Papers

Stable polynomials are the natural multivariate generalization of real rooted univariate polynomials. While their definition is purely algebraic in nature, they have deep connections to combinatorics. One such connection is the support theorem proved by Brändén in 2007, showing that the support of any stable polynomial is a jump system, and hence that the support of any multiaffine stable polynomial is a $\Delta$-matroid. In this work, we generalize this result, showing that coefficients of multiaffine real stable polynomials give rise to oriented $\Delta$-matroids, and coefficients of multiaffine stable polynomials over Puiseux series give rise to valuated $\Delta$-matroids.

The basic question in perturbation analysis of Markov chains is how small changes in their transition kernels affect their stationary distributions. Classical perturbation bounds typically require the kernel error to be much smaller than $1/\tau$, where $\tau$ is a mixing or relaxation time. Although this scaling is sharp for Markov chains in general, we investigate a general "square-rooting" phenomenon in which one-step errors of order roughly $1/\sqrt{\tau}$ can be sufficient for local updates. We proved a form of this phenomenon in Lin, Liu and Smith (2025) under strong assumptions. Here we substantially weaken these assumptions, and prove this phenomenon occurs using three distinct approaches. First, block factorization applies under structural assumptions on the stationary measures of both chains. Second, approximate block-update arguments extend the result to statistically relevant Markov chain Monte Carlo (MCMC) settings, where structural guarantees are available for the exact posterior and its associated sampler, but not for the perturbed posterior. Third, we use direct calculations for a class of models with hard constraints where neither general result is directly available. We illustrate these results in three MCMC settings and show how they directly inform the tuning of approximate MCMC algorithms.

Probabilistic shaping (PS) is a well-known method to achieve improved performance upon a regular quadrature amplitude modulation (QAM) by taking the target constellation and making the distribution of underlying points non-uniform. It has been extensively studied over the years for the additive white Gaussian noise (AWGN) and Rayleigh fading channels. However, the potential of probabilistic shaping in the multiple-input and multiple-output (MIMO) setting needs further investigations. In this paper, we prove that if shaped symbols follow Maxwell-Boltzmann distribution, the optimal maximum a posteriori (MAP) detection is equivalent to the case of uniform constellation with a simple preprocessing step. Our approach has multiple benefits. It enables to utilize the same processing chain for both shaped and unshaped system, which simplifies the receiver architecture. This technique can be applied for both linear MMSE and nonlinear (near-)MAP demapper types. In addition, the complexity of adaptive methods such as sphere decoding can be reduced.

Let $G$ be a simple, simply connected algebraic group scheme defined over $\mathbb{F}_{p}$, and let $G_{r}$ be the $r$th Frobenius kernel. Donkin's famous Tilting Module Conjecture purports that a given indecomposable injective $G_{r}$-module can be realized as the restriction of a specific tilting module. The conjecture was first stated in 1990 and withstood proof for nearly 30 years. The authors discovered a counterexample in 2019. This paper provides some indication about the rich mathematics surrounding the Tilting Module Conjecture that also discusses an older conjecture by Humphreys and Verma. New versions of the Tilting Module Conjectures are formulated given multiple families of new counterexamples. Finally, through a new calculation, the authors present further evidence that the Tilting Module Conjecture should hold for reductive algebraic groups with an underlying root system of Type $\mathrm{A}_{n}$ for all fields of characteristic $p > 0$.

Recent developments in the US knowledge economy have created a significant growth in datacenter loads, with two major consequences for power generation. First, to compensate for growth in load, datacenters are encouraged to bring their own generating units. Second, in search of the remaining pools of dispatchable generation, utilities are increasingly turning to subtransmission and distribution level generating assets. Coupled with increasing loads and the resulting tighter grid conditions, both trends are likely to create a need to commit a large number of localized generating units under grid constraints. We propose an extension of Relax-and- Round Unit Commitment (RRUC) that is capable of committing generating units for larger problems faster than conventional methods, while staying within intertemporal and spatial MVA constraints. We demonstrate the performance of RRUC using synthetic congestible test systems ranging from 100 to 20,000 buses. RRUC consistently finds low cost solutions, independent of the problem size, and its run time increases sub-quadratically in the number of buses. RRUC can solve the 100 bus system in less than a second and the 20,000 bus system 7 minutes. In contrast, a leading state of the art solver cannot find a feasible solution to the 100 bus system in 15 minutes.

