Papers

Diffusion and Gaussian-interpolant flow-matching samplers approach data through a terminal noise floor $\varepsilon$, a singular limit for manifold-supported or rank-deficient data. We study two properties of a complete sampler specification, comprising its update rule, time grid, and terminal rule. Asymptotic preservation (AP) means a stable and consistent zero-noise discretization with a step count bounded independently of $\varepsilon$. Uniform accuracy (UA) of order $p$ means that, at numerical resolution $h$, the endpoint $W_2$ error is $O(h^p)$ with a floor-independent constant. Bounded log-noise stepping fails AP because its step count diverges. Stopping a stable base solver at a positive switching scale $a$ and appending one map fitted to the analytic normal mode restores AP. On smooth compact boundaryless manifolds, the standard map has exact-input error $O(a^2-\varepsilon^2)$ and sharp zero-floor error $\Theta(a^2)$. A base solver with a floor-uniform order-$p$ estimate on the resolved interval retains that order when $a=O(h^{p/2})$, provided the terminal transfer factor remains bounded. Along exact trajectories, the posterior-mean identity $D(x(\sigma),\sigma)=x(\sigma)-\sigma x'(\sigma)$ cancels the linear terminal defect and enables higher-order fitted maps. A three-evaluation Hermite construction is uniformly third order for exact switching-scale input over $0\le\varepsilon\le a$, and a seven-evaluation construction is fourth order at zero. We classify representative diffusion and flow-matching specifications by AP and UA. On EDM and Rectified Flow checkpoints, a paired decomposition separates base-integration from terminal-completion error and predicts held-out same-seed endpoint errors.

We prove the remaining conjectural distribution-equivalence of mesh patterns of length 2 from 2019. This completes the classification of distribution-equivalence and Wilf-equivalence classes for mesh patterns of length 2. Consequently the numbers of such classes are 105 and 46, respectively. We approach this by constructing a marked occurrence bijection, in the spirit of the cluster method for consecutive patterns.

Future networks must serve massive populations of devices that sense and communicate simultaneously under short-packet constraints, yet the fundamental limits of integrated sensing and communication (ISAC) in the finite-blocklength multiple-access regime remain largely undiscovered. This paper closes this gap from a source-channel coding perspective. We prove that satisfying a sensing-distortion constraint is information-theoretically equivalent to a source-coding requirement, which collapses sensing and communication into the joint recovery of a single effective payload within a coded multiple-access framework. Building on this equivalence, we derive a finite-blocklength achievability bound together with a Fano-sum many-user converse and a genie-aided single-user converse, yielding a tight characterization of the minimum energy per bit and the rate-sensing tradeoff. Numerical results reveal that the energy price of sensing fidelity grows almost linearly in dB per decade of distortion tightening and is significantly amplified by the multiple-access load, and that joint encoding of the effective payload strictly outperforms an optimized orthogonal two-phase scheme, demonstrating a genuine integration gain of ISAC at finite blocklength.

The curve shortening–DeTurck flow introduced by Elliott and Fritz (IMA J. Numer. Anal. 37(2): 543–603, 2017) is a reparameterization of the curve shortening flow via the DeTurck trick. For corresponding discrete schemes, although the $H^1$-error estimate has been established, the optimal $L^2$-error estimate remains open due to technical difficulties. In this paper, we prove optimal $L^2$-error estimates for linearized Euler and Crank–Nicolson discretizations combined with finite elements of degree $k\ge 1$ in space. Moreover, we provide an extrinsic approach to derive the curve shortening–DeTurck flow. Numerical experiments are presented to illustrate the mesh distribution properties and to verify the convergence rates in both space and time.

We give a Fourier criterion for the Toeplitzness of bounded operators on Fock spaces, where Toeplitzness means representability as a Toeplitz operator with a bounded measurable symbol. For a Toeplitz operator, the anti-diagonal restriction of its canonical kernel is the Fourier transform of the Gaussian-weighted symbol. Consequently, Fourier inversion of this anti-diagonal restriction recovers the unique bounded symbol whenever such a representation exists. As applications, we characterize the Toeplitzness of weighted composition operators and generalized Volterra-type operators.

