Papers

Both the uniform Roe algebras and the quasi-local algebras encode the large scale geometry of metric spaces. Whether these algebras coincide is a question which goes back to Roe and has only been recently solved by Ozawa. We propose a dynamical point of view on this problem. Given a set $X$, every map $h\colon X\to\mathbb{R}$ induces a one-parameter group $\sigma_h$ of automorphisms of $\mathcal{B}(\ell_2(X))$, given by conjugation by the diagonal unitaries $e^{ith}$, and we show that the operators which are continuity points of $\sigma_h$ are precisely the uniform Roe algebra of the pseudo-metric induced by $h$. If we moreover assume that $X$ is a uniformly locally finite metric space and $h$ is allowed to range over all coarse maps, this gives dynamical characterizations of both algebras: $\mathrm{C}^*_{ql}(X)$ consists exactly of the operators which are continuous for all such flows, while $\mathrm{C}^*_u(X)$ consists exactly of norm limits of those which are analytic. The regularity conditions lying between continuity and analyticity then give rise to $\mathrm{C}^*$-algebras between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_{ql}(X)$ which are invariant under bijective coarse equivalence, and we show that this scale is not degenerate: if $X$ is a coarse disjoint union of expander graphs, the algebra generated by operators which are analytic on a strip for every diagonal flow sits strictly between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_{ql}(X)$.

In this paper, we investigate weakly persistent centers for several classes of planar cubic differential systems, including systems with a nondegenerate linear center and systems with a nilpotent center. Necessary parameter relations are obtained by computing singular point quantities or quasi-Lyapunov constants, while their sufficiency is established by suitable analytic constructions. For the general complex cubic system, we obtain necessary and sufficient conditions for the origin to be a weakly persistent center, and the corresponding result for the associated general real cubic system follows by imposing the conjugacy relations between the complex coefficients. We further characterize the weakly persistent isochronous centers of the general complex cubic system by combining the weakly persistent center conditions with the vanishing of the linearization quantities and constructing time-preserving analytic linearizations.

We study the noise threshold for positive quantum capacity for the qubit depolarizing channel. We explore analytically the action of the qubit depolarizing channel on the symmetric subspaces of the input qubits, in the limit of asymptotically many uses of the channel. We observe the emergence of a bosonic Gaussian channel. Furthermore, the codes previously developed for the depolarizing channel can be translated to codes for the emergent Gaussian channel, and it is easier to further optimize these codes for the simpler emergent Gaussian channel. Translating these codes back to the depolarizing channel leads to extremely good input states for the coherent information of the depolarizing channel producing new lower bounds on the noise threshold for positive capacity. In addition to improved lower bounds on the threshold, this newly found link between depolarizing noise and Gaussian channels offers a novel perspective contributing to our understanding of these symmetric codes.

The Finitistic Dimension Conjecture is one of the most important homological conjectures in the representation theory of finite-dimensional algebras. Recently, Gélinas arXiv:2004.04828 proposed the notion of "delooping level" to study the finitistic dimension Conjecture. This article reports a misconception between the delooping level and the tensor product of algebras. Namely, for finite-dimensional algebras $A$ and $B$, the difference $\operatorname{dell}(A\otimes_{\mathbf{k}}B) - \left(\operatorname{dell}(A)+ \operatorname{dell}(B)\right)$ can be arbitrarily large or negative.

We study deterministic first-order bilevel optimization under weak lower-level convexity, allowing nonconvex lower-level objectives and without assuming strong convexity, the Polyak-Łojasiewicz condition, or an error-bound property. We consider a $\delta$-relaxed Moreau-gap constraint, with $\delta>0$, for the lower-level stationarity condition and propose an inexact variable-smoothing penalty method (IVSP) for computing its approximate Karush-Kuhn-Tucker (KKT) points. For any fixed relaxation level $\delta$, under a standard extended no-nonzero-abnormal-multiplier constraint qualification (ENNAMCQ), we prove finite stabilization of the adaptive penalty parameter and an overall $\widetilde O(\varepsilon^{-3})$ first-order complexity for computing an $\varepsilon$-KKT point. Notably, we give verifiable sufficient conditions for ENNAMCQ covering convex lower-level objectives without nonconstant affine segments (including the strictly convex case), and a class of nonconvex sample-reweighting models. The positive relaxation avoids the intrinsic constraint-qualification degeneracy of the exact Moreau-gap constraint while achieving an $\mathcal O(\sqrt{\delta})$ lower-level near-stationarity guarantee. Numerical experiments on synthetic and real-world bilevel learning problems illustrate the practical performance of IVSP.

