identifies two important factors in resolving rapid oscillations: the initial slope scale and the distribution of partition points induced by the networks. Motivated by this analysis, we propose RepNN, a reparameterized neural network model with ReLU or tanh activations designed for high-frequency and multiscale problems. The key idea is to reparameterize the weights and biases in the first hidden layer, which enables effective control of the initial slope scale and provides an appropriate distribution of the initial partition points. Furthermore, treating the reparameterized weights and biases as trainable parameters allows the DNN to achieve adaptive frequency scaling during training. In addition, we derive quantitative estimates for the output and slope magnitudes of the reparameterized DNN to guide the initialization of the proposed method. Numerical experiments, including multiscale one-, two-, and four-dimensional function approximations, forward and inverse PDE problems in combination with physics-informed neural networks (PINNs), and operator learning for an earthquake problem using real data, demonstrate that RepNN improves the predicted accuracy of vanilla DNNs in capturing highly oscillatory features. These results indicate that RepNN provides an effective and flexible approach for overcoming spectral bias and applying DNNs to multiscale problems.

Recently, Nguyen, Seymour and Scott verified the small quasi-kernel conjecture for split digraphs, and initiated the study of quasi-kernels in break digraphs. Following their research, we introduce a weighted half-neighborhood property for hereditary classes of oriented graphs and show that it gives a \(2n/3\) bound of small quasi-kernel for break digraphs. We also record two stronger \(n/2\) results for special classes of break digraphs. Finally, using the same framework we also prove that every digraph on \(n\) vertices has a quasi-kernel \(Q\) with \(|N_D^+[Q]|\ge \sqrt n\).

Neural ordinary differential equations (neural ODE) gained attention in safety critical settings such as continuous-time controllers for cyber-physical systems and classifiers integrated into automated decision pipelines, raising the question whether their behavior can be formally verified. Existing tools dedicated to neural ODE provide only a single reachability call without iterative input-set refinement, limiting the precision of their verdicts to whatever one reachability call can deliver. We present TNODEV, the first formal verifier for neural ODE that integrates a falsification checker, a fast interval-based reachability backend based on continuous-time mixed monotonicity, a verification and refinement loop with three input-set splitting heuristics, and a parallel scheduler in a single end-to-end pipeline. TNODEV supports safe-set inclusion verification on pure neural ODE, neural ODE in closed loop with a neural network controller and general neural ODE (GNODE), with the safe set specified either as an interval or as the half-space intersection induced by a target classification label. We evaluate TNODEV on a range of benchmarks across safe-set inclusion and classification-robustness properties, including a direct reachability comparison against NNV 2.0 and CORA and a verification comparison against NNV 2.0 on MNIST general neural ODE classifiers.

Decision-making under uncertainty is ubiquitous in environmental project planning. Environmental processes such as a streamflow discharge often present a subexponential memory, where the autocorrelation persists for a long time. In addition, optimization problems driven by environmental processes encounter the issue of model ambiguity because of a lack of sufficient data for model identification. To facilitate decision-making for the management of aquatic environments (e.g., flood mitigation, water abstraction for hydropower generation), we formulate a unified distributionally robust stochastic optimization problem based on a mixed moving average (MMA) process. The MMA process is a superposition of infinite-dimensional affine stochastic processes that is seemingly complex, but the affine property helps with the formulation and computation of the optimization. Our problem is based on a convex objective with a nonsmooth conditional value-at-risk measure. We present a convergent regularization to obtain its smooth and strictly convex counterpart. The model ambiguity is represented as a distortion of the probability density of the target dynamics, and it is penalized by a divergence with which the optimization problem remains convex and becomes computable. As a case study, we apply the optimization problem to two cases with identified parameter values. The performance of the optimized dynamics is evaluated through a statistical simulation. This paper serves as a multidisciplinary work covering both the theory and application of distributionally robust optimization.