Papers

In this contribution, we investigate 2-dimensional model problems of three coupled equations. We assume that the underlying Hamiltonian presents a symmetric intersection between three eigenvalues of Dirac's type with a mass function that vanish along a curve. We investigate the existence of edge modes for these models: such functions are solutions of the semiclassical associated evolution problem that are asymptotic to a coherent state with zero energy and break the correspondence principle by propagating inside the crossing set and not along a classical trajectory.

Euler's Gradus Suavitatis (1739) assigns a dissonance value to a musical interval p/q by the formula G(p/q) = 1 + \Omega^(p) + \Omega^(q), where \Omega^(n) = \sum_i e_i(p_i - 1) sums the weighted prime exponents of n. We propose the simpler asymmetric formula f(p/q) = p + \Omega^(q), which treats numerator and denominator differently and performs comparably on standard consonance data. We also show that, under a model in which harmonics are integer-indexed and counted uniformly up to a fixed truncation level, Gradus is equivalent to a weighted harmonic coincidence count with weights w(n) = \Omega^(n), connecting it to Galileo's earlier pulse-coincidence model (1638). The formula naturally generates a coprime integer triangle T(n,k) = n + \Omega^(k), whose rightmost diagonal gives the two-stage dissonance of the superparticular (consecutive-harmonic) intervals. The formula f admits a simple two-stage interpretation in terms of harmonic context and partial recognition, which we offer as a speculative perceptual hypothesis.

For $p>1$ and a $p$-energy on a volume doubling metric measure space, we prove the Poincaré inequality under the elliptic Harnack inequality, two-sided capacity bounds, and additional geometric and analytic assumptions. Combining this with our previous results, we obtain, under the same structural assumptions, the equivalence between the elliptic Harnack inequality together with two-sided capacity bounds and the conjunction of the Poincaré inequality and the cutoff Sobolev inequality.

Let ${\ell}$, p $\ge$ 5 be primes such that p | (${\ell}$ -1). Let $\Delta$ > 0 be the fundamental discriminant of a real quadratic field in which ${\ell}$ splits. We denote by h - ${\ell}$ ($\Delta$) the order of the minus part (for the Galois action) of the ray class group of Q( $\sqrt$ $\Delta$) of modulus ${\ell}$. In this paper, we study the indivisibility of h - ${\ell}$ ($\Delta$) by p, and prove that under the assumption that this set is non-empty. This lower bound is made unconditional if ${\ell}$ = 2p + 1, i.e. if p is a Sophie Germain prime. Our result can be viewed as being in the continuity of the results of Kohnen-Ono, Ono, Byeon, Beckwith etc. regarding the class numbers of quadratic fields, in the sense that we rely on techniques from the theory of half-integral weight modular forms. Significant difficulties however arise in our study, as we have to study Eisenstein congruences for cuspforms of weight 3 2 , and use a generalized Shimura correspondence of Baruch-Mao. Combined with the results of Lecouturier-Wang, our result has implications eg. for the 5-part of BSD for even quadratic twists of X 0 (11).

We study some basic properties of spaces which satisfy the usual axioms of a metric space but without the symmetry axiom. We realised that such a study is needed in view of the relatively recent appearance of several papers on natural asymmetric metrics to which the available theories do not apply. One prominent example is Thurston's metric on the Teichmüller spaces of hyperbolic surfaces of finite type, introduced by Thurston in 1985, with several variants and generalisations. Another asymmetric metric is the earthquake metric, also introduced by Thurston and about which several basic questions remain open. Other asymmetric metrics we consider here include the Funk metric, the Apollonian metric and the left Hausdorff metric. We are particularly interested in questions of completeness and completion and in associated notions of boundary at infinity for these asymmetric metrics. In this paper, after discussing several examples, we prove results such as a Banach fixed point theorem, an Arzelà–Ascoli theorem and a Hopf–Rinow theorem adapted to this asymmetric setting. We introduce a property we call “convergence-symmetry” which turns out to be crucial in the study of metric completeness of some asymmetric metrics. This property is stronger than a property which was formulated by Herbert Busemann around 1970, and which we call the “Busemann condition”. Every convergence-symmetric metric space has a unique minimal completion which is convergence-symmetric. This does not hold for spaces satisfying Busemann's condition. Several examples we consider satisfy Busemann's condition but are not convergence-symmetric. We have included throughout the paper a certain number of open questions.

