Papers

We introduce the weighted Laplace space $H_w$, an RKHS of Laplace transforms on $(0,\infty)$, and study spectral measures in its dual space $H'_w$. For conforming FEM discretizations of the Dirichlet Laplacian on bounded Lipschitz domains, we prove the dual-norm inequality $\|\mu_h\|_{H'_w} \leq \|\mu\|_{H'_w}$, where $\mu = \sum_k\delta_{\lambda_k}$ and $\mu_h = \sum_k \delta_{\lambda_{k,h}}$. The proof combines min-max monotonicity of FEM eigenvalues with a heat-trace representation of the dual norm. We then analyze $H_w$-adapted rational approximation of shifted symbols $\phi(x)=(x+\kappa^2)^{-\beta}$ and give a conditional transfer principle for estimates proved in the corresponding weighted Laplace pre-image norm. Via dual pairing, the norm inequality yields uniform bounds for finite spectral sums and related transformed observables.

The dimer spectral transform gives algebro-geometric action-angle coordinates for cluster integrable systems. Fock gave an explicit formula for its inverse. We study Fock's inverse map in the tropical limit where Harnack spectral curves degenerate to nodal curves. We determine the leading order asymptotics of the classical objects appearing in Fock's formula and show that they are governed by their tropical counterparts. This yields a tropical inverse spectral transform on tropical spectral data. We show that Fock's inverse spectral transform commutes with tropicalization, and also derive a tropical version of Fay's trisecant identity.

We prove the exact asymptotics of the multidimensional normalized $L^1$ Nikolskii constant \[ \mathcal L^*(d)=\Bigl(\frac{\pi}{2}+o(1)\Bigr)2^{-d}, \quad d\to\infty. \] In addition, for each fixed dimension we obtain asymptotics for the positive zeros of the extremal function $\varphi_d$, and determine their limiting distribution as $d\to\infty$. The lower bound follows from the construction of an admissible function and its asymptotic analysis. For the upper bound, we represent $x^{d+1}\varphi_d(x)$ as the product of two solutions of the equation $u''+V_d(x)u=0$, compare this equation with a Bessel model, and analyze relative canonical products.

This paper studies the isolated calmness property of solution mappings arising from convex regularized linear inverse problems. We mainly establish a tangent-cone characterization of this property. For convex piecewise linear-quadratic regularizers, the isolated calmness coincides with the solution uniqueness, leading to simple verification procedures. In particular, for analysis sparsity regularization, our condition can be verified by linear programming.

Schwartz-MacPherson classes of degeneracy loci of symmetric and skew-symmetric maps. Together with the known formulas for ordinary linear maps, this completes the stable SSM theory for the three classical types. We interpret these formulas probabilistically, in terms of the endpoint random partition of a stochastic rational six-vertex model: the skew classes are probabilities of Maya-dimer events, and the symmetric classes are Chebyshev moments of the random 2-core.

We introduce LatentGamma, a differentiable operator on the unit cube $[0,1]^M$ designed as a smooth surrogate of Tarski's immediate consequence operator $\TP$ associated with a definite Horn program $P$ on $M$ atoms. The operator is built as a five-stage composition combining sigmoidal gating, softmax routing, and a residual update that enforces monotonicity by construction. We establish four theoretical results. First, the iterated sequence is coordinate-wise non-decreasing and bounded by $\ind$, hence converges to a fixed point of LatentGamma; the operator itself is lattice-monotone on $[0,1]^M$. Second, our constructive expressivity theorem shows that for every definite Horn program $P$ there exists a closed-form parameter assignment $\theta^*(P)$ and an explicit time bound $T_{\max}(P)$ such that the iterated sequence exactly reproduces $\TPinf(F_0)$ for every initial fact set $F_0$ throughout the window $[D(P, F_0), T_{\max}(P)]$, where $D(P, F_0) \leq M$ is the derivation depth. The window $T_{\max}$ is large in practice ($\geq 10^5$ for sparse programs) and reflects the finite-time nature of computation by smooth sigmoidal gates. Third, the oracle is robust to Gaussian noise on its body and head logits, with explicit non-asymptotic bounds. Fourth, we prove a matching information-theoretic lower bound on the parameter count. We provide a complete numerical validation on programs ranging from $M = 33$ to $M = 504$ atoms: oracle accuracy reaches $1.0000$ on $2000$ test cases with zero false-positive and false-negative rates, and the empirical noise tolerance $\sigma_{\max}$ scales precisely as the union bound $RM \cdot \Phi(-10/\sigma_\eta)$ predicts.

