Papers

In this paper, we study nonasymptotic $L^p$ error bounds for interval length and conditional coverage in split conformalized quantile regression (CQR). Our bounds rely on local regularity conditions and accuracy guarantees for the estimated quantiles. We further instantiate our bounds for quantile regression with sparse ReLU neural networks. We also consider covariate shift, where the calibration and test covariates have different distributions, and derive nonasymptotic bounds for this setting. We obtain matching minimax upper and lower bounds in expectation for two constructed fixed-score calibration benchmarks under known covariate shift. The bounds match for every $p\in[1,\infty]$ in the scalar problem and for finite $p$ in the $K$-threshold problem; for the latter, a high-probability minimax lower bound holds for every $p\in[1,\infty]$.

Radical truncation forgets information about a module. We study the ambiguity created by this loss: which nonisomorphic modules remain indistinguishable after all proper radical truncations have been fixed, how the resulting families are organized, and what information can be recovered from them. We introduce terminal ambiguity geometry to make this residual indeterminacy into a geometric object. The remaining ambiguity is governed by the terminal radical layer, and we classify terminal completions by Grassmannian orbit spaces, with projective ambiguity appearing as the rank-one case. This classification gives criteria for rigidity and nonuniqueness and, in the split case, determines the dimensions of the completion spaces and gives exact finite-field counts of nonisomorphic terminal completions. We also establish reconstruction and symmetry results for the rank-one projective geometry. Under suitable rank hypotheses, the ambiguity geometry recovers the corresponding residue division-algebra data, terminal species, and terminal radical layer, while its automorphisms are described by classical projective semilinear groups. Thus the paper studies both sides of radical truncation: the ambiguity produced by the information it forgets and the extent to which that information can be recovered from the resulting geometry.

In this short paper, we prove the bi-exactness of all wreath-like product groups $G\in\mathcal{WR}(A,B\curvearrowright I)$, whenever $A$ is amenable, $B$ is bi-exact, and the action $B\curvearrowright I$ has amenable stabilizers. As applications, we obtain solidity for the associated group von Neumann algebras and, by combining our result with existing rigidity theorems, new examples exhibiting the McDuff superrigidity.

We establish a function field analogue of a recent conjecture of Andrews–Petsche. Our main result characterizes abelian dynamical Galois groups to be precisely the isotrivial ones, whenever the degree of the polynomial is smaller than $p$, the characteristic of the field. The proof of this characterization is achieved in four independent steps as follows: $$\text{Abelian} \implies \text{Finite ramification} \implies \text{PCF map} \implies \text{Isotrivial map} \implies \text{Isotrivial pair}, $$ and works more generally for maps with a superattracting fixed point. We observe that this chain of implications is sharp in the degree: as soon as one reaches $p$ there are new non-isotrivial examples coming from Drinfeld modules. We propose a full conjectural classification in all degrees, taking into account all of the new exotic examples coming from Drinfeld modules and their associated Lattès maps.

In 1959, Snapper showed that the Euler characteristic of the tensor powers of a line bundle on a normal projective scheme is a polynomial, later named the Snapper polynomial. Positivity of coefficients of Snapper polynomials implies various notions of positivity of line bundles, which we study through the lens of magic positivity and real-rootedness. We introduce zonotopal classes in the Grothendieck $K$-ring of vector bundles of the toric variety for any loopless matroid, and prove that their Snapper polynomials are magic positive. Our proof realizes such a Snapper polynomial as a weighted independence polynomial of the Dilworth truncation along certain lines of the matroid. As a consequence, their coefficients are positive, and their $h^{\ast}$-polynomials are real-rooted. In the realizable case, this polynomial is the multigraded Hilbert polynomial of the wonderful variety embedded in a product of projective lines. We introduce analogous line bundles on the Deligne–Mumford–Knudsen moduli space $\overline{\mathcal M}_{0,n}$ and prove that their Snapper polynomials are magic positive. For cotangent line bundles whose first Chern classes are distinct $\psi$-classes, which are not zonotopal, we nonetheless prove that their $h^{\ast}$-polynomials are real-rooted, whereas their Snapper polynomials are magic positive if and only if $n\leqslant7$. More generally, we introduce saturated and weakly saturated $K$-classes of matroids, which furnish a sufficient and a necessary condition for magic positivity of Snapper polynomials in terms of their dragon Hall–Rado polymatroids.