Information criteria such as WAIC and WBIC extend model selection to singular learning machines, but they are usually derived for equilibrium posteriors. We formulate a finite-time analogue of WBIC by replacing the equilibrium posterior with an effective ensemble generated by learning dynamics under a resource constraint. When this ensemble admits an analytic effective potential, singular learning theory yields a resource-dependent real log canonical threshold. The resulting estimator gives a computable thermodynamic contribution to time-bounded MDL and identifies the finite-time singular complexity relevant to the structural information measured by epiplexity.

The Brody distribution, originally a phenomenological interpolation between Poisson and Wigner level-spacing statistics in quantum chaos, is calibrated here as a quantitative measure of short-range exclusion in 2D spatial point processes. Two results form the core. First, the 2D complete-spatial-randomness baseline is recalibrated to $\beta=0.96\pm0.15$, correcting the inappropriate 1D Poisson reference. Second, an empirical $\beta$–$r_{\text{excl}}$ calibration is validated against the effective hard-core radius with Spearman $\rho=0.988$. The framework is demonstrated on 58 manufactured surfaces (10 materials, 10 processes), phase-extracted interferometric profilometry of a certified roundness standard, and 2D binary embeddings of prime numbers. A sparse-integer control proves the prime $\beta=2.15$ signal is genuinely arithmetic ($\Delta\beta=+0.68$ over random-integer control), while a Cantor-embedding null result ($\beta=1.40$, TOST $p<0.01$) demonstrates that 2D exclusion is embedding-created rather than intrinsic. Density-thinning experiments establish that $\beta$ captures exclusion strength rather than point density, while absolute values are density-dependent. A distinct CSR baseline for binary fields at low fill fraction is identified, with a decision table provided. The $\beta$–$r_{\text{excl}}$ calibration, the CSR baseline correction, and the control protocols together constitute a calibrated measurement framework for reproducible characterisation of short-range exclusion in 2D spatial point processes.

We consider the inverse problem of determining an inclusion contained in an elastic body undergoing time-harmonic oscillations at fixed frequency by local Cauchy data. Both the body and the inclusion are made by different homogeneous linearly elastic isotropic materials, with piecewise constant mass densities. Under mild a priori regularity assumptions on the unknown inclusion and with no spectral hypothesis on the frequency, we establish a logarithmic-type stability estimate by local Cauchy data. We also provide a stability result in terms of the so local Dirichlet to Neumann map for sufficiently small frequency.

We study mirror flows as Banach-space counterparts of Hilbertian gradient flows and show that they play an essential role in $p$-Laplacian eigenvalue problems on metric measure spaces. Under a Rellich–Kondrachov type compactness assumption and for $p\ge2$, we establish a Ljusternik–Schnirelman type existence theorem without assuming $C^1$-regularity of the associated energy. More precisely, we prove that every element $\lambda$ of the Krasnoselskii spectrum is an eigenvalue of the $p$-Laplacian $\Delta_p$; that is, there exists a nontrivial solution $f$ to $\Delta_p f=-\lambda |f|^{p-2}f$.We also investigate the large time behavior of the corresponding mirror flow and prove its convergence to an eigenfunction when the eigenvalue is simple and isolated.

