Papers

We study conditions under which the metric projection onto a strictly convex body can be approximated by the projections onto polygonal approximations (in the Hausdorff-Pompeiu sense) of the strictly convex body itself. We provide a necessary condition by proving that the union of all the vertices of the polygonal approximations must be dense in the boundary of that set. To showcase our result, given a superellipse (Lamé curve) of order p>1, we prove that a sequence of polygons obtained by radially projecting the torsion subgroup of the unit circle onto the superellipse, converges to the superellipse, in the Hausdorff-Pompeiu metric. Moreover, this leads to a globally convergent new algorithm useful in approximating the metric projection onto a superelliptic disk (the convex hull of a superellipse).

Schauder Orlicz-type estimates are derived for weak solutions to second-order linear elliptic equations in divergence form with lower-order terms. The Orlicz setting $X=L^\psi$ is treated first. Under suitable assumptions on the Young function $\psi$ and on the coefficients, the optimal associated space for the lower-order datum is identified. An a priori estimate in $W^{1,\psi}$ is then obtained. The discussion is next extended to rearrangement-invariant Banach function spaces. A class $(\mathcal C)$ is introduced to characterize the spaces $X$ for which a corresponding associated space $Y$ yields Schauder-type estimates. Lorentz spaces are finally examined as concrete examples.

The second Eulerian numbers are defined via the descent enumerator of Stirling permutations, a class of permutations introduced by Gessel and Stanley. We give a simple and conceptual proof of two identities relating the Bernoulli numbers and the second Eulerian numbers. We rely on a recent Ehrhart-theoretic idea of Ferroni.

We investigate Fourier multipliers associated with the Strichartz Fourier transform on the Heisenberg group. In particular, we establish Hörmander-type $L^{p}-L^{q}$ boundedness results for the range $1<p\leq 2\leq q<\infty$. The analysis is based on deriving suitable analogues of the Hausdorff-Young and Paley inequalities for the Strichartz Fourier transform, followed by interpolation arguments to obtain the desired multiplier estimates. As an application, we study the local well-posedness of certain nonlinear partial differential equations. Furthermore, we establish an $L^{p}$-boundedness theorem for Fourier multipliers associated with the Strichartz Fourier transform for the full range $1<p<\infty$.

Let $({X}, \omega)$ be a compact $n$-dimensional Kähler orbifold, the stabilizer groups of which are abelian and have rank at most two. Let ${E}$ be an orbi-ample vector bundle of rank $2$ over ${X}$ and let $H$ be a Hermitian metric on ${E}$ such that the curvature form of $\det H$ is $-2\pi \sqrt{-1} \omega$. We show that a certain weighted sum of Bergman kernels for ${Sym}^i {E} \otimes \det({E})^{k+j}$ as $i$ and $j$ vary over a finite set admit an asymptotic expansion. This extends a similar result for cyclic Kähler orbifolds.

In this paper, we introduce the notion of $t$-tone edge coloring. A $t$-tone edge $k$-coloring of a graph $G$ assigns to each edge of $G$ a set of $t$ distinct colors from $\{1,\dots,k\}$ such that any two edges at distance $d$ share fewer than $d$ common colors. The $t$-tone chromatic index of $G$, denoted by $\tau'_t(G)$, is the minimum integer $k$ for which $G$ admits a $t$-tone edge $k$-coloring. We focus on the case $t=2$ and establish several upper bounds on $\tau'_2$. In particular, for every graph $G$ with maximum degree $\Delta(G)\ge2$, we prove that $\tau'_2(G)\le 6\Delta(G)-4$, improving the corresponding bound derived from the vertex analogue. We also show that every tree $T$ with $\Delta(T)\ge3$ satisfies $\tau'_2(T)=2\Delta(T)$. Furthermore, every planar graph $G$ satisfies $\tau'_2(G)\le \max\{41,3\Delta(G)+5\}$, while every outerplanar graph $G$ satisfies $\tau'_2(G)\le \max\{14,3\Delta(G)\}$. For subcubic graphs $G$, the vertex analogue yields $\tau'_2(G)\le12$. We improve this bound to $11$ for claw-free subcubic graphs and to $10$ for $2$-degenerate subcubic graphs. Finally, we propose two conjectures concerning optimal bounds for cubic and $K_4$-free cubic graphs, and establish them for series-parallel subcubic multigraphs and subcubic outerplanar graphs, respectively.

