Papers

We call a unital associative algebra $A$ trace residually finite-dimensional if its elements are separated by finite-dimensional representations and its cocenter $A/[A,A]$ is separated by the corresponding trace functionals. For RFD algebras $A$ and $B$, we prove that the trace radical of $A*B$, the common kernel of all finite-dimensional trace functionals, is the direct sum of the trace radicals of $A$ and $B$, which implies that $A*B$ is trace RFD if and only if both $A$ and $B$ are trace RFD. The proof combines an explicit cocenter decomposition with a construction of finite-dimensional representations whose traces detect nonzero classes of words of length greater than one.

Particle Swarm Optimization (PSO) is a global optimization algorithm defined by an interacting set of particles evolving over the search space. Heuristically motivated, its theoretical analysis remains limited due to the second-order, stochastic, and highly nonlinear nature of the dynamics. In this paper, we connect classical PSO stability analysis under the stagnation assumption with more recent mean-field methods, providing new quantitative estimates for the time-discrete algorithm. We study in particular a regularized PSO model without memory, with non-degenerate noise by adding a noise floor to the original model. Studying such a surrogate model allows us to identify quantitative conditions under which the dynamics is stable and converges toward a small neighborhood of a global minimizer. We do so by first studying the Schur stability of the linearized dynamics, then analyzing the convergence properties of a nonlinear mean-field system via a Laplace principle, and finally establishing a quantitative error bound for the mean-field approximation of order $N^{-1/2}$.

We introduce heavy-Wigner random tensors and establish the convergence of a generating family of invariant linear forms in their entries together with a central limit theorem. The relevant group actions arise naturally in the study of tensor eigenvalues. Then, we compute the limiting spectral measure of the matrix obtained by contraction of a heavy-Wigner tensor and we derive two applications. The first one concerns the adjacency tensor of a Erdős-Rényi hypergraph. We identify the local weak limit of the hypergraph and study the limiting spectral distribution of the matrix contraction. The second application concerns Lévy tensors whose entries belong to the domain of attraction of an $\alpha$-stable law.

We prove that at most six pairwise non-overlapping infinite unit cylinders can simultaneously touch a unit ball. Six is achievable, so the answer to Kuperberg's question is exactly six. The proof is computer-assisted in the following sense: the statement is reduced to a finite case analysis with 2,954,984 cases, and in each case a program verifies that one of three inequalities between explicit polynomials with rational coefficients holds throughout a box. Each case is an elementary arithmetic check, and the complete list of cases is provided with the paper.

This paper develops a canonical decomposition–reconstruction theory for balanced residuated partially ordered semigroups. The starting point is the intrinsic local-unit map $\tau(x)=x\backslash x=x/x, $ whose values are positive idempotents. The primitive fibres of this map are generally too fine to be compatible with multiplication and residuals: the local units of $xy$, $x\backslash y$, and $x/y$ need not be determined by the local units of $x$ and $y$. We therefore construct the residual coherentization $\mathcal C_{\mathrm r}(\mathbf M)$, the finest quotient of the positive-idempotent skeleton on which these three local-unit outputs are well defined at quotient level. The blocks of $\mathcal C_{\mathrm r}(\mathbf M)$ define the canonical components, while the quotient skeleton records the target component for products and residuals. Together with the component algebras, the product-shadow maps $x\mapsto xq$, and the residual-shadow maps $y\mapsto y/q$, these data reconstruct the original algebra. The final part compares this construction with subsemilattice-steady visibility decompositions. At every finite stage of the induced iterative decompositions, the partition obtained from residual coherentization is finer than the partition obtained from any subsemilattice-steady visibility choice. Equivalently, each component produced by the subsemilattice-steady construction is a union of residual-coherent components.