We compute the Fefferman boundary measure and the associated Fefferman–Szegő kernel for the egg domains $$ E_{n,m}=\{(z,w)\in\mathbb{C}^{n-1}\times\mathbb{C}:\ |z|^2+|w|^{2m}<1\}. $$ The kernel is given both by an orthogonal monomial expansion and by a closed form in a natural auxiliary finite-type variable; its diagonal weak-boundary exponent recovers the integer $m$. For $n\ge2$, the associated Fefferman–Szegő metric has constant scalar curvature only in the ball case $m=1$, and the Kähler–Einstein, constant Ricci-spectrum, and Bergman-proportionality statements follow as corollaries of the same calculation.

This paper introduces a constructive definition of generalized Fourier transforms based entirely on ordinary truncated Fourier integrals and ordered dual-domain limits, within the framework of improper Riemann integration and classical analysis. The proposed truncate-and-generalized-limit (t.g.l.) formulation does not require test-function spaces, Lebesgue measure theory, or duality pairings in its proofs: the forward and inverse transforms are defined directly through finite-domain truncation of the target function, followed by successive ordered limits in the time and frequency domains. As consequences of this constructive definition, the formulation provides a unified treatment of non-decaying, oscillatory, and locally singular functions beyond the classical L1(R) setting; reveals an inherent asymmetry between the forward transform, interpreted as a first-order generalized-limit family, and the inverse transform, which requires frequency-domain truncation to generate pointwise reconstruction through Dirichlet-type oscillatory localisation; and clarifies the distinction between the t.g.l. approach and distribution theory, where generalized Fourier transforms are introduced through duality pairings rather than constructed from ordinary integrals. The inversion formula is established rigorously for two concrete admissible classes using only the classical Dirichlet convergence theorem. Several examples confirm that the framework covers constants, polynomials, periodic functions, singular kernels, and chirp signals within a single constructive scheme.

Seeking to bridge the structural divide between a permutation's cycle notation and its one-line notation, Berman and Tenner introduced a novel notion of permutation pattern known as the arrow pattern. Recently, Archer and Laudone initiated a systematic study of arrow pattern avoidance, leaving behind three intriguing conjectures. In this paper, we resolve all three conjectures. First, we enumerate all six subclasses of permutations that simultaneously avoid a classical pattern of length 3 and a fixed arrow pattern of length 3, thereby confirming the first two conjectures. Second, we settle the third conjecture (which involves a different arrow pattern) by providing two independent proofs. These proofs rely on a restriction of Biane's bijection to non-nesting involutions and Krattenthaler's bijection from 321-avoiding permutations to Dyck paths, respectively.

We discuss lower-bound constructions for several no-dimensional theorems of combinatorial geometry in Banach spaces. The common mechanism is the Maurey–Pisier theorem: the supremal Rademacher type of a Banach space forces finite-dimensional \(\ell_p\)-structures, and standard-coordinate configurations in these model spaces give lower bounds for the error terms. For the Helly approximation property the relevant type is the type of the dual space. For colorful Radon, colorful Tverberg, selection, and weak \(\varepsilon\)-net statements the relevant type is the type of the original space. We show that the powers appearing in the no-dimensional Helly, Radon, Tverberg, and selection estimates are optimal at the supremal-type exponent. If the supremal type is attained, the known upper estimates coming from the corresponding type inequalities have the best possible order. We also include endpoint statements for spaces of trivial type. In this case the error terms in the Helly, Radon, Tverberg, and selection statements cannot tend to zero. Finally, we prove an endpoint obstruction for no-dimensional weak \(\eps\)-nets in spaces of trivial type. For every fixed cardinality bound, one can find a finite set in the unit ball for which no approximate weak \(\eps\)-net of that size exists below a fixed positive radius. The proof combines the simplex example in \(\ell_1^N\), the Lovász theorem on the chromatic number of Kneser's graph, and finite representability of \(\ell_1^N\) in spaces of trivial type.