Managing autonomous vehicle platoons requires a delicate balance between string stability and rigorous safety. This challenge is intensified by aggressive transients, such as highway merging. Although Constant Time Headway (CTH) spacing is the industry standard for Cooperative Adaptive Cruise Control, it lacks formal safety guarantees during significant velocity deviations. This letter proposes a computationally efficient control framework that considers a linear time-invariant model for the dynamics of each vehicle, while ensuring formal transient safety and stability. By introducing a spacing policy that naturally converges to CTH at steady state, we establish platoon safety as an inductive property. We derive a non-linear and saturated control law for the lead follower and provide sufficient initial conditions to guarantee velocity non-negativity and safety throughout the platoon for any CTH-based followers' control law. Numerical examples indicate the proposed methodology may be applicable even under non-nominal setups.

We study spectral concentration bounds for kernel matrices as approximation of the corresponding kernel integral operator. Results are established under weak assumptions on the data setting and the reproducing kernel relying only on a Mercer condition and a local Weyl law. This allows us to deal with key features of kernel matrices, such as large multiplicities, large effective dimension, and heavy-tailed distributions. Our results apply to infinite dimensional principal component analysis, manifold learning, and Bayesian nonparametric statistics. We illustrate this via two prototypical examples: The heat kernel on the sphere and a wavelet prior from Bayesian nonparametrics.

A matching $M$ in a graph $G = (V, E)$ is a set of edges such that no two edges in $M$ share a common vertex. A matching with maximum cardinality is called a maximum matching and its cardinality is the matching number $\gamma(G)$. The spectral radius of $G$ is the maximum absolute eigenvalue of its adjacency matrix. This article addresses the Brualdi-Solheid problem–the determination of extremal spectral radii within specific graph classes–for the class $\mathcal{U}_{n,\gamma}$ of simple connected unicyclic graphs on $n$ vertices with matching number $\gamma$. We specifically characterize all graphs that achieve the minimum spectral radius in $\mathcal{U}_{n,\gamma}$ for matching numbers $\gamma \in \left\{ 1, 2, 3, \lfloor \frac{n}{2} \rfloor \right\}$.

We prove the global existence of segregated weak solutions of a one-dimensional degenerate cross-diffusion system with independent drifts, which is endowed with a Wasserstein gradient flow structure. We argue by a Lagrangian formulation written in terms of the (pseudo-)inverse for the cumulative mass function of the sum of the species, which we solve by a Minimising Movement Scheme in the setting of $L^2 \cap BV_{\mathrm{loc}}$. This Lagrangian problem gives rise to a parabolic PDE similar to a $p$-Laplace equation, with typical range $p \in (-\infty,1)$. We employ monotonicity methods \emphà la} Minty–Browder to obtain strong convergence and pass to the limit $\tau \to 0$ in the time-step of the discrete scheme. Our contribution simultaneously treats all porous medium degeneracies, the log-entropy, and fast diffusions of index $\alpha \in (\frac{1}{3},1)$.

Deep neural networks have achieved strong performance in medical image classification, but often work like black-box. Commonly used post-hoc interpretation methods often provide heuristic visualizations whose relationship to the classifier's predictive distribution is indirect. This work introduces a local sensitivity analysis framework based on the input-dependent Fisher Information Matrix (iFIM) of a trained classifier. The iFIM characterizes how the classifier's predictive distribution changes under infinitesimal perturbations of the input image. By using a Gram-matrix formulation, the nonzero eigenspectrum of the iFIM can be recovered without explicitly forming the full image-dimensional Fisher matrix. The leading iFIM eigenspace is then used to project an input image into a high local-sensitivity component and its orthogonal component. These components provide a model-intrinsic description of local predictive sensitivity, rather than a conventional pixel-wise attribution heatmap or a causal segmentation of task-relevant anatomy. The framework is evaluated on controlled and clinical medical image classification tasks using multiple classifier architectures. Perturbation-based experiments show that high-sensitivity iFIM components are more strongly coupled to changes in predictive confidence and classification performance than lower-sensitivity complementary components. The results support the iFIM framework as a principled tool for analyzing local decision sensitivity and for complementing existing attribution-based interpretability methods in medical imaging.

