Papers

Let $(X,d,p)$ be a noncollapsed pointed Gromov–Hausdorff limit of complete $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, where $n\ge4$. We assume that, on each bounded ball, the integral of the $3/2$ power of the small-ball volume deficit relative to the hyperbolic comparison volume is $O(r^3)$ as $r\rightarrow0$. We prove that the metric singular set has Hausdorff dimension at most $n-3$ and sigma-finite $(n-3)$-dimensional Hausdorff measure, thus confirming a particular case of codimension-three regularity conjecture \cite[Conjecture 2.4]{Naber2020}.

In this paper, we study a stochastic optimal control problem for forward-backward stochastic control systems with quadratic convex generator and unbounded terminal condition, where the control domain is not necessarily convex. Since the existing bounded mean oscillation (BMO) martingale approach used in the setting of quadratic BSDEs with bounded terminal conditions is not directly applicable to the present setting, we introduce a new probability measure, under which all subsequent analysis is then carried out. Finally, a global stochastic maximum principle is obtained.

A central conjecture in the regularity theory of harmonic maps asserts that the singular set of a stationary harmonic map has codimension at least three. In this paper, we construct counterexamples to this folklore conjecture. We construct stationary harmonic maps whose singular sets $S$ satisfies $\dim S=n-2$ and $\mathcal{H}^{n-2}(S)=0$ . The target is a smooth compact manifold diffeomorphic to $\mathbb{S}^2 \times \mathbb{T}^3$, with metric arbitrarily close in $C^\infty$ to the standard product metric.

We consider moving-average iteration for fixed complex matrices and place the theory in the general context of Krasnoselskii and Mann iteration. Necessary and sufficient spectral conditions for the asymptotic stability of moving-average iteration are presented in dependence on the fixed window width. For the corresponding region of stability in the complex plane explicit formulas are derived and it is shown that the region is open, convex and semi-algebraic. The regions are shown to be strictly increasing in the window length and the limiting stability region is determined as the window size tends to infinity.

Cross-border FM spectrum coordination involves protecting foreign broadcasting services while preserving domestic coverage, amid increasingly large radio-planning datasets containing thousands of transmitters and millions of transmitter-pixel relationships. In such scenarios, conventional optimisation approaches become computationally demanding due to the high dimensionality of the associated power-control problem. This paper proposes an interference-driven clustered optimisation framework for large-scale FM spectrum coordination. The proposed method exploits the observation that violations of foreign-service protection are typically dominated by a limited subset of transmitters. Protected services are therefore analysed to identify dominant interferers and quantify their impact on interference. These relationships are represented through an interference graph from which optimisation-oriented transmitter clusters are extracted. The clusters decompose the global power-control problem into smaller optimisation tasks solved with clustered simulated annealing, followed by a global refinement that captures residual inter-cluster interactions. Coverage and interference are evaluated using frequency-dependent protection criteria and a dynamic strongest-service assignment model. To enable operational-scale planning, the framework uses sparse matrices and GPU-accelerated computations. Tests on realistic cross-border FM coordination scenarios show that the clustering strategy greatly reduces optimisation complexity and runtime while maintaining foreign-service protection and domestic coverage. The method also yields an interpretable ranking of transmitters that contribute most to harmful interference, supporting optimisation and spectrum planning.

We introduce a new model for a system of interacting bosons placed in an array of sites. At its core is a nonlinear, nonlocal evolution equation, which we have dubbed the Schrödinger-Dirichlet equation. The construction is closely related to the Bose-Hubbard model and to a specific type of generalized bosons. In contrast to conventional mean-field closures, the resulting nonlinear dynamics need not preserve product structure and can generate entanglement from initially separable states. The relevant methods of analysis are based on harmonic analysis for the multiplicative group of positive rationals.