We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most $\delta$ and $d=\dim A$, the optimal dimension-only modulus of continuity is $h_2(\delta)+\delta\log(d^2-1)$ up to $\delta=1-d^{-2}$ and $2\log d$ thereafter, where $h_2$ denotes the binary entropy. When $\dim B\ge d$, this bound is tight for every $\delta\in[0,1]$. The key proof idea was developed with the assistance of ChatGPT 5.6 Sol, building on and adapting the tight classical proof of Alhejji & Smith [IEEE ISIT (2020)], which follows a conceptually different approach.

Completely positive (CP) matrices are ubiquitous in modern science and technology with applications in optimization, graph theory, and quantum entanglement. Recently, Johnston et al. [Linear Algebra and its Applications, 2022] have cast CP matrices into the framework of quantum resource theories, where CP states serve as free states and CP-preserving channels act as free operations. This work addresses several questions raised in their work. Specifically, we provide the necessary and sufficient conditions of CP-preserving channels in small dimensions, which are necessary in higher dimensions, and discuss the resource quantification via the trace distance of non-negativity. By constructing an explicit counterexample, we demonstrate that the trace-distance measure of non-negativity violates strong monotonicity. We also provide an alternative proof that every CPDNN channel $\Phi:\MM_n\to \MM_2$ is CPCP. Additionally, we show that any unital CPDNN map $\Phi:\MM_2\to \MM_n$ is also CPCP.

The goal of this paper is to consider symplectic singularities of the moduli space of $G$-Higgs bundles over a compact Riemann surface with genus at least $2$. Furthermore, when $G$ is semisimple, we also study its symplectic resolution.

We present a distinctive approach to parameterizing the risk neutral distribution. Using parsimonious and interpretable parameters, the model provides direct and localized control over the shape of the implied volatility curve. It captures a wide variety of shapes, including those with local concavity. Empirical results demonstrate accurate calibration across a quarter million curves from a two-year Standard and Poor's 500 index option dataset. The fitted parameters exhibit stable patterns across tenors, enabling term structure interpolation and dynamic process construction without static arbitrage.

In this expository article we review a remarkable bijection $\phi_n$ due to Loehr and Remmel from the set of parking functions of size n into itself, which sends the bistatistic (dinv, area) into the bistatistic (area, pmaj). The only novelty of the present work is our definition of $\phi_n$, which is more direct than the original one, hence easier to compute and to work with.

We classify all corefree fusion systems on a Sylow 3-subgroup of the sporadic groups $\mathrm{Fi}_{24}'$ and $\mathrm{M}$. The program to classify of all corefree fusion systems on Sylow $p$-subgroups of sporadic groups has been running for many years, comprising of work done in several papers by numerous authors. We complete this program for $p$ odd. In the process, we expand the theory of proto-essential subgroups and describe algorithms that make the process of finding all corefree fusion systems on a $p$-group better by several orders of magnitude.

Finding a set of vertices in a graph with no edges between them, INDSET, is a well-known NP-complete problem. The queen graph of a chessboard is constructed by taking vertices as the tiles of the chessboard and drawing edges between two tiles if a queen can move from one to the other. We call INDQUEENS the independent set problem on a queen graph where the chessboard is a polyomino. We prove that INDQUEENS on polyominoes is NP-complete, proving a conjecture of Langlois-Rémillard–Müßig–Roldán. As our reduction is parsimonious, we can further prove that it is #P-complete. We furthermore prove that INDROOKS on polyominoes is #P-complete, despite being solvable in polynomial time.

In this paper we establish global $W^{2,p}$ estimates for the convex potentials of quadratic optimal transport between bounded convex domains with continuous positive densities. All the assumptions are optimal. The main new ideas include a blow-up analysis that allows for lower-dimensional collapse of the limiting source measure and reduces the limiting problem to a transport problem on its affine hull, and a good-bad scale decomposition in which rigidity controls the good scales while a counting argument shows that the proportion of bad scales tends to zero.