At a 2021 AIM workshop, Guo conjectured that the positive square energy s+ = E+_2 should inherit the familiar edge-addition monotonicity of the spectral radius, rho(G + uv) >= rho(G). That conjecture was subsequently shown to fail at p = 2. Tang, Liu, and Wang then introduced positive p-energy, proved nonmonotonicity for every 1 <= p < 3, and in version 3 of their preprint (26 March 2025) explicitly conjectured that monotonicity should hold for p >= 3. We disprove this conjectured high-exponent extension completely: for every real p > 2 there are infinitely many connected graphs G and nonedges uv such that E+_p(G + uv) < E+_p(G). Together with the Tang-Liu-Wang counterexamples below 3, this shows that no exponent p >= 1 restores the spectral-radius-style monotonicity: positive p-energy can decrease under the addition of an edge for every real p >= 1. The construction is a chain of clique blocks joined by regular bipartite graphs. An equitable quotient converges to Q = I + cA(P_k). For noninteger p, a binomial-series sign argument for a fractional power of I - cA(P_k) gives the required negative endpoint entry. At integer exponents, choosing c across the first spectral threshold leaves exactly one negative eigenvalue, and path locality forces the positive spectral contribution to have negative sign. We also give a fully rational 38-vertex certificate at p = 4 and determine the complete failure interval of a fixed 17-vertex counterexample at p = 3.

We study generative bootstrap processes obtained by resampling from fitted generative distributions. We establish necessary and sufficient conditions for their conditional weak convergence to the $P$-Brownian bridge, formulated in terms of finite-dimensional conditional weak convergence and conditional asymptotic equicontinuity. We further provide general sufficient conditions and a Giné-Zinn-type characterization relating this convergence to the $P$-Donsker property. As applications, we verify these conditions for prominent classes of generative models, including triangular normalizing flows, flow matching, score-based diffusion models, and Wasserstein generative adversarial networks.

Understanding how humans respond to incentives, both at the individual and collective levels, is crucial to the design of effective policies. Within the continuous-time stochastic framework for large interacting populations, mean field games (MFGs) model populations of non-cooperative agents, whereas mean field control (MFC) describes the fully cooperative benchmark, interpreted in our setting as fully altruistic behavior. Mixed-individual MFGs interpolate between these two extremes through a parameter governing the degree of altruism. A central challenge for regulators and policymakers, however, is that intrinsic altruism levels and other private structural parameters, such as individual labor costs, are typically unobservable. To address this challenge, we develop an inverse learning framework for mixed-individual MFGs. Our approach enables the recovery of (latent) altruism and labor cost levels from noisy observations, with experiments demonstrating the feasibility and accuracy of our method. These findings underscore the promise of inverse MFG methodologies for uncovering latent preference structures in large populations, with important implications for incentive design, empirical behavioral modeling, and data-driven policy analysis.

Richard Stanley asked whether every tree $T$ is determined up to isomorphism by its chromatic symmetric function $X_T$. We approach this open problem by regarding $X_T$ as a polynomial in the power-sum symmetric functions $p_1, p_2, \dots$ and studying the invariant $\Phi_T = \left.\frac{\partial {X_T}}{\partial p_1}\right\rvert_{p_1 = 0}$. We prove that $\Phi_T$ distinguishes, among proper trees, every proper tree in which any non-leaf vertex with two or more non-leaf neighbors has a number of leaf neighbors occurring at no other non-leaf vertex. We give an equivalent formulation of Stanley's question in terms of $\Phi_T$. Finally, we show that counts of spanning forests with at most $t$ edges cannot distinguish all $k$-vertex trees when $t<\left\lfloor k/2\right\rfloor$, for every $k\ge4$.