Hernandez and Leclerc introduced the notion of monoidal categorification of cluster algebras. We define similarly the notion of monoidal categorifications of generalized cluster algebras: an abelian monoidal category $\mathcal M$ is said to be a monoidal categorification of a generalized cluster algebra $\mathcal A$ if the Grothendieck ring of $\mathcal M$ is isomorphic to the upper generalized cluster algebra $\mathcal A^{\mathrm{up}}$, and if cluster monomials (resp. cluster variables) of $\mathcal A$ correspond to classes of real simple (resp. real prime simple) objects of $\mathcal M$. Let $\varepsilon$ be a root of unity such that $\varepsilon^{2\ell}=1$ for some $\ell\in\mathbb{Z}_{\geq 2}$. Denote by $\mathcal{C}_{\varepsilon}$ the category of finite-dimensional modules of the restricted quantum loop algebra $U_\varepsilon^{\res}(L\mathfrak{sl}_k)$ at root $\varepsilon$ of unity, and let $\mathcal{C}_{\varepsilon, \xi}$ be a full subcategory of $\mathcal{C}_{\varepsilon}$ determined by a bipartition $\xi: I \to \{0,1\}$ of the Dynkin diagram. For $k=3$, Gleitz conjectured that the Grothendieck ring of $\mathcal C_{\varepsilon,\xi}$ is isomorphic to a generalized cluster algebra of rank $2\ell-2$, and that generalized cluster monomials correspond to classes of simple modules. This conjecture is a special case of a more general conjecture of Fraser. In this paper, we prove the first part of Gleitz's conjecture. More precisely, for $k=3$ and arbitrary $\ell\ge2$, we prove that the Grothendieck ring of $\mathcal C_{\varepsilon,\xi}$ is isomorphic to a generalized cluster algebra of rank $2\ell-2$. We also classify the real Kirillov–Reshetikhin modules of $U^{\mathrm{res}}_\varepsilon(L\mathfrak{sl}_3)$ and obtain mutation sequences for the real Kirillov–Reshetikhin modules from the initial seed of the generalized cluster algebra.

Cizma and Linial asked for a classification of the metrizable theta graphs. We solve both their problem and its strict analogue. For $a\le b\le c$, the theta graph $\Theta_{a,b,c}$ is metrizable if and only if $a\le 2$ or $(a,b,c)=(3,3,3)$, and it is strictly metrizable if and only if $a\le2$. Thus $\Theta_{3,3,3}$ is precisely the exceptional theta graph that is metrizable but not strictly metrizable. The negative directions follow from the known obstructions $\Theta_{3,3,4}$ and $\Theta_{3,3,3}$ together with topological-minor closure. The positive direction is constructive. Consistency turns the possible detours through a length-two arm into compatible Ferrers relations, which are represented by one-dimensional potentials; all resulting shortest-path comparisons have positive slack. The exceptional ordinary-metrizable graph $\Theta_{3,3,3}$ is handled by a two-threshold weak Ferrers representation. The proof is structural, yields rational edge lengths algorithmically, and uses no enumeration of path systems.

We investigate normality and $\phi$-normality criteria for harmonic mappings in the unit disk. A new sufficient condition for normality involving extended spherical derivatives is established. We further prove a Zalcman-Pang type rescaling lemma for $\phi$-normal harmonic mappings and derive new $\phi$-normality criteria as applications. In addition, we introduce $\phi$-normal families of harmonic mappings and obtain corresponding Lappan-type characterizations. In particular, we show that for sense-preserving harmonic mappings the relevant test set may be taken to consist of only three points.

We prove that every $n$-dimensional lattice simplex $P$ whose lattice length $L(P)\ge n-1$ is integrally closed. As an application, we obtain a simple criterion for the projective normality of ample line bundles on $\mathbb{Q}$-factorial toric Fano varieties with Picard number one. We further obtain a refinement of this result in terms of the invariant $\Gamma_{P}$.

