Papers

Kanade and Russell initiated a family of conjectural Rogers–Ramanujan type identities of moduli $9$ and $12$, which ultimately comprised five modulo $9$ identities and eleven modulo $12$ identities. The eleven modulo $12$ conjectures were subsequently settled through the work of Bringmann, Jennings-Shaffer, and Mahlburg and of Rosengren. In this paper, we prove all five modulo $9$ Kanade–Russell sum-product identities, four individual generalized Nahm-sum dual identities, and a product formula for the natural dual companion of the fifth Kanade–Russell identity, which is expressed as a linear combination of two negative-mixed-term generalized Nahm sums. The first three individual dual identities settle Conjecture~3.6 of Wang and Wang, while the fourth proves the corresponding conjecture of Li and Wang. Our results also connect directly with the recent Dynkin-diagram framework of Sun and Wang for generalized Nahm sums. They identified the rank-two pairs $(T_1,G_2)$ and $(G_2,T_1)$ as unresolved cases whose modularity would follow, respectively, from the first modulo $9$ Kanade–Russell identity and its Wang-Wang dual. The present results prove precisely these two required identities and hence establish the corresponding modularity statements unconditionally.

We study direct products of monoids with zero and give a criterion under which their factors can be recovered from the multiplicative structure alone. While classical decomposition theory encodes direct products through factor congruences, central elements, and refinement properties, we give a concrete multiplicative reconstruction mechanism. If the factors have no nontrivial complemented central idempotents, then the coordinate idempotents are precisely the atoms and coatoms of the complemented-central-idempotent poset, and their multiplicative stabilizers are precisely the coordinate factors and cofactors. It follows that every isomorphism between such products is monomial, yielding the corresponding wreath-product description of automorphism groups. More generally, every product decomposition is obtained by grouping the original factors; in particular, strict refinement follows. We apply these results to multiplicative monoids of directly indecomposable unital rings, including connected commutative rings, and to arithmetic examples arising from residue-class rings.

We study the correspondence proposed in \cite{critical-structures} between inner functions modulo post-compositions with automorphisms of the unit disk and cyclic subspaces of the weighted Bergman space $A^2_1$. The correspondence sends an inner function $I$ to the invariant subspace $[I']$, and in the opposite direction, assigns to a non-zero function $H \in A^2_1$ the Liouville map $I_H$ associated to the canonical solution of the Gauss curvature equation $\Delta u = |H|^2 e^{2u}$. We prove that $I'_H$ generates the same cyclic subspace as $H$. Combined with the results in \cite{critical-structures}, this shows that these two mappings are inverses of one another, and hence the correspondence $I \to [I']$ is a bijection.

Given a simple graph $G = (V, E)$ on $v$ vertices and two distinct vertices $x, y \in V$, the co-degree $c_{x,y}$ associated to the pair $\{x, y\}$ is the number of their common neighbors in the graph $G$. The co-degree sequence of $G$, denoted by $\gamma(G)$, is the list of all the co-degrees associated to all the possible pairs of distinct vertices, arranged in non-increasing order. In this paper we consider the following problem, which can be viewed as a generalization of a result by Erdős and Gallai as well as of the Erdős, Rényi and Sós' friendship theorem: given an integer $v\geqslant 2$ and a sequence $\gamma$ of nonnegative integers arranged in non-increasing order, establish if there exists a simple graph on $v$ vertices having $\gamma$ as its co-degree sequence and, in case of positive answer, provide such a graph. We provide a full answer to this problem for the class of planar $C_4$-free graphs.

While people have studied $m$-ary partitions of an integer $n$ and studied conjugation of partitions of $n$, these topics are rarely mixed because the $m$-ary property is almost always lost after conjugation. In a previous work, Flowers and Lockard investigated $m$-ary partitions of $n$ whose conjugates were also $m$-ary. We generalize that previous work by studying $m$-ary partitions whose conjugates are $q$-ary, where $m$ and $q$ may be distinct. We provide a family of operators on these partitions that can be used to generate all such partitions uniquely and associate a unique polynomial with each partition based on the sequence of operators used to generate it. Using the generating operators and modular arithmetic we explore many examples and families of $m$-ary partitions whose conjugates are $q$-ary.