We study random planar quadrangulations in which each block – i.e., each simple (without multiple edges) component – is assigned a weight $u$. We derive an explicit expression for the distance-dependent two-point function, defined as the generating function of such block-weighted maps with two marked edges at a fixed graph distance. Using contour integral representations in two variables combined with a delicate saddle-point analysis, we compute the associated distance profile in the scaling limit of large quadrangulations. We recover the known phase transition at $u = 9/5$, characterized by distinct scaling exponents and different scaling functions, below, at, and above criticality. We also discuss the block distance profile, where the two marked edges are conditioned to lie within the same block.

We compare closed-loop and open-loop Nash equilibria in a finite-horizon stochastic linear-quadratic $N$-player game with decoupled state dynamics and interaction through the state costs. We introduce a block-diagonal reference game and a nearby perturbed game, and study each under both information structures. The closed-loop and open-loop problems lead to different Riccati systems, but for the reference game their induced equilibrium state and control processes coincide exactly. We then prove solvability and stability under weak perturbations of the state costs, yielding quantitative bounds between the corresponding equilibria. In particular, when the perturbation decreases sufficiently fast with the population size, the closed-loop and open-loop equilibria of the perturbed game become asymptotically equivalent. The comparison is carried out directly at the finite-player level, without requiring exchangeability or a mean-field limit.

We study the relation between non-anticipative functional calculus and compatible families of controlled paths. For a non-anticipative functional satisfying horizontal Lipschitz regularity, we show that the iterated vertical derivatives generate compatible higher-order controlled Taylor expansions along Hölder controls, with level-dependent remainder exponents. We derive a rough-integration criterion from these estimates and show that, for $\gamma-$Hölder controls with $\tfrac{1}{2}\geq \gamma>\sqrt{2}-1$, the first-order remainder estimate is recovered. Our main result is a converse representation theorem. We consider non-anticipative functionals $G_0,\ldots,G_p$ which satisfy compatible higher-order controlled Taylor estimates along $\gamma$-Hölder paths. Under natural continuity and compatibility assumptions, we prove that they can be represented as the iterated vertical derivatives of the base functional: $ G_j=\nabla_\omega^jG_0,\qquad j=1,\ldots,p.$ Thus the Gubinelli coefficients of a compatible controlled family are symmetric and uniquely determined by its base functional; in particular, the Gubinelli derivative is identified with the vertical derivative introduced in functional Itô calculus, giving the coefficient hierarchy an intrinsic path-space differential structure. We show that this class of compatible coefficient families is stable under admissible non-anticipative functional composition and derive the corresponding functional chain rule. As an application, we obtain well-posedness for a class of path-dependent rough differential equations with Volterra memory, and identify the Gubinelli derivative of the resulting path-dependent rough coefficient.

In this paper, we study the existence, uniqueness, and quantitative estimates for weak solutions to linear elliptic Dirichlet problems of the form \[ -\operatorname{div}(\gamma \nabla u)+\langle \nabla\phi+\mathbf{H},\nabla u\rangle+(c+\alpha)u=f \quad\text{ in }U, \quad\; u=0 \quad\text{on }\partial U, \] where $U\subset \mathbb{R}^d$ is bounded, $\gamma\in[1,\infty)$ is a constant, $\phi\in H^{1,2}(U)\cap L^\infty(U)$, $\mathbf{H}\in L^p(U,\mathbb{R}^d)$ for some $p \in (d, \infty)$, and $c\in L^1(U)$ with $c\ge0$. A key feature of this setting is that the drift contains the low-regularity term $\nabla\phi$, which is only assumed to belong to $L^2(U,\mathbb{R}^d)$, while the zero-order coefficient is merely integrable. We prove that, even under these rough assumptions, well-posedness and quantitative energy and $L^2$ estimates remain valid. In addition, by using a previously established interpolation result, we characterize a trade-off between the integrability of the source term $f$ and that of the zero-order coefficient $c$, and show that well-posedness together with the corresponding quantitative estimates hold under these interpolated assumptions.

