Papers

We prove that every action of a countably infinite discrete amenable group on a nonempty compact Hausdorff zero-dimensional space has dynamical comparison, without assuming minimality, freeness, or metrizability. More precisely, strict inequalities under all invariant probability measures imply comparison in the clopen type semigroup with an order-unit remainder. For minimal Cantor actions, the clopen type semigroup is cancellative and almost unperforated, and its canonical map onto the positive cone of the coinvariant group is an isomorphism of ordered monoids. These results answer Melleray's cancellation question and the amenable case of his comparison question. For minimal Cantor actions, we also prove goodness of the associated affine evaluation map and obtain a minimal homeomorphism of the same Cantor space with the same invariant probability measures. We identify the precise obstruction to recovering clopen equidecomposability from affine evaluation alone. This obstruction is given by the infinitesimal subgroup: for every nonempty proper clopen set $A$, it parametrizes the topological-full-group orbits of clopen sets having the same affine evaluation as $A$. Combining comparison with the theorems of Kerr and Szabó, we prove that every free action of a countably infinite discrete amenable group on a nonempty compact metrizable space with the topological small boundary property is almost finite. This settles Naryshkin's almost-finiteness conjecture for finite-dimensional compact metrizable spaces. If the action is also minimal, its reduced crossed product is $\mathcal Z$-stable, has nuclear dimension at most one, and satisfies all the regularity conditions in the Toms–Winter conjecture.

This paper investigates a non-orthogonal multiple-access (NOMA)-assisted multi-user wireless-fed pinching-antenna system (Wi-PASS). A multi-antenna base station (BS) simultaneously serves one direct user and wirelessly feeds a full-duplex amplify-and-forward relay equipped with a directional horn receiver. The relay injects the NOMA waveform into a dielectric waveguide, and one position-adjustable pinching antenna serves two additional users. Under maximum-gain zero-forcing transmission, ideal successive interference cancellation, and an additive residual self-interference model, we minimize the total consumed power by jointly optimizing the BS powers, relay amplification factor, NOMA power coefficients, decoding order, and pinching-antenna position. For a fixed position and decoding order, a variable transformation reduces the resource-allocation problem to a strictly convex scalar problem and yields a closed-form global solution. The position-dependent decoding order partitions the waveguide into finitely many intervals, and the derivative of the optimized power is governed by a quadratic polynomial on each interval. Hence, the globally optimal position is found by evaluating a small finite candidate set. Simulations at 28~GHz over 1000 random user topologies show that the proposed architecture consistently requires the lowest consumed power among direct-transmission, array-fed, no-PASS, and equal-time orthogonal multiple-access (OMA) benchmarks. At target signal-to-interference-plus-noise ratios of 20, 25, and 30~dB, it reduces the average power by 19.2%, 21.1%, and 21.7%, respectively, relative to equal-time OMA, while preserving its advantage as the BS-relay distance and residual SI increase.

Let \(\{X_n:n\ge0\}\) be a controlled branching process with controls \(\phi_n\), and let \(N_n=\phi_n(X_n)\) be its progenitor count. We study large deviations of \(X_{n+1}/N_n\), \(N_n/X_n\), and \(X_{n+1}/X_n\), on their natural positive-denominator events, in supercritical and critical regimes. We treat generation-wide random slopes and asymptotically deterministic first-order slopes. For asymptotically affine random-slope controls, an affine recursion and a random environmental product yield a sharp Bahadur–Rao–Petrov upper-tail asymptotic. A positive Cramér tilt identifies the prefactor via a nontrivial normalized limit, and conditional on the upper-deviation event the three ratios concentrate around their environmental centers. With individual-sum controls, the process is an exact branching process in a random environment with immigration (BPREI). Motivated by this connection, we establish sharp upper large-deviation asymptotics for BPREI and transfer its positive-transform and harmonic-moment profiles to the controlled process. In the nondecreasing supercritical case, harmonic moments exhibit an exact environmental–boundary–persistence trichotomy. In a centered finite-variance critical class, finite-horizon-minimum methods yield exact \(n^{-1/2}\)-scale limits for all positive harmonic orders and for the three ratio-deviation probabilities. For deterministic first-order slopes, the environmental-product mechanism disappears. Matched lower and upper probability-generating-function envelopes transfer auxiliary exact-recursion profiles to controls without an exact branching recursion, producing distinct supercritical and critical harmonic-moment trichotomies. Thus the location of first-order randomness determines the lower-tail mechanism, while immigration may change a sharp constant, a phase boundary, or the decay exponent.