We establish near-optimal higher-order oracle bounds for smooth monotone variational inequalities. For fixed $p\ge2$, let $F$ be monotone on a known compact convex set $X$ of diameter at most $D$, with $\operatorname{Lip}(D^{p-1}F)\le L_p$. Each feasible query returns the complete jet $(F,DF,\ldots,D^{p-1}F)$, and the goal is to find $x$ with tangent residual $\operatorname{dist}(0,F(x)+N_X(x))\le\varepsilon$. Writing $Q=L_pD^p/\varepsilon$, we improve the $\widetilde O_p(Q^{1/p})$ upper bound of Chen et al. to $\widetilde O_p(Q^{2/(3p-1)})$ via a dimension-independent deterministic algorithm that returns an explicit tangent-residual certificate. We prove a matching $\Omega_p(Q^{2/(3p-1)})$ lower bound for arbitrary adaptive deterministic algorithms and randomized algorithms with per-instance success probability at least $2/3$, without span or tensor-update restrictions. Hence the high-dimensional worst-case oracle complexity is $\widetilde\Theta_p((L_pD^p/\varepsilon)^{2/(3p-1)})$. The same method applies to smooth convex–concave minimax problems, improving the fixed-geometry accuracy exponent from $4/(3p+1)$ to $2/(3p-1)$ and matching the known lower-bound exponent.

This work presents an interpretable data-to-control framework for actuator-constrained flutter suppression by combining higher-order dynamic mode decomposition (HODMD), dynamic mode decomposition with control (DMDc), and model predictive control (MPC). The primary case is a pitch-plunge typical section with a trailing-edge flap and rational unsteady aerodynamics. HODMD identifies the dominant frequency and growth rate in subcritical, near-critical, and post-critical regimes from four measured structural channels. With 2% RMS sensor noise, the near-critical growth-rate error is 1.8 x 10^-3 s^-1, substantially lower than with standard DMD without delay embedding. A five-state real reduced-order model is formed from the HODMD subspace, and its flap-input matrix is estimated from a small-amplitude PRBS record using DMDc. For an independent chirp input, the normalized error over all channels is 2.4 x 10^-3. The constrained MPC stabilizes the true post-critical plant for an initial pitch perturbation of 8 degrees with a flap limit of 1 degree, whereas saturated LQR designed from the same model reaches the prescribed validity limit. Three supporting studies assess transferability. SU2 simulations show HODMD frequency recovery consistent with the available spectral resolution; OpenFOAM dynamic-mesh calculations show nearly linear articulated-flap moment authority from 2 to 8 degrees with frequency-dependent gain; and a two-way SU2 Python-FSI case demonstrates bounded disturbance rejection using a proportional-derivative baseline. These studies support modal identification, actuator authority, and closed-loop feasibility, but are not presented as CFD-level MPC validation. The framework provides a reproducible route from early-time measurements to constrained aeroelastic control.

We develop a discrete design theory for cofactor reference forms that preserve the complete finite-element elasticity tangent. A fixed continuous potential generates the coefficient, while the variation space and quadrature determine compatibility. We classify the exact local potential spaces for all-order tensor elements, anisotropic spaces, cubic serendipity and quadratic tetrahedra. At a stress-free Mooney-Rivlin state with a common positive material-coefficient sum, a convex loss expresses the attainable quadrature-point material margin. The optimum is attained; under stated space and sampling assumptions, its ideal value is attained precisely when the physical identity potential is compatible. A conforming fixed-candidate example proves a strict margin loss, linear in curvature amplitude and uniform in cell size, under displacement enrichment. Explicit positive candidates accompany this restriction. A Gram identity gives the sharp constant-coefficient defect at a prescribed margin in the stated norm. Compatible and reference-defect-corrected forms retain the original tangent through a deformation-independent external-boundary Hessian; equal potential traces give equal exact assembled gauges. Quadratic tensor geometry admits a stress-free physical coercivity bound uniform in mesh size and displacement order under explicit material, shape, quadrature and boundary assumptions. Certified curved tetrahedral shear families with zero recovered cell pressure establish finite-deformation regimes, while a loaded pressure-curvature counterexample identifies their limitation. The theory and observations distinguish local exactness, material certificates, complete-reference coercivity and unchanged equations.

Volume quadrature can change nonlinear finite-element constraints while preserving their reference-state derivatives. We connect an explicit determinant defect to feasible-set geometry and singular mechanical response. For affine tensor elements of coordinate degree $p\ge3$ with $n\ge p+1$ Gauss points per coordinate, determinant volume is exact precisely when $2n\ge3p$. Below that threshold we construct a boundary-fixed cubic defect at every order. The same directions yield a full-space cube-root residual–distance bound under explicit cell-support, coefficient and physical-norm assumptions, with mesh-uniform upper constants at fixed order. With all cell-pressure equations retained, the volume Jacobian gains rank at nearby feasible states despite agreement through second derivatives at rest; the cube-root exponent is sharp on each fixed mesh. A general localized-minimum theorem shows that the first reduced compatibility term contributes its weighted square to the leading energy in a joint small-load, large-bulk limit. Cubic and quadratic defects therefore produce sextic and quartic terms. The full cubic-element interior space has an exact normal form and sharp local error exponents. Curved quadratic tetrahedra supply the second-order contrast, a sparse rational witness and an exact four-Jacobian volume formula. Finite-strain tensor calculations illustrate normalized response separation, with explicit stationary-point, extreme-bulk and pressure-recovery qualifications. The constructive correction preserves exact cell volumes, so quadrature feasibility remains distinct from physical volume preservation.

