Papers

Let $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n$ be the eigenvalues of a simple graph $G$ of order $n$. The HL-index of $G$ is defined by $R(G)=\max\|\lambda_h|,|\lambda_\ell|\}$ with $h=\lfloor(n+1)/2\rfloor$ and $\ell=\lceil(n+1)/2\rceil$.In this paper, we prove that if $G$ is $ K_4$-minor-free or $ K _ {2,3} $-minor-free, then $R(G)\leq\sqrt{5}-1$ with equality attained by an infinite family of outerplanar graphs.Moreover, we show that $R(G)\leq\sqrt{d-2}$ for triangle-free graphs with maximum degree at most $d$ and average degree at most $(d-2)(d^2-2d+2)/(d^2-3d+5)$.

We derive the four principal asymptotic rate-distance tradeoffs for binary codes—Plotkin, Elias–Bassalygo, and the two McEliece–Rodemich–Rumsey–Welch (MRRW) bounds—from one theorem, the “pretty good criterion.” If the bit error rate under the pretty good measurement (PGM)—the quantum analog of posterior sampling—of a binary-input output-symmetric classical–quantum (cq) channel lies below $\delta$, then every length-$n$ binary code, linear or nonlinear, of relative distance $\delta$ has rate at most the channel's capacity, up to an $O(n^{-1/2})$ correction. Rate–distance bounds thereby reduce to a channel design problem, wherein the task is to minimize channel capacity subject to the posterior bit error rate constraint. Via the pretty good criterion, the binary erasure channel (BEC) yields Plotkin, the binary symmetric channel (BSC) yields Elias–Bassalygo, the pure-state channel (PSC) yields the first MRRW bound, and a masked PSC yields the second MRRW bound exactly. This framework is then instantiated with new channels to improve upon the MRRW bounds. Specifically, the mixed-qubit channel (MQC), a mixed-state version of PSC, strictly improves the first MRRW bound at every $0 < \delta < \frac{1}{2}$, while the masked mixed-qubit channel (2MQC) strictly improves the second MRRW bound throughout the same interval.

To each VOA of strong CFT type, we associate a non-associative algebra which generalizes Griess algebras of One-Zero (OZ) VOAs. We compute the fusion law of Ising vectors in this algebra using techniques from Miyamoto, leveraging the module theory of the corresponding Virasoro VOA. This new class of algebras encapsulates the Chayet-Garibaldi algebras, a family of algebras constructed from absolutely simple linear algebraic groups that includes the $3876$-dimensional algebra for $E_8$.

Model learning by an eavesdropper is treated as an estimation problem in a federated environment. The Fisher Information Matrix for the eavesdropper's estimation problem is driven to singularity through a signaling design; this ensures that the eavesdropper cannot learn the model. Herein, the innovation of prior designs is that model shifts are designed to maximize the difference in the model learned by Eve and the central server while satisfying a transmission power constraint for the agents. Two shift schemes are provided. MaxModShift outperforms a prior ModShift design while requiring lesser transmission power. Compared to a noise injection scheme, MaxModShift performs better while requiring a lower bandwidth secret channel and a reduced average power consumption.

A discrete entropy inequality is the principal nonlinear stability estimate available for systems of conservation laws, and evaluating it presupposes a physically admissible state. So far, however, the two have been secured separately. Entropy-stable schemes are almost always semi-discrete, are built around one selected entropy pair, and take for granted the positivity of density and pressure that makes the entropy well defined in the first place, whereas bound-preserving limiters keep the solution admissible but deliver no entropy estimate. For the special relativistic Euler equations, the two cannot be separated at all, since the conservative-to-primitive map is implicit, and an inadmissible state therefore has no entropy to correct. Here we construct high-order discontinuous Galerkin and finite volume schemes that, to our knowledge, for the first time, are entropy stable in the fully discrete sense for an arbitrary prescribed finite family of convex entropy pairs, a property we call multi-entropy stability, and are provably admissible wherever an entropy is evaluated. All of this is achieved by a single cellwise projection, and neither conservation nor the design order is lost. The construction rests on relativistic causality, which bounds every characteristic speed by the speed of light. Consequently, the numerical viscosity can be fixed once for all states and all equations of state, and one two-point building block then serves the whole entropy family. Since only the convexity of the admissible set and this speed bound are used, the same route remains open for related systems. Finally, in computations with four equations of state, the schemes retain high-order accuracy close to vacuum, produce no inadmissible state in strong shocks, near-vacuum shock–vortex interaction or jets with Lorentz factor above $70$, and confirm the monotone decay of every enforced discrete entropy.

