Papers

We study increasing propagation of chaos for the critical Curie- Weiss model (i.e. the mean-field Ising model at critical inverse temperature 1, with no external field). We give a simple way to see that for windows of size k(N) of smaller order than sqrt{N} we still have propagation of chaos (reproving earlier results, see e.g.[1]), while for k(N) of order sqrt{N} the propagation of chaos breaks down. The law of a single spin converges to \pi, the Bernoulli law with parameter 1/2. If k(N) = alpha sqrt{N}, we give an explicit formula for the limiting distance in total variation of the law of the first k(N) spins with respect to the k-fold product of pi, as a function of alpha. For even larger window sizes, the distribution of the spins has, in the thermodynamical limit, maximal distance to the k-fold product of pi. One of the ingredients of the proof is a result about the unimodality/non- unimodality of the law of the number of positive spins among the first k = k(N) spins, which may be of independent interest.

In this paper, we study the emergent collective dynamics of multi-agent systems driven by infinite distributed fading memory of Volterra type. We establish a unified theoretical framework covering both first-order opinion consensus dynamics and second-order velocity alignment flocking kinematics. By introducing Dafermos past-history transformations, the governing integro-differential systems are reformulated into dynamical systems on an extended product Hilbert spaces. For first-order dynamics, we prove that fading memory inherently provides a hidden dissipative mechanism, guaranteeing unconditional global exponential consensus with or without instantaneous communication forces. For second-order dynamics, we obtain unconditional exponential flocking for the pure fading memory system and we give a sufficient condition for flocking when an instantaneous interaction is also present. In particular, this condition is always satisfied, namely the flocking occurs unconditionally, when the influence function has a divergent tail.

Let A be a d-by-n integer matrix of rank d, let beta belong to the d-dimensional complex vector space, and consider the A-hypergeometric system defined by the ideal H_A(beta), with relation lattice L equal to the integer kernel of A. No homogeneity, pointedness, or positivity assumption is imposed on A. We study directed formal logarithmic solutions simultaneously over all exponent lattices in the Euler fiber over beta, which form an uncountable family when L has positive rank. For a fixed L-generic direction, negative-support strata are described by rational sign polyhedra, and their directional boundedness is characterized by recession cones. On each exponent lattice, normalization reduces the infinitely indexed coefficient equations to a finite system of Euler, transit, and directional vanishing relations, represented by a finitely presented directional coefficient module. We prove that only finitely many exponent lattices contribute nonzero directed solution spaces and that every contributing coefficient module has finite length. Macaulay inverse-system duality then identifies the dimension of the all-lattice directed formal logarithmic solution space with the length of the all-lattice coefficient module. We also give an explicit formal-series realization and a polynomial colon-ideal criterion for all realizable lowest exponents.

For a finite extension of global fields $K/k$, the norm map $N_{K/k} : K^\times \to k^\times$ extends to a map on idèle groups. The Hasse norm principle holds if every element of $k^\times$ that is a norm everywhere locally is also a norm globally. In this paper, we study the statistics of the Hasse norm principle in a setting that is out of reach in the number field context, namely that of $A_4$-quartic extensions. We show that failures of the Hasse norm principle are generally rare for $A_4$-quartic extensions of global function fields $\mathbb{F}_q(t)$. We achieve this by introducing a decorated Hurwitz space parametrising the failures of the Hasse norm principle and then using the Chebotarev density theorem to estimate their frequency.

Let $X=\{x_1,\ldots,x_N\}\subset\Sph^{d-1}$ be a spherical $t$-design, $t\ge2$, and let $P_X\subset\R^d$ be its Minkowski polytope. If $h_i=h_{P_X}(x_i)$, then \[ P_X^\circ=\conv\{x_i/h_i:1\le i\le N\}, \] so the design is the radial projection of the polar vertex set. This yields exact degree-dependent moment identities and quantitative control of the unweighted polar moments from the Hausdorff sphericity of $P_X$. Our main results concern self-polarity. If \[ P_X^\circ=cUP_X,\qquad U\in O(d), \] we obtain the structured slack factorization \[ A=c\,hh^T-X^TU^TX,\qquad \rank A=d+1, \] whose zero pattern records the facet–vertex incidences. For node-transitive designs, the common incidence level equals the inradius-to-circumradius ratio $r/R$. Combining this with the one-dimensional moment problem underlying the Fazekas–Levenshtein covering bound, we prove \[ \frac rR\ge\eta_{t,d}, \] with equality forcing the twisted inner-product rows to realize the corresponding Gaussian or Gauss–Radau quadrature rule; in particular, the quantities $N\lambda_k$ become integer incidence multiplicities. We further prove a quantitative near-equality theorem: if \[ \delta=\frac rR-\eta_{t,d} \] is small, then each row is $O_{d,t}(\sqrt\delta)$-close in $W_1$ to the extremal quadrature measure, yielding near-incidence rigidity and an arithmetic stability gap. As applications, we obtain three-dimensional rigidity and identify the regular simplex and the $24$-cell as the regular self-polar examples.