We consider the wave equation with variable wave speed in Euclidean space, with point sources modeled by Dirac delta initial displacement data. There are three special subsets of the whole space: (1) the source set where the delta initial conditions are supported, (2) the unknown set where the wave speed is not known a priori, and (3) the receiver set where the waves are measured. The inverse problem is to reconstruct the wave speed uniquely in the unknown set from an unlabeled collection of waves generated by point sources and measured in the receiver set. We use propagation of singularities and sharp finite speed of propagation to reduce this data to geometric travel-time data, whose form depends on how the three sets lie in relation to each other. We give three scenarios where this procedure leads to unique determination of the wave speed.

We prove that, on a bounded uniformly convex domain of class $C^{3,1}$, the first Robin eigenfunction is strictly log-concave and the Robin torsion function is strictly $1/2$-concave for all sufficiently large Robin parameters. This means that Conjecture 1.1 of Andrews, Clutterbuck and Hauer holds for uniformly convex $C^{3,1}$ domains in any dimension, and settles an open problem posed by Crasta and Fragalà when the domain has $C^{3,1}$ regularity. Our proof derives suitable uniform $C^2$-decay estimates by maximum principle arguments, thereby removing the imposed higher-order boundary regularity assumption required previously.

We study Gallai decomposition for groups equipped with two-sided invariant partial orders. The key algebraic step extends to arbitrary binary relations compatible with the group operation: if all left and right translations preserve a binary relation $\rho$, then every least strong module $S_{\rho}(\e,g)$, $g\ne\e$, is a subgroup. For a partial order this subgroup is convex. Thus the robust modules through the identity of an ordered group form a canonical chain of convex subgroups, with each canonical factor $H/H^-$ prime, totally ordered, or equality-ordered. We characterize exactly the subgroups that are modules, show that they form a complete sublattice of the subgroup lattice, establish overlap and inheritance results for arbitrary subgroups, and prove compatibility with quotients by normal strong subgroups. For $N$-free ordered groups the prime factors disappear. Using the robust-module decomposition of cographs, we characterize all two-sided invariant $N$-free partial orders by reduced admissible two-coloured subgroup chains, with totally ordered and equality-ordered canonical factors; the order is determined by the first nontrivial factor of each element. We also determine how the canonical decomposition restricts to arbitrary subgroups, characterize finite width and prove width divisibility for subgroups, and show that every reduced two-coloured chain is realized by an $N$-free ordered abelian group whose canonical factors are isomorphic to $\mathbb Z$.

This paper deals with two kinds of Bäcklund transformations for scalar hyperbolic partial differential equations. We prove that both these types of transformations map solutions of a Darboux integrable equation into solutions of, generally speaking, another but also Darboux integrable equation. The latter fact can be used to roughly check the completeness of a list of Darboux integrable equations. To illustrate this, we apply the above transformations to several equations from a well-known list of Darboux integrable equations and, as a result, obtain a Darboux integrable equation which is absent in this list, but is already known at present. As a generalization of the last equation, we construct a family of Darboux integrable equations that is parametrized by three arbitrary functions, each of which depends on two arguments. This family is probably new.

The Fourier-cosine (COS) method of Fang and Oosterlee (2008) recovers densities and computes expectations semi-analytically from characteristic functions (ch.f.s). In higher dimensions, both expansion synthesis and COS coefficient-tensor construction scale exponentially with dimension. We address this curse of dimensionality with two tensor-train (TT) methods. A straightforward TT decomposition of the COS coefficient tensor gives COS-TT, which removes the exponential cost of online synthesis but not of offline decomposition. Our main contribution, COS-TT-CHF, instead compresses a ch.f. sample tensor and maps it to the original Fourier–cosine coefficient tensor through an alternative COS representation. Apart from the model-dependent cost of one ch.f. evaluation, both offline and online costs scale linearly with dimension, while error grows only algebraically. COS-TT-CHF introduces two additional errors: frequency-domain truncation and discretization errors. Both are controlled by parameter-selection rules from theoretical error analysis. We further propose an approach for expectations of nonseparable functions of additive scalar aggregates by computing the ch.f.s of the aggregates using tensorized COS methods, reducing the original multidimensional problem to a one-dimensional COS calculation. Closed-form formulas are derived for the TT-contraction integrals arising when the aggregate is a weighted sum, as in European basket option pricing. Experiments under geometric Brownian motion (GBM) and variance gamma (VG) show that, with the parameter-selection rules, COS-TT-CHF remains accurate through 150 dimensions for GBM and 100 for VG within a 30-minute offline computation budget on a laptop. As a by-product, our Fourier-cosine TT constructions yield cosine-basis functional TTs (FTTs) under weaker assumptions than in spectral FTT literature.