We compute the special values ($p$MLFVs) of the $p$-adic multiple $L$-functions introduced by Furusho, Komori, Matsumoto, and Tsumura at tuples of positive integers. Furusho and Jarossay show that the special values can be expressed as an infinite sum of cyclotomic multiple harmonic values (CMHVs) with coefficients given by cyclotomic multiple Bernoulli numbers (CMBNs). We provide an explicit formula for CMBNs in terms of twisted multiple Bernoulli numbers (TMBNs), which are special values of generalized Euler-Zagier-Lerch type complex multiple zeta functions at tuples of non-positive integers. As a result, we obtain that these $p$MLFVs can be expressed as infinite sums of CMHVs, with coefficients given by the special values of the complex functions at tuples of non-positive integers.

In this paper, we establish lower bounds for extreme values of derivatives of Dirichlet \(L\)-functions in the range \(1/2<\sigma<1\). Compared with the work of Aistleitner, Mahatab, Munsch, and Peyrot in 2019, our result shows that derivatives of Dirichlet \(L\)-functions can attain extreme values of the same order of magnitude as the original \(L\)-functions when \(1/2<\sigma<1\) holds.

In linear regression, the signs of coefficients convey the direction of covariate effects and are central to empirical interpretation. In high-dimensional settings, however, the abundance of candidate covariates introduces substantial model selection uncertainty. We study the deliberate manipulation of coefficient signs through the inclusion of a carefully chosen auxiliary variable, a practice we term SHAVE (Sign Hacking with Auxiliary Variable Exploration). We show that, conditional on the outcome and variables of interest, there exists a set of auxiliary-variable realizations with positive Lebesgue measure that lead to sign reversals upon inclusion. Moreover, with high probability, such variables can be found when many auxiliary candidates are available, leading simultaneously to reversed signs, inflated $t$- and $F$-statistics. Simulation studies and an empirical application corroborate these theoretical findings. We further propose detection strategies for SHAVE when augmented or independent datasets are available, as SHAVE has important implications for reproducibility, $p$-hacking, and research integrity.

Let $S_N = \sum_{j=1}^N j^{-3}$ and $R_N = \zeta(3) - S_N$. The simple continued fraction of $\sqrt{R_N}$ has partial quotients of generic size $O(N)$. We prove that at the sequence of indices $N_k = (Q_{2k+1}-1)/2$, where $Q_{2k+1}$ are companion Pell numbers, the continued fraction begins \[ \sqrt{R_{N_k}} = \bigl[0;\; M_k-1,\; 1,\; 6M_k^3+12M_k-2,\; 1,\; \ldots\,\bigr], \] with $M_k = P_{2k+1}$ (Pell numbers), and the third partial quotient grows cubically while generic ones are linear. We determine all partial quotients through the fifth: \begin{align*} \PQ_0 &= M_k - 1, & \PQ_2 &= 6M_k^3 + 12M_k - 2, & \PQ_4 &= \Bigl\lfloor\frac{10M_k - 261}{261}\Bigr\rfloor, \PQ_1 &= 1, & \PQ_3 &= 1, & \PQ_5 &= \Bigl\lfloor\frac{261}{r_k}\Bigr\rfloor + \epsilon_k, \end{align*} where $r_k = (10M_k) \bmod 261$ satisfies the recurrence $r_{k+1} \equiv 6r_k - r_{k-1} \pmod{261}$, and $\epsilon_k = -1$ at the $k$ with $r_k \mid 261$ (the two residue classes $k \equiv 57, 62 \pmod{60}$), and $\epsilon_k = 0$ otherwise. All six formulas follow from the Euler–Maclaurin expansion of $1/\sqrt{R_{N_k}}$, carried to sufficient precision, combined with the Pell identity $Q_{2k+1}^2 - 2M_k^2 = -1$. The delicate first step, $\PQ_0 = M_k - 1$, is proved by rationalizing the irrational factor $\sqrt{2}$ in the Euler–Maclaurin expansion; we complement this proof with a heuristic derivation via Gosper's bihomographic continued-fraction algorithm that exposes the underlying mechanism. All claimed results have been formalized in LEAN with the aid of Aristotle.