We study a moving-average process with (not necessarily symmetric) Laplace innovations under the constraint of positivity. In the three nondegenerate parameter regimes $0<\theta<1$, $\theta>1$, and $\theta<0$, we prove convergence of the conditioned finite-dimensional distributions and identify the limit as a Doob $h$-transform. The regimes lead to qualitatively different limiting dynamics: the invariant law is supported on the positive half-line for $0<\theta<1$, the conditioned chain is confined to the negative half-line for $\theta>1$, and the dynamics are genuinely two-sided and governed by $q$-trigonometric functions for $\theta<0$. In each case, the persistence exponent, sharp persistence asymptotics, the defining eigenfunction, and the unique invariant distribution are obtained explicitly.

Symmetric powers of matroids were introduced by Lovász and Mason. The subject has recently attracted renewed interest through its connection with the realizability of tropical linear spaces by tropical ideals. Anderson proved that the tropical linear space associated with a matroid $M$ is the variety of a tropical ideal if and only if $M$ has a $d$th symmetric power for every positive integer $d$. He also asked whether the Vámos matroid $V_8$ has a second symmetric power. We answer this question in the negative by proving that $V_8$ has no second symmetric power. By Anderson's characterization, the associated tropical linear space cannot be realized as the variety of a tropical ideal.

The work is devoted to the study of Dirac operators with MIT bag boundary conditions in Euclidean domains with compact Lipschitz boundaries in arbitrary dimensions. It is shown that such operators are self-adjoint on suitable definition domains and can be recovered as the norm-resolvent limits of Dirac operators in the whole space with a large mass term outside the domain, under the assumption that an associated Robin-Laplacian eigenvalue has a prescribed asymptotic behavior with respect to a parameter in the boundary condition. This assumption is shown to hold for a class of non-smooth domains, which includes convex domains and, more generally, domains that can be "locally convexified" by suitable diffeomorphisms. To the best of our knowledge, this represents the first infinite mass interpretation for the MIT bag model in the sense of resolvent convergence for non-smooth domains. Most results are extended to the generalized MIT bag boundary conditions with the help of the recently established congruence transform.

In this article, we consider the affine Grassmann code $C^{\mathbb A}(3,6)$, obtained from the affine open cell ${\mathbb A}^9$ of the Grassmannian $G_{3,6}$. We exploit the representation of codewords as linear combinations of minors of all sizes of a generic $3\times3$ matrix and classify them according to the largest size of a minor occurring with a nonzero coefficient. Using this classification, we determine all possible Hamming weights of codewords of $C^{\mathbb A}(3,6)$ and, for each weight, compute the number of codewords attaining that weight. Consequently, we obtain the complete weight spectrum of the affine Grassmann code $C^{\mathbb A}(3,6)$.

We study estimation of a constant conditional variance $\sigma^2$ in nonparametric regression with a $d$-dimensional random design. This is an important problem, and similar questions arise in causal inference. The regression function is $\beta_b$-Hölder smooth, the design density is $\beta_g$-Hölder smooth and bounded above and away from zero, and we consider the nonparametric regime $\beta_b>1$ and $d>4\beta_b$. Set $\beta_g^\star=\beta_b(1-4\beta_b/d)/\{1+2\beta_b/d+8(\beta_b/d)^2\}$. We give an estimator whose mean squared error is upper bounded by $Cn^{-4(\beta_b+1)/(d+4)}$ in the low-regularity regime when $0<\beta_g\leq\beta_g^\star$. The low-regularity branch is based on a new two-scale construction: the covariate space is partitioned into cells, the local polynomial trend is projected out within each suitable cell, and the squared normalized contrast from one eligible close pair per cell is averaged across cells. In the high regularity regime when $\beta_g>\beta_g^\star$, a higher-order influence function estimator of Robins, Li, Tchetgen Tchetgen, and van der Vaart (2008) provides the rate $Cn^{-8\beta_b/(d+4\beta_b)}$. We also give an all-pairs ridge extension, which achieves the same two-scale rate, and evaluate the methods alongside a range of existing estimators in simulations.