The well-known open problem of Bell and Skandera asks whether a real-rooted polynomial $f(t)$ with positive integer coefficients and constant term one is the $f$-polynomial of a simplicial complex. Mu and Welker proved that if the recursive decomposition $f(t)=g(t)+th(t)$ satisfies the corresponding coefficient inequality $h_i<g_i$, then this open problem has an affirmative answer. Mu and Welker also conjectured that the real-rootedness of $f(t)$ implies that of $g(t)$ and $h(t)$. We give counterexamples to the conjecture of Mu and Welker for every degree at least three, and prove that the assertion holds in degrees $1$ and $2$. Moreover, each polynomial we construct is the $f$-polynomial of a simplicial complex.

PASS provide a flexible waveguide-based architecture for reconfiguring wireless propagation environments and creating geometry-dependent radiating apertures. This paper investigates a near-field multicast ISAC system enabled by a lossy multi-waveguide PASS, where a base station transmits a common message to multiple communication users while simultaneously sensing one or multiple targets. The PA positions along the waveguides and the feed-domain transmit covariance matrix are jointly designed to improve sensing accuracy under multicast communication constraints. We first develop a near-field multicast ISAC signal model that accounts for waveguide attenuation, equal-radiated-power operation, geometry-dependent free-space propagation, and monostatic sensing. Then, we derive the FIM for target parameter estimation and obtain a compact projected-Jacobian representation by exploiting the block-diagonal PA transfer structure. This representation reveals how the PA geometry and transmit covariance jointly affect the CRB. Based on this structure, we further characterize the per-waveguide and cross-waveguide FIM contributions, the loss-aperture tradeoff, the identifiability condition, and the communication-sensing phase conflict. To minimize the CRB, we formulate a joint PA-position and transmit-covariance optimization problem subject to a multicast rate constraint, a feed-power budget, and PA deployment constraints. An alternating optimization algorithm is developed, where the covariance subproblem is solved as a semidefinite program and the PA-position subproblem is handled by waveguide-wise block coordinate descent.

We prove finite time blow-up from smooth, radial, compactly supported initial data for the energy supercritical defocusing nonlinear Schrödinger equation $i\partial_tu+\Delta u-|u|^{p-1}u=0$ in $\mathbb R^d$, for $(d,p)\in\{(3,27),(4,9),(5,7)\}$. The solutions are asymptotically self-similar, their $L^\infty$ norm grows at the scaling rate $(T-t)^{-1/(p-1)}$, and their critical Sobolev norm $\dot H^{s_c}$ diverges. This closes the conjecture of Bourgain \cite{Bourgain2000} on global well-posedness and scattering for the defocusing energy supercritical equation in dimensions three and four. The proof relies on a shooting argument and is computer-assisted.

Let $p>5$ be an odd prime, put $n=(p-1)/2$, and let $\left(\frac{\cdot}{p}\right)$ be the Legendre symbol. Define $D_p^{\pm}$ to be the following determinants \[ \begin{aligned} D_p^{\pm}&:=\det\left[(i\pm j)\left(\frac{i\pm j}{p}\right) \right]_{1\leq i,j\leq n}. \end{aligned} \] In this article, we give explicit formulas for these determinants modulo $p$ in terms of Bernoulli numbers. As a consequence of Reinhart's recent counterexample to the Ankeny–Artin–Chowla conjecture, we find a prime $p$ such that $D_p^{\pm} \equiv 0 \pmod p$. This gives a negative answer to a question of the Sun.