We study an infinite family of strongly coupled four-dimensional $\mathcal N=2$ superconformal field theories (SCFTs) distinguished by a trivial Higgs branch, known as the $(A_2,D_{3m+1})$ Argyre-Douglas theories. We propose that their associated vertex operator algebras (VOAs) are the doublet algebras $\mathcal A(4m+2)$, an infinite family of non-rational vertex operator superalgebras with a remarkably simple strong generating set of only three fields. We provide several highly nontrivial checks of this proposal. In particular, we reproduce the four-dimensional conformal anomalies $a$ and $c$ from the VOA and analytically prove the exact equality between the Schur index of the SCFT and the supercharacter of the VOA. A key ingredient is a diagonal-gauging realization of the $(A_2,D_{3m+1})$ theories in terms of two simpler building blocks, which makes the Schur-index computation tractable. Remarkably, we find that the resulting Schur index of the $(A_2,D_{3m+1})$ theory coincides with that of $\mathcal N=4$ $SU(2)$ Super-Yang-Mills theory, up to an overall prefactor and an appropriate identification of fugacities. We also discuss a generalization to the two-parameter family $(A_{2s},D_{(2s+1)m+1})$, whose members likewise have trivial Higgs branches and admit diagonal-gauging realizations. A particularly interesting subfamily is $(A_{2s},D_{2s+2})$, for which the four-dimensional conformal anomalies coincide, $a=c$. As a byproduct, we also show that the doublet VOA $\mathcal A(4m+2)$ is not classically free for positive integer $m $, and we discuss its implications for the Schur and Macdonald indices. Our results reveal a systematic connection between Higgsless SCFTs, diagonal gauging, and strongly finite but non-rational vertex operator algebras.

In this paper, we consider the Smagorinsky model for the Navier-Stokes equations within a virtual element framework. Under the standard assumption of small data, we prove the existence and uniqueness of a solution. Assuming more regularity to the solutions, we derive the known convergence rates $h$ for the a priori error estimates of the Smagorinsky model in two dimensional domains. We additionally prove that divergence-free virtual discretizations provide improved convergence orders, with weaker regularity assumptions than in the finite element literature. We conclude the paper with numerical results that corroborate the theory.

Let $(M^n,g)$ be complete, let $u>0$, and assume $\operatorname{Ric}_g-\alpha\frac{\Delta_g u}{u}g\ge (n-1)\kappa g$. We introduce mixed radial balls associated with the conformal metric $u^{2\alpha}g$. For these mixed radial balls, we obtain a space-form-sharp model comparison and polynomial weighted volume growth without pointwise bounds on $u$. As applications, we give a radial derivation of the spectral Bonnet–Myers and sharp volume theorem of Antonelli–Xu, and a short volume growth derivation of the stable Bernstein theorem in $\mathbb{R}^4$.

We show that the first-order theory of the Farey graph is $1$-ample but not $2$-ample. Along the way, we prove that it weakly eliminates imaginaries and observe that forking is trivial.