We show that every irreducible linear differential equations on the Riemann sphere with regular and/or unramified irregular singularities can be transformed into either the trivial equation or a Fuchsian system of $E_8$-fundamental spectral type by a sequence of invertible transformations consisting of confluences, unfoldings, Laplace transformations, gauge transformations and Möbius transformations. The $E_8$-spectral type is uniquely determined by the original equation. A Fuchsian system of $E_8$-fundamental spectral type has three singular points, 0, 1, and $\infty$. The degrees of the minimal polynomials of the residue matrices at 0 and 1 are two and three, respectively, and the sum of the maximal dimensions of eigenspaces of the three residue matrices is not greater than the size of the matrices. The result follows from the correspondence between spectral types of Fuchsian systems and roots of a star-shaped Kac-Moody root system.

The known proofs of the gradient estimate for complex Hessian equations rely on the second order estimate of Hou-Ma-Wu and on a Liouville theorem of Dinew-Kolodziej. In this paper, we give a direct proof of the gradient estimate for concave fully nonlinear elliptic equations on compact Hermitian manifolds, which uses neither the Hou-Ma-Wu estimate nor the Liouville theorem. Instead, the proof combines a stability estimate with a comparison argument against nearby smooth admissible functions. We also give an independent proof of the complex Hessian estimate on compact Kähler manifolds, based on Dirichlet Green's functions, under additional assumptions on the operator.

Let $N$ be a square-free, positive integer and let $p\ge 5$ be a prime dividing $N$. In this work, we construct semistable models of Atkin–Lehner quotients of $X_0(N)$ over the Witt ring $W(\overline{\mathbb{F}}_p)$ by taking the corresponding quotients of the Deligne–Rapoport model of $X_0(N)$. We give an explicit description of the special fiber of the models and obtain genus formulae for the Atkin–Lehner quotients of $X_0(N)$.

We give a new proof of the classification of unipotent representations for split finite groups of Lie type. To do this, we first construct an (infinity, 2)-categorification of Lusztig's map from the Hecke algebra to the asymptotic algebra. Then we use the theory of weights to show that it induces an isomorphism after applying trace of Frobenius.

Let $\mathcal{F}$ be a nonempty family of connected bipartite graphs, each with at least two edges. For a graph $G$, a proper vertex coloring of $G$ is $(2,\mathcal{F})$-avoiding if no member of $\mathcal{F}$ occurs bichromatically, and $\chi_{2,\mathcal{F}}(G)$ denotes the minimum number of colors in such a coloring. A B-coloring of $G$ is a proper edge-coloring in which every $4$-cycle is rainbow, and $q_B(G)$ denotes the minimum number of colors in a B-coloring of $G$. For a fixed connected bipartite graph $F$ with at least one edge and bipartition classes $X_F$ and $Y_F$, define $k(F)=\min\{|I|:I\subseteq X_F\text{ or }I\subseteq Y_F,F-I\text{ is a forest}\}$. Let $m\ge2$ be the minimum number of edges in a member of $\mathcal{F}$. We prove that if $k(F)\le m-2$, then every $F$-free graph $G$ of sufficiently large maximum degree $\Delta$ satisfies $\chi_{2,\mathcal{F}}(G)=O((\frac{\Delta^m}{\log\Delta})^{\frac{1}{m-1}})$, which gives a positive answer to Chuet's Problem A and C in a sharp sense, thereby extending the results of Chuet [arXiv:2603.23379] from frugal colorings to $(2,\mathcal{F})$-avoiding colorings. For B-colorings, put $k=k(F)$, $h=|V(F)|$, and $s=\min\{|X_F|,|Y_F|\}$. We prove that every $F$-free graph $G$ of sufficiently large maximum degree $\Delta$ satisfies \[ q_B(G)\le \begin{cases} \Delta+\Delta^{1-\eta}+1, & \text{if }s\le2,\\ (4h-2)(\Delta-1)+1, & \text{if }s\ge3\text{ and }k\le1,\\ C\frac{\Delta^{2-\frac{1}{k}}}{\log\Delta}, & \text{if }k\ge2, \end{cases} \] where $\eta>0$ and $C>0$ depend only on $F$. For $k\le1$, the linear order is best possible, and for $k\ge2$, the bound is nearly sharp. To prove these results, we develop a common reduction of the coloring problems to $P$-perfect matching problems in auxiliary hypergraphs and apply the forbidden-submatching theorem of Delcourt and Postle.