For a finite nonempty poset \(F\), the normalized probabilistic powerdomain \(\Vone(F)\) is an RB-domain exactly when \(F\) is a finite rooted tree. We extend this classification to arbitrary nonempty dcpos from the viewpoint of forbidden structure. The principal-ideal chain condition is expressed by the absence of a lower fork, i.e. a triple \((x,y,t)\) with \(x\leq t\), \(y\leq t\), and \(x\parallel y\). A useful point is that any dcpo $P$ without lower forks is continuous. For normalized valuations the least element remains necessary, and we prove \[ \begin{aligned} \Vone(P)\text{ is RB} \Longleftrightarrow \Vone(P)\text{ is a pointed bc-domain} \Longleftrightarrow P\text{ has a least element and contains no lower fork}. \end{aligned} \] For subprobability and extended valuations, the analogous classifications hold without the pointedness assumption on \(P\).

In the theoretical study of nonequilibrium non-Hermitian systems, obtaining exact analytical solutions for their nonadiabatic dynamics is highly desirable yet often challenging. In this work, we identify a class of non-Hermitian quantum systems where this difficulty can be substantially reduced. Employing the Wei-Norman approach, we show that for a spin-$S$ subject to a general non-Hermitian time-dependent drive, the matrix elements of the evolution operator can be expressed in closed analytical forms (via Jacobi polynomials) in terms of the corresponding spin-$1/2$ model. This approach is straightforward and accessible to nonspecialists in Lie algebra. As an application, we investigate a specific nonequilibrium non-Hermitian phenomenon known as defect freezing, i.e., the existence of excitations in the adiabatic limit, in spin-$S$ extensions of the $\mathcal{PT}$-symmetric Su-Schrieffer-Heeger model under linear quenches. We derive exact analytical expressions for the momentum-resolved excitation probabilities and the total excitation densities. Our results reveal that defect freezing occurs exclusively in momentum sectors that traverse the $\mathcal{PT}$-symmetry-broken region – and thus pass through a pair of higher-order exceptional points (EPs) – during the quench; notably, the excitation density exhibits a singularity at a critical value of the non-Hermiticity parameter. This work enriches the analytical toolkit for nonadiabatic dynamics in multi-level non-Hermitian systems and provides quantitative, testable predictions for defect freezing across higher-order EPs, possibly accessible on platforms such as electric circuit networks and photonic lattices.

This paper investigates energy-efficient clustering in user-centric cell-free massive MIMO networks, addressing the access point clustering and power allocation problems via a mixed-integer fractional program. We propose a framework for energy-efficient clustering and power allocation with a graph-based structured search and describe its optimum solution via an exhaustive search. We also develop the Graph-Based Steepest Ascent (GBSA) algorithm, which combines a graph-based structured search along with continuous power allocation via fractional programming. The proposed GBSA algorithm achieves linear per-iteration complexity while reaching energy efficiency close to the global optimum, outperforming competing techniques and offering a scalable solution for future networks.

We show that trace reconstruction on n-bit strings is possible using a quasipolynomial number of traces, for any retention probability p that is at least inverse polylogarithmic in n.

We study the fundamental classification problem of computing a separating hyperplane for a binary-labeled dataset of size $n$ with normalized $d$-dimensional features. Letting $\Phi \in \mathbb{R}^{n \times d}$ denote the feature matrix and $\gamma$ the margin of the maximum-margin separating hyperplane, we present a randomized algorithm that solves this problem in $\tilde{O}(\gamma^{-2/3}\, \operatorname{nnz}(\Phi) + \gamma^{-2(\omega+1)/3})$-sequential running time (work), $\tilde{O}(\gamma^{-2/3})$-parallel (computational) depth, and accesses $\Phi$ only through $\tilde{O}(\gamma^{-2/3})$-matrix-vector queries (matvecs). We also present a second, faster randomized algorithm with a $\tilde{O}(\gamma^{-2/3}\, \operatorname{nnz}(\Phi) + \gamma^{-2})$-sequential running time that uses $\tilde{O}(\gamma^{-2/3})$-matvecs to $\Phi$, but achieves only $\tilde{O}(\gamma^{-4/3})$-parallel depth. Both algorithms match the near-optimal deterministic matvec complexity recently established by Kornowski and Shamir [2025], Karmarkar et al. [2026] and achieve improved sequential runtime and parallel depth, albeit at the expense of using randomness.

We derive a quasi-incompressible Cahn-Hilliard-Darcy phase-field model (qCHD) with the logarithmic Flory-Huggins free energy density function for two-phase flows in porous media. The model satisfies an energy-dissipation law. In the formal sharp interface limit, the qCHD model gives rise to the classical Muskat's problem. By exploiting estimates of the pressure from Darcy's equations, we establish global existence of weak solutions in both 2D and 3D to the qCHD model.