We study formal deformations of the natural $\osp(n|2)$-action, $n\geq 3$, on the superspace $\Sc^n_d=\bigoplus_{k\geq 0}\Fc^n_{d-\frac{k}{2}}$ of symbols of linear differential operators on weighted densities over $\R^{1|n}$. Starting from the first cohomology space computed in \cite{10}, we compute the cup-product $\Hd^1\vee \Hd^1\to \Hd^2$ which carries the quadratic obstructions. The answer is governed by the $\osp(n|2)$-invariant operators $A_k=\eta_1\cdots\eta_n\partial_x^{k-1}$: the two cocycles $h_k$ and $\widetilde{h}_k$ spanning the off-diagonal part of $\Hd^1$ are exactly the two derivatives of the coboundary of $A_k$ with respect to the two weights. Consequently, all the products of two off-diagonal classes and all the products of two diagonal classes vanish, and the whole obstruction is carried, for each $k$, by a single non-trivial 2-cocycle $\Omega_k$. If $2d\notin\N$ the space $\Hd^1\vee\Hd^1$ is identically zero, so every infinitesimal deformation is integrable. If $2d=m\in\N$ we obtain exactly $m$ quadratic integrability conditions, $\tau_{2-n-k}(t_k-\widetilde{t}_k)+\tau_k\widetilde{t}_k=0$, $1\leq k\leq m$, and we prove that they are also sufficient: no condition of order $\geq 3$ occurs and the versal deformation is of degree one in the parameters. In particular every integrable formal deformation is equivalent to its infinitesimal part.

Determining the dimension of the $C^1$ cubic spline space $S_3^1(\mathcal{T})$ on an arbitrary nondegenerate planar triangulation has remained unresolved since the 1970s. Schumaker's lower bound includes a local correction $\sigma$ for singular interior four-stars, and it was conjectured that this bound is always attained. We disprove this conjecture by constructing a one-parameter family of nondegenerate realizations of a fixed 18-triangle complex, with only the central vertex moving as $v_6(t)=(t,0)$ on the admissible interval $I=(-3/4,24/55)$. The family exhibits three distinct cases. For $t\in I\setminus\{1/5,3/83\}$, the lower bound is attained and $\dim S_3^1(\mathcal{T}(t))=33$. At $t=3/83$, the central four-star is singular, $\sigma=1$, and the resulting dimension 34 is exactly accounted for by the classical local correction. At $t=1/5$, however, all interior vertices are nonsingular and $\sigma=0$, yet $\dim S_3^1(\mathcal{T}(1/5))=34>P_{\mathcal{T}(1/5)}(1,3)=33$. The smoothing-cofactor calculation shows that the dependence at $t=3/83$ is confined to the central vertex block, whereas the dependence at $t=1/5$ couples all seven interior vertex cycles even though every individual block has full row rank. A complementary Bernstein–Bézier calculation gives the same dimension profile. Thus the singular-four-star correction does not capture every geometry-dependent contribution to $\dim S_3^1(\mathcal{T})$; genuinely global compatibility must also be taken into account.

We study the Cauchy problem for the Benjamin–Ono–Zakharov–Kuznetsov equation on $\mathbb R^2$. Following the strategy of Kenig and Ziesler, we establish new maximal-function estimates adapted to the BO–ZK equation and use them to implement the Kenig–Koenig method. As a result, we improve the best previously known isotropic result of Nascimento (2020), lowering the local well-posedness threshold from $s>5/4$ to $s>19/16$. At the BO–ZK endpoint, the resulting isotropic data class also contains the anisotropic $E^{5/4+}$ class of the preceding theory. On bounded subsets of $H^s(\mathbb R^2)$, the lifespan may be chosen so that $ T\gtrsim_s \bigl(1+\norm{u_0}_{H^s}\bigr)^{-8}. $ The proof combines a sharp dyadic mixed maximal-function estimate with an anisotropic local-smoothing mechanism that exploits the complementary behavior of the longitudinal and transverse group velocities. In particular, transverse dispersion compensates for the degeneration of longitudinal smoothing near the characteristic region. Together with refined short-time Strichartz estimates and a modified energy argument, these ingredients close the nonlinear estimates at the stated regularity. Existence, uniqueness, and continuous dependence on the initial data are then established in the corresponding solution class. The resulting threshold reflects the present optimization of the method and is not claimed to be sharp.

We consider reaction-diffusion eigenproblems with oscillatory diffusion and reaction coefficients. The reaction coefficient magnitude is large: the corrector equation identified by periodic homogenization involves both the diffusion and the reaction operators. We study the numerical approximation of this problem using the Multiscale Finite Element Method (MsFEM). This now classical method is a finite element type method that performs a Galerkin approximation of the oscillatory problem on a specific, problem dependent, basis set. The basis functions are precomputed in an offline stage. Inspired by homogenization theory and using some filtering ideas, we show how to define these basis functions in order to obtain an efficient method. The comprehensive set of numerical experiments that we present, in periodic and non-periodic cases, for the scalar-valued version of the problem (which is then self-adjoint) and for the vector-valued version of the problem (which is then in general non self-adjoint), demonstrates the performance of the approach. Some theoretical arguments complement the numerical observations.