We call an $n$-tuple of $2 \times 2$ matrices a “\textit{$2 \times 2$ matrix $\Gamma_n$-contraction}” if its joint eigenvalues (joint spectrum for a commuting tuple of matrices) are contained in $\Gamma_n$. In this short note on $\Gamma_n$-contraction, we produce two families of $2 \times 2$ matrix $\Gamma_n$-contractions whose fundamental operators do not satisfy the commutativity conditions and \begin{equation*} \begin{aligned} F_iF^*_{n-j} - F_jF^*_{n-i} = F^*_{n-j}F_i - F^*_{n-i}F_j, \quad 1 \le i, j \le n-1, \end{aligned} \end{equation*} which shows that the commutativity as well as the conditions on fundamental operators mentioned above are sufficient, but not necessary, for the existence of $\Gamma_n$-isometric dilation.

We construct two toric manifolds of complex dimension four, which are not homotopy equivalent but have isomorphic integral cohomology rings.

Let $f$ be a primitive holomorphic cusp form of arbitrary weight, level and nebentypus. We prove $\int_0^T|L(1/2+it,f)|^6\,dt\ll_{f,\epsilon}T^{2+\epsilon}$, extending Jutila's level-one theorem. The new ingredient is a large sieve comparing the stationary phases in the Booker-Milinovich-Ng transformation at different heights. Applications include moments of order $T^{1+\epsilon}$ on every line to the right of the critical line, $\Omega(T^{4/35-\epsilon})$ simple zeros, zero density estimates, and bounds for cubic-field power sums, shifted convolution sums and the general divisor problem.

How much time is fundamentally required to extract the syndrome of a stabilizer code on hardware with a given connectivity graph? Previous bounds answer through a chosen list of checks (Delfosse, Beverland, and Tremblay, 2021) or through the global code parameters (Baspin, Fawzi, and Shayeghi, 2023). Here, we show that neither determines the interaction that a particular hardware cut requires. We therefore define the intrinsic cut cost, $\chi$, directly from the stabilizer group, independent of the generating set used to describe the code. It is computable from the check matrix alone, and equals the mutual information across the cut in the maximally mixed code state. We show that every exact extraction circuit, however adaptive, needs at least $\chi/2$ two-qubit operations across the cut, and alternatively with free local processing exactly $\chi$ shared Bell pairs. Guided by this intrinsic cut cost, we construct annular surface codes with constant-depth local extraction that give a counterexample to Corollary 1 of Delfosse, Beverland, and Tremblay. Furthermore, for a fixed quantum Tanner family, $\chi$ grows at least linearly with the size of every sufficiently small region, giving a depth guarantee at each corresponding bottleneck in every layout. This strengthens the global perspective of Baspin, Fawzi, and Shayeghi and, together with our matching square-grid construction, yields a tight space–depth tradeoff. Our results connect the interaction required by syndrome extraction to hardware communication capacity and thus provide a way to assess unavoidable circuit costs before committing to a particular layout.

The finite basis problem for small semirings differs from its semigroup counterpart even in order three. Recent work classifies several additive types of four-element additively idempotent semirings, including all 386 semirings whose additive reduct is a chain. We consider all four-element semirings with commutative addition, in the binary signature without named constants. Combining the existing classifications with structural reductions and polynomial normal forms, we obtain 2284 finitely based and 57 nonfinitely based isomorphism types among the 2341 types. The nonfinitely based cases consist of 45 additively idempotent semirings and twelve with nonidempotent addition. The positive arguments give explicit bases or finite constructions with specified bounds. The negative arguments use cited results, term retractions and a hypergraph obstruction. A complete catalogue records the applicable result for each representative; separate correspondence tables identify precisely the cases supplied by the earlier classifications.