This paper is devoted to the study of strong solutions for the compressible Navier-Stokes/Allen-Cahn system in bounded domain $\Omega\subset\mathbb R^3$, allowing for the presence of initial vacuum. A characteristic of this system is the strong coupling between density and the Allen-Cahn equation, which leads to strong degeneracy in vacuum regions. Under a compatibility condition on the initial phase-field variable, we establish the local existence and uniqueness of strong solutions for $0\le\rho_0\in W^{1,q}$ with $q\in(3,6)$, $u_0\in H_0^1$ and $\chi_0\in H^2$. Owing to time-weighted estimates, no compatibility condition is required for the velocity, but these estimates introduce a singularity in proving uniqueness. We then establish a criterion for the possible breakdown of such a local strong solution at finite time in terms of blow-up of the quantities $\|\nabla u\|_{L_t^{1} L_x^{\infty}}$, $\|u\|_{L_t^{2} L_x^{\infty}}$ and $\|\nabla \chi\|_{L_t^{2} L_x^{\infty}}$.

Matrix measures induced by vector norms are widely used in contraction theory of nonlinear dynamical systems. A natural and important robustness question is whether negativity of a matrix measure is preserved under arbitrary nonnegative diagonal damping. Matrix measures with this property have been called admissible. We show that an induced matrix measure is admissible if and only if the underlying norm is orthant-monotonic. Equivalently, these are precisely the induced matrix measures satisfying $\mu(D) = \max_i\{d_{ii} \}$ for every nonnegative diagonal matrix $D$. We further show that this class is maximal for uniform preservation of contraction under nonnegative diagonal perturbations. The result gives a new geometric characterization of admissibility and clarifies the role of orthant-monotonicity in additive $D$-stability and diffusion-induced instability.

We consider the ODE/IM correspondence for the value $c=-2$ of the Virasoro central charge (free-fermion point) and the associated quantum KdV model $-$ the quantization of the second hamiltonian structure of the classical periodic KdV model. We prove that the ODE/IM correspondence is complete (in the sense of V. Bazhanov, S. Lukyanov, and A. Zamolodchikov), namely that any solution of the Bethe equations coincides with the spectrum of a rational extension of the (quantum) harmonic oscillator. To this end, on the ODE side we consider Crum$-$Darboux transformations of the harmonic oscillator and the associated Laguerre Wronskians, which are remarkable special functions parametrized by pairs of partitions which we study in depth. As a further result, on the IM side, we diagonalize explicitly the first three hamiltonian operators of quantum KdV (in the free field representation): the eigenstates are Schur functions and the eigenvalues are shifted symmetric functions on partitions. We give two applications of this result: i) we prove that the eigenvalues are given by the evaluation of the Newton symmetric polynomials at the poles of the associated monster potentials, as further conjectured by V. Bazhanov, S. Lukyanov, and A. Zamolodchikov; ii) we show that these hamiltonian operators also belong to the algebra of hamiltonian operators obtained by quantizing the first hamiltonian structure of the classical periodic dispersionless KdV model.

We characterize the potential V (x) that minimizes the fundamental spectral gap of weighted Schrödinger operators on the interval [0,{\pi}] subject to Dirichlet boundary conditions, under the constraint that the potential V (x) is convex and that the weight function w(x) is concave.