We study a family of functionals involving a product of logarithmic capacity and torsional rigidity. Using several geometric inequalities, we identify the discs as the only maximizers among planar convex sets for certain values of the parameters. Finally, we prove regularity of maximizers in the non-critical case. This is done by establishing a shape-differentiability theorem for the logarithmic capacity in the class of planar convex bodies.

We study policy optimization for discrete-time robust $\mathcal{H}_\infty$ control with static output-feedback, and present the first feasibility-preserving algorithm with a deterministic, non-asymptotic complexity guarantee. This problem naturally leads to a nonsmooth and nonconvex optimization over the set of stabilizing feedback gains. We first establish several structural properties of the $\mathcal{H}_\infty$ cost. In particular, we show that the cost is weakly convex on every convex subset of a sublevel set. For the state-feedback case, we further establish a weak Polyak–Łojasiewicz inequality, which ensures that every stationary point is globally optimal. Building on these properties, we develop a proximal bundle method for $\mathcal{H}_\infty$ policy optimization. The proposed method can be viewed as an implementable approximation of the proximal point method and uses only function value and subgradient information. We show that all iterates remain stabilizing and establish a deterministic non-asymptotic complexity bound of $\mathcal{O}(\max\{\eta^{-4},\epsilon^{-2}\})$ for finding an $(\eta,\epsilon)$-stationary point. Numerical experiments illustrate our theoretical results.

Using degenerations of rational maps and local asymptotics of periodic multipliers, we prove that for every $d\ge 2$, the multipliers of any $2d-2$ distinct periodic orbits of degree $d$ rational maps are algebraically independent over $\mathbb C$, provided that at most $d$ of the selected orbits are fixed points. This condition is sharp because of the Holomorphic Index Formula that relates the multipliers of the $d+1$ fixed points. The result of this paper removes the additional restrictions on periods present in earlier work. The proof proceeds by induction on the degree, using a one-hole degeneration for the induction step and a three-hole degeneration in an exceptional degree four case.

We study continuous-time full-order dynamic output-feedback $H_\infty$ policy search, a nonconvex and nonsmooth problem. Direct policy search is a central paradigm in reinforcement learning and continuous control, but rigorous guarantees remain scarce in robust output-feedback settings. The $H_\infty$ problem is a canonical benchmark because it captures disturbance attenuation and robustness while exposing the hard nonsmooth geometry of policy-space optimization. We prove that on the exact identity-gauge slice of the extended convex lift, $\varepsilon$-stationarity yields $O(\varepsilon)$-suboptimality on compact exact slices, which in turn yields convergence-rate guarantees for nonsmooth policy-search methods. This result addresses the finite-time optimality-gap question raised by Guo and Hu [2022] in the more general dynamic output-feedback $H_\infty$ policy-search setting. We further use the established value equivalence supplied by extended convex lifting to formulate a nonstrict-feasibility bisection method with one final strict-feasibility recovery step, yielding an explicit $\varepsilon$-optimal stabilizing controller. These results provide a quantitative and algorithmic strengthening of prior qualitative optimality theory for nonsmooth $H_\infty$ policy search.