We formulate a conjectural generalization of the arithmetic Siegel-Weil formula, relating intersection multiplicities of cycles on unitary Rapoport-Zink spaces to central derivatives of local Whittaker functions. The new aspect of the conjecture is that it incorporates the action of certain Hecke correspondences on the Rapoport-Zink space. We verify the conjecture in a low-dimensional case. A significant portion of the paper is devoted to generalizing the classical theory of representation densities for local Hermitian spaces, as this is needed for explicit computation of the relevant Whittaker functions. We expect this generalization to be of independent interest.

We study microscopic spectral correlations between two nearby parameter values in the three non-Hermitian Gaussian bulk classes: $A$ (unconstrained), $AI^\dagger$ (complex symmetric), and $AII^\dagger$ (complex self-dual, with each degenerate doublet counted once). Parameter dependence is modelled by stationary matrix Ornstein–Uhlenbeck evolution, whose microscopic time scale is $N^{-1}$. For every positive integer replica number $n$, we derive exact finite-$N$ auxiliary-field integral representations for joint characteristic-polynomial moments and obtain their bulk asymptotics as $N\to\infty$ with $n$ fixed. With the spectrum normalized to the unit disk, correlations near a bulk point $z_0$, between spectral positions $z_1,z_2$ at times $t_1,t_2$, depend only on $s=N|z_1-z_2|^2+N(1-|z_0|^2)|t_1-t_2|$. Adopting the static Hermitian/non-Hermitian replica continuation yields explicit two-time kernels; for $AI^\dagger$ and $AII^\dagger$ these are replica conjectures and, to our knowledge, the first microscopic two-time kernels proposed for these classes. They determine density correlations, cross-time eigenvalue-count covariances, and equal-time number variances. For a disk of mean eigenvalue count $y$, the latter obey $\mbox{Var}\mathcal N_X(D_y)=\kappa_X\sqrt y+ \beta_X/\sqrt y+o(y^{-1/2})$, with explicitly given constants for each class. Exact perturbation theory further identifies eigenvector nonorthogonality as the mechanism of short-time spectral diffusion and yields a basis-independent condition number for a two-dimensional Kramers eigenspace in class $AII^\dagger$. Direct simulations agree quantitatively with its conjectured inverse-gamma law.

We prove the dynamical uniform boundedness conjecture for unicritical polynomials $z^d + c$ over every number field $K$ when $d \geq 4$, and over every number field $K$ not containing $\mathbb{Q}(\sqrt{-3})$ when $d = 3$. Furthermore, our unconditional bounds on perodic points are effectively computable. In the remaining cases, we show that the dynamical uniform boundedness conjecture is implied by the boundedness of Mordell-Weil ranks over $K$ for a specific family of elliptic curves when $d = 3$ and for Jacobians of dynatomic curves when $d = 2$.

This is a mostly expository paper on the construction of theta functions on the Siegel and Hermitian half-spaces using the machinery of the Weil representation, especially theta functions weighted by polynomials. When the polynomial in question satisfies suitable homogeneity and harmonicity conditions, the resulting theta function is holomorphic. Non-holomorphic Siegel and Hermitian theta functions can be constructed in a similar way, using a more general class of polynomials.

We study the mixing time of a systematic-scan analogue of Kac's walk that was proposed as a fast surrogate for Haar-distributed orthogonal matrices in randomized high-dimensional algorithms and was conjectured to approach Haar measure after only logarithmically many sweeps. We show that this conjectured speed-up does not occur for convergence of the full matrix law to Haar measure in Frobenius Wasserstein distance. At fixed normalized accuracy, the mixing time lies between order $n/\log n$ and order $n$ sweeps; at fixed absolute Frobenius accuracy, the corresponding bounds are between order $n$ and order $n\log n$. More strongly, below the scale $n/\log n$, the normalized Wasserstein distance remains asymptotically at its extremal value. We also show that the output law is singular with respect to Haar measure for fewer than $n/2$ sweeps. Thus the sampler may provide effective application-specific randomization without exhibiting the much faster full-Haar mixing.