In undersampled inverse problems (such as sparse-view computed tomography), only a small number of measurements are collected, which reduces radiation exposure and acquisition time and cost, and can also address the inaccessibility of certain acquisition arrangements. Most learning-based methods for such problems require supervision in the form of fully sampled measurements and ground-truth images, which are costly or even infeasible to acquire. To overcome this issue, we propose Exact Learning for Inverse Problems from Partial Self-supervision (ELIPPS), an incomplete self-supervised training paradigm for undersampled inverse problems that requires neither ground-truth images nor fully sampled measurements. ELIPPS learns solely from incomplete forward measurements on a fixed incomplete supervision set. Our theory shows that when the data distribution is invariant under certain transformations, minimizing a masked empirical risk is equivalent to minimizing the full self-supervised risk, i.e. the risk against the complete, noise-free measurement. The equivalence is exact for arbitrary equivariant hypothesis classes and holds for signal-dependent noise such as the pre-log Poisson statistics of low-dose tomography, and for the log-transformed count model up to a quantifiable bias. When the underlying coverage condition is only approximately satisfied on a discrete grid, we give a stability estimate in terms of the associated frame constants and the irreducible error of the inverse problem. We realize ELIPPS for computed tomography by exploiting rotation and reflection invariance. In our experiments, ELIPPS substantially outperforms naive masked supervision and reaches the same order of accuracy as a reference model trained with full clean measurements.

Let $T$ be a bounded self-adjoint operator on a separable infinite-dimensional right quaternionic Hilbert space, and let $\sigdS(T)$ be the set of right eigenvalues of finite type. We prove that Weyl's theorem holds for $T$: $$\sigeS(T)=\sigS(T)\setminus\sigdS(T),$$ which completes the inclusion obtained in the general case. The proof proceeds through a single three-way criterion: a point of $\sigS(T)$ lies outside $\sigeS(T)$, and equally lies in $\sigdS(T)$, exactly when it is isolated with a spectral atom of finite rank. The natural route through Riesz projections is not available as it stands, because a Riesz projection need not be orthogonal; we show that for a self-adjoint operator it in fact is, the Riesz projection of an isolated spectral part coinciding with the corresponding spectral projection, so that the algebraic and geometric multiplicities agree. A $2\times2$ quaternionic matrix shows that this last identity fails without self-adjointness, and a normal operator with genuinely spherical $S$-spectrum shows what has to change in the normal case.

Remote source monitoring is a key use case for Internet of things (IoT) applications, calling on efficient communications protocols to ensure timely data delivery. In this paper, we consider a practically inspired scenario in which an IoT device reports sensor readings to a gateway over a correlated, lossy channel, receiving feedback on the transmission outcome only after some delay. For this setting, we investigate practical threshold-based reporting strategies for monitoring a Markov process, and evaluate two approaches: a reactive strategy where the node awaits feedback after an update, and a proactive strategy that preemptively transmits a second update. Leveraging Markov renewal processes, we provide an exact performance analysis in terms of the mean squared error (MSE). Our study characterizes the fundamental trade-off between estimation error and communication rate, revealing that proactive transmissions significantly improve MSE even under strict IoT rate constraints.

Rigorous error analysis for numerical approximations of parabolic equations with hysteresis remains limited, even for the widely used Prandtl–Ishlinskii model of play type. In this work, we establish an $O(h+\tau)$ error bound for an implicit Euler $P_1$ finite element discretization. The analysis requires neither higher-order temporal regularity nor additional spatial regularity of the hysteresis variables. For the temporal discretization, we exploit a convex subgradient-flow structure in a weighted Hilbert space and estimates of residual monotone terms to obtain first-order convergence. For the spatial discretization, only the diffusive field is restricted to the finite element space, and a constraint-preserving comparison yields an $O(h)$ semidiscrete estimate. The analysis is developed for play-type Prandtl–Ishlinskii operators formulated directly on a spatial Hilbert space, encompassing the canonical pointwise extension as well as more general spatially structured constraints.