Predicting the initiation and propagation of cracks in heterogeneous wellbore systems under complex in-situ conditions remains challenging. We present a hybrid phase-field fracture framework to model crack growth in heterogeneous wellbore systems with weak interfaces. The framework is first validated against benchmark problems with available analytical and numerical solutions. Subsequently, numerical experiments are conducted to isolate the effects of interface strength and casing eccentricity on crack growth. The results show that casing eccentricity strongly influences both the pressure at crack initiation and the resulting crack paths, reducing the crack initiation pressure by up to 30% relative to the concentric configuration. For the examples considered here, beyond a critical eccentricity threshold of 50%, localized shear stresses drive the nucleation of inclined cracks in the rock formation in addition to radial cracking – a failure mode absent in concentric configurations. For sufficiently weak interfaces (i.e., interfaces with 30% of the toughness of the surrounding bulk material), radially propagating cracks in the cement sheath are deflected along the interface rather than penetrating into the rock formation. This deflection delays stress relaxation within the sheath, promotes the nucleation of additional radial cracks, and increases the risk of sustained casing pressure and wellbore failure. Finally, a three-dimensional simulation reveals depth-dependent crack nucleation, stress-shadow effects that suppress full-depth crack growth, and crack coalescence along the cement-rock interface – phenomena that are fundamentally inaccessible under plane-strain assumptions – demonstrating the applicability of the framework to realistic heterogeneous wellbore systems.

We present a domain-extension framework for output-feedback boundary stabilization of reaction-diffusion equations on arbitrary bounded Lipschitz domains, including non-convex and multiply connected geometries. The plant is posed on an irregular domain whose boundary has actuated and uncontrolled portions. Just as backstepping transforms the plant dynamics into a stable target system, the method embeds the plant in a target domain, such as a ball or a rectangle, where a stabilizing design is already known. Every boundary portion through which the extension proceeds must carry actuation and the complementary collocated measurement. Uncontrolled portions are allowed when they are shared with the target boundary and have the same boundary-condition type. The gap between the two domains is filled with a virtual copy of the plant dynamics, coupled to the plant through interface conditions, and the concatenated state evolves exactly as the known closed loop on the target domain. Well-posedness and exponential stability of the physical state follow by restriction. The offline design data are inherited from the target design and are closed-form for constant-coefficient plants on balls and rectangles. Online simulation of the virtual PDE has the same computational character as a full-order PDE observer, a standard component of output-feedback designs. A new explicit Neumann-actuated backstepping law on n-balls enlarges the available target designs. Output feedback is obtained by lifting the target-domain observer, driven by a collocated interface measurement relayed through the virtual domain. Numerical experiments on star-shaped, horseshoe, and multiply connected domains, with a partitioned plant/controller implementation and a shared-wall cavity, test the designs.

We study the mixed local-nonlocal operator $\mathcal{L}_{p,q} := -\Delta_p + (-\Delta)_q^s$ in the isocritical regime $p^* = q_s^*$, i.e. $1 - N/p = s - N/q$, under which both operators become critical for the same nonlinearity. We consider \[ -\Delta_p u + (-\Delta)_q^s u = |u|^{p^*-2}u \qquad \text{in } \mathbb{R}^N, \] with $N \geq 2$, $1 < p < N$, $0 < s < 1$, $1 < sq < N$. In this regime the energy space reduces to $\mathcal{D}_0^{1,p}(\mathbb{R}^N)$, and both best Sobolev constants enter the variational structure simultaneously. We prove: $(i)$ existence of a nonnegative radial ground state via Nehari manifold methods and a double-threshold concentration-compactness analysis; $(ii)$ a logarithmic energy estimate, weak comparison principle, and strong maximum principle for all admissible exponents; $(iii)$ a weak Harnack inequality; and $(iv)$ sharp two-sided decay $U(x) \asymp |x|^{-(N-p)/(p-1)}$ for positive radial solutions, matching the fundamental solution of the $p$-Laplacian.