We establish three averaging principles for distribution-dependent stochastic reaction–diffusion equations with rapidly oscillating coefficients on the torus $\mathbb T^d$, $d\le3$. First, solutions converge in mean square, uniformly on finite time intervals, to solutions of the averaged equation. Under a contraction condition, both the original and averaged equations admit unique bounded entire solutions whose mean-square distance vanishes uniformly for all $t\in\mathbb R$. At the level of probability laws, the original nonautonomous equation possesses a family of pullback attractors, whereas the averaged equation has a global attractor; the former converge upper-semicontinuously to the latter, uniformly over the coefficient hull. As an application, we present a class of stochastic reaction–diffusion models motivated by large-scale interacting systems.

We study regularity, Harnack inequalities, Liouville rigidity, and principal eigenvalues for viscosity solutions of singular or degenerate fully nonlinear elliptic equations $\Phi(x,|\nabla u|)F(D^2u)-H(x,\nabla u)+c(x)|u|^{i(\Phi)}u=h(x)$ in a bounded domain $\Omega$, where $\Phi$ describes the singular or degenerate dependence on the gradient and $H$ is a Hamiltonian. We first prove global $C^{1,\gamma}$ regularity for the Dirichlet problem without lower-order terms. The argument combines a global $L^\infty$ estimate from the Alexandroff–Bakelman–Pucci inequality, boundary barriers yielding a global Lipschitz bound, and a compactness-based iterative approximation scheme. We next establish an additive Harnack inequality for nonnegative viscosity solutions by sliding from below a cusp function of the form $-|x|^{1/2}$. Under an additional homogeneity assumption, this yields the classical Harnack inequality and Liouville-type theorems in $\mathbb{R}^n$. Finally, under suitable homogeneity and comparison assumptions, we develop a generalized Dirichlet principal-eigenvalue theory for the full operator. We prove the existence of principal eigenfunctions and characterize the associated eigenvalues through maximum and minimum principles. These results provide a unified framework for global regularity, Harnack estimates, Liouville rigidity, and principal eigenvalues for a broad class of singular and degenerate fully nonlinear equations with Hamiltonian terms.

We study the isomorphism groupoid $\mathcal{T}(M)$ of $\theta$-almost twisted Poisson ($\theta$-atP) structures on a smooth manifold $M$, focusing on the internal structure of its morphisms. A morphism in $\mathcal{T}(M)$ is a $C^\infty(M)$-linear isomorphism $\Phi:\gO^1(M)\to\gO^1(M)$ that simultaneously intertwines the anchor maps and the $(H,\theta)$-twisted Koszul brackets associated with two $\theta$-atP structures. Every such morphism induces a canonical isomorphism in $\theta$-atP cohomology. We define a classifying functor $$ \Delta : \mathrm{Mor}(\mathcal{T}(M)) \longrightarrow (Z^1_{\mathrm{dR}}(M) ,+), \qquad \Delta(\Phi)=\theta' - \theta, $$ which is additive under composition and partitions the morphisms into two complementary families: the sub-groupoid $\mathcal{T}_{\mathrm{fix}}=\ker\Delta$ of isomorphisms preserving $\theta$, and the family $\mathcal{T}_{\mathrm{mod}}$ of isomorphisms shifting $\theta$. We describe each element in this partition.

Given a category $\calC$ and a center, that is a collection of pairs $(d_i, N_i)$ consisting of a morphism $d_i$ and a sieve $N_i$ over its codomain, the dilatation of $\calC$ is a new category $\calC'$ in which every $n \in N_i$ factors, uniquely and functorially, through $d_i$. This paper presents the theory of dilatations of categories through a full formalization of the construction and its main theorems in the Lean~4 proof assistant, on top of the Mathlib library. An appendix collects a systematic dictionary between the mathematical statements and the Lean declarations that formalize them.

We prove the K-polystability of the general Fano complete intersection of arbitrary multidegree and dimension, and the K-stability of the general Fano complete intersection that is not isomorphic to projective space or a quadric hypersurface. We prove analogous results for certain smooth weighted complete intersections.