The $F$-invariant in cluster algebras is a combinatorial invariant that unifies the $E$-invariant from additive categorification and the $\mathfrak{d}$-invariant from monoidal categorification. In this paper, we study its refinement, the partial $F$-invariant, and establish its mutation formula under changes of the initial seed. As an application, we prove a conjecture of Reading, which asserts that the non-compatible cluster variables can be separated by sign-coherence of $g$-vectors upon varying the initial seed. We further show that, for cluster monomials, the partial $F$-invariants coincide with both the partial $E$-invariants for reachable decorated representations of quivers with potentials and the pole orders of normalized $R$-matrices (or partial $\mathfrak{d}$-invariants) for finite-dimensional reachable simple modules over quantum affine algebras. As consequences, we obtain a combinatorial formula for the pole orders for reachable simple modules in terms of $q$-characters; we verify the conjectural explicit formula for the pole orders between Kirillov–Reshetikhin modules.

We consider the lowest eigenvalue $\lambda(b)$ of the magnetic Neumann Laplacian in the unit disk, for a constant magnetic field of strength $b>0$. We prove that $\lambda$ is strictly increasing on $(0,+\infty)$. This means that strong diamagnetism holds at every field strength, and not only at large ones. We also show that the normalized energies at the successive crossings of angular-momentum branches form a strictly increasing sequence; combined with the strong-field asymptotics, this gives the global bound $\lambda(b)<\Theta_0 b$, where $\Theta_0$ is the de Gennes constant. These results settle the three conjectures formulated by Helffer and Léna for the disk. As a consequence, the local, or spectral, critical field $H_{C_3}^{\mathrm{loc}}$ of Ginzburg–Landau theory is, in the disk, uniquely determined for every value of the Ginzburg–Landau parameter, and not only for large ones. We also give a second proof of the bound $\lambda(b)<\Theta_0 b$, independent of the first and of the results of Helffer and Léna, by a direct variational method: trial states built from the de Gennes ground state for large fields, constant trial states for small fields, and, on the remaining bounded field interval, finite-dimensional spaces of polynomial trial states certified by finitely many exact computations in rational arithmetic. That proof uses no asymptotic input. It yields in addition an explicit upper bound for $\lambda(b)$, valid above an explicit field strength, whose two leading terms are those of the strong-field asymptotics.

Toward modeling computational concurrency using the language of central idempotents in monoidal categories, we give an explicit construction of the pushout of strict symmetric monoidal categories with strong monoidal functors between them. We show that any braided strong monoidal functor strongly preserves central idempotents, and as a corollary, that the lattice of central idempotents of a symmetric monoidal category is isomorphic as a meet-semilattice to that of its strictification.

Two combinatorial rules are known for the Grassmannian Grothendieck polynomial $G^{(\beta)}_\lambda$: a sum over set-valued tableaux of shape $\lambda$, due to Buch, and a sum over the efficient cells of a cellular decomposition of the Gelfand-Zetlin polytope $GZ(\lambda)$, due to E. Presnova and the author. All coefficients in both sums equal $1$. We construct an explicit bijection between the two indexing sets which matches the summands term by term, carrying the number of excess entries of a tableau to the dimension of the corresponding cell; in particular the two rules are equivalent, either being deducible from the other. The efficiency condition on cells turns out to be the column-strictness of tableaux. We then transport Yu's square-root crystal operators to the cells and find that they respect dimension, along a double $i$-string the cells alternate between two consecutive dimensions, but not incidence: consecutive cells of such a string need not share a point, already for $\lambda=(2,1,0)$.