We consider nonhomogeneous kinetic equations that involve a free transport operator and a diffusion of porous medium type acting on velocities. The main novelty is a gradient flow interpretation of dynamics driven by an interplay of conservative and dissipative effects. We rely on a notion of discrepancy adapted to a phase space of positions and velocities, built upon second-order characteristics obeying Newton's laws. The equation appears as the steepest descent of the free energy functional. We also prove that approximate solutions constructed with an implicit Euler scheme converge to a solution of the kinetic equation. Thus, we generalise to a family of nonlinear kinetic equations the celebrated JKO scheme in mass transport theory. Most of our results are new even in the case of the linear Vlasov-Fokker-Planck equation.

We consider mixed-integer linear-quadratic generalized Nash equilibrium problems, i.e., games in which each player solves a mixed-integer program subject to linear constraints in her own and rivals' strategies as well as an objective which is quadratic in her own strategies and bilinear in her own and rivals' strategies. For this class of games, we study the question of the existence of rational equilibria assuming rational input data. We distinguish four subclasses according to the presence of player-quadratic terms in the objective and rival-dependent constraints. As our main result, we completely settle the rationality question for all four subclasses, i.e., we show that only player-linear games without player-quadratic terms and without rival-dependent constraints admit rational equilibria – if the game admits equilibria at all. All other three classes contain instances with irrational equilibria only.

This paper addresses real-time state estimation for a nine-compartment nonlinear COVID-19 epidemic model with two co-circulating strains, a super-spreader subpopulation, vaccination with waning immunity, hospitalization, and mortality. Time-varying transmission and vaccination rates are known inputs from a companion calibration, leaving the reconstruction of all nine states from three routinely reported observables: hospitalizations H, fatalities F, and vaccinated stock V. The contributions are theoretical rather than in the filter recursion. First, a Lie-derivative observability analysis yields, via a six-step derivation, the closed-form determinant |det(O9)| = delta_w * gamma_a^2 * kappa * rho2 * w1^2 * (delta_i - delta_p)^2 * |r1 - r2|, showing the level-2 codistribution is rank-deficient at the calibrated symmetric parameters (delta_i = delta_p, r1 = r2); the third Lie derivative restores full rank 9, with r2 the symmetry-breaking parameter. Second, an EKF is designed on the Euler-discretized dynamics with a closed-form 9x9 Jacobian and Joseph covariance update. Third, local exponential mean-square boundedness of the error is proved as a full theorem via the Reif-Gunther-Yaz-Unbehauen hypotheses, exploiting the bilinear drift and linear output to obtain a global-radius quadratic remainder bound that extends to bilinear-drift, linear-output systems. Fourth, the noise covariances are designed from calibration residuals and assessed by NEES and innovation-whiteness tests. All experiments use synthetic measurements from the calibrated model, so reported RMSE values (0.07%-2.72%) are methodology benchmarks, not predictive accuracy. A parameter-mismatch study shows measured and directly-coupled channels stay accurate under model error up to +/-30% while indirectly observed states degrade gracefully. The framework provides the state-feedback basis for future Model Predictive Control.

For higher-order discretizations of explicit dynamics problems, Isogeometric Analysis (IGA) has several favorable properties as compared to classical Finite Element Analysis (FEA). While FEA produces spurious modes at orders beyond linear, this is not the case for IGA. Consequently, fewer degrees of freedom are required for comparable accuracy, larger timesteps can be taken, and the method is more robust for nonlinear problems. If outlier modes are removed, the timestep even becomes virtually independent of the order. In this paper, we investigate how these advantages apply to the poroelastic continuum model. We consider both a primal formulation, wherein our variables are the displacement of the matrix material, the fluid displacement, and the pressure, as well as a reduced form wherein the pressure is eliminated. For our discretizations, we employ divergence-conforming spline spaces. Conforming spline spaces for the fluid displacement ensure inf-sup stability for the primal form, as well as a correct null space in the reduced form. Furthermore, we prove and demonstrate that the two formulations coincide when both displacements are discretized with conforming spline spaces. Through spectral analysis, we find that the aforementioned benefits of IGA do carry over directly to the context of poroelasticity. In 1D, we split the discrete spectrum into fast and slow waves. When normalized against an analytical solution, each of these sub-spectra closely resembles results known in elasticity. Consequently, when poroelasticity is discretized with outlier-free IGA, the timestep is essentially independent of the order. We show this timestep scaling in 2D as well.