In this paper, we aim to present the notion phi-(k,n)-absorbing delta-primary hyperideals in a Krasner (m,n)-hyperring G.

Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations. Transformer-based neural operators are of particular interest, since attention can learn long-range dependencies in the computational domain. However, standard attention has two major limitations when applied to PDEs: it scales quadratically with the number of computational nodes, and it lacks an explicit bias toward local interactions. To address these issues, we introduce Local Linear Transformer (LLT) for PDE operator learning. The architecture combines linear global attention with local spatial mixing, and incorporates coordinate and geometry information. We evaluate LLT on several PDE problems, including elasticity, plasticity, airfoil flow, pipe flow, and Darcy flow. The reference data for these problems span finite-element, finite-volume, and finite-difference discretizations on structured and unstructured meshes. Compared with other neural-operator and transformer baselines from prior studies, LLT achieves competitive or lower relative $L_2$ error across these problems. On matched structured discretizations, wall-clock time per training iteration is reduced by factors of 1.8 to 2.5 relative to Transolver. We also scale the approach and apply it to a three-dimensional car aerodynamics dataset with 32,186 unstructured mesh points per sample. Together, these results indicate that LLT provides an accurate and computationally efficient operator for PDE problems across discretizations, mesh types, and problem settings.

We introduce an interpolatory subdivision scheme for control polygons that take values in the three-dimensional Heisenberg group, the simplest noncommutative model geometry. The scheme keeps existing points at every refinement step and inserts new ones by a coordinate rule whose central correction comes from the group law. The two horizontal coordinates are refined by the classical four-point scheme of Dyn, Gregory and Levin, while the central coordinate acquires a closed-form correction built from a signed area of neighbouring horizontal data. Our main finding concerns the regularity of the limit curve. The horizontal part is exactly the classical four-point limit and inherits its smoothness. The central part behaves very differently. We prove that it converges to a continuous limit that belongs to the Zygmund class, with a logarithmic modulus of continuity. Under an explicit and verifiable condition on the central forcing, this logarithmic bound is sharp, because the scaled first differences then grow linearly with the refinement level, and the limit fails to be continuously differentiable. The effect is confirmed numerically. The correction is harmless at any single refinement step, but its repeated injection at every scale is what impacts smoothness. The example serves as a caution for nonlinear and group-valued subdivision, where a geometrically natural correction can impact regularity.

The generalized Serre question asks whether every algebraic vector bundle on a topologically contractible smooth affine complex variety is trivial. We give an affirmative answer for rational threefolds. More generally, for a topologically contractible smooth affine complex threefold $X$, we prove that $\text{CH}^2(X)=0$ whenever $X$ admits a smooth projective compactification whose Chow group of $0$-cycles is supported on a curve. This uncovers the link between the generalized van de Ven question, Bloch's conjecture and the generalized Serre question for threefolds. We also prove that every Koras-Russell threefold is rational and therefore has only trivial algebraic vector bundles, hence answer a question of Koras and Russell.

Modern supply chains span diverse operational environments, ranging from e-commerce distribution networks to customized production-to-order manufacturing lines. Across these settings, operational efficiency depends on coordinating two highly interdependent stages: order preparation and downstream delivery. Although these stages are traditionally managed in isolation, real-world fulfillment systems must satisfy stringent delivery expectations under dynamic stochastic order arrivals. To bridge this gap, we introduce the Dynamic Order Fulfillment Problem (DOFP), a new problem class unifying logistical challenges previously studied separately. We model DOFP as a Markov decision process whose state and decision spaces are partitioned into preparation and delivery sub-spaces, linked by synchronization constraints. While recent approaches attempt to optimize both fulfillment stages simultaneously over myopic rolling horizons, our framework isolates and optimizes the downstream delivery policy, treating preparation strictly as a state-level constraint filter. To solve this, we develop the Decomposition-Driven Framework with Value Function Approximation (DDF-VFA), which utilizes a novel policy-level decomposition. This design partitions the search into a delivery-stage master problem and a preparation-stage compatibility subproblem, iteratively refined via feedback loops. DDF-VFA executes this strategy by combining a large-neighborhood search over partial delivery decisions with a neural-network value function approximation for the cost-to-go. Numerical illustrations on two example variants using real-world datasets show that DDF-VFA consistently outperforms benchmarks that optimize the two stages independently or jointly without decomposition. Finally, the framework naturally scales to accommodate additional real-world complexities such as batched or multi-stage preparation.