Let $(M^n,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$. We show that the existence of a conformal metric with positive $Q$-curvature $Q_g$ and positive scalar curvature $R_g$ is equivalent to the positivity of both the Yamabe invariant $Y(M^n,[g])$ and the Paneitz operator $P_g$. For $n=5$, this equivalence confirms a conjecture of Gursky-Hang-Lin (2016, IMRN). Furthermore, assuming $Y(M^n,[g])>0$, $Q_g\ge 0$, and $Q_g\not\equiv 0$, we prove that both $R_g$ and $P_g$ are positive which resolves a problem of Hang-Yang (2016, CPAM). As a corollary, we show that the hypotheses of Gursky-Malchiodi (2015, JEMS) are equivalent to those of Hang-Yang (2016, CPAM).

Twisted rational maps arise naturally in relative algebraic dynamics: if a rational self-map preserves a fibration, then the induced map on the generic fiber is usually not an ordinary rational self-map over the function field, but a twisted one. This suggests that twisted rational maps form a natural framework for studying relative dynamics. In this framework we extend the theory of dynamical degrees, the numerical invariants measuring the asymptotic complexity of a dynamical system: the defining limits exist, are independent of the choice of polarization, and are birational invariants. We also identify the relative dynamical degrees of a semi-conjugacy with the dynamical degrees of the induced twisted rational map on the generic fiber, and prove the corresponding mixed degree formula. Finally, using the spectral interpretation of dynamical degrees and valuative methods at infinity, we prove an algebraicity result for the first dynamical degree of twisted endomorphisms of affine varieties.

High-dimensional kernel density estimation suffers from the curse of dimensionality. This study proposes a hybrid density estimator defined as the product of a joint density over a pre-specified subset of dimensions and marginal densities for the remaining variables, instead of estimating the full high-dimensional density directly. Under this framework, given the observed data, we select a subset of variables for joint density construction to improve estimation accuracy, while modeling the remaining variables through their respective marginal densities. This construction involves a trade-off between the approximation error induced by simplifying the dependence structure and the variance reduction achieved by lowering the dimension of the joint density. We employ a genetic algorithm to efficiently identify such variable subsets. Theoretical and numerical results demonstrate that the proposed estimator can outperform conventional full-dimensional kernel density estimation when this trade-off is appropriately balanced.

Given a closed subgroup $H\subseteq G$ and a Hamiltonian $H$-space $Y$, one can construct an induced Hamiltonian $G$-space. In this paper we investigate this construction in the setting where $Y$ is endowed with a compatible complex structure. Our aim is to develop symplectic induction in this framework and to establish the corresponding Kähler analogues of the classical results. The main idea is to replace the cotangent bundle $T^*G$, appearing in the standard construction of $\operatorname{Ind}$, with the Grauert tube of $G$. We focus on Lie groups admitting a globally defined Grauert tube and study the resulting pseudo–Kähler induction procedure.

Optimal transport and Gromov–Wasserstein distances are useful tools for comparing probability measures and metric measure spaces, but their balanced formulations force all mass to be matched. This constraint is often too strong for data with outliers, missing parts, or only partial overlap. In this paper, we develop entropic partial optimal transport for Gaussian mixture models and define a partial mixture Gromov–Wasserstein distance. For the finite entropic partial optimal transport problem, we prove the existence and uniqueness of the minimizer and establish quantitative large-penalty estimates. Moreover, the resulting entropic partial component couplings induce continuous partial transport plans through Gaussian optimal maps. We analyze their large-penalty and subsequent zero-entropy limits and construct the associated displacement interpolations and barycentric projection maps. In addition, by identifying each Gaussian mixture with a finite metric measure space of Gaussian components, we establish the metric property and large-penalty limit of the partial mixture Gromov–Wasserstein distance. Finally, numerical experiments on synthetic Gaussian mixtures and point clouds illustrate the effects of the penalty and entropic regularization and the robustness of partial matching to outliers.

We develop a classical model-theoretic representation of partial Boolean algebras and use it to formulate quantum-like uncertainty without abandoning classical propositional logic. Given a surjection from the initial formulæ\ of a propositional language onto a partial Boolean algebra $(\mathcal V,\Pi)$, we construct a consistent theory $\mathcal T_g$ whose core—the ordered set of equivalence classes of initial formulæ—is isomorphic to $(\mathcal V,\Pi)$. Its models are characterized as upward-closed clusters, yielding a model-theoretic formulation of KS-colourability: an $n$-dimensional partial Boolean algebra is KS-colourable exactly when the induced theory has a model meeting every pre-frame in one primitive formula. For finite-spectrum observables, uncertainty is defined by the number of locally admissible atomic outcomes. Dispersion-free models are characterized by the singleton pre-frame condition. With an additional measurement-update postulate, a finite example shows how measurement of an incompatible observable can destroy sharpness, providing a model-theoretic form of back-action. A $12$-vertex partial Boolean algebra is KS-colourable, whereas a rigorously constructed $140$-vertex, four-dimensional partial Boolean algebra is not, as shown by a parity argument; its incidence structure is isomorphic to the Peres $24$-ray, $24$-basis configuration in $\mathbb R^4$. Macroscopic interpretations show that these phenomena arise from the organization of propositions and models rather than from nonclassical deduction. Finally, we relate certain models to the probability-one propositions of pure states and density operators, while emphasizing that such certainty models do not determine the full quantum state.