Let $\widetilde G$ be a nonlinear double cover of the real points of a connected, simply connected, semisimple complex group. In previous work, we introduced $\prod_{\rho/2}^s(\widetilde G)$, the set of genuine small representations with infinitesimal character $\rho/2$ in the simply laced setting, and showed that, when $G$ is split, these representations are exhausted by the Kazhdan–Patterson lift of the trivial representation. In this paper, we consider nonsplit real Spin groups of type $D_n$. For each $\rho/2$-compatible genuine central character, we determine the irreducible constituents of $\mathrm{Lift}_G^{\widetilde G}(\mathbb C)$. We also describe explicitly the relevant genuine small representations in terms of Langlands parameters. The nonsplit case exhibits a new phenomenon: the lift of the trivial representation need not exhaust $\prod\nolimits^s_{\rho/2}(\widetilde G)$. For the families considered here, equality holds only for $\widetilde{\mathrm{Spin}}(2p+2,2p)$; in the other two families the lift is a proper subset, and in one case it is zero.

In 1928 H. Cartan stated a conjecture about holomorphic curves in $\mathbb{P}^n$ parametrized by the unit disc, omitting $n+2$ hyperplanes in general position. He proved it for $n=2$. In 1996 the first-named author constructed counterexamples for all $n\geq 3$, and proposed a modified form of Cartan's conjecture which he proved for $n=3$. In this paper we prove this modified conjecture in all dimensions. As a byproduct we prove a quantitative version of the classical theorem that analytic functions are linearly dependent if and only if their Wronskian determinant is zero. We interpret our results in terms of the Kobayashi–Royden pseudometric on the complement of $n+2$ hyperplanes, and on smooth algebraic varieties in the algebraic torus.

Let $\mathfrak R=\nabla(-\Delta_D)^{-1/2}$ be the Riesz transform associated with the Dirichlet Laplacian on the upper half-space $\mathbb R^n_+$, and let $H^1_r(\mathbb R^n_+)$ denote the Hardy space of restrictions. We establish an endpoint theory for commutators with $\mathfrak R$ on $H^1_r(\mathbb R^n_+)$. For every $b\in \mathrm{BMO}(\mathbb R^n_+)$, the commutator $[b,\mathfrak R]$ extends boundedly from $H^1_r(\mathbb R^n_+)$ to $L^{1,\infty}(\mathbb R^n_+)$. We then prove the sharp strong-endpoint characterization \[ [b,\mathfrak R]:H^1_r(\mathbb R^n_+)\to H^1_r(\mathbb R^n_+) \quad\Longleftrightarrow\quad b\in \mathrm{BMO}^{\log}(\mathbb R^n_+). \] Moreover, the $\mathrm{BMO}^{\log}$ norm is quantitatively equivalent to the sum of the usual $\mathrm{BMO}$ norm and the $H^1_r$ operator norms of the component commutators $[b,\mathfrak R_j]$. The logarithmic condition records the distance to the boundary and is therefore intrinsic to the Dirichlet geometry.