Feature and sample screening reduce the cost of machine learning by eliminating irrelevant features and noninformative samples, respectively. Although recognized as primal-dual counterparts, their relationship remains informal and model-dependent. Viewing screening and duality as transformations of objective functions, we introduce Fenchel-Rockafellar (FR) representations, a class of convex problems encompassing the Lasso and SVM that is closed under both transformations. We then prove that feature and sample screening form an equivariant pair: dualization followed by feature screening is equal to sample screening followed by dualization.

We consider generalized principal eigenvalues of one-dimensional nonlocal dispersal operators with a drift of fixed sign. A principal difficulty in this non-self-adjoint setting is the absence of a variational characterization of the principal value. In the symmetric drift-free problem, the critical nonlocal-to-local limit can be treated through a quadratic variational structure and Sobolev-seminorm approximation as Berestycki-Coville-Vo \cite{BCV}. The drift destroys this structure and, on a bounded interval, introduces at the same time a one-sided inflow–outflow boundary geometry. Replacing the missing variational argument by estimates which remain stable under singular rescaling is therefore a central technical issue. Our approach is direct : we work with a single generalized principal value and prove the maximum principle, simplicity in the positive cone and the scaling limits from the equation itself, without passing through auxiliary generalized principal eigenvalues analogous to $\lambda_1'$ and $\lambda_1''$ in Berestycki–Rossi \cite{BR}. The replacement mechanisms are directional Harnack inequalities and coefficient barriers, direct–adjoint identities, logarithmic Collatz–Wielandt transforms, Fourier coercivity of zero extensions and scale-dependent localization. At the critical diffusive scale, Fourier compactness and an exact energy–transport identity recover the missing inflow Dirichlet condition and identify the local Dirichlet limit without a Rayleigh quotient. For variable coefficients we determine the complete small-range phase diagram on bounded intervals and on the line, and we obtain sharp large-range three-term asymptotics by a rank-one reduction and a corner Laplace analysis in the advective travel-time variable. In the homogeneous whole-line problem the critical correction is of order $\sigma^2$.

We show the number of connected components of the Hilbert scheme of orthogonal Grassmannians under certain condition, and use this result to describe the geometry of the Hilbert scheme. Subsequently, we determine the Nef cone of the Hilbert scheme by identifying curves dual to the generators of its Neron-Severi group. Our results generalize those of ordinary Grassmannians by Seong, and our approach adapts his proof technique alongside relevant Schubert calculus.

We study the parity-perturbed Hofstadter $Q$-recursion $$ \widetilde Q(1)=\widetilde Q(2)=1,\qquad \widetilde Q(n)=\widetilde Q(n-\widetilde Q(n-1)) +\widetilde Q(n-\widetilde Q(n-2))+(-1)^n, $$ and the associated two-variable Dirichlet series $$ Z_{\widetilde Q}(s,t)=\sum_{n\ge1}n^{-s}\widetilde Q(n)^{-t}. $$ The estimate $\widetilde Q(n)=n/2+O(n/\sqrt{\log n})$ gives the exact domain of absolute convergence $\operatorname{Re}(s+t)>1$. With $w=s+t$, we separate the universal term $2^t\zeta(w)$ and derive exact transport, frequency-position, and dyadic renormalization identities. The main result concerns $t=-1$. For $E(n)=2\widetilde Q(n)-n$ and $A(X)=\sum_{n\le X}E(n)$, the binary-arch clock yields $$ A(X)=X\log_2X+X\Omega\!\left(\log_2\frac{3X}{32}\right) +O\!\left(\frac{X}{\sqrt{\log X}}\right), $$ where $\Omega$ is an explicit continuous periodic function. This continues the normalized correction to $\operatorname{Re}w>0$ and yields a boundary resonance lattice: a double resonance at $w=0$ and simple resonances at $2\pi i m/\log2$. After subtracting the full-slice order-$X$ skeleton, we analyze the negative-even arch channel. Its companion-forest layers have a weak Gaussian limit, and a canonical subsequence realizes the optimal $n/\sqrt{\log n}$ pointwise scale with an explicit signed constant. The negative-arch mass satisfies $$ A_r=\frac{512}{9\sqrt{2\pi}}\frac{16^r}{\sqrt r} \left(1-\frac{13}{16r}+O(r^{-2})\right). $$ We do not claim a full-slice continuation across $\operatorname{Re}w=0$.