We study the optimization landscape for infinite-horizon discounted Markov decision processes (MDPs) with general state and action spaces under structured stationary policy classes. A general weighted policy-iteration approach to establishing global convergence guarantees for policy gradient methods requires closure under weighted policy improvement at every policy, a property that may fail even when the policy class contains an optimal policy. To address this issue, we propose weaker conditions that guarantee the absence of suboptimal stationary points and establish the Polyak–Lojasiewicz–Kurdyka (PLK) condition for the policy gradient objective with a finite concentrability coefficient. We also establish the PLK condition from a policy-improvement bound that holds at every state, without a concentrability assumption. Our general results encompass settings covered by the earlier framework when the common standing assumptions hold for the same policy class and parameter domain. We further verify our proposed conditions for two operations models: inventory systems with Markov-modulated demand and stochastic cash-balance problems. For both models, the Bellman equation yields approximate convexity of the Q-value functions in the action variable, with deviations controlled by the first-order stationarity measure. These estimates establish exponent-one and, under additional curvature assumptions, exponent-two PLK conditions, which, together with Lipschitz continuity of the policy gradient, imply an $\mathcal{O}(1/\epsilon)$ iteration complexity and linear convergence, respectively, for projected gradient descent using exact policy gradients. To the best of our knowledge, we provide the first non-asymptotic convergence rates for solving infinite-horizon discounted inventory systems with Markov-modulated demand and stochastic cash-balance problems using policy gradient methods.

We establish independence of $q$ as a necessary condition for coefficientwise nonnegativity of normalized principal specialization ratios of monic Macdonald polynomials. For arbitrary pairs of partitions, we characterize independence of $q$. For denominator partitions with largest and smallest parts differing by at least two, every ratio independent of $q$ is a power of $t$. We classify the nonnegative ratios under addition of maximal rows and columns of fixed height. For rows of length at least two, the classification is equivalent to rectangular complementation. For arbitrary initial partitions, the generating polynomials of successive row ratios at the smallest number of variables admissible for all stages have simple negative real zeros with lower bounds for the ratios of consecutive absolute zeros. We derive a reciprocity criterion, bounds for gamma coefficients, and the corresponding Jack limit. For generalized staircases, we determine shortest horizontal and conjugate vertical strip sequences and obtain terminating basic hypergeometric representations.

For a class of growth processes of Laplacian type in the plane, we suggest an interpretation as a “gradient descent” in the space of smooth closed curves. More precisely, we show that boundary of a growing domain moves along a gradient of a certain functional in the space of curves. In the simplest cases this functional is $\log (1/r)$, where $r$ is the external conformal radius of the growing domain.

We introduce and study the family of super adaptable monoids, which includes, among many others, all conical refinement monoids that are either finitely generated, or countable, primely generated, and regular. Our main result shows that any countable super adaptable monoid can be realized by a von Neumann regular ring, thus answering affirmatively the Realization problem for this class. The result is new even for the subclass of countable, primely generated, regular conical refinement monoids, which we show can be realized by both a von Neumann regular ring and a purely infinite, real rank zero C*-algebra. To prove this result, we develop a systematic framework that connects three fundamental structures: abstract monoids, separated graphs (a specific class of colored graphs), and von Neumann regular rings. By establishing functorial connections among these classes, we translate several algebraic properties of monoids into combinatorial and ring-theoretic notions. This allows us not only to prove the main realization result, but also to obtain several characterizations of super adaptable monoids. As a particular application, these can be used to provide a complete characterization of which regular monoids arise as graph monoids. To our knowledge, this work offers the first self-contained treatment in the literature where the complete realization process —from abstract monoids through I-systems and separated graphs to regular rings and C*-algebras— is developed sequentially and in full detail within a single text.

We study real-analytic, planar, maps, at a neighbourhood of an elliptic fixed point. We construct normal forms for them, when they admit a real-analytic first integral. We prove that a second first integral, when it exists, forces their linearisation. We consider both the resonant and the non-resonant case, whether the maps preserve a symplectic form or not. The second first integral is allowed to be smooth, real-meromorphic or real-analytic.