We formulate a framework of Floquet algebraic tomography for finite-dimensional non-Hermitian monodromy matrices from observable trace sequences $\zeta_n^{(O)}={\rm Tr}(OM^n)$. Since these sequences are constrained by the characteristic polynomial of $M$, the inverse problem is a finite-dimensional algebraic reconstruction problem rather than a generic exponential fit. We organize the reconstruction through the observable resolvent, spectral determinant, and Dirichlet spectral data, separating the common spectral skeleton from observable-dependent dressing. Cayley–Hamilton and Hankel methods recover the similarity-invariant spectral data, while multi-observable and Liouville-space extensions connect the construction to realization theory and tomography reconstruction. We further clarify the limits of identifiability from restricted observable algebras: the data determine a visible representative, micromotion can enlarge the sampled visible operator space, and exact symmetries impose residual invisible sectors. Two examples, a driven transmon qutrit and a finite non-Hermitian Floquet SSH chain, demonstrate leakage-induced visibility expansion, observable-dependent phase response, EP-accessible branch geometry, and disorder/probe-dependent observable-dimension readouts.

Time series driven by unobserved latent states frequently exhibit abrupt jump discontinuities whose timing and magnitude cannot be predicted from observed history alone. Classical jump-diffusion models offer a principled mathematical framework but assume rigid parametric forms, while recent neural jump models operate on fully observed trajectories without inferring the hidden states that govern the dynamics. We propose Deep ZakaiJ, a latent-state model for partially observed jump-diffusion systems that embeds the Zakai nonlinear filtering equation into a neural encoder–decoder architecture. The encoder recursively updates a belief over the latent state via Strang splitting into three interpretable substeps: prior propagation, diffusion innovation, and jump innovation, yielding a differentiable, first-order-accurate approximation of the exact filtering evolution. The decoder is a structured jump-diffusion model explicitly conditioned on the filtered belief, preserving the separation between continuous dynamics and discontinuous shocks. On synthetic, financial, and oceanographic datasets, Deep ZakaiJ improves distributional forecasts while remaining competitive in point accuracy, achieving calibrated predictive intervals and recovering interpretable latent structure in synthetic and qualitative case studies.

Doob fixed-time conditioning enables the sampling of rare trajectories of Markov processes by modifying the drift so that reaching a prescribed target at a given time is guaranteed. We study the statistics of this conditioned path ensemble through the moment generating function in the weak-noise large deviation regime. Since the Doob drift is rarely available in closed form, we reinterpret the conditioned ensemble as the original process post-selected on the terminal constraint, thereby avoiding explicit construction of the Doob transform. This viewpoint then yields an optimal-control representation for the leading exponential contribution to the generating function, expressed as a variational principle with terminal boundary conditions set by the Doob end-point constraint. We illustrate the framework with two analytical examples and with an application to heat dissipation of a minimal model of biomolecular folding.

Recently, Qiu, Xu, Ye and Yu proved that for product system of finitely many minimal systems, the maximal $\infty$-step pro-nilfactor of the system is the topological characteristic factor. In this paper, we extend the result to polynomials.

Triorthogonal codes are a basic ingredient in magic-state distillation, as they enable transversal implementation of logical $\mathsf{T}$ gates. As a non-Clifford operation, the $\mathsf{T}$ gate enables universal fault-tolerant quantum computation when combined with Clifford operations. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk-polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find novel nontrivial triorthogonal codes (degenerate and non-degenerate) that are not necessarily constructed from triply even binary linear codes. By imposing automorphism-group constraints, the formulation also recovers the known [[49,1,5]] and [[95,1,7]] triorthogonal codes. We also evaluate the performance of the obtained high-distance triorthogonal codes over the dephasing channel using different decoding methods, showing that automorphism ensemble decoding (AED) and guessing random additive-noise decoding (GRAND) are promising options.