Consider a constraint satisfaction problem $\mathbf{C}$ on finitely many independent random variables with dependency graph $G$. Let $p_a$ be the violation probability of a constraint $a\in \mathbf{C}$ and $N_G^2 (a)$ the set of constraints at distance one or two from $a$ in $G$. Suppose that, there exists $x\in (0,1)^{\mathbf{C}}$ such that, for a sufficiently small universal constant $c > 0$, and for all $a \in \mathbf{C}$, \[ p_a \leq c \cdot x_a \prod_{b\in N_G^2(a)}(1-x_b). \] Under the above analog of the asymmetric Lovász Local Lemma, we give an FPRAS for the probability that all constraints are satisfied, and an approximate sampler, running in polynomial expected time, for the product distribution conditioned on this event. The degree of the polynomial in the running time is independent of the domain sizes, constraint sizes, or degree of the dependency graph. Up to the choice of the constant $c$, our condition on $p_a$ matches known hardness results. Our work builds on the method of Liu, Wang, Yin, Zhang, and Zhou, who obtained an FPRAS for the probability of satisfaction in the setting of the symmetric Lovász Local Lemma. Our sampling result is new even in this special case.

We study closed range and essential norms of bounded composition operators $C_\varphi$ on weighted Dirichlet spaces $\mathcal{D}_{\alpha}$. For $-1<\alpha<0$, we first consider maps whose images are obtained by removing a compact subset from a simply connected subdomain of $\mathbb{D}$. In this setting, closed range is equivalent to the Reverse Carleson property and the Reverse Carleson Disk Condition for the counting measure $\mu_{\varphi,\alpha}$, uniform weighted area density of the image, $\mathbb{T}\subset\overline{\varphi(\mathbb{D})}$, and the inclusion of an outer annulus in the image. We also obtain a closed range characterization under the Uniform Tail Condition. For all $\alpha>-1$, we establish two-sided essential norm estimates in terms of local averages of the generalized Nevanlinna counting function and an exact formula using boundary tail integrals. If the normalized counting density has vanishing oscillation at the boundary, we obtain exact formulas in terms of its boundary limsup, Berezin transform, and local averages.

M. E. Zieve conjectured that a candidate datum over the sphere is realizable whenever the gcd of the parts of every branch partition is one and it satisfies the lcm condition. We prove that this conjecture reduces to its three-branch-point case. More precisely, if every Zieve-admissible triple is realizable, then every Zieve-admissible datum with at least four branch points is realizable. In particular, Zieve's conjecture implies the prime-degree conjecture posed by A. L. Edmonds, R. S. Kulkarni and R. E. Stong. Generative-AI disclosure. OpenAI's GPT-5.6 Sol and GPT-6 Astra assisted throughout the research and preparation of this article, including the exploration of reduction strategies, proof development and checking, consistency checks on notation and terminology, and manuscript drafting and revision. The authors reviewed all AI-assisted material with careful scrutiny and take full responsibility for all mathematical claims and for the final text.

Let $(\mathcal X, d, \mu)$ be a space of homogeneous type and let $\rho$ be an admissible function on $\mathcal X$. In this paper, we introduce a new class of singular integral operators associated with $\rho$, including a wide range of operators arising in harmonic analysis. For such an operator $T$ and a function $b$ belonging to suitable localized $\mathrm{BMO}$ spaces associated with $\rho$, which are strictly larger than the classical space $\mathrm{BMO}(\mathcal X)$, we establish the boundedness of the commutator $[b, T]$ from the Hardy space $H^1_\rho(\mathcal X)$ into $L^{1,\infty}(\mathcal X)$, $L^1(\mathcal X)$, and $H^1_\rho(\mathcal X)$. Moreover, the boundedness of $[b,T]$ on $H^1_\rho(\mathcal X)$ is characterized by necessary and sufficient conditions. We then apply this to investigate the boundedness of commutators of singular integrals in various settings, including Schrödinger operators on stratified Lie groups and Laguerre operators of convolution type. Our results are new even for Schrödinger operators on $\mathbb R^n$.

In this paper we find conditions on the PT-symmetric non-self-adjoint Dirac operator L(Q) with matrix-valued coefficients Q whose entries are PT-symmetric periodic functions for which the number of gaps in the real part of its spectrum is finite.