In these notes, we introduce the 50-year-old $K(\pi, 1)$ conjecture alongside Coxeter and Artin groups. Roughly speaking, the conjecture states that the complement in $\mathbb{C}^n$ of a "symmetric" configuration of hyperplanes is a $K(\pi, 1)$ space. Our end goal is to present a proof of the conjecture in the so-called spherical case, where only a finite number of hyperplanes are removed, through methods from combinatorial topology. This proof draws inspiration from the original proof of the spherical case, which is a special case of a celebrated 1972 theorem by Pierre Deligne.

Consider $k$ pairwise disjoint residue classes $a_i \pmod{m_i}$. We prove that \[ \max_{1\leq i<j\leq k}\gcd(m_i,m_j) \gg k\exp\!\left(-(2+o(1)) \sqrt{\frac{\log k}{\log\log k}}\right). \] The proof uses a complete graph whose edges are colored by the gcds of the corresponding moduli, together with a structural lemma, a sieve-theoretic partition, Möbius inversion, and the discrete Fourier transform.

Majority dynamics is a two-opinion process in which each vertex repeatedly updates to the majority opinion among its neighbors. We study this process on a resampled sparse binary stochastic block model in the assortative regime. At each time step, a graph is sampled from the current opinion partition: vertices with the same opinion are joined with probability $\alpha=a\log N/N$, while vertices with differing opinions are joined with probability $\beta=b\log N/N$, where $a>b>1$. Let $B_t$ and $R_t$ denote the blue and red camps at time $t$. We show that the weighted advantage $\widetilde{\Delta}_t =b|B_t|-a|R_t|$, rather than the unweighted advantage $\Delta_t=|B_t|-|R_t|$ alone, governs the pace to unanimity. Our results, which hold with high probability as \(N\to\infty\), identify three regimes for blue unanimity under the initial blue advantage, i.e., $\Delta_0>0$: constant time, subpolynomial time, and polynomial time. First, when $\widetilde{\Delta}_0 \gtrsim -N/\sqrt{\log N}$, blue unanimity occurs within three updates. Second, when $\widetilde{\Delta}_0 < 0$ and $|\widetilde{\Delta}_0| = o(N)$, blue unanimity occurs within $N^{o(1)}$ updates. Furthermore, when $\widetilde{\Delta}_0 < 0$, $|\widetilde{\Delta}_0| = O(N)$, and $\Delta_0\gg\sqrt{N/\log N}$, blue unanimity still occurs within $N^{I_0+o(1)}$ updates, where \[ I_0= \left(\mathbf{ReLU}\Big(\sqrt{a\frac{|R_0|}{N}}-\sqrt{b\frac{|B_0|}{N}}\Big)\right)^2, \] and $\mathbf{ReLU}(x)=\max\{x,0\}$. Conversely, away from the weighted threshold, when $|B_0|/|R_0|\le a/b-\kappa$ and $\Delta_0>0$, $N^{I_0 - o(1)}$ updates are necessary for blue unanimity. Our analysis relies on detailed estimates for one-vertex flip probabilities in sparse binomial differences, which could be of independent interest.

Persistent cohomology provides a principled framework for constructing nonlinear coordinates that reflect the topology of data. However, these topological coordinates can be severely distorted by non-uniform sampling density, limiting their applicability to real-world data. While density-robust circular coordinates have recently been developed, the extension to spherical coordinates remains an open challenge: unlike the circular case, spherical coordinates are obtained through a nonlinear variational problem for sphere-valued maps, to which existing density-correction mechanisms are not directly applicable. In this paper, we introduce the first density-robust construction of spherical coordinates from persistent cohomology. Rather than modifying the coordinate optimization itself, we extend a subsampling-and-alignment framework for circular coordinates to $S^2$, which first computes spherical coordinates on approximately uniform subsamples obtained by rejection sampling and then combines them into a global consensus map. The principal mathematical difficulty is the alignment of independently computed sphere-valued coordinates. We formulate this challenge as a spherical Procrustes problem and establish approximation guarantees for a computationally tractable Euclidean relaxation. Our resulting construction is robust to non-uniform sampling and retains the accuracy of classical spherical coordinates under uniform sampling. Moreover, by computing persistent cohomology only on fixed-size subsamples, our approach avoids the quartic memory bottleneck of the classical spherical coordinate pipeline and scales to substantially larger datasets. We conduct experiments on synthetic data and demonstrate accurate coordinate recovery under severe sampling bias and scalability to datasets of 10,000 points.