Recent work has constructed Abelian Chern-Simons theories as categorical TQFTs, allowing us to naturally incorporate categorical defects and construct defect extensions of Abelian Chern-Simons TQFTs. We first identify the Turaev-Viro realizations of Abelian Chern-Simons theory in the center and doubled pointed modular cases, clarifying the distinction between single bulk realizations and canonical doubled ones. Then we obtain Turaev-Viro partition functions in terms of Reidemeister-torsion densities. Alternatively, the Alterfold construction supplies the associated topological boundaries, domain walls, and condensation sectors, establishing an explicit Alterfold/Chern-Simons dictionary. We show that the finite quadratic module is the invariant controlling the bulk theory, its topological symmetries, orientation-reversal invariance, and defects. We further show that multicomponent Abelian BF theory arises as the extended TQFT of an off-diagonal Abelian Chern-Simons theory, placing it naturally within the same extended framework. Finally, we demonstrate that recently proposed Abelian Chern-Simons dualities do not define a genuine TQFT duality. These results provide a concrete model for defects in Abelian topological orders and suggest a route toward the non-Abelian case.

Noncommutative resolutions of AS-Gorenstein isolated singularities are investigated by Li–Shen–Wu. However, establishing their existence and constructing such resolutions are generally difficult, even when they exist. In this paper, we study conditions under which a commonly graded AS-regular algebra serves as a noncommutative resolution of an AS-Gorenstein isolated singularity. We investigate projective modules over a noetherian commonly graded AS-regular algebra whose endomorphism rings admit resolutions by the underlying regular algebra. This leads to a more general definition of noncommutative resolutions of balanced Cohen–Macaulay isolated singularities. We show that the existence of such resolutions is equivalent to the existence of cluster tilting modules over balanced CM isolated singularities. The corresponding noncommutative analogue of the Bondal-Orlov conjecture is established in dimensions $2$ and $3$. As an application, we study Hopf actions on commonly graded AS-Gorenstein algebras and investigate noncommutative resolutions of invariant rings. We present three examples of noncommutative resolutions, including one in which the noncommutative isolated singularity is not connected graded.

We construct an explicit involution on Dyck paths satisfying $\defc\le\min(\area,\dinv)$ that interchanges area and dinv, where $\defc=\binom n2-\area-\dinv$ and $n$ is the semilength. In addition, we give an involution on Dyck paths with $\defc \le 2n-8$ that interchanges area and dinv. Finally, we give an explicit partition formula for the portion of the $q,t$-Catalan polynomial of total degree at least $\binom n2-2n+8$, thereby proving a conjecture of Lee and Li~\cite[Conjecture~4]{LeeLi11}. Altogether our results give a combinatorial explanation of $q,t$-Catalan symmetry in the region $\defc\le\max(2n-8,\min(\area,\dinv))$. Our constructions rely on a number of new combinatorial objects developed here and three main external tools: the dual Dyck insertion of \cite{Hawkes26}, the Garsia–Milne involution principle~\cite{GarsiaMilne81},~\cite{Doyle19}, and the partition bijection of ~\cite{LoehrWarrington09}.

We establish stability principles under compact gluing for the reverse square-root inequality and the Riesz transform. Below a common Sobolev dimension, the reverse estimate passes from finitely many model manifolds to their connected sum. A boundary-compatible Poisson parametrix also proves the Neumann reverse inequality for every $1<p<\infty$ for real symmetric uniformly elliptic divergence-form operators on exterior Lipschitz domains. For the forward transform, we glue a bounded-core model and an independent whole-space end model that agree with the coefficients on overlapping regions. A homogeneous Fredholm construction then resolves the form-valued error. The resulting range is controlled by the two model exponents, with the additional restriction $p<n$ for Dirichlet data. More broadly, our method illuminates the interplay between the bounded interior and exterior problems, and provides a flexible framework for studying other boundary value problems.

Let $(a,b,c,d) = \left(2mn, 2mp, m^2 - n^2 - p^2, m^2 + n^2 + p^2\right)$ be a primitive Pythagorean quadruple and let $S=\langle a, b, c, d\rangle$ be the numerical semigroup generated by $a,b,c,$ and $d.$ For convenience, we also let $Q = n^2 + p^2, \delta = \gcd(n,p),$ and $n = \delta n_0$ for some $n_0 \in \mathbb{Z}$. In this paper, we determine the Frobenius number of $S$ and derive an explicit formula in terms of $m$, $n$, and $p$ with the assumption that $m\geq 2Q$ for $\delta = 1$ and $m \geq \frac{2Q}{\delta} - 1$ for the remaining cases. The proof is based on an explicit complete residue system modulo $2mn$, a normalization procedure for arbitrary semigroup elements, and a lift-orbit description of boundary representatives.