Return panels, covariances, and large feature matrices evolve one observation or one entry at a time, yet downstream models require an up-to-date low-rank factorization $A_t \approx U_t \Sigma_t V_t^\top$ on every tick – a regime where full SVD is prohibitive and existing alternatives sacrifice either singular vectors, singular values, or long-horizon stability. We present a practical, metric-driven study of Brand-style incremental SVD, built around a unified engine that handles row appends, column appends, rank-1 entry updates, and metrics tracking within a single framework, with two core contributions. For rank-1 entry updates, we derive an explicit projection-based rule $U'\Sigma'(V')^\top = P_U(\widehat{A} + \delta\,e_ie_j^\top)P_V$ that keeps rank fixed while discarding only the out-of-subspace remainder in a quantifiable way, turning Brand's rank-suppression heuristic into an operational scheme. We then treat refresh scheduling as a first-class design axis, systematically comparing periodic, error-threshold, angle-threshold, and adaptive-rank policies on the accuracy-latency frontier. A unified framework tracks error ratios, principal angles, explained variance, and per-update runtime on long synthetic streams and a multi-asset ETF factor model for covariance and portfolio-risk estimation. With a sensible rank and refresh cadence, incremental SVD matches full-SVD accuracy within a few percent at a fraction of the cost, scaling to high-frequency regimes where batch SVDs are infeasible.

Pipedreams and bumpless pipedreams are two combinatorial models that compute double Grothendieck polynomials. While studying matrix Schubert varieties, Pechen Pipedreams and marked bumpless pipedreams are two combinatorial models that compute double Grothendieck polynomials. While studying matrix Schubert varieties, Pechenik, Speyer, and Weigandt defined the Rajchgot code, denoted by $\mathsf{rajcode}(\cdot)$, which captures the leading monomial of the top-degree component of a Grothendieck polynomial. Combinatorially, their result implies that there exists a unique pipedream (or marked bumpless pipedream) with row weight $\mathsf{rajcode}(w)$ and column weight $\mathsf{rajcode}(w^{-1})$. A construction of such a pipedream was subsequently given by Chou and Yu. In this paper, we resolve the marked bumpless pipedream version of this problem by providing an explicit algorithm.

Among the many proofs of Turán's classical theorem, one particularly surprising proof due to Li and Li uses ideals in polynomial rings to record missing edges. Motivated by their proof, Mubayi proposed a hypergraph analogue, conjecturing that an ideal generated by multipartite $3$-graphs coincides with a differentiated diagonal-vanishing ideal. If true, this conjecture would imply the extremal-number part of Mubayi's classical hypergraph extension of Turán's theorem. We disprove this conjecture throughout the nontrivial parameter range. We then give an alternative algebraic proof of Mubayi's extremal formula for the family $\mathcal{K}_{\ell}^{(r)}$ of clique expansions, using monomial cover ideals and a Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients.

If a self-map $\sigma \colon \mathcal{X} \rightarrow \mathcal{X}$ has a dynamical zeta function with nonzero radius of convergence $1/\Lambda$ and the Cesàro mean $B$ of $ \# \mathrm{Fix}(\sigma^k)/\Lambda^k$ exists and is positive, we show a large deviation principle for the number of prime orbits occurring in the decomposition of a general orbit of length $\leq X$ (an element of the free abelian monoid generated by the prime orbits or, equivalently, a prime orbit of a finite multiset in $\mathcal{X}$) with speed $B \log X$ and universal rate function equal to that of the Poisson distribution with unit mean. We also show a large deviation principle for more general strongly additive functions. The proof uses asymptotic results on the total number of general orbits, as well as a weak analogue of Mertens's second theorem, that may be of independent interest. The theory applies, for example, to endomorphisms of algebraic groups over finite fields, additive cellular automata, and automorphisms of some solenoids.