We develop a geometric framework for distance-based formation control that separates the evolution of inter-agent distances from its realization by compatible node motions, reducing the stability problem to the edge space. We show that local exponential convergence of the edge dynamics implies local exponential convergence of the formation, and that stability is certified by spectral properties of a linear edge operator. We introduce a hierarchy of generic spectral properties — weak admissibility, admissibility, and strong admissibility — that provide necessary conditions for local exponential stability. Specializing to directed sensing, we obtain a necessary and sufficient spectral condition for local stability at an arbitrary target, together with a quadratic sufficient certificate. These conditions reveal that stability depends jointly on the graph orientation and target geometry, and show that persistence is neither necessary nor sufficient for local convergence. We show that every generically rigid graph admits an admissible orientation and, for acyclic orientations, we give an exact combinatorial characterization of admissibility. Finally, the quadratic certificate leads to a semidefinite program for synthesizing stabilizing edge gains.

In a 2021 paper, Jankiewicz, Norin, and Wise asked whether there exists a finite $4$-connected graph of girth at least four and nonnegative Charney–Davis curvature such that no $4$-connected ordinary subgraph admits a legal system. We construct such a graph by starting from the hexagonal prism and attaching three $K_{3,4}$-based caps along pairwise disjoint induced $4$-cycles. The key structural input is a restriction theorem showing that a legal system on an induced-$4$-cycle amalgam restricts to each side, so the obstruction carried by the negatively curved prism survives the attachments. The resulting $33$-vertex graph is $4$-regular and $4$-connected, has girth four and Charney–Davis curvature one, and, by $4$-regularity, is its own unique $4$-connected ordinary subgraph.

Differential privacy provides a mathematical framework for guaranteeing privacy for sensitive data. In quantum information processing, the interaction of privacy constraints with quantum resources such as entanglement remains a question of interest. Given that the utility of many protocols, and often the presence of a quantum advantage, relies on quantum resources such as entanglement, it is crucial to understand when a privacy requirement for a quantum channel is compatible with the channel's ability to preserve entanglement. We study this question for quantum local differential privacy (QLDP). Our main result shows that every $\varepsilon$-QLDP channel with a $d$-dimensional input is entanglement-breaking whenever $\varepsilon\leq\log\frac{d}{d-1}$. We also prove an approximate version for $(\varepsilon,\delta)$-QLDP, where channels in the same high-privacy regime are close in diamond norm to an entanglement-breaking channel. We further prove a composition result for a collection of private quantum channels having entangled inputs and global measurements in the high-privacy regime. Finally, we apply our results to private quantum learning theory. We prove that any learning protocol using arbitrary quantum memory on copies of the output of an entanglement-breaking channel can be simulated by a protocol that measures the corresponding unprocessed input copies one at a time while storing only classical information. Combining this result with our high-privacy entanglement-breaking theorem, we show that under sufficiently private local noise, a learning protocol with quantum memory for purity testing and bipartite product testing is subject to the sample complexity lower bounds for protocols with single-copy measurements on the noiseless tasks. We also obtain stronger sample complexity lower bounds when a single highly private channel acts on the entire multipartite input.

Gao and Xie conjectured that the coefficients of the ordinary, unnormalized inverse Kazhdan-Lusztig polynomial of every matroid are log-concave with no internal zeros. We give counterexamples and prove locality and composition formulas for deletions of projective charts. For every finite field and every nonempty finite set S contained in {2, 3, ...}, these formulas yield simple 3-connected matroids whose negative internal Turán determinants occur exactly at S. Their coefficients are positive and strictly decreasing. The polynomial degree can be prescribed above max S, and all selected Turán ratios can tend to zero simultaneously. The universal first-index inequality remains open.