This paper addresses a fundamental question in non-convex optimization: How should a stochastic optimizer allocate computation between global exploration and local exploitation? We study this question in continuous time, using Langevin dynamics for global exploration and deterministic gradient flow for local exploitation. The simplest algorithm in this class is the best-state Langevin–gradient method: several Langevin trajectories explore the objective landscape, the best state encountered is retained, and a single gradient trajectory then refines this state to high terminal accuracy. Our main objective is to characterize the computational work required to achieve a prescribed accuracy with prescribed confidence. We develop low-temperature approximations for this probably approximately correct (PAC) work–accuracy tradeoff, both for the best-state method and for related Langevin schemes. Global exploration is governed by energy barriers, through spectral-gap and Eyring–Kramers asymptotics. The resulting approximations expose the competing effects of temperature, computational budget, and the number of exploratory trajectories, and quantify their dependence on the confidence parameter. In contrast, terminal accuracy is largely separated from global exploration: once a suitable region of attraction has been reached, gradient flow requires only $O(\log(1/\varepsilon))$ additional work to reach accuracy at least $\varepsilon$. Numerical experiments on challenging objectives serve to compare algorithms and to test the quantitative predictions of the asymptotic theory. They identify regimes in which the low-temperature approximations provide useful guidance, as well as regimes exhibiting substantial temperature sensitivity and potentially significant high-dimensional limitations.

We develop a rigorous renormalization group (RG) formalism on Wiener space. We introduce an RG operator $\mathcal R$ acting on probability measures over continuous paths, whose fixed point is the Wiener measure. We linearize $\mathcal{R}$ around the Brownian fixed point and analyze the spectral structure of the resulting operator $\mathscr{L}$. Its eigenvectors are too singular to be realized as honest measures on path space, and we therefore develop a generalized spectral theory within the framework of white noise analysis (Hida calculus). The eigenvectors of $\mathscr{L}$ are realized as Hida distributions, satisfying $\mathscr{L} U=\lambda U$ in the weak sense. This analysis yields structural results such as a spectral gap and a diagonal-concentration property of the eigenvectors. Moreover, we identify a family $\{\mathfrak{U}_n\}_{n \geq 0}$ of eigenvectors with eigenvalues $\lambda_n = 2^{1-n/2}$, consisting of Wick polynomials of white noise formally expressed as $\mathfrak{U}_n = \int_0^1 {:}\dot W(t)^n{:}\, dt$, and rigorously constructed as Hida distributions. We then use this spectral structure to uncover a finer, second layer of universality in the Donsker invariance principle: not only is the convergence of random walks towards the Brownian scaling limit universal, but the entire hierarchy of leading corrections to the Brownian limit is universal as well, governed by the eigenpairs $(\lambda_n, \mathfrak{U}_n)$. We also show, at a formal level, that the framework developed here extends to Gibbs-type perturbations of the Brownian fixed point, in the spirit of self-interacting quantum field theories. In particular, we recover the standard irrelevant/marginal/relevant classification, usually obtained in the physics literature by power-counting, solely from the spectral analysis of $\mathscr{L}$.

In this paper, we develop and analyze techniques for recovering a linear image $Bx$ of an unknown signal $x$ from indirect noisy observation $\omega=Ax+\xi$. It is a priori known that $x\in \cX$, a given convex compact set, and that $x$ is $s$-sparse—has at most $s$ nonvanishing entries. The proposed estimates belong to a large family of recovery routines by $\ell_1$-minimization. However, unlike the classical result describing performance of such estimates, we do not make any special (and hard to check) assumptions about the sensing matrix $A$ such as nullspace or Restricted Isometry condition and the like. As a consequence, parameters of the estimates and the upper bounds on their risks are not available in a closed analytic form, but are delivered instead by efficient computation as solutions to explicit convex optimization problems.

We show that on a compact, connected, Riemannian manifold without boundary, every isotopic to the identity diffeomorphism can be written as a finite composition of optimal mass transport maps. This result represents a non-linear infinite dimensional generalization of Ballantine's factorization results.