The cyclic codes $\mho(q,m,h)$ introduced by Ding, Li, and Xia form a nonbinary generalization of punctured binary Reed–Muller codes. Ding, Li, and Xia established the bounds $(q^{h+1}-1)/(q-1)\leq d(\mho(q,m,h))\leq 2q^h-1$ and asked whether the BCH lower bound is always exact. This paper proves that, for every prime power $q$, every $m\geq 2$, and every $1\leq h\leq m-1$, the minimum distance is $d(\mho(q,m,h))=(q^{h+1}-1)/(q-1)$. The upper bound is obtained by an explicit projective-subspace construction. For any $(h+1)$-dimensional $\F_q$-subspace $V$ of $\F_{q^m}$, the set $V^{[q-1]}=\{x^{q-1}:x\in V\setminus\{0\}\}$ supports a codeword of weight $(q^{h+1}-1)/(q-1)$. Its membership in $\mho(q,m,h)$ follows from a vanishing lemma for subspace power sums and the digit-sum estimate $s_q((q-1)a)\leq(q-1)\wtq(a)$. The constructed codeword meets the BCH lower bound and therefore determines the exact minimum distance.

\noindent We study residual-polynomial acceleration of the proximal point method (PPM) for maximal monotone inclusions, with Anderson acceleration (AA) as the prototypical adaptive scheme. We answer three questions exactly. (i)~The minimax complexity over all adaptive methods is precisely $d_0/(K+1)$ per $K$ resolvent evaluations. The upper bound is attained by the averaged-reflection estimator; the matching lower bound uses an explicit skew-adjoint instance with resolvent eigenvalues at the roots of $u^{K+1}=-1$ and $\csc^2$-distributed masses, on which every degree-$K$ polynomial method satisfies $\|r(y_K)\|\ge 1/(K+1)$. The optimal polynomial is uniquely the Fejér kernel, and the same instance certifies a per-step floor. (ii)~A sharp phase transition separates regimes: Jackson-kernel polynomials achieve $O(d_0/(K^2 s))$ when the spectral floor $s$ satisfies $sK\to\infty$, while at the critical scale $s\asymp 1/K$ the barrier is exactly $1/(K+1)$. The picture extends to normal operators and the nonlinear family $M=S+N_C$. (iii)~On linear problems AA-PPM needs no safeguarding; on nonlinear problems certification of the $O(1/k)$ envelope requires exactly two oracle evaluations per iteration, and this factor is optimal. We also correct and complete the theory for structured problems—affine, strongly monotone, piecewise-affine, and Hölderian growth—and confirm all predictions numerically.

We introduce a Morse theoretical approach to the construction of the Yang–Mills measure on the space of connections of a compact Riemannian surface. This provides a direct continuous version of this measure which was previously obtained through lattice approximations by Chevyrev in the case of the flat torus and by one of the authors and Nohra for general compact Riemannian surfaces. The starting point is the new notion of a Morse gauge together with the resolution of random cohomological equations associated to Morse–Smale vector fields. This is achieved by improving exponential convergence to equilibrium results for Morse–Smale gradient flows that were obtained by two of the authors in the context of the study of Ruelle spectra and by Jia, Stewart and Sverak in the context of simplified models from fluid mechanics. Combining these random solutions with the data given by the Morse complex, we introduce a free Yang-Mills measure on space of connections and, using classical tools from stochastic differential equations, we show how to make sense of holonomies for random connections along a large class of curves. Finally, by setting a proper conditioning of this free measure through these random holonomies, we define the Yang–Mills measure and we compute its partition function together with the law of random holonomies with respect to this measure, recovering the formulas from the works of Migdal, Witten and Lévy.