In this paper, we study a stochastic Susceptible-Infected-Removed (SIR) model where the infection and the recovery rates depend on individual covariates for susceptibility and infectiousness of the infector and the infectee. Such models allow explicit nonlinearity in the incidence term. They are also important from a practical perspective, as they allow for the incorporation of individual heterogeneity into the epidemic process. Statistical estimates for crucial epidemiological parameters, such as the basic reproduction number, herd immunity threshold, could be vastly different, and even biased, when the population heterogeneity is ignored in the mathematical model. We describe our epidemic model as an Interacting Particle System (IPS) of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures. Our main mathematical contributions are a Functional Law of Large Numbers (FLLN), which approximates the empirical random measure of the IPS by means of a deterministic measure-valued function, and the propagation of chaos phenomenon, which establishes asymptotic independence of the particles as the population size goes to infinity with an explicit construction of McKean–Vlasov type Kac's “nonlinear process”. We also briefly mention how the propagation of chaos phenomenon leads to a product-form likelihood function, which forms the basis of the so-called Dynamic Survival Analysis (DSA) method for parameter inference based on sparse data.

We study the one-dimensional viscous Burgers equation on $(0,L)$ with the conservative boundary conditions $u_x=-u^2$. On the half-line, solutions approach a nonlinear self-similar profile, whereas on a bounded interval they converge to a nonconstant equilibrium. We describe explicitly how the dynamics passes between these states at the critical diffusive scale $t=cL^2$. For compactly supported initial data of mass $M$, the rescaled solution converges as $L\to\infty$ to an explicit crossover profile $\Phi_c$. In similarity variables, $\Phi_c$ converges to the half-line profile $f_M$ as $c\downarrow0$; after rescaling to domain variables, it converges to the interval equilibrium as $c\to\infty$. We also determine the sharp onset of confinement: \[ \lim_{c\downarrow0}-c\log|\Phi_c(0)-f_M(0)|=1\qquad(M\ne0). \] Moreover, on compact sets in similarity variables, the interval and half-line solutions differ in $C^k$ by at most $C_{k,\varepsilon}\exp(-(1-\varepsilon)L^2/t)$, uniformly in $L$, and the exponential constant is optimal. Thus we identify not only the transition scale $L^2$, but also the profile governing the crossover and the sharp rate at which the remote boundary becomes visible. We finally discuss the conclusions that persist for space-dependent diffusivity and illustrate the three asymptotic regimes numerically.

Pate proved ordinary irreducible-immanant permanental dominance through order $13$ and identified $(4,4,3,3)$ as the sole remaining order-$14$ case, with $(5,4,3,3)$ and $(3^5)$ forming the order-$15$ frontier. These three cases are settled here; consequently $d_\lambda(A)/f^\lambda\le \operatorname{per}(A)$ for every partition $\lambda\vdash n$ with $n\le15$ and every complex Hermitian positive-semidefinite matrix $A$. The argument also yields results beyond this finite frontier: an exact four-term bridge for $(4,4,3,3)$, the uniform family $(m,4,3,3)$, a two-parameter family $(a,b,3,3)$ for $a\ge b\ge4$ and $5a\ge8b$, and a long-first-row criterion for arbitrary fixed tails. These results arise from explicit specializations of Pate's $W$-function positivity framework using partial swaps, Young projectors, Pieri–content identities, and branching data. For $(3^5)$, an exact Farkas certificate shows that the central-projector partial-swap cone is insufficient; a branching-refined one-swap construction escapes this obstruction and yields a positive $106+19$-witness certificate. Boundary-compression and node-moving results further describe the reach and limitations of the local-filter method. All finite certificates are checked by exact integer or rational arithmetic and are supplied as ancillary material. The order-$14$ bridge is additionally formalized and kernel-checked in Lean 4 for all complex Hermitian positive-semidefinite matrices, including the exact coefficient normalization and the deduction of $(4,4,3,3)$ permanental dominance from four explicitly stated Pate inequalities.

We examine the generalized Hilbert (matrix) operators $$ H_{g,\gamma} : (a_n) \mapsto \sum_{n=1}^{\infty} \bigg(\frac{k}{n}\bigg)^{\gamma} \frac{g_k a_n}{n+k} $$ on the $\ell^p$ spaces, $1<p<\infty$, where $g=(g_n)$ is a sequence and $-1/p<\gamma <1-1/p$. Given a partition of the natural numbers $\bigcup_j I_j = \mathbb N$, we encode $g$ to our investigation via its $\ell^p$-means $G_j$ over the sets $I_j$. We prove that if $I_j$ increases exponentially, the data $(G_j)$ characterizes boundedness, but is insufficient to determine the exact value of the essential norm. Nonetheless, if the growth rate of $(I_j)$ is subexponential, the corresponding data $(G_j)$, given that the tail converge, is sufficient to determine the exact value, in which case we also calculate it. Moreover, it is shown that the boundedness of $(G_j)$ is sufficient, but not necessary (and necessary, but not sufficient) to ensure $H_{g,\gamma}$ is bounded if the underlying partition $(I_j)$ increases subexponentially (and superexponentially, respectively). Using the dual operator, we also prove that the essential norm of $H_{g,\gamma}$ is comparable with $\limsup_j G(j)$ when $(I_j)$ grows exponentially.