Nonstabilizerness (magic) is the quantum resource that, together with stabilizer operations, makes universal quantum computation possible. However, quantifying nonstabilizerness is difficult. The standard measure sums over exponentially many Pauli operators, and for quantum codes no quantitative theory has been available. A structural observation removes this obstacle: the Clifford sector carries no magic and can be removed by a Clifford transformation, leaving an order-three object whose magic is an exactly evaluable fourth moment. Removing this inert background yields closed forms for several code families: a formula for all Dicke states, reducing a 100-qubit case from $4^{100}$ terms to a short binomial sum; a bound for cubic-phase codes (twisted quantum doubles and non-Abelian topological order), saturated only by the $D_4$ code; and a cyclic/zero criterion for group multiplication states. For codeword-stabilized (CWS) codes, the companion paper \cite{Liu26arXiv} carries this reduction to its conclusion: the most magical codes are exactly the Sidon sets, and the nonstabilizerness bounds the code's transversal non-Clifford power. Together these results reduce the computation of nonstabilizerness for a broad class of codes to finite classical counting problems, and identify the maximally magical codes among them.

We show that, for some absolute constants $c_1,c_2>0$, every $(n,d,\lambda)$-graph with $\lambda\le c_1 d$ contains an induced cycle of length at least $c_2n\log(d/\lambda)/d$. This is best possible up to the values of $c_1,c_2$. Our techniques include a multi-scale algorithmic analysis, an adapted depth-first exploration procedure, a link to percolation theory, and estimates for random row-and-column extraction in symmetric matrices.

Let $f(w,k)$ be the maximum size of a $w$-uniform family containing no sunflower with $k$ petals. We introduce a recursive construction for sunflower-free families and use it to obtain a general lower bound on the exponential growth rate of $f(w,k)$. We also prove a general upper bound for $3$-uniform families with at least four petals. Our results give $39\le f(3,4)\le49$, $f(3,5)\le146$, $153\le f(3,6)\le255$, $259\le f(3,7)\le474$, and $54\le f(4,3)\le83$. In addition, we prove that the maximum size of an intersecting $4$-uniform family containing no sunflower with three petals is $27$. The upper bounds $49$ and $83$ are computer-assisted. The finite lower bounds come from explicit constructions.

There is a close relationship between divided differences and hypergeometric series, and recent studies have shown that divided differences can be used effectively to derive terminating hypergeometric identities. In this paper, we apply this approach to terminating zero-balanced hypergeometric series. Using explicit product evaluations arising from divided differences, we give alternative proofs of zero-balanced ${}_3F_2(1)$ and ${}_4F_3(1)$ summation formulas and then extend the argument to the general terminating ${}_{r+1}F_r(1)$ case. We also combine the Lagrange representation with the Leibniz rule for divided differences to derive a finite convolution transformation for a terminating ${}_{r+2}F_{r+1}(1)$ series. Thus, the divided-difference approach not only reproduces the known zero-balanced summation formulas but also yields a finite convolution transformation for terminating hypergeometric series, together with its zero-balanced specialization.

We prove that, for every sufficiently large modulus $q\not\equiv2\bmod4$, at least $3/8-o(1)$ of the primitive Dirichlet characters $\chi$ modulo $q$ have $L(1/2,\chi)\ne0$. We also establish stronger proportions for moduli $q$ satisfying certain arithmetic conditions.

Investigating the descendant tree T(Delta) of 3-groups G of maximal nilpotency-class, with root Delta=SmallGroup(27,3), we show for the first time that an entropy-value H(P_r)=sum P_r(G)|log(P_r(G))| can also be determined for the real probability-distribution P_r(G) of all terminal vertices outside of the mainline on this infinite digraph. These vertices arise as automorphism groups Gal(F(3,infinity,K)/K) of maximal unramified pro-3-extensions F(3,infinity,K) of real-quadratic number fields K=Q(d^1/2), d>0. For the groups G of another descendant tree T(N), with root N=SmallGroup(729,45) and unbounded coclass, which has been studied by L. Bartholdi and M. R. Bush, we compare the entropy H(P_r) of the real measure P_r(G) with the entropy H(P_i) of the imaginary measure P_i(G).

We prove that the focusing and defocusing continuum Calogero–Moser equations are globally well-posed in $H^s_+(\mathbb R)$ for every integer $s\geq 3$. In the focusing case, this requires the initial data to satisfy the mass condition $$\|q_0\|_{L^2}^2<2\pi.$$ Our proof is based on a parabolic regularization of the equation, uniform a–priori estimates for the regularized solutions, and a Bona–Smith type approximation argument.