We study Bochner–Riesz summability for a one-dimensional model of noncompact manifolds with ends. Each end has an effective Euclidean dimension, and these dimensions may differ from one end to another. The central point is the contrast between the symmetric and asymmetric cases. When the end dimensions agree, the model follows Herz's radial theory for the Euclidean Laplacian. When they are unequal, a Fefferman-type obstruction appears, analogous to the ball multiplier obstruction in higher-dimensional Fourier analysis, even though the model itself is one-dimensional. We give a complete characterisation of the \(L^p\)-boundedness of the corresponding spectral projections and Bochner–Riesz means. In the asymmetric case, the boundedness range contains an additional restriction depending on the difference between the end dimensions. This restriction is absent from Herz's radial model and shows that Bochner–Riesz summability on spaces with unequal ends is governed not only by the maximal end dimension, but also by the interaction between the ends.

Let $S$ be a semigroup with zero. This paper studies the zero-divisor digraph $\overrightarrow{\Gamma}(S)$ and the extended zero-divisor digraph $\overrightarrow{\Gamma}_{\!E}$, and introduces the annihilator digraph $\overrightarrow{\mathrm{AG}}(S)$ via left and right annihilators. The diameter bound when every zero-divisor is nilpotent, and the sinks and the sources of $\overrightarrow{\Gamma}_{\!E}(S)$ being identical to those of $\overrightarrow{\Gamma}(S)$ is demonstrated. The conditions in which $\overrightarrow{\Gamma}(S)=\overrightarrow{\Gamma}_{\!E}(S)=\overrightarrow{\mathrm{AG}}(S)$ holds are established, and the connectedness, diameter, girth, and vertex degrees of $\overrightarrow{\mathrm{AG}}(S)$ are bounded when every zero-divisor is nilpotent or two-sided. The extended zero-divisor digraph is connected if and only if the zero-divisor digraph is connected, and it contains a directed cycle if and only if the zero-divisor digraph does. The knit degrees of $\overrightarrow{\Gamma}(S)$, $\overrightarrow{\Gamma}_{\!E}(S)$, and $\overrightarrow{\mathrm{AG}}(S)$ are computed. For a unital ring $R$, the equality $\overrightarrow{\Gamma}_{\!E}(R)=\overrightarrow{\Gamma}(R)$ is characterized by nilpotency indices and one-sided annihilator conditions; it holds for the full matrix ring $M_{n}(F)$ over a field $F$ if and only if $n=2$, and $\overrightarrow{\mathrm{AG}}(M_{n}(F))$ is connected and contains a directed cycle. Moreover, for an artinian noncommutative ring $R$, it was proved that $\overrightarrow{\Gamma}(R)$ is connected if and only if $\overrightarrow{\Gamma}_{\!E}(R)$ is connected if and only if every one-sided identity element of $R$ is a two-sided identity of $R$.

We derive an explicit discrete energy identity for rational time discretizations generated by the first-subdiagonal Padé approximants of the exponential for solving linear seminegative problems. This work extends the diagonal Padé energy laws in [Z. Sun, Y. Wei, and K. Wu, SIAM J. Numer. Anal., 60 (2022)] to the first-subdiagonal family. The main new ingredient is an explicit Cholesky-type factorization of the energy coefficient matrix associated with the semi-inner-product terms in the discrete energy identity. The construction and proof of this factorization are nontrivial, since the matrix entries are alternating sums of Padé coefficients and the triangular factor has a parity-dependent factorial structure. We prove the factorization by reducing it to scalar rational identities and establishing them through finite product reductions and telescoping summations. Together with a \beta-coefficient cancellation, the factorization yields an exact discrete energy law that recovers the classical unconditional contractivity for linear seminegative problems. Numerical experiments adapted from the diagonal Padé energy-law setting illustrate the predicted order and verify the discrete dissipation identity.