We study codes correcting adversarial insertions and deletions with list size $L$ fixed independently of the block length. We derive new achievable-rate bounds for binary codes and upper bounds over every fixed alphabet of size $q\ge2$, retaining explicit dependence on $L$. We establish a combinatorial reduction that trades $L$ units of insertion budget for one unit of deletion budget in the decoding guarantee, without changing the code or increasing the list size. Consequently, asymptotic bounds for mixed errors with insertion fraction $\gamma$ and deletion fraction $\delta$ follow from insertion-only lower bounds at $\gamma+L\delta$ and deletion-only upper bounds at $\delta+\gamma/L$. For binary unique decoding, we strictly improve Levenshtein's classical asymptotic rate lower bound for every deletion fraction $0<\delta<1/2$ for which the classical rate expression is nonnegative. At $\delta=0.1$, the lower bound increases from approximately $0.162009$ to $0.180431$, a relative increase of about $11.37\%$. Our framework also yields insertion and deletion lower bounds for every fixed list size. The existence proofs combine the Lovász local lemma with sampling from words having a specified number of runs, where a run is a maximal block of equal symbols. Generating functions provide refined bounds on the probability that $L+1$ sampled words share an allowed received word. We also derive a Levenshtein-type upper bound by run counting and, separately, a higher-order Elias bound using intersections and unions of the position sets used to embed $L+1$ codewords in a common supersequence. The latter recovers Yasunaga's asymptotic unique-decoding bound at $L=1$ and strictly improves the Haeupler–Shahrasbi–Sudan insertion bound for every fixed $L$ and $0<\gamma<q-1$. Numerical comparisons quantify the gains and remaining gaps.

Families of plateaued Boolean functions with pairwise disjoint Walsh supports are useful in secondary constructions of cryptographic Boolean functions. Of particular interest are maximum-cardinality families whose members admit no nonzero linear structures. To the best of our knowledge, the previously known general construction attaining both properties is spectral (Hodžić et al., IEEE Trans. Inf. Theory 65(9): 5865–5879, 2019). In that work, explicit algebraic normal forms are not generally provided, and no general method is established for prescribing a common algebraic degree for all family members. In this paper, we present two new explicit algebraic constructions within a unified framework, one based on linear functions and the other on partially linear functions with bent components. Let $p\geq 2$ and $q\geq 0$ satisfy $q<2^p-p-1$, and set $m=p+q$. Both constructions yield maximum-cardinality families of $2^{q+1}$ $(q+1)$-plateaued Boolean functions with pairwise disjoint Walsh supports. No member admits a nonzero linear structure, and every member has an explicit generalized Maiorana–McFarland representation. The first construction produces functions in $m+p+1$ variables and realizes any prescribed common algebraic degree $3\leq d\leq p+1$, provided that $q<\sum_{i=2}^{d-1}\binom{p}{i}$; its maximum attainable degree $p+1$ is optimal. The second construction produces functions in $n+p+1$ variables, where $n>m$ and $n-m$ is even, and realizes any prescribed common algebraic degree $3\leq d\leq p+(n-m)/2$, provided that $q<\sum_{i=2}^{\min\{d-1,p\}}\binom{p}{i}$; its maximum attainable degree $p+(n-m)/2$ is next-to-optimal.

A graph $G$ is deletion-induced-saturated for $H$ if $G$ has an edge, contains no induced copy of $H$, and deleting any edge of $G$ creates an induced copy of $H$. We prove, with finite certificate verification, that a finite graph $H$ admits such a finite graph $G$ if and only if $H$ is not complete. This resolves the deletion conjecture of Fan, Hajebi, Hajebi and Spirkl. The main step transfers suitable free amalgamations to finite extensions using a local lifting theorem of Auinger, Bitterlich and Otto. A second criterion treats edge addition by protecting specified nonedges and then taking a maximal induced-$H$-free completion. Structural results of Bonamy, Groenland, Johnston, Morrison and Scott reduce the remaining targets to dense templates and a finite hereditary class. Two uniform constructions in halved cubes handle the dense templates. The finite part is supported by exhaustive coverage certificates, structural certificates and explicit hosts, including a circulant graph on $30$ vertices.