Quantization schemes based on randomized rotations have recently received renewed attention, including the roles of MMSE and unbiased reconstruction scalings. In this note, we point out the connection to classical results in statistical signal processing and communication theory. Specifically, the two reconstruction scales used in the EDEN line of work admit a natural interpretation as finite-dimensional, realization-dependent counterparts of the Wiener and unbiased coefficients in the classical CDEF formulation. At finite blocklength, the CDEF +1 relation holds pointwise for each rotation realization as an exact geometric (Pythagorean) identity, but does not hold after averaging the distortions over the rotation. The classical SNR relation $\sf{SNR}_{\rm MMSE}=\sf{SNR}_{\rm MMSE,U}+1$ is recovered as $d\to\infty$: once the overall scale is handled separately, the empirical coordinate statistics of a randomly rotated vector approach their i.i.d. Gaussian counterparts, and the rotation-dependent quantities concentrate. Importantly, EDEN goes beyond this classical correspondence: for every finite $d$, its Haar-rotation formulation guarantees exact conditional unbiasedness, a stronger property than the second-order notion of unbiasedness in CDEF. We further comment on two distinct roles random rotations play in quantization: one is approximate Gaussianization of the coordinates; the other is decorrelation of reconstruction errors across quantization branches.

We construct a numerically flat holomorphic vector bundle of rank two on a compact complex threefold satisfying the ordinary $\partial\bar\partial$-lemma and prove that it admits no holomorphic connection. This gives, in particular, a negative answer to a question posed by Cao–Deng–Matsumura. In contrast, for compact simply connected complex manifolds $X$, we prove that the answer is affirmative if $H^1(X,\mathcal O_X)=0$, and hence if the Frölicher spectral sequence of $X$ degenerates at $E_1$.

We study which compact flat manifolds of dimensions $3 \leq n \leq 6$ occur as cusp cross-sections in commensurability classes of non-compact arithmetic hyperbolic $(n+1)$-manifolds. Using the classification of low-dimensional flat manifolds together with the relationship between their holonomy representations and rational quadratic forms, we determine the possible arithmetic commensurability classes for every flat manifold in these dimensions. We show that every flat $n$-manifold for $n=3,4,5$ occurs as a cusp cross-section in infinitely many distinct arithmetic commensurability classes. In dimension six, the same holds with exactly eight exceptions: eight non-orientable flat $6$-manifolds occur in a unique arithmetic commensurability class. We also show that, for every $3 \leq n \leq 6$, there is a single commensurability class of arithmetic hyperbolic $(n+1)$-manifolds containing every flat $n$-manifold as a cusp cross-section.

We determine the optimal order of the scaling parameter in the metric cotype $2$ inequality for complete Alexandrov spaces of nonnegative curvature. It grows linearly with the dimension $n$ of the discrete torus. This answers Question 17 of Eskenazis, Mendel, and Naor. In their sign-vector normalization on $\bigl(\mathbb{Z}/(2m\mathbb{Z})\bigr)^n$, the square of the optimal metric cotype $2$ constant for this class lies between $\max\{1,n/m\}$ and $e\lceil n/m\rceil$ for every dyadic $m\geq 2$. The upper bound follows from a dyadic Bernoulli-thinning argument using only the Lang–Schroeder–Sturm inequality. Snowflake universality of the fixed Wasserstein space $\mathcal{P}_2(\mathbb{R}^3)$ gives the $n/m$ lower bound.

We study optimal input design over a finite horizon for linear dynamical systems. The goal is to minimize a weighted inverse-covariance (information) criterion subject to an energy budget. The set of covariances achievable by causal policies is convex but lacks a tractable explicit description, ruling out projection-based methods. We show that Frank–Wolfe applies naturally: each linear minimization subproblem is a budget-constrained finite-horizon linear quadratic (LQ) problem, solvable by a Riccati recursion and one-dimensional bisection over a Lagrange multiplier. Using smoothness of the objective over the feasible set, we establish an $\mathcal{O}(1/M)$ convergence rate for the objective value, while strong convexity yields an $\mathcal{O}(1/\sqrt{M})$ rate for the iterates. We further extend the framework to input design for system identification with unknown dynamics and adaptive online LQR, and illustrate the approach numerically.