Let G = UT(4; p) be the group of all 4 ? 4 unitriangular matrices over the fi?nite fi?eld Fp, where p is a prime. Using the six-parameter form of the elements of G, we describe the commutativity relation explicitly and use it to analyse the struc- ture of the graph. We prove that the reduced commuting graph is connected and has diameter 3. We also determine the size of its maximal cliques, chromatic number, independence number, per- fectness, etc. by decomposing the graph into cosets, layers, and direction parts.

We study the discrete diffusive exchange-driven growth (EDG) equations on a bounded smooth domain of arbitrary dimension subject to homogeneous Neumann boundary conditions. The system belongs to the class of infinite systems of semilinear partial differential equations with nonlinear source terms of quadratic type. Global-in-time existence of solutions is established for separable exchange kernels of the form \(K_{i,j}=b_i a_j\), where the donor rates exhibit at most linear growth while the receiver rates are sublinear. The analysis is based on a uniform Fisher information estimate obtained from an entropy-entropy dissipation identity. This estimate yields renormalized solutions to a truncated system with finitely many species. A compactness argument then enables passage to the limit in the exchange operator, leading to the existence of global-in-time solutions for the full system.

A score-driven filter multiplies its scaled log-likelihood score by a scale parameter. We call this coefficient the gain and learn it online. Given the current state and realised scaled score, each admissible gain selects a reachable next state and predictive density. A scalar gain moves along a line; diagonal gains control coordinatewise transmission and may change direction. We evaluate gain selection using a one-step predictive Kullback-Leibler objective. In the scalar unscaled case, the negative consecutive-score product is a stochastic gradient; the positive product used in accelerated recursions is a descent direction. Positive scalar score scaling changes only the effective learning rate. Strictly increasing, continuously differentiable gain links with positive derivative induce mirror-descent geometry, while persistence adds a Bregman pull towards a reference gain. Under convexity, compactness, integrability, and schedule conditions, projected and discounted mirror updates satisfy dynamic-regret bounds relative to time-varying, current-information comparators. Simulations isolate score scaling, link geometry, persistence, and coordinatewise gains. Across twelve equity indices, the bounded discounted-logistic gain records a lower out-of-sample mean negative log score than the constant gain in eleven markets, although market-level evidence is mixed. It also avoids the extreme transients of the numerically capped exponential-link benchmark.

In this article we study the spectral problem related to the fractional Laplace-Beltrami equation on $(\mathbb{R}^d,g)$ and establish its equivalence with the classical anisotropic Helmholtz equation. The proof is based on Seeley's construction of complex powers of elliptic operators and the pseudodifferential symbolic calculus. As an application, we describe the related fixed-frequency inverse scattering problem of recovering the metric from the scattering amplitude.

We prove that no randomized integral or fractional algorithm for online vertex cover under general vertex arrivals achieves a competitive ratio strictly below $1+\sqrt{e}/2\approx1.824360635$, even on bipartite graphs and against an oblivious adversary. This improves the previous lower bound of approximately $1.753$. Our proof extends the complete-bipartite alternating construction of Wang and Wong to an arbitrary number of alternations. The resulting adversary is described by a monotone integral recurrence. If the recurrence never violates the competitive budget, its iterates converge to an integrable fixed point; classifying all such fixed points forces the excess ratio to be at least $\sqrt{e}/2$. A truncated discrete recurrence and a Riemann-sum argument convert every strict continuous violation into a finite, algorithm-dependent but realization-oblivious input. We also exhibit a critical fixed point showing that $1+\sqrt{e}/2$ is the exact limit of this homogeneous complete-bipartite recurrence, rather than a numerical artifact.

We develop a primal–dual interior-point method for nonsymmetric conic optimization based on a conjugate-free scaling matrix. The scaling is obtained from a single-secant BFGS update of the primal barrier Hessian. In contrast to multi-secant BFGS scalings, it does not require conjugate-barrier derivatives. This feature is important for high-dimensional nonsymmetric cones, where conjugate-barrier derivatives may be unavailable in closed form or expensive to compute. We embed the conjugate-free scaling in a homogeneous self-dual predictor–corrector framework. Using a split central-path neighborhood that separately controls the conic variables and the scalar homogeneous variables, we prove that the scaling matrix remains uniformly comparable to the primal barrier Hessian. This comparison bound is used to prove neighborhood preservation and to show that the complementarity measure and the linear residual decrease at a uniform rate. Consequently, the method attains an iteration bound of $\mathcal{O}(\sqrt{\nu}\log(1/\varepsilon))$, improving the $\mathcal{O}(\nu\log(1/\varepsilon))$ bound of Badenbroek and Dahl [Optim. Methods Softw., 37 (2022), pp. 1027–1064] and matching the best-known complexity order for interior-point methods. Numerical experiments on instances involving the operator perspective epigraph cone and the quantum relative entropy cone show that the method is competitive with QICS, a specialized solver for conic models arising in quantum information.