We introduce a physics-inspired continuous relaxation framework that yields substantially improved solutions for NP-hard combinatorial optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), binary sparse coding, and planted-solution Ising models. By parameterizing discrete binary variables as continuous wave-like states on the complex unit circle, we inherently smooth highly non-convex energy landscapes. We show that representing binary variables as complex phases reveals an implicit regularization mechanism that promotes convergence toward discrete states. Extracting this mechanism yields significant improvements even within standard real-valued optimization frameworks, using this regularizer explicitly. Empirically, this regularization yields vastly higher ground-state convergence rates than standard real-valued alternatives. Our models achieved zero error in large-scale 160x160 QUBO tasks under severe noise (sigma=0.25), and outperformed traditional algorithms (OMP and LASSO) in underdefined sparse coding with perfect recovery at sigma=0.15. The solver's robustness was further validated by recovering exact ground-state configurations in 8 out of 11 rigorously engineered planted-solution benchmarks.

We study a finite-horizon stochastic control criterion for non-convex optimization in which Brownian exploration is balanced against a quadratic control cost. Rather than emphasizing the classical Hopf–Cole representation, we isolate the exact drift selected by the criterion and reorganize it in a form adapted to optimization. The key object is the conditional terminal law of the optimal process. We show that this law is a Gibbs measure for a proximally penalized energy, yielding three exact representations of the drift: potential, averaged-gradient, and barycentric. We then analyze two asymptotic regimes relevant for optimization. As terminal time is approached, the drift recovers a scaled gradient-descent field. In the low-temperature regime, assuming a unique global minimizer, the conditional terminal law concentrates on it even in the presence of nonglobal local minima, and the drift converges to an affine attraction field toward it. In the nondegenerate case we also derive Laplace asymptotics for the drift, the value function, and the covariance of the conditional terminal law. Finally, we record a simple gradient-free discretization suggested by the barycentric formula.

We study a singularly perturbed Dirichlet problem for the $p$-Laplacian with competing superlinear terms, \[ -\varepsilon \Delta_p u = a(x)|u|^{q-2}u - b(x)|u|^{\gamma-2}u, \qquad u|_{\partial\Omega}=0, \] where $1<p<q<\gamma<p^*$, $a\geq 0$, $b\geq\sigma_b>0$, and $\varepsilon>0$ is small. By means of the nonlinear Rayleigh quotient method, we introduce two critical parameter values, $\varepsilon^*$ and $\varepsilon_e^*$, related respectively to the Nehari manifold and to the zero energy level. We prove the nonexistence of nontrivial weak solutions for $\varepsilon>\varepsilon^*$, and the existence of at least two positive weak solutions for $0<\varepsilon<\varepsilon_e^*$; one of them is a ground state. The main result describes the asymptotic behaviour of ground states as $\varepsilon\to 0^+$. If, in addition, $a\geq\sigma_a>0$, then every family of positive ground states $u_\varepsilon$ converges in measure in $\Omega$ to the explicit profile \[ \bar u_0(x) = \left(\frac{a(x)}{b(x)}\right)^{1/(\gamma-q)}. \] Moreover, \[ u_\varepsilon\to\bar u_0 \quad\text{strongly in }L^r(\Omega), \qquad 1\le r<\gamma, \] and \[ u_\varepsilon\rightharpoonup\bar u_0 \quad\text{weakly in }L^r(\Omega), \qquad 1<r\le\gamma . \]

Let $n, m \ge 4$. We study the Zappa–Szép products $G = HK$ with $H = \langle x\rangle \rtimes \langle y\rangle \cong SD_{2^n}$ and $K = \langle z\rangle \rtimes \langle w\rangle \cong SD_{2^m}$, together with the fixed generators $x, y, z, w$, and we describe the parameter tuples which encode the mixed commutators of these generators, according to the cores of $\langle x\rangle$ and $\langle z\rangle$ in~$G$. First, when both $\langle x\rangle$ and $\langle z\rangle$ are normal in~$G$, we show that the admissible parameter tuples are exactly the solutions of an explicit system of six polynomial congruences. Second, when the cores $\langle x\rangle_G$ and $\langle z\rangle_G$ are arbitrary subgroups of $\langle x\rangle$ and $\langle z\rangle$, under the simplifying assumption $[x, z] = 1$, and for generating systems for which all mixed commutators lie in $\langle x\rangle\langle z\rangle$, we obtain an analogous classification by twelve congruences together with two order conditions; this is the semidihedral counterpart of the Hu–Yu classification~\cite{HuYu2025} for dihedral groups $D_{2n}$ and $D_{2m}$ with $n, m$ odd. In contrast with that case, we further construct an explicit exact product with both cores non-trivial and $[x, z] \ne 1$, so that the semidihedral setting contains additional families of exact products.