For every $1\leq p<\infty$ and even integer $m\geq4$, we determine the optimal order of the $p$-moment torus inequality for $L_1$: it is $m^p n^{(1-p/2)_+}+n$, with comparison constants independent of $p,m,n$ after taking $p$-th roots. Consequently, for every $2\leq q<\infty$ and $1\leq p\leq q$, the corresponding metric cotype inequality holds at the sharp scale $m=O(n^{1/q})$. In particular, the quadratic inequality holds with $m=O(\sqrt{n})$, answering the $L_1$ sharp metric cotype question of Mendel and Naor. The proof combines finite cut representations with a nonlinear smoothing estimate and an exact ternary rounding identity.

We construct wavelet orthonormal bases for nonexpansive dilations and integer translations that do not admit a wavelet set. In addition, these orthonormal bases do not satisfy the Calderón sum formula. In particular, we show the existence of a wavelet basis for a dilation matrix with determinant one.

We study unbiased estimation of scalar-valued polynomial functionals of quantum states from independent copies. We establish an equivalence between the first-order marginal of a permutation-invariant finite-copy observable and the functional gradient. We then prove that, among unbiased permutation-invariant estimators, the quantum U-statistic is the unique extension to an arbitrary number of copies. We further derive a universal variance expansion in which the leading $1/n$ term is determined by the variance of the functional gradient, while higher-order contributions are of order $O(1/n^2)$. This leading variance coincides with the multiparameter quantum Cramér–Rao limit, establishing asymptotic efficiency of quantum U-statistics. We also characterize the higher-order scaling at points where the variance of the first-order gradient vanishes. As an application, we analyze the Bures $\chi^2$-divergence and show that a spectral lower bound on the reference state is sufficient but not necessary for bounded-variance estimation. Beyond asymptotic variance, we establish variance-sensitive exponential concentration bounds, deriving a closed-form Bernstein-type inequality to capture finite-sample tail behaviour, and establish the Moderate Deviation Principle to characterize the intermediate asymptotic regime.

In this paper, we continue the development of the foundations of a quaternionic function theory that is associated to a fractional proportional Cauchy-Riemann type operator acting on quaternionic functions. We introduce a new class of fractional hyperholomorphic functions adapted to affine transformations of the domain, and study the corresponding quaternionic right module structure. In particular, we establish versions of the Stokes, Borel-Pompeiu and Cauchy formulas for this function theory. A covariance property of the fractional operator under affine transformations is proved, showing that the composition with an affine map, weighted by a suitable quaternionic factor, preserves the class of fractional hyperholomorphic functions. The result is shown to include the case considered previously by the authors, and extends their covariance property to the setting of affine transformations.

We consider the space $\mathcal{D}'^E_L(U)$ of all distributions on the open set $U$ whose wave front set measured with respect to a Banach space $E$ lies in a closed conic subset $L$ of $U\times(\mathbb{R}^n\backslash\{0\})$. The Banach space $E$ only satisfies mild technical assumptions. We show that the pullback by a diffeomorphism $f:O\rightarrow U$ is a topological isomorphism $f^*:\mathcal{D}'^E_L(U)\rightarrow\mathcal{D}'^E_{f^*L}(O)$. In the case when $E$ is the $L^p$-Sobolev space $W^{r,p}(\mathbb{R}^n)$ of order $r\in\mathbb{R}$, we study the pullback by a smooth map $f:O\rightarrow U$ of constant rank.

In this paper we derive a Probably Approximately Correct (PAC)-Bayesian error bound for partially observed linear time-invariant (LTI) stochastic dynamical systems in state-space form with inputs and sub-Gaussian noise. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. The bound derived in this paper relates the expectation of prediction errors with the prediction error generated by the model on the data used for learning. In addition, we show that it can also be used to derive bounds for the parameter estimation error. In turn, this allows us to provide finite-sample error bounds for the prediction error and parameter estimation error for a wide class of system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

We establish new well-posedness results for transport and flow equations driven by contact vector fields on Carnot groups. The velocity fields are assumed to have horizontal Sobolev regularity, namely Sobolev regularity only along the horizontal directions determined by the stratified geometry of the group. In the broader sub-Riemannian setting, results of this type were previously known only for Heisenberg groups. Our proof relies on the theory of renormalized solutions as originally introduced by DiPerna and Lions in the Euclidean setting.