Let $\Omega\subset\mathbb C^n$ be a smoothly bounded convex domain of finite type. We characterize absolute summability on its Bergman spaces and estimate approximation errors in the summing norm. For $1<p<\infty$ and $1\le q,r<\infty$, we characterize absolutely $r$-summing Carleson embeddings $A^p(\Omega)\to L^q(\mu)$ by diagonal operators formed from normalized masses on McNeal balls. For $1<p,q<\infty$ and $1\le r<\infty$, we obtain a corresponding big Hankel criterion using local $L^q$ distances to holomorphic functions, with the appropriate volume weights. Both criteria give upper and lower norm estimates. The proofs combine uniform local nuclear estimates, kernel synthesis, and an analytic decomposition at two fixed scales. Local Taylor polynomials give explicit rank and error bounds for the embeddings. These bounds transfer to general Hankel operators when $p=2$ or $q=2$. Suitable atomic embeddings have the same best approximation errors as their diagonal models, up to constants. A family of Hankel operators contains a complemented copy of the diagonal ideal and has matching approximation rates for power sequences in the Hilbert case. For positive Toeplitz operators on finite-type ellipsoids, we compute exact spectral asymptotics in three regimes, including the critical logarithmic factor and the leading constants for geometric symbols. These asymptotics give optimal approximation rates in the summing norm. Applications to composition and Volterra operators include the target endpoint $q=1$.

We describe what a typical covering of a closed 2-orbifold looks like on a small scale. Specifically, we prove that a uniformly random covering of degree n contains in expectation $\Theta(n^0)$ short closed geodesics, potentially $\Theta(n^0)$ pairs of nearby order-2 cone points, in general $\Theta(n^{\frac{1}{m}})$ cone points of order m, and with high probability no small regions with more complicated topology. This generalises many results of Magee and Puder (2023) from orientable surfaces to possibly non-orientable orbifolds, and answers some of the questions raised by Puder and Zimhoni (2024).

We prove that every smooth Fano threefold of characteristic p>5 is quasi-F-split. Moreover, if p>11, then it is F-split.

We construct a polynomial invariant of planar curves via skein relations, which generalizes Arnold's numerical invariant $\operatorname{St}$.

Let $X$ be a smooth minimal projective variety of general type of dimension $n\geq 4$ over $\bb C$. We prove that $|\frac{n^2+n}{2}K_X|$ is base point free. This gives a new bound for the pluricanonical case of Fujita Freeness Conjecture for nef and big canonical divisors.

Motivated by Mader's conjecture on connectivity keeping trees, we study trees obtained from paths by adding one pendant vertex, as well as related problems in triangle-free graphs. For an integer $m$ and $1\leq i\leq m-1$, let $P_m^+(i)$ denote the tree obtained from a path of order $m-1$ by adding one pendant vertex adjacent to its $i$th vertex. We prove that, for positive integers $k,m,1\leq i\leq m-1$, every $k$-connected graph $G$ with $\delta(G)\geq \lfloor \frac{3k}{2}\rfloor+m-1$ contains a subgraph $T\cong P_m^+(i)$ such that $\kappa(G-V(T))\geq k$. This confirms Mader's conjecture for all pendant extensions of paths. For highly connected triangle-free graphs, a connectivity keeping result for paths was obtained in [J. Combin. Theory Ser. B, 174 (2025), 190-206]. Let $(X,Y)$ be the bipartition of $P_m^+(i)$. We further prove that every $k$-connected triangle-free graph $G$ with $\delta(G)\geq k+\max\{|X|,|Y|\}+[P_m^+(i)\text{ is bad}]$ contains a subgraph $T\cong P_m^+(i)$ such that $\kappa(G-V(T))\geq k$, where we use Iverson's convention for $[P_m^+(i)\text{ is bad}]$. This extends the corresponding result for paths to pendant extensions of paths.