In positive degree, equivariant Gromov-Witten invariants of Calabi-Yau fivefolds are expected to admit an interpretation in terms of M2-branes, while we conjecture that constant maps are governed by the corresponding supergravity index. We make the first expectation precise by proposing a modular interpretation of M2-branes supported on several smooth curves meeting at an $n$-fold point. Together, the two pictures yield conjectural formulas for the Gromov-Witten invariants, which we translate into closed formulas for pointed and unpointed quintuple Hodge integrals. We show that these conjectures imply a K-theoretic Gromov-Witten/Pairs correspondence for local curves in degree one and link the generating series of constant maps to Donaldson-Thomas theory of points. We also prove the conjectures in two limits of the equivariant parameters. Along the way, we obtain new closed formulas for certain triple Hodge integrals.

We develop a spectrum-level graded $K$-theory for real super Banach algebras. Our construction is categorical and homotopy theoretic, in the style of algebraic $K$-theory: the graded $K$-theory spectrum is obtained by a (co)fiber sequence from the $(\infty,1)$-categorical group completion of topological groupoids of finitely generated projective graded modules, rather than from spaces of Fredholm operators or Kasparov cycles. We define a connective spectrum $k^{\mathrm{ABS}}_A$ refining the Atiyah–Bott–Shapiro construction as a cofiber, together with its periodification $K^{\mathrm{gr}}_A$, and show that both are lax symmetric monoidal and functorial in bimodules, not merely in homomorphisms. The failure of graded $A$-modules to present all cocycles in $K^{\mathrm{gr}}_0(A)$ is shown to be purely a $\pi_0$-phenomenon on $k^{\mathrm{ABS}}_A$. We obtain a natural equivalence $K^{\mathrm{gr}}_{A\widehat{\otimes} \mathrm{Cl}_{p,q}} \simeq \Sigma^{p-q}K^{\mathrm{gr}}_A$, which links topological Bott periodicity with the Morita equivalence between $\mathrm{Cl}_8$ and $\mathbb{R}$. Restricting to invertible finite-dimensional semisimple super algebras yields a symmetric monoidal functor $\operatorname{Pic}(\operatorname{Bim}(\mathrm{sBan}_{\mathbb{R}})^{\mathrm{fd}}) \to \operatorname{Pic}(\mathrm{Mod}(KO))$ which splits off the bottom three Postnikov layers of $\operatorname{Pic}(\mathrm{Mod}(KO))$, giving a direct link between super division algebras and invertible $KO$-modules. We also give spectral refinements of Karoubi's and van Daele's graded $K$-groups, with explicit comparison equivalences, therefore connecting to $KK$-theory. We also provide an extensive general treatment for $K$-theory of ungraded topological rings that might be of independent interest. In particular, we characterize connective topological $K$-theory of ungraded Banach algebras by a universal property.

Recent work has shown that sub-Gaussian random variables are dominated in convex order by a sharp multiple of a Gaussian. We study the analogous question for the sub-Gamma class underlying Bernstein's inequality, with the Laplace law as the majorant. Here the moment generating function is controlled by a Bernstein-type bound over a finite range of frequencies, rather than by a purely quadratic bound. We derive a variational formula for the optimal multiple, show that it is strictly larger than the natural scale $\sigma \vee \alpha$, and prove that this value is sharp, being attained by an asymmetric two-point distribution. We then turn to the sub-exponential class, which has a quadratic bound as in the sub-Gaussian case, but only over a bounded range as in the sub-Gamma case. Interestingly, the sub-exponential class displays a different behavior: the optimal multiple is exactly $\sigma \vee \alpha$, but is attained only when $\alpha \leq \sigma$; when $\alpha > \sigma$, the constant remains sharp, but equality cannot hold for any non-affine convex function. Both results rely on a sharp finite-range variant of the Kearns-Saul inequality, which is of independent interest.