The totally asymmetric simple exclusion process (TASEP) with open boundaries is a finite Markov chain describing particles hopping between adjacent sites on a one-dimensional lattice with left and right boundary transitions governed by parameters $\alpha$ and $\beta$. The multispecies TASEP is a higher-rank generalization in which particles have different species. Multiline queues were introduced by Ferrari and Martin (2007) to compute the stationary distribution of the multispecies TASEP on a circle. It has remained an open problem to find a combinatorial formula for the stationary distribution of the multispecies open-boundary TASEP. Using Kirillov–Reshetikhin crystals of type $C$, we construct type $C$ multiline queues and a corresponding Ferrari–Martin pairing algorithm that projects them to TASEP configurations. This yields a combinatorial formula for the stationary distribution of the multispecies open-boundary TASEP for the $\alpha=\beta=1$ specialization.

We prove same-space norm inflation for the three-dimensional hypodissipative Navier–Stokes equations with dissipation $(-\Delta)^\alpha$, $0<\alpha<1$. Let $2\le p<\infty$ and \[ 0<s<1-2\alpha+\frac3p. \] In the Besov case, let also $1\le q\le\infty$. There exist divergence-free $C_c^\infty(\mathbb R^3)$ initial data that are arbitrarily small in $W^{s,p}$, respectively in $B^s_{p,q}$, while the corresponding unique local smooth solution becomes arbitrarily large in the same space in arbitrarily short time. The proof adapts the anisotropic vortex-ring mixing mechanism to fractional dissipation; the strict scaling-supercritical gap makes both the curvature error and the nonlocal dissipative error perturbative.

Fix an o-minimal expansion $\mathcal{R}=(\mathbb{R},+,\cdot,0,1 <,...)$ of the real ordered field. Given $C^1$ functions $f_1,...,f_k$, $g_1,...,g_k:M\to\mathbb{R}$ on a definable cell $M$, let $h_{i,0}$ denote $f_i$ and $h_{i,1}$ denote $g_i$. Suppose that for all $\tau\in 2^{[k]}$, $H_{\tau}=(h_{1,\tau(1)},..,h_{k,\tau(k)}) :M\to \mathbb{R}^k$ is regular and proper on $M$, and that for all $i\in[k]$, $\{f_i=0\}$ and $ \{g_i=0\}$ are connected, and $\{f_i=0\}\cap \{g_i=0\}=\emptyset$. We show that then there exists a sequence $(\square_{i,\epsilon}:i\in[k],\epsilon \in \{0,1\})\in\{\leq ,\geq \}^{[k]\times\{0,1\}}$ such that the enclosed region $\underset{i\in[k]}{\bigcap}\{ f_i\square_{i,0} 0\}\cap \{ g_i\square_{i,1} 0\} $ is compact.

For a compact subset of an aspherical symplectic manifold, we show that the symplectic cohomology with support on it remains invariant under suitable perturbations of the symplectic structure.

This paper shows that the substructural logic without the identity principle A->A (i.e., subreflexive logic) has principled sound and complete semantics and supports a variety of applications. This decidable generalization of propositional logic naturally interprets implication as robust consequence. Subreflexive logic is proved to admit syntactic cut elimination. Heyting and Boolean semialgebras are introduced as generalizations of Heyting and Boolean algebras and are shown to provide complete algebraic semantics without inadvertently reintroducing reflexivity. Semi-adjunctions on semi-categories and (identity-free) (co-)units are defined to give complete semi-categorical semantics. In the classical case, denotational set semantics that interpret implication as robust material implication are proved complete for subreflexive logic.