In the first part of this work we showed that, within a Hilbert plane deprived of the Side-Angle-Side axiom, the Side-Angle-Angle criterion, together with a ray correspondence principle [RCT], the existence of angle bisectors [AB], the congruence of supplements of congruent angles [SA], and the Pons Asinorum [PA], suffices to reconstruct SAS. We left open the question of whether [PA] is genuinely required alongside the other three principles, noting only a qualitative asymmetry in the nature of the principles involved. In this second part we answer this question: we show that \begin{equation*} SAA,\ [RCT],\ [AB] \;\vdash\; [PA], \end{equation*} so that [PA] is redundant among the hypotheses of our main theorem, which improves to \begin{equation*} SAA,\ [RCT],\ [AB],\ [SA] \;\vdash\; SAS. \end{equation*} The proof adapts an argument recently given by Donnelly, who reconstructs SAS from SAA together with an angle addition axiom and the existence of angle bisectors.

Mitra, Offen, and Sayag introduced distinguished multisegments and multisegments of Speh type, and proposed that every distinguished multisegment is of Speh type, together with a related duality hypothesis (see A. Mitra, O. Offen, and E. Sayag, Klyachko Models for Ladder Representations, Documenta Math. 22 (2017), 611-657). They proved these statements for sets of segments and when at most two segments share an endpoint. We develop a combinatorial theory of relevant decompositions and prove that relevance is preserved under the involution $\Delta\mapsto\Delta^\vee$ together with reversal of standard order. Hence, $\mathfrak m$ is distinguished if and only if $\mathfrak m^\vee$ is distinguished, so the two hypotheses are equivalent. We define $S_{\mathfrak m}(\Delta)=\sum_{t\ge0}(-1)^t\mathfrak m(\nu^t\Delta)$ and show that $\mathfrak m$ is of Speh type if and only if $S_{\mathfrak m}(\Delta)\ge0$ for every segment $\Delta$. Using rigidity properties, we obtain a peeling theorem and a numerical tameness condition, and prove both hypotheses for every tame multisegment, extending the previously known classes. Finally, we construct a five-segment multisegment that is not distinguished, although every standard order with non-increasing endpoints admits a non-trivial relevant decomposition. Thus, endpoint-ordered witnesses alone cannot prove the hypothesis in general. The remaining case reduces to multisegments for which both $\mathfrak m$ and $\mathfrak m^\vee$ are non-tame.

In this paper, we establish a quantum Cheeger inequality for primitive KMS-symmetric quantum Markov semigroups in terms of projection conductance. We discuss both projection conductance and classical conductance for graph-based KMS-symmetric quantum Markov semigroups. We show that hypercontractivity and the logarithmic Sobolev inequality hold for primitive KMS-symmetric quantum Markov semigroups. We also present applications of the quantum Cheeger inequality to logarithmic Sobolev inequalities, hypercontractivity, and complete modified logarithmic Sobolev inequalities.

An induced matching is a matching that forms an induced subgraph. A graph is $t$-color-critical if removing some induced matching of size $t$ lowers its chromatic number, but removing any $t-1$ vertices does not. Let $F$ be a $t$-color-critical graph with $\chi(F)=r+1$. For sufficiently large $n$, Simonovits determined the unique edge-extremal $F$-free graph on $n$ vertices. Recently, Zheng, Li and Li [Linear Algebra Appl.\ 730 (2026) 546–565] conjectured that, for $t\ge 2$ and $r\ge 3$, the join $K_{t-1}\vee T_{n-t+1,r}$ uniquely maximizes the signless Laplacian spectral radius among all $n$-vertex $F$-free graphs when $n$ is sufficiently large. In this paper, we prove this conjecture. In contrast to the usual spectral arguments, our proof of this conjecture relies on two techniques of a rather different flavour. Our first technique is an analogue of Zykov symmetrization for the signless Laplacian matrix. Our second technique is an induction on $n$, from which we obtain the lower bound on the smallest entry of the Perron vector of a signless Laplacian spectral extremal graph rather than a structural statement.