We give a characteristic-function formulation of Kunchenko's stochastic-polynomial construction for settings in which raw moments may fail to exist. In the finite-variance trigonometric case, the coefficients of the Kunchenko normal system are expressed through the characteristic function and its derivative. In the moment-free case, empirical characteristic functions on a fixed finite frequency grid define a bounded discrepancy geometry that remains meaningful for Cauchy, symmetric stable, and other heavy-tailed laws. We prove well-definedness and finite-grid almost sure consistency of this empirical characteristic-function geometry. We introduce the associated minimum-CF-distance estimator and establish its identifiability, strong consistency, and asymptotic normality on a fixed grid, with a covariance built from bounded trigonometric moments that stays finite even for Cauchy and stable laws; refining the grid increases the optimal-weight information monotonically to the Fisher information, so the estimator is asymptotically efficient in the dense-grid limit. We also relate bounded sine scores to weak stochastic-polynomial estimating equations. A small Lean 4 / Mathlib supplement checks selected deterministic identities underlying the bounded-score construction; convergence arguments and statistical interpretation remain outside the formalization.

In this paper, we prove that the continuous orbit equivalence class of a one-sided topological Markov shift contains a one-sided topological Markov shift whose topological entropy is greater than an arbitrary prescribed positive real number, and also contains a one-sided topological Markov shift whose topological entropy is less than an arbitrary prescribed positive real number.

In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we introduce appropriate notions of singular Griffiths/Nakano positivity on complex spaces and establish various properties of these notions. By applying these results, we provide $L^2$-Dolbeault fine resolutions and cohomological isomorphisms, and $L^2$-existence theorems. As an application, we obtain Nakano-Nadel vanishing theorems on weakly pseudoconvex complex spaces.

Let $\Lambda$ be a Legendrian in the contact boundary of a Liouville domain $\Omega$. We explain how the non-existence of Reeb chords with endpoints on $\Lambda$ of length up to $a$ enables one to embed $D_{\epsilon}T^{*}\Lambda\times D(a)$ into $\Omega$ in an exact way. As in earlier work of Zhengyi Zhou, we use the Viterbo restriction map to deduce a contradiction in certain cases. In particular, we show that if $M$ admits a submersion from a product of spheres (e.g., the $n$-torus), then all compact Legendrians $\Lambda\subset ST^{*}M$ admit a Reeb chord for every choice of contact form $ST^{*}M$. The obstruction we use in this case is based on the idea of inverting the degree-$n$ classes in cohomology, and is similar to the notion of string point invertibility introduced by Egor Shelukhin.

Starting with a coupled discrete reaction–diffusion formulation for the lit and latent order books with non-uniformly sampled event times and meta-order source terms we show how two familiar market-microstructure regularities can emerge from this framework: the long-memory of trade signs associated with the Lillo–Mike–Farmer (LMF) theory and the square-root law (SQRL) of meta-order impact. This uses the locally linear order book and constant participation-rate execution in the front dynamics to reduce the dynamics to a Volterra equation whose leading-order solution then yields the well know result of concave impact trajectory, and a completion impact proportional to the square root of the meta-order size. We then use the interface representation to show how heavy-tailed Pareto meta-order lengths generate power-law trade-sign autocorrelations through the source term. These are familiar derivations, what is slightly different here is that we reinterpret these known derivations to make it clear that LMF law is an event-time sign-memory statement, whereas the square-root law is a physical-time viability statement where subordination can alter the calendar-time impact trajectories depending on the mappings and interpolation used to set continuum operational time.