We study the minimum time for perfect state transfer under polynomial Hamiltonians with bounded degree and spectral width. For a strongly cospectral pair and width bound $W$, the optimum, when finite, is an integer multiple of $\pi/W$, determined by integer interpolation with prescribed parities. For equally spaced supported eigenvalues with alternating signs, we give degree bounds under which every minimizer is affine, and sharp asymptotics for each fixed exact degree. Near-minimizing phase polynomials satisfy a quantitative Chebyshev stability estimate. We determine the optimal transfer time for every degree bound on hypercubes of odd prime dimension. For complementary vertices of $J(2m,m)$, the optimal time at fixed spectral width grows exponentially in $m$ throughout an interval of feasible degrees. We also construct polynomial Hamiltonians showing that every feasible degree $m-t$ with $t=o(m)$ admits subexponential transfer time.

We prove that every extremal Kähler–Ricci soliton on a Fano manifold is Kähler–Einstein. This solves the problem of Calamai and Petrecca in full generality.

We prove the face-number conjecture of Hibi and Li: for every finite poset, the order polytope has no more faces of any given dimension than the chain polytope. More generally, the face numbers increase weakly along the admissible family of chain-order polytopes when one order coordinate is replaced by a chain coordinate.

The inverse scattering problem for the one-dimensional nonlinear Dirac equation \begin{align*} \begin{cases} i(u_t+u_x)+v=\mathcal{N}_1(u,v),\\ i(v_t-v_x)+u=\mathcal{N}_2(u,v) \end{cases} \end{align*} is studied. We assume that the unknown nonlinearities $\mathcal{N}_j(u,v)$ ($j=1,2$) of the equation belong to $C^\infty(\mathbb{C}^2;\mathbb{C})$ and satisfy $(\partial_\mathbf{z}^\alpha \mathcal{N}_j)(\mathbf{z})=O(|\mathbf{z}|^{\max\{ 5-|\alpha|,0 \}})$ ($|\mathbf{z}|\to 0$) for any multi-index $\alpha \in (\mathbb{N} \cup \{ 0 \})^4$. Here, $\mathbf{z}=(z_1,z_2)\in \mathbb{C}^2$, $ \partial_\mathbf{z}^\alpha= (\partial_{z_1},\partial_{\overline{z_1}},\partial_{z_2},\partial_{\overline{z_2}})^\alpha, $ and $\partial_{z_1},\partial_{\overline{z_1}},\partial_{z_2},\partial_{\overline{z_2}}$ are the Wirtinger differential operators. Under some additional assumptions on $\mathcal{N}_j$, we establish a reconstruction formula for $(\partial_\mathbf{z}^\alpha \mathcal{N}_j)(0)$ ($|\alpha|\ge 5$) using the knowledge of the scattering operator for the equation. Although such results were established for the nonlinear Schrödinger equation [Sasaki 2024] and the nonlinear Klein-Gordon equation [Sasaki 2025], the methods in [Sasaki 2024, 2025] are not directly applicable to the nonlinear Dirac equation due to its system nature. Specifically, it is quite difficult to prove the invertibility of a matrix associated with the input data. To overcome this difficulty, we introduce a new method based on the massless limit for the free solutions.

We prove that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive Kähler–Einstein manifold exists for all time, remains embedded, and converges smoothly and graphically to a minimal Lagrangian, under natural exactness and regularity assumptions. We also study the generalised Lagrangian mean curvature flow of Behrndt in Kähler manifolds which are almost-Einstein in the sense that the Ricci form satisfies $\rho = C\omega + ndd^cf$, and which satisfy $C>0$. With analogous assumptions on the flow, we obtain subconvergence to an $f$-minimal Lagrangian submanifold, with an upgrade to smooth graphical convergence in the case that $(M,g,f)$ is analytic. The proof proceeds by first reducing the flow to a weighted curve shortening flow on a compact two-dimensional orbifold. We then establish both a Grayson-type description of finite-time singularities and a long-time subconvergence theorem for weighted curve shortening flow in the orbifold setting. Finally, we use a Łojasiewicz–Simon inequality argument to upgrade to smooth convergence of the full flow.