Ring-like communication graphs appear in UAV formations, cyclic patrols, perimeter monitoring, and other multi-agent tasks in which agents exchange information mainly with neighboring vehicles along a closed route. When measurement and actuation noise are persistent, a useful augmentation should improve both the convergence rate of consensus and the steady-state disagreement level. This paper studies the addition of a single weighted chord to a connected weighted cycle. The central observation is that a chord is not just a generic rank-one edge update: it splits the cycle into two complementary resistance arcs, and this resistance split governs both the algebraic-connectivity gain and the Kirchhoff-index reduction. We first derive exact chord-induced effective-resistance and Kirchhoff-index update formulas, giving a closed-form coherence objective. We then prove that, under bounded conductances and small resistance discrepancy, near-antipodal resistance-balanced chords are near-optimal for algebraic-connectivity improvement; an i.i.d. bounded-conductance model yields the same conclusion with high probability. Finally, because the best convergence-rate chord and the best coherence chord need not coincide, we formulate the design as a finite Pareto problem and introduce RBAPS and AW-RBAPS, two resistance-balanced screening rules that retain only linear or near-linear candidate sets. Numerical experiments show that AW-RBAPS remains effective beyond the formal moderate-heterogeneity regime and approximates the exhaustive Pareto front with mean hypervolume ratio $0.9987$ while evaluating about $10.1\%$ of admissible chords.

The Normalized Maximum Likelihood (NML) codelength, or stochastic complexity, represents a principled criterion for universal coding. While recent coarea-based formulations provided a calculation method for smooth models, this framework collapses for the non-smooth estimators ubiquitous in modern machine learning (e.g., Lasso, Sparse SVMs). In this work, we provide a rigorous framework for computing the NML for regular path-differentiable Lipschitz (PDL) estimators. By applying classical geometric measure theory and bridging the coarea formula with conservative Jacobians, we prove that the stochastic complexity for non-smooth models is well-posed and theoretically consistent with the outputs of modern Automatic Differentiation. To compute this quantity exactly, we introduce the Propose-and-Project Metropolis-Hastings (PDL-PPMH) sampler, a geometric MCMC algorithm capable of traversing the non-differentiable level sets of the maximum likelihood estimator. We theoretically justify its components, including a stochastic tangent space proposal and a provably convergent non-smooth projection solver. We demonstrate the method's robustness by sampling from a high-dimensional Lasso posterior ($P=2000$), while simultaneously quantifying the computational scaling that governs the trade-off between exactness and mixing time. Crucially, we empirically demonstrate that our exact NML criterion provides a highly data-efficient alternative to cross-validation, achieving statistically indistinguishable predictive optima without requiring data splitting. Altogether, our work paves the way for the theoretical analysis of the NML codelength for regular non-smooth models.

Let $\mu_p$ be the generalized Gaussian distribution on $\mathbb{R}^n$ with density $e^{-\frac{|x|^p}{p}}$ multiplied by a constant depending on $p\ge 1$ and $n$, and $\alpha_p(n)$ be the largest number such that the Brunn-Minkowski type inequality $$\mu_p(\lambda K+(1-\lambda) L)^{\alpha_p(n)} \geq \lambda \mu_p(K)^{\alpha_p(n)}+(1-\lambda) \mu_p(L)^{\alpha_p(n)}$$ holds for all convex bodies $K,L$ in $\mathbb{R}^n$ containing the origin and $\lambda\in[0,1]$. In this paper, the new lower and upper bounds for $\alpha_p(n)$ are found, and their asymptotically optimality as $n\to +\infty$ is proved.