We study large deviations for rare clusters in sparse systems of moving particles. In the regime \(nr_n^d\to0\) and \(\rho_{k,n}=n^kr_n^{d(k-1)}\to\infty\), we prove a large deviation principle for the empirical measure of isolated \(k\)-particle trajectory clusters. The speed is \(\rho_{k,n}\), and the rate function is the relative entropy \(h(\,\cdot\,\mid\tau_k)\), where the finite reference measure \(\tau_k\) explicitly incorporates the underlying path law. As consequences, we derive free-energy variational formulas for bounded interactions and a hard-core constraint and identify the corresponding minimizing cluster law. The normalized optimizer provides the basis for Metropolis–Hastings sampling of interacting trajectory clusters. As an application motivated by chain formation and swarming in active particle systems, we calibrate the resulting stochastic model to experimental trajectories of magnetic micromotors and find that the fitted velocity scale varies systematically with particle size and magnetic forcing.

Colored partitions constitute a well-established and extensively studied area of research with numerous generalizations arising from imposing restrictions on the allowed colors of the parts. In particular, restricted colored partitions have received considerable attention owing to their rich combinatorial and arithmetic properties. Motivated by these developments, we consider a colored overpartition function $\overline{\mathfrak{C}}_{m,r,s}(n)$, which enumerates the overpartitions of $n$ such that each part divisible by $m$ can be assigned any of $r$ colors, while each part not divisible by $m$ may be assigned any of $s$ colors. We investigate an asymptotic formula and several arithmetic properties of this function, including its density properties.

Based on second-order inertial dynamics, we develop two accelerated Lagrange multiplier (LM)-based optimization methods: Linear LM-based second-order flow method and linear LM-based variable and operator splitting method for unconstrained non-convex optimization problems. Relaxed and adaptive variants of both methods are proposed to prevent degeneration of the Lagrange multiplier and to enable automatic adjustment of the time step size. Within the novel second-order LM-based optimization framework, the associated pseudo-energy functional is shown to dissipate monotonically along the iterative trajectory, and convergence of the generated iterates to stationary points is rigorously established. Extensive numerical experiments on function optimization and partial differential equation benchmark problems, including applications as optimizers for physics-informed neural networks (PINNs) and deep operator networks (DeepONets), demonstrate that the proposed methods effectively escape local minima, achieve high accuracy, and exhibit strong robustness with respect to the choice of the initial learning rate during neural network training.

The large-distance behaviour of a sandwich gravitational wave can be approximated by a continuous but not necessarily smooth profile, providing us with a simplified description of particle motion. Our approximate model is consistent with the Carroll symmetry. Our strategy is illustrated by the Poschl-Teller profile. The limits of the sandwich approximation are discussed.

We establish a multiplicity-one refinement of the non-archimedean type-II Howe correspondence: whenever the small theta lift is non-zero, it occurs in the big theta lift with Jordan-Hölder multiplicity one. Furthermore, we apply this result alongside the Godement-Jacquet zeta integral to compute big theta lift for Arthur-type representations. Based on these computations, we also propose a conjecture regarding the irreducibility of the big theta lift.

We study the critical exponent of additive codes through an integer polymatroid associated with the code. We give a coding-theoretic proof of Whittle's Critical Theorem in this setting, a geometric description of the critical exponent in terms of $h$-projective systems, and general bounds, including an analogue of Kung's girth bound. We then study additive codes whose polymatroid is the hypergraphic polymatroid of a hypergraph $H$. For these codes the critical exponent turns to be determined by the weak chromatic number of $H$. If the code is faithful, then the minimum folded Hamming weight of the dual code is equal to the Berge girth of $H$. If $H$ is connected, the minimum distance is equal to the edge-connectivity of $H$. As a consequence, for $h\geq2$ we determine all such codes with connected $H$ that attain the Singleton bound, that is, all faithful hypergraphic additive quasi-MDS codes. We specialise the Griesmer and linear programming bounds to hypergraphic codes, we derive a lower bound on the minimum distance from the Laplacian eigenvalues of the weighted $2$-section of $H$, and we compare all these bounds computationally.