In this paper, we apply the resonator method to exhibit large values at the central point of the families of cubic and quartic Dirichlet $L$-functions of prime moduli under the generalized Riemann hypothesis.

An edge card of a graph is obtained by deleting one edge, and a class edge-reconstruction number asks for the fewest carefully selected cards that identify the graph when its class is known. We determine the sharp universal bound for maximal planar graphs. Two selected cards always suffice, and the octahedral graph shows that two can be necessary; some maximal planar graphs are already identified by one card. The argument exploits the fact that deleting a flippable edge leaves a single quadrilateral whose two diagonals give the only possible maximal-planar completions. Degree information then rules out the competing completion, with a separate argument for graphs containing a vertex of degree three. This settles a problem posed in a 2010 survey on reconstruction numbers.

For $n$-dimensional simultaneous best Diophantine approximations in an arbitrary norm induced by an inner product, for $n\geq2$ we prove $q_{k+2^n}\geq q_k+\min\{q_{k+2^{n-1}},2q_{k+1}\}$. This yields $g_n(\alpha):=\liminf_{m\to\infty}(q_m)^{1/m}\geq\varphi^{1/2^{n-1}}, \text{ for } \ \varphi = \dfrac{1 + \sqrt{5}}{2}.$ Consequently, $\displaystyle G(n):=\inf_{\alpha\in\mathbb R^n\setminus\mathbb Q^n}g_n(\alpha)\geq\varphi^{1/2^{n-1}}$ and $\underline{\mathcal D}_n(\alpha)\leq\left\lfloor 2^{n-1}\frac{\log2}{\log\varphi}\right\rfloor+1$, where $\underline{\mathcal D}_n(\alpha)$ is a quantity related to multidimensional generalizations of the three-distance theorem. In particular, $g_2(\alpha)\geq\sqrt\varphi, \, g_3(\alpha)\geq\sqrt[4]{\varphi}, \, \underline{\mathcal D}_2(\alpha)\leq3, \, \underline{\mathcal D}_3(\alpha)\leq6.$

We investigate kinetic Fokker–Planck equations with rough divergence-free drifts in anisotropic mixed Lebesgue spaces. The divergence-free structure yields an energy cancellation that remains effective under kinetic localization. Combining this cancellation with anisotropic regularization estimates and a De Giorgi iteration, we establish local boundedness for weak subsolutions and, consequently, global well-posedness for the Cauchy problem in a scaling-supercritical regime.

We prove that the regular truncated octahedron uniquely minimizes surface area among all parallelohedra of fixed volume. Equivalently, every three-dimensional parallelohedron $P$ satisfies \[ \frac{\mathcal H^2(\partial P)}{|P|^{2/3}} \ge \frac{3(1+2\sqrt3)}{4^{2/3}}, \] with equality if and only if $P$ is similar to the regular truncated octahedron. Among the non-truncated Fedorov types we prove a stronger sharp bound, attained uniquely by the regular rhombic dodecahedron.

Gaussian graphical models (GGMs) are essential tools for interpretable structure learning. However, in high-dimensional, small-sample regimes, the available data is often insufficient for the maximum likelihood estimator to exist. Colored Gaussian graphical models (CGGMs) mitigate this limitation by imposing symmetry constraints through graph coloring, which reduces the required sample size. This minimal number of observations needed to guarantee that the estimator exists almost surely is defined as the maximum likelihood threshold (MLT). Here, we address the computation of the MLT for CGGMs by focusing on its geometric formulation: finding the minimum rank of a sample covariance matrix such that its projection lies almost surely within the interior of the cone of sufficient statistics. We establish a unified theoretical framework, extending results from uncolored to colored models and introducing new symbolic algorithms. Furthermore, we present a computational study integrating sampling with topological data analysis (TDA) to investigate the local geometry of the cone of sufficient statistics. Our results demonstrate the potential of TDA to overcome the computational bottlenecks of traditional symbolic algebraic methods, particularly Groebner basis computations, in analyzing the likelihood geometry of CGGMs.