We establish Chernoff-type bounds for the largest eigenvalue of sums of Hermitian random matrices generated by a time-inhomogeneous Markov chain. Our primary regime assumes a compact state space and contractivity of each Markov kernel in Wasserstein distance, i.e., positive Ollivier-Ricci curvature. This assumption is convenient to verify in inhomogeneous settings and is satisfied by several chains of practical interest, such as stochastic gradient descent on strongly convex smooth objectives. We also develop analogous bounds for noncompact state spaces under a notion of spectral gap for inhomogeneous chains introduced by Saloff-Coste and Zuniga. Finally, we illustrate the utility of our results through an analysis of the Elo rating system, a popular method for ranking players in sports analytics, under a dynamic version of the Bradley-Terry-Luce model.

The geometrically defined wide class of time-like surfaces in $\mathbb R^3$, admitting real asymptotic lines is considered. A fundamental theorem of Bonnet-type is obtained for these surfaces. It states that a surface in this class is determined (up to a motion) by four invariant functions, satisfying some natural PDEs. Then canonical parameters are defined for these surfaces and it is proved that such a surface is determined (up to a motion) in canonical parameters with only two invariant functions (which in particular can be the Gauss and the mean curvature), satisfying a partial differential equation, equivalent to the Gauss equation.

We prove global Lipschitz estimates for Brenier maps between probability measures on $\mathbb{R}^n$ whose densities belong to the family $$ \rho_{U,\,p}=Z_{U,\, p}^{-1}\exp(-\Theta_p(U)), \qquad \Theta_p(t)=p\log\Bigl(1+\frac{t}{p}\Bigr), \qquad p\in[n,+\infty], $$ with finite normalization constant $Z_{U,\, p}$, and with the convention $\Theta_{\infty}(t)=t$. We allow different parameters for source and target, $d,D\in[n,+\infty]$, with $d\le D$. Our global estimate is uniform in $n,d,D$, and in the case $d=D<+\infty$, it improves the bounds of arXiv:2404.05456 by removing their exponential dependence on the dimension. We also prove localized estimates inside fixed balls $B_R$ whose constants are stable under the limits $d,D\to+\infty$ and they allow us to recover Caffarelli's celebrated contraction theorem with sharp constants.

A planar dual billiard is a planar curve $\gamma$ equipped with a family $(\sigma_P)|_{P\in\gamma}$ of projective involutions of the projective lines $L_P$ tangent to $\gamma$ at $P$ that fix $P$. A dual billiard is called rationally integrable, if there exists a rational function $R(x,y)$ of two variables (called first integral) whose restriction to each tangent line $L_P$ is $\sigma_P$-invariant. In the previous author's paper it was shown that rationally integrable dual billiards exist only on conics punctured at $k$ points, $0\leq k\leq 4$. Their classification given there includes standard examples with quadratic integrals, defined by conical pencils, and an infinite family of exotic examples with minimal degree of integrals being any even number greater than two. In the present paper we study a question stated by Dmitry Treschev on dynamics and conservation laws in the exotic examples. We study the dynamics acting on the two-dimensional phase space: the complex algebraic surface consisting of pairs $(Q,P)$, where $Q\in\mathbb{CP}^2\setminus\gamma$, $P\in\gamma$ and the line $QP$ is tangent to $\gamma$ at $P$. We show that the phase space is fibered by invariant algebraic curves along which $Q$ lies in a level curve of the integral of the dual billiard. For each example we find the type of a generic level curve of the integral and of the corresponding fiber. We show that a generic level curve is rational in most of examples and elliptic in two examples. We present formulas for an invariant area form on the phase space and for invariant holomorphic differentials on invariant curves. This yields a formula for holomorphic differentials on the elliplic level curves of the integral.