Finite free convolution is obtained by averaging characteristic polynomials over Haar unitary conjugation. We ask when two maximal abelian $\ast$-subalgebras of the complex matrix algebra ${\sf M}_n$ can be placed in finite free position: that is, when their relative position realizes this averaging exactly for every pair of elements, one from each subalgebra. Writing such a pair as ${\sf D}_n$ and $U{\sf D}_nU^*$ with $U$ unitary, we characterize finite free position, for either additive or multiplicative convolution, by the condition $|\det U[I,J]|^2=\binom{n}{r}^{-1}$ for every $1\le r\le n$ and all $I,J$ with $|I|=|J|=r$. We show that this condition holds if and only if $n\le3$ and $\sqrt{n} U$ is a complex Hadamard matrix. To quantify the failure of exact realization for $n\ge 4$, we introduce the uniform-minor discrepancy $\delta_r(U)$. We identify it with the mean-square error in the $r$-th coefficient of finite free multiplicative convolution for two diagonal matrices whose diagonal entries are independent and uniformly distributed on the unit circle. We establish the symmetry $\delta_r(U)=\delta_{n-r}(U)$ and the monotonicity $\delta_1(U)\le\delta_2(U)\le\cdots\le \delta_{\lfloor n/2\rfloor}(U)$. For flat unitaries, we derive an explicit formula for $\delta_2$, yielding $\delta_r(U)\ge\frac{n-3}{2n}$ for $2\le r\le n-2$. Equality for $r=2$ holds precisely when the entrywise square of $\sqrt{n}U$ is also complex Hadamard.

We disprove a conjecture of Liu, Luo, and Luo (2020) concerning the Landau radius for bounded planar harmonic mappings. We construct an explicit harmonic mapping $F_M$ satisfying the standard normalization conditions, which fails to be locally univalent strictly within the classical holomorphic Landau disk for all $M > M^* \approx 4.451$. By combining this counterexample with the lower bound established by Ponnusamy, Kalaj, and Vuorinen (2014), we demonstrate that the exact asymptotics of the harmonic Landau radius is $\frac{\pi}{8M}$.

For any $t\in [0,1/2)$, consider the process $\eta_t:[0,1/2]\mapsto\mathbb{R}$ defined by $\eta_t(s)=\Pi(t+s)-\Pi(t)$, where $\Pi$ is the directed geodesic from $(0,0)$ to $(0,1)$ in the directed landscape. Let $0\leq t<u<1/2$. We show that the laws of $\eta_t$ and $\eta_u$ are mutually absolutely continuous if and only if $t>0$. This proves Conjecture 14.5 in Dauvergne, Ortmann, and Virág, 2022 in the affirmative.

In this paper, we study the exact distributions of the age of information (AoI) and peak AoI (PAoI) in a bufferless setting in which time-stamped updates, or processed tasks, generated by one or several information sources according to a Poisson process for each source, are submitted to a shared pool of $c$ homogeneous servers with exponentially distributed service times, i.e., the so-called M/M/c/c or the Erlang loss system. We consider three update management policies which come into play when a new update arrives to find all the servers busy: the arriving update is blocked (non-preemptive, NP) as in the Erlang loss system, or it preempts a randomly chosen update of its own source in service (preempt at random, PR), or the stalest such update (preempt the stalest, PS). All three policies preserve the same birth-death structure of the server occupancy process associated with the underlying Erlang loss system. However, their AoI and PAoI distributions can be very different. The approach we take is the absorbing Markov chain (AMC) method, in which a single AoI cycle, rather than the entire sample path of the system, is modeled by an absorbing Markov chain. Via the AMC method, we derive the exact distributions of AoI and PAoI in matrix-exponential form. The method extends to multiple sources sharing the same pool of servers, with the AoI of a given source being affected by the remaining sources only through their aggregate update rate. Numerical examples illustrate the implications of our findings, including distribution-based server provisioning under age violation constraints.