Conventional wireless protocols such as Hybrid Automatic Repeat Request (HARQ) rely on frequent and timely feedback, which becomes impractical in low-feedback regimes including non-terrestrial networks and massive IoT. This limitation is particularly critical in heterogeneous multi-user systems with unknown and asymmetric channels, where a single weak user can dominate the overall completion latency. We propose Overhearing-driven Non-Orthogonal Multiple Access (ONOMA), a novel cross-layer transmission scheme that minimizes latency without requiring instantaneous or statistical CSI at the transmitter. ONOMA integrates Random Linear Network Coding (RLNC) with symbol-aware NOMA and explicitly exploits overhearing and acknowledgment timing. In the first phase, users overhear RLNC transmissions, and the relative timing of acknowledgments is used to implicitly infer channel strength ordering. In the second phase, symbol reconstruction enables interference-free decoding for strong users, effectively decoupling user latencies. An adaptive power allocation policy is derived from acknowledgment timing-based channel estimates. Analytical and simulation results show that ONOMA outperforms TDMA, multicast, FDMA, inter-session, and classical NOMA, reducing completion time by up to 34% in two-user and 50% in larger asymmetric networks.

For the discontinuous Galerkin code Trixi$.$jl we implement capabilities for adaptively coupling arbitrarily many domains with different physics. The coupling of the systems is realized through the exchange of boundary information and user-definable coupling functions. This gives us the ability to couple systems that do not share a single variable or have a very different number of variables. This is particularly useful when we have a hierarchy of systems. Our implementation is such that we can adaptively select the model of the coupled systems in the course of the simulation. This allows for highly dynamical scenarios as can be found e.g. in astrophysics. The criteria for adaptivity can be user defined and tailored to the physical problem. Compared to computing the most complex model in the entire simulation domain, our method of coupled systems of high and low complexity leads to a significant reduction in computational time with very little overhead.

In this paper, we study state-feedback fixed-time stabilization of continuous-time perturbed linear systems with nilpotent matrices in the presence of unknown large constant input/measurement delays with known bounds. By using an additional artificial delay, we design a feedback that for a precisely known linear system with a nilpotent matrix drives the state to zero in fixed time (which is larger than the bound of the unknown delay) and achieves input-to-state stability with a large exponential decay rate in the presence of additive disturbances and small system matrix perturbations and nonlinearities. For perturbed single integrators, fixed-time stabilization is achieved in the case of fast-varying unknown measurement delays with known bounds. The results are extended to discrete-time systems. Numerical simulations illustrate the efficiency of the results.

In this expository note we discuss the example of R.~Alweiss giving a construction of a set $S \subset \mathbf{N}$ which intersects every nil-Bohr set, but which is not a set of $2$-recurrence.

An arbitrary-order family of implicit two-derivative one-step methods is constructed in an anchored active-stage formulation. At each information node the method uses both the vector field and its first total time derivative, enriching the local Hermite data without increasing the number of unknown stage states. With the known initial value retained as an anchor and $s$ unknown active stages, $2s$ Hermite moment conditions yield global order $2s$. The two remaining coefficients in each stage row are fixed by the second-subdiagonal Padé approximant $[s-1/s+1]_{e^z}$. For every ordered real node set, a Padé–Hermite basis theorem proves that the closure is unique, preserves all moment conditions, and gives $\det(I-zA-z^2\widehat A)=Q_s(z)$ and $R_s(z)=P_s(z)/Q_s(z)$. Hence the coupled stage system has no hidden poles and the accepted one-step map is $L$-stable (and therefore $A$-stable) for every positive integer $s$. Exact symbolic verification is reported through $s=6$, and high-precision computations confirm orders $2,4,6,$ and $8$ for the first four members. At equal active-stage count, comparisons with Gauss–Legendre and Radau IIA methods demonstrate the combined high-order accuracy and strong stiff damping of the construction over broad step-size ranges.