We prove local rigidity of every smooth conformally Euclidean metric for the anisotropic Calderón problem on smooth compact domains $M\subset\mathbb R^n$, $n\ge3$. Any smooth Riemannian metric $g$ sufficiently close to a fixed background $g_0=e^{2c}\mathsf e$ in $H^s(M)$, with integer $s>n/2+1$, having the same Dirichlet-to-Neumann map satisfies $g=\Phi^*g_0$ for a smooth diffeomorphism $\Phi$ fixing the boundary pointwise.

We study the Einstein-Vlasov system near the Schwarzschild spacetime in spherical symmetry. In particular, we consider the case when the matter is supported on bounded geodesics. In this regime, the decay of the matter is driven by phase mixing rather than dispersion. For the linearized problem, we obtain quantitative phase mixing; the key new ingredient proved here is a monotonicity property of the period function. For the nonlinear coupled problem, we prove phase mixing up to the timescale expected from the nonlinear echo mechanism; this is achieved by analysis on dynamical action angle variables and the integral estimates with the vector field method. This construction is made possible by our gauge choice, an isotropic coordinate system together with a foliation of prescribed mean curvature, in which the Einstein equations gain derivatives over the matter due to its elliptic nature.

Knot theory has provided a rich topological taxonomy for band structures, but its reach is fundamentally limited: knot invariants classify only 1D nodal lines at gap closure, and cannot encode the full dispersion or rich Fermi surface structure of realistic materials. Here we show that knotted graphs (knots that admit graph-like intersections in 3D space) - which have so far been elusive in condensed matter literature - provide a unified topological language for classifying the entire band dispersion, and even the eigenstate topology in some contexts. We propose a new framework beyond the existing Yamada polynomials that can topologically characterize the intricacies of realistic Fermi surfaces completely, crucially including how their multiple disconnected pieces are nested. This yields the Yamada set, a boundary-resolved extension which organizes the full topological evolution across energy into a Yamada sequence: a compact dispersion-level fingerprint directly tied to experimental signatures of Lifshitz transitions. Our framework is demonstrated with DFT-based band structures of real materials. Beyond dispersion-level classifications, this framework can be extended to non-Hermitian exceptional surfaces, where Berry-curvature flux further equips the knotted-graph skeleton with a directed Abelian edge flow that also captures the eigenstate topology.

We construct a class of integrable chiral fermionic quantum field theories with interactions that depend on both space and time. The construction starts from a unitary, regular, difference-form two-body scattering matrix satisfying braiding unitarity and the Yang–Baxter equation. Each freely moving right- or left-moving particle carries a characteristic coordinate that remains constant along its trajectory, and the scattering matrix depends on differences of these coordinates. The many-body amplitudes form a discrete connection over the coordinate-ordering sectors. The Yang–Baxter equation gives local path independence, while periodic winding on a spatial circle produces quantum Knizhnik–Zamolodchikov (qKZ) difference equations expressing global flatness. Once a reference amplitude satisfying the exchange and qKZ constraints is specified, these transport equations determine the exact wavefunction in every fixed-particle-number sector. Independent reparametrizations of the right- and left-moving characteristics give couplings that depend on both space and time. For the chiral $SU(2)$ Gross–Neveu realization, we solve the qKZ equations for this full characteristic family using off-shell Bethe vectors. The resulting amplitudes are written in the characteristic labels, and their pullbacks give exact many-body wavefunctions for the corresponding space-time-dependent couplings. The two-body amplitude is a ratio of Gamma functions that resums the phase shifts from successive scattering processes. Equal-slope affine maps give a spatially homogeneous specialization in which the coupling depends only on time, and the Bethe ansatz with constant coupling is recovered when the qKZ step vanishes.