We study a parabolic obstacle partial integro-differential equation (PIDE) with a dynamically moving bilateral free boundary. This type of problem arises in the mathematical modeling of speculative asset bubbles with Lévy jump processes. We investigate the existence of viscosity supersolution barriers within the class of functions exhibiting linear asymptotic growth ($O(|g|)$ at infinity) across three distinct parametric regimes. Our intention is to determine when such a barrier can be constructed by analyzing the balance between the stabilizing local drift, defined by the discount rate $r$ and mean-reversion $\rho$, and the non-local jump dispersion, characterized by the large-jump intensity $\lambda$ and Lipschitz constant $L_\gamma$. First, when $r+\rho > \sqrt{\lambda}L_\gamma$, we prove the global existence of non-negative viscosity supersolutions. Second, in the deficit regime ($r+\rho < \sqrt{\lambda}L_\gamma$), we construct non-negative supersolutions for every finite horizon $T>0$. However, by utilizing an asymptotic slope envelope, we prove that these barriers cannot be bounded by a fixed, pre-determined linear growth ceiling $C_{\max}$ across arbitrarily large horizons; rather, the required linear growth constant must inflate exponentially as the horizon length increases. Finally, at the exact critical boundary ($r+\rho = \sqrt{\lambda}L_\gamma$), we show the existence of a supersolution with a uniform spatial growth bound, provided an additional spatial no-crossing condition holds on the negative tail.

This paper shows that the optimal gain of the risk-sensitive linear quadratic regulator (LQR) problem is invariant under input randomization when the cost penalizes the realized noisy input. This may appear counterintuitive at first glance because certainty equivalence does not hold for risk-sensitive LQR and input randomization inflates the effective process noise. Nonetheless, the gain is preserved because the input noise enters not only the system dynamics but also the cost functional, and its total effect eventually vanishes. Consequently, the optimal gain and the Riccati recursion need not be recomputed, and the increment in the optimal cost can be readily evaluated in closed form. This result facilitates the use of risk-sensitive LQR in applications that employ input randomization for privacy or exploration, such as dynamic watermarking for replay attack detection.

We introduce farthest-cell triplet entropy, the conditional Shannon entropy of the farthest-prototype label given three random prototypes. For independent queries and prototypes, its estimator records only the farthest label, not coordinates or numerical distances. The statistic is bounded by $\log 3$, is invariant under common strictly increasing transformations of the dissimilarities, and has an exact mutual-information interpretation. In high-dimensional isotropic radial models $X_d=R_dU_d$, the Euclidean ordering reduces to scores $\lambda_d\xi_{i,d}-Z_i$, where $\lambda_d=\sqrt d\,\operatorname{sd}(R_d)/\mathbb{E} R_d$ and the $Z_i$ are independent standard Gaussian variables. This gives angular-dominated, intermediate, and radial-dominated entropy limits $\log 3$, $H_{\infty}(\lambda;F)$, and $0$. In hyperbolic space of curvature $-\kappa_d^2$, the same master curve appears at $\lambda_{d,\mathbb H}=\sqrt d\,\tau_d A(s_d)$, where $\tau_d=\operatorname{sd}(R_d)/\mathbb{E} R_d$, $s_d=\kappa_d\mathbb{E} R_d$, and $A(s)=s\coth s$. With a calibrated radial law and $\tau_d$, and a monotone operating interval, entropy inversion identifies the scale-invariant target $s_d^2$; absolute curvature requires an external length unit. CPU simulations give Euclidean and hyperbolic master-curve RMSEs of $0.0164$ and $0.0209$. Inversion from observed synthetic latent coordinates has a median relative error in $\kappa$ of $6.6\%$, while angular anisotropy increases this error to $68.9\%$. Thus the entropy statistic is comparison-based, whereas curvature recovery remains model-calibrated, is not graph-only, and is not robust to anisotropy.

We introduce a stream-function formulation for the Surface Stokes-Helfrich model and study an Arbitrary Lagrangian-Eulerian (ALE) Surface Finite Element (SFEM) discretization in space and a semi-implicit discretization in time of these equations. In contrast with special cases of a surface Stokes model on stationary or prescribed evolving surfaces the surface pressure remains as an unknown and needs to be reconstructed in each time step. As for the surface Stokes model on a prescribed evolving surface the formulation also requires an additional gradient potential. A detailed computational comparison with an ALE-SFEM discretization in space and a similar semi-implicit discretization in time of the corresponding velocity-pressure formulation is explored on a representative problem formulation for simply-connected surfaces, demonstrating the validity of the approach.