Recently, Garvan and Sarma studied sums of non-overlined parts in overpartitions and partitions without repeated odd parts. In this paper, we study the corresponding statistic for overlined parts. We define $\mathrm{OSOME}(n)$ as the sum of all odd overlined parts minus the sum of all even overlined parts in the overpartitions of $n$. We obtain a closed form for its generating function and use it to prove congruences by elementary $q$-series methods, the theory of half-integral weight modular forms, and known congruences for the overpartition function.

We study the quantum $K$-theory of the flag variety $G/B$. For each minuscule fundamental weight $\varpi$, we construct an explicit relation in the torus-equivariant quantum $K$-theory $QK_T(G/B)$. The relation can be regarded as a quantum deformation of the character of the irreducible representation with highest weight $\varpi$.

We study two aspects of the abelian sandpile model on comb lattices. First, we prove that the infinite-volume limit of the stationary measures is supported on the saturated configuration. We then investigate the shape of avalanches induced by adding a particle at the origin in finite boxes of size $(2n+1)\times(2n+1)$ around the origin. In the stationary distribution on these finite boxes, we show that avalanches reach the boundary along the vertical teeth with probability tending to $1$, while the horizontal spread is of order $\sqrt{n}$. Finally, we establish that the single-source limit shape for abelian sandpiles on the comb lattice agrees with the corresponding limit shapes for the divisible sandpile, internal diffusion-limited aggregation (IDLA), and rotor-router aggregation models. This establishes limit shape universality on the comb lattice.

We prove that every finite-dimensional Busemann \(G\)-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere \(L=S(c,r)\), we show that every embedded arc contained in an exact distance level is a homotopical \(Z_2\)-set in \(L\). It follows that \(L\) has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for \(L\times\mathbb R\), and a local avoidance argument at the center yields DDP for the ambient \(G\)-space. Since finite-dimensional Busemann \(G\)-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.

We interpret the convolution constructions of [arXiv:2604.26357] in Hodge-theoretic terms, extending the Hodge defect calculation of [arXiv:2609.19341]. For Hodge–Tate modules on $\mathbb C^*$ smooth outside $1$, the defect is the action of an element of the Deligne–Terasoma transport algebra expressed through the Drinfeld associator. For convolutions of iterated Kummer extensions, comparison of mixed Hodge structures produces a beta factor and yields the regularized double shuffle relations.

We establish the Varadhan small-time asymptotics for the quenched and annealed heat kernels of one-dimensional diffusions in a random environment with generator $\mathcal L_W f(x)=e^{-\rho(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $a$ and $\rho$ are continuous in space and satisfy a local exponential moment condition. We assume that the law of the intrinsic coordinate map $\Lambda_W$ has compact support $\mathscr L$ consisting of strictly increasing functions. Let $q^W(t,x,y)$ and $q(t,x,y)=\mathbb E[q^W(t,x,y)]$ denote the quenched and annealed heat kernels with respect to Lebesgue measure, respectively. We prove that, for almost every environment $W$, $\lim_{t\downarrow0}t\log q^W(t,x,y)=-\frac12|\Lambda_W(y)-\Lambda_W(x)|^2$, and that $\lim_{t\downarrow0}t\log q(t,x,y)=-\frac12\min_{\Lambda\in\mathscr L}|\Lambda(y)-\Lambda(x)|^2$. Both limits hold uniformly on compact subsets of $\mathbb R^2$. The framework includes Brox diffusion, formally described by $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion.

Let $A$ and $B$ be $C^*$-algebras, not necessarily unital, and let $T:A_+\to B_+$ be a positively homogeneous order isomorphism. We first prove that $T$ is additive and extends uniquely to a bounded real-linear order isomorphism between the self-adjoint parts of $A$ and $B$. Passing to the biduals, we then obtain a representation theorem for positively homogeneous order isomorphisms. Finally, we study surjective maps between the positive cones of $C^*$-algebras, again not necessarily unital, which preserve the norm of the arithmetic mean. This extends a previous result by removing the assumption that at least one of the algebras is unital.