This paper studies the asymptotic growth of tensor powers in affine Hecke categories, or equivalently of powers of Kazhdan–Lusztig basis elements in affine Hecke algebras. We prove general bounds in arbitrary affine type, determine precise asymptotics in affine type A1, and establish corresponding results for several natural families in affine type A2.

A closed positively curved space curve is determined by its curvature and signed torsion up to an orientation-preserving rigid motion; that sign is the only place the ambient orientation enters. We ask what survives its loss, for closed embedded curves in $\mathbb R^3$ with $\kappa>0$ compared pointwise in a common arclength label. The answer is governed by the branch invariant $c(\tau)$, the number of components left by the infinite-order zero set of $\tau$: the smooth signed lifts of $|\tau|$ number exactly $2^{c(\tau)}$, and reduce to $\{\tau,-\tau\}$ precisely when $c(\tau)\le1$. Hence a given unsigned datum is carried by at most $2^{c(\tau)}$ classes modulo $SE(3)$, and by a single $E(3)$-orbit when $c(\tau)\le1$. Both extremes occur: for arbitrary knot types $K_1,\dots,K_m$ there is a datum with fibre exactly $2^m$ classes modulo $SE(3)$, realising all connected sums of the $K_i$ and their mirrors; under a chirality hypothesis these are $2^m$ knot types. Conversely, curves with only simple torsion zeros are open and dense, hence residual, among parametrised $C^r$ embeddings ($r\ge4$), and each is determined up to $E(3)$ by its datum. No uniform quantitative form of this rigidity exists; but on each stratum $\Delta=\inf_s\sqrt{\tau^2+(\tau')^2}\ge\delta>0$ with uniform $C^5$ and curvature bounds the orbit distance obeys a log-Lipschitz bound, whose optimal constants diverge as $\delta\downarrow0$ on the strata containing a fixed exact ambiguous pair. The engine is a one-dimensional inverse estimate for the signed square root, logarithmically optimal at that level.

For an edge $uv$ of a finite simple graph $G$, its imbalance is $|d_G(u)-d_G(v)|$, and the imbalance multiset $M_G$ consists of the imbalances of all edges of $G$. Kozerenko and Skochko conjectured that $M_G$ is graphic whenever every edge has positive imbalance. We prove this conjecture. The main ingredient is the following capacity bound: for every set $A$ of $k$ edges, \[ \sum_{e\in E(G)\setminus A}\min\{k,\operatorname{imb}_G(e)\} \ge k\max\{\Delta-k,0\}, \] where $\Delta$ is the maximum degree of $G$. This bound yields all Erdős–Gallai inequalities directly; a parity computation completes the proof.

We study the existence of continuous maps with nonzero Brouwer degree between partial flag manifolds. We prove that every continuous map between distinct complex or quaternionic partial flag manifolds has degree zero if either the domain or the codomain is not a Grassmannian. This establishes, for such partial flag manifolds, the analogue of the results of Ramani-Sankaran and Sankaran-Sarkar for complex and quaternionic Grassmannians, as well as an algebraic-geometric result of Paranjape and Srinivas for complex Grassmannians. As a consequence, we prove that complex and quaternionic partial flag manifolds are cohomologically rigid; that is, their rational cohomology rings determine their homeomorphism types. We conjecture that complex and quaternionic partial flag manifolds are homologically rigid.

Recently, Chen, Liu, and You (An extremal theorem for positive curvature of graphs, arXiv:2607.02297) proved an extremal theorem for positive Lin–Lu–Yau curvature. They further proposed the similar problems for other discrete curvature. In this paper, we prove a sharp extremal theorem for non-negative Bakry–Émery curvature with non-normalized Laplacian: every graph of order \(n\geq 7\) with more than \[ \binom {n}{2}-\lfloor{\frac{n}{2}\rfloor}-2 \] edges satisfies $CD(0,\infty)$, and this threshold is optimal.