An orthogonal signed graph is a connected signed graph whose signed adjacency matrix has pairwise orthogonal rows. They are closely related to Hadamard matrices, maximal arrangements of equiangular lines, bipartite Ramanujan graphs, and the remarkable resolution of the Sensitivity Conjecture. Orthogonal signed graphs of degree at most 4 have been completely classified, and partial results were known for degree 5. We complete the classification of orthogonal signed graphs of maximum degree 5. We also provide several new infinite families of 6- and 8-regular orthogonal signed graphs.

For the minimum-compliance shape optimization problem it was established recently [Duke Math.\ J. 172 (2023), 43–103] that a sequence of approximately optimal shapes of exact volume \(\e\) converges to a limiting generalized shape, which is represented by a possibly diffuse probability measure, as \(\e \todown 0\). This generalized shape minimizes the so-called light-structures compliance functional, which — as conjectured by Bouchitté — is expressed in terms of a relaxed integrand, reflecting that optimal shapes must only involve “lower-dimensional” structures, like the Michell-type trusses that are well known in the engineering domain. However, the aforementioned result only applies for the simplest isotropic quadratic energy density, that is, the squared Frobenius norm, in dimensions two and three. The present work extends the relaxation result to the entire class of (isotropic or anisotropic) elasticity tensors and to any dimension, thus solving Bouchitté's conjecture in full generality. Since the previous work relied on explicit formulas for the relaxed integrands (due to Allaire, Kohn, and Strang), which are not available in the general case, several new arguments and substantial modifications of the original approach are required. We also significantly simplify and streamline other parts of the original argument, in addition to clarifying some related statements in the literature.

For coprime integers $a$ and $b$, it is known that exactly one of the two Diophantine equations $$ ax+by\ =\ \frac{(a-1)(b-1)}{2} \qquad\text{and}\qquad 1+ax+by\ =\ \frac{(a-1)(b-1)}{2} $$ admits a nonnegative integer solution, and that this solution is unique. We first generalize this result by replacing the right-hand side with an arbitrary integer $m$ and its complement $ab-a-b-m$. This framework enables us to study the existence and uniqueness of nonnegative integer solutions to $$ ax+by\ =\ \frac{(a-1)(b-1)}{k} \qquad\text{and}\qquad 1+ax+by\ =\ \frac{(a-1)(b-1)}{k}, $$ where $k$ is a fixed positive integer. We then obtain explicit results when $a$ and $b$ are consecutive Fibonacci numbers. Finally, we examine the original pair of equations in several particular settings, including when $b\equiv \pm1\mod a$, when $b$ is replaced by a higher power, and when the parameters are squared.

In this work we study the maximal ratio $I$ between the white noise integrated over a set and the perimeter of the set, which can be seen as a random isoperimetric problem, and appears in the random-field Ising model (Ding-Wirth) and min-max optimal matching (Leighton-Shor). In the planar case considered here, such a ratio is scale-invariant and therefore the problem is critical. As such it requires an ultraviolet cutoff, which we impose by restricting to polygonal sets with side-length at least 1 contained in the ball $B_L$. Our main result establishes the leading-order asymptotics $\mathbb{E}I\approx i\ln^{3/4}L$ for some $i\in(0,\infty)$ and superconcentration at scale $O(\ln^{-1/4}L)$. This is done by relating to a simpler $(1+1)$-dimensional action, studied by the last two authors and C. Wagner. Such a connection is found by a classical geometric linearization of the perimeter to the Dirichlet energy, justified by Ried-Wagner large-scale regularity theory, and allows a coarse-graining and scale-by-scale iteration argument.