Random circuit sampling of bitstrings from a Haar-random quantum state is widely believed to be classically intractable, and has therefore been implemented as a primary benchmark for demonstrating quantum advantage. Here, we challenge this premise by proposing an efficient classical frozen-tree sampling algorithm that exploits the conditional scale invariance of Haar-random quantum states [Oh, arXiv:2602.19448]. The frozen-tree sampler draws bitstrings of $n$ qubits in $O(n)$ time per sample. Moreover, its output probability $p_F(x)$ is statistically identical to the probability $p_C(x)$ of a random quantum circuit, since both are independent instances of the same Dirichlet distribution. Consequently, no statistical test acting on samples alone can distinguish the classical frozen-tree sampler from a quantum random circuit. The claimed quantum advantage of random circuit sampling therefore does not withstand scrutiny: its hardness lies not in sampling from the Dirichlet distribution, which is classically efficient, but in identifying a specific circuit realization.

Given the action of a group $G$ on a set $X$, the set of all $G$-equivariant functions, i.e., those satisfying $f(g\cdot x)=g\cdot f(x)$ for all $g\in G$ and $x\in X$, forms a monoid under composition. In this work we study their generating sets. First, we propose bounds for the cardinalities of the generating sets of their group of units, denoted by $\operatorname{Aut}_{G}(X)$. Subsequently, using so-called orbital infiltrations, certain transformations that turn out to be indispensable and provide relevant structural information about the monoid, we determine conditions on the group $G$, the set $X$, and the action that prevent the whole monoid $\operatorname{End}_{G}(X)$ from admitting a finite generating set.

We develop a Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors \(C_{\mu\nu}{}^\lambda\) and \(B_{\mu\nu}{}^\lambda\). Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map, \( \mathcal D(X,Y)Z=XZY-\frac12\{YX,Z\}, \) whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations \((B+C)\) and \((B-C)\) generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.

We consider minimisers in the homotopy class of a homeomorphism $f_0:R\to S$ between analytically finite Riemann surfaces with minimal $L^p$- conformal energy \[ \mathsf{E}_p(f:R,S)=\int_R \IK^p(z,f)\; d\sigma_R(z). \] The problem was first raised by Ahlfors in his celebrated proof of Teichmüller's theorem-the case $p=\infty$, but the existence, topological regularity and analytic regularity of these $L^p$ minimisers remained unknown for all $1<p<\infty$. Ahlfors established weak existence for $p\geq 2$. Here we prove that for all p, $1\leq p<\infty$, such minimisers exist, are unique and are diffeomorphisms. They are quasiconformal but not diffeomorphic at $p=\infty$.

Bergeron's conjecture states that, if $1\leq a<b<c<d$ are integers with $ad=bc$, then one has the coefficient-wise inequality ${\binom{b+c}b}_q \ge {\binom{a+d}a}_q$ among two Gaussian polynomials. It originated in algebraic combinatorics and is wide open. The corresponding inequality for binomial coefficients (i.e., the case $q=1$) must be known to experts, but we could not find it in the literature. We give two proofs, each generalizing the statement in a separate direction. Binomial coefficients (and Gaussian polynomials) are fundamental combinatorial objects, and so one naturally hopes to see a combinatorial proof of this inequality. However, this seems hard to come by. We nevertheless give a combinatorial proof of a special case.

We study the top singular value for a sum of $m$ independent $n \times n$ random matrices, each of which is a product of $N$ i.i.d. $n\times n$ Gaussian matrices. Our main conceptual observation is that when $m,n,N\rightarrow \infty$, the top singular value coincides with the partition function in a random energy model at the inverse temperature $\beta=\sqrt{2(N-1)/(n\log m)}$, with energies depending on the ratio $N/n$. We provide several non-asymptotic results making this approximation precise.