The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential $\xi$. We consider the case where $\{\xi(z): z\in \mathbb{Z}^d\}$ are independent and identically distributed Pareto random variables with parameter $\alpha$, and assume that the solution is initially localised at the origin. We establish the full asymptotic behaviour of the total mass of the solution as time tends to infinity in the heaviest-tailed regime $\alpha\in(d,2d)$. In particular, we find qualitatively different behaviour in dimension $d=1$ and in dimensions $d\ge 2$.

For Hermitian matrices $A$ and $B$, we prove that $(A\join B)\oplus(A\meet B)$ is unitarily equivalent to $A\oplus B$, where the join and meet are taken in Olson's spectral order. Taking traces answers a question of Bourin and Lee: for positive semidefinite matrices, $\Tr(A\join B)=\Tr(A+B)$ if and only if $A\meet B=0$. Iterating the direct-sum identity gives a trace formula for a finite family of positive matrices. The trace of the spectral supremum equals the trace of the sum precisely when the ranges form an algebraic direct sum. For each fixed $1<p<\infty$, the spectral supremum and the sum have equal Schatten $p$-norms if and only if the ranges are pairwise orthogonal. Finally we establish an elegant formula for the Frobenius inner product and the spectral order.

Motivated by questions arising in noncommutative Iwasawa theory, let $R$ be a (not necessarily commutative) ring, $T \in R$ a regular central element, and $M$ a pseudo-null $R$-module. We investigate necessary and sufficient conditions under which the quotient $M/TM$ is pseudo-null as an $R/TR$-module. We first give necessary and sufficient conditions in terms of associated prime ideals when $R$ is commutative. Then we apply this to the case when $R$ is a Krull domain and obtain a precise relationship between the characteristic ideals of $M/TM$ and the $T$-torsion submodule $M[T]$. We give necessary and sufficient conditions in terms of $\operatorname{Ext}$ groups when $R$ is a noncommutative ring and then compare them with the criterion when $R$ is commutative. Lastly, we give an application to noncommutative Iwasawa theory. Roughly speaking, if `big' dual fine Selmer group is pseudo-null, then `most' specialized dual fine Selmer group is pseudo-null.

This paper constructs explicit closed-form Green's functions for the Helmholtz equation on annular and spherical-shell domains with two independent Robin impedances on the inner and outer boundaries. Graf's and the hyperspherical addition theorems reduce each angular mode to an explicit $2\times2$ linear system, and the resonance spectrum is governed by a characteristic determinant bilinear in the two impedances. This bilinearity has a direct inverse-problem consequence: two resonant frequencies of one non-radial angular mode generate at most two candidate impedance pairs via an explicit quadratic equation, a third resonance selects the physical pair, and an exact reflection-symmetry obstruction identifies where the radial spherical mode cannot recover the impedances. Spectrally, we prove the branches positive, simple and strictly increasing in both impedances; derive first-order asymptotics at the four corners of the impedance plane, with coefficients given by boundary masses and normal derivatives of the limiting eigenfunctions, and an explicit mixed second-order coefficient at the Dirichlet–Dirichlet corner, which for the radial mode evaluates in closed form to $2\pi/(R_2-R_1)^3$; establish a low-frequency resonance-free band, with a second-order threshold expansion explicit in dimension three and a rational approximation accurate over the whole impedance range; and prove the universal high-frequency spacing law with a shell-curvature correction. Jacobian-based sensitivity and conditioning criteria are included. The kernels and spectra are computable to machine precision, all asymptotic regimes are confirmed numerically, and the kernels provide reference solutions for finite-element and boundary-element validation.

A technical condition arising in the analysis of atomistic models for dislocations is studied. Key estimates for the far-field strain behaviour proved in Ehrlacher, Ortner, Shapeev (2016) rely on a summation-by-parts argument for linearised forces, under conditions of sufficient decay and of vanishing net force in an infinite system. In particular, the latter condition is required for an application of this theory to the standard far-field dislocation predictor, but the vanishing of the associated force sum has not been fully justified. Here, the missing verification of this condition is provided and its role in determining the far-field decay of the corrector is clarified. It is moreover shown that for more general physically compatible predictors, the linearised force sum need not vanish, leading to slower decay and potentially divergent finite-domain approximations.