We prove that the cone $C(\Sigma)$ in $\mathbb R^4$ over a suitable minimal Möbius band $\Sigma\subset \mathbb S^3$ is locally mass-minimising with coefficients in $\mathbb Z_2$. This proves a conjecture by Brian White in the context of models of boundary singularities of Plateau's problem in ambient dimension $4$ or higher. The key insight is that the cylindrical slices $C(\Sigma)\cap \{(z,w)\in \mathbb R^2 \times \mathbb R^2 : |w|=r\}$ are minimal surfaces in a flat rank-one translation quotient of $\mathbb R^3$. In fact, they correspond to non-orientable minimising helicoids previously studied by Antonio Ros. We also show that, up to rigid motions, the minimising cones $C(\Sigma)\times \mathbb R^{n-4}\subset \mathbb R^n$ are the only non-planar minimal hypercones with boundary $\mathbb R^{n-2}$ which are smooth away from the boundary that satisfy the minimal cylindrical slice property.

The best constructive lower bounds on the capacity of the binary deletion channel come from random codes with independent run lengths, yet the two strongest such bounds, due to Drinea and Mitzenmacher and to Venkataramanan, Tatikonda, and Ramchandran, have been evaluated mainly for geometric or low-parameter run-length laws. We let a learning algorithm choose the run-length law freely, which raises three challenges. First, the Drinea-Mitzenmacher functional is an infinite sum over the ways deletions merge runs; we show that it depends on the law only through its mean, its full-deletion probability, and bilinear forms in the law and its renewal weights, so that gradients of truncations are exact and every truncation can only lower the bound. Second, for non-Markov inputs the output is no longer Markov and the Venkataramanan-Tatikonda-Ramchandran analysis breaks down; we show that the residual length of the current input run turns the output into a hidden Markov chain, which extends the bound to every finite-support law. Third, its correction term counts only output runs formed from three input runs; we prove a larger correction that accounts for every output run formed by several input runs, which improves the published bound even for truncated geometric laws. Certified by interval arithmetic, the new bounds exceed all previously published deterministic lower bounds at every tabulated deletion probability, by up to $6.86\times10^{-3}$ bits per channel use and 6.8%. At large deletion probability the learned laws concentrate on run-length clusters with survivor counts about two standard deviations apart, like a pulse-amplitude constellation.

We study the finite spectra of the syllogistic logic with cardinality comparisons $S^\dagger(card)$. Since the language does not contain conjunction at the sentence level, we consider the spectrum of a theory $\Gamma$, namely the set of positive integers $n$ for which $\Gamma$ is satisfiable in an $n$-element model. We give a complete classification of these spectra: besides the empty set, every $S^\dagger(card)$-spectrum is either an eventual tail of the positive integers or an eventual tail of the positive even integers. We then consider the extension of $S^\dagger(card)$ by Boolean connectives at the sentence level and show that the collection of its spectra forms the topology generated by the spectra of the original language.

We study the squared Rayleigh-residual functional on complex projective space and its negative gradient flow. For an arbitrary complex matrix, every positive-residual critical point has a strictly descending tangent direction; consequently, the local minima are precisely the eigenlines. For normal matrices, we describe the critical set in terms of circles through spectral points and identify the amplitude dynamics with a replicator equation whose payoff matrix has rank at most four. This gives explicit logarithmic first integrals and a reduced representation of interior trajectories. We compute transverse escape rates near critical components and analyze how polynomial reweighting changes unstable coordinates. Finally, we distinguish local saddle trapping from concentration of Haar-distributed initial states and derive the density of the filtered ensemble. Numerical examples with random spectra and a Jordan block illustrate convergence, nonmonotone Rayleigh motion, and the improvement of finite-time spectral exploration by low-degree random polynomial filters.

We establish Dirichlet–Neumann bracketing for the Dirichlet eigenvalues of $\psi(-\Delta)$ on bounded Lipschitz domains, where $\psi$ is an arbitrary complete Bernstein function. The eigenvalues lie between $\psi$ applied to the corresponding Neumann and Dirichlet eigenvalues of the Laplacian. Both inequalities are strict whenever $\psi$ admits no meromorphic continuation to $\mathbb C \setminus \{0\}$. The proof uses quadratic forms, operator monotonicity, and an analysis of equality in resolvent comparisons. Applying the bracketing to intervals and balls gives a unified proof of simplicity of interval eigenvalues and antisymmetry of second eigenfunctions in balls under the same condition on $\psi$. For fractional powers, these recover results of Fall, Ghimenti, Micheletti and Pistoia for the interval, and of Fall, Feulefack, Temgoua and Weth and, independently, Benedikt, Bobkov, Dhara and Girg for the ball. The argument extends these conclusions to a broader class of nonlocal operators.

In DNA data storage, racetrack memories, and packet networks, data are written as many short strands of fixed length. The receiver knows where each strand begins and ends, and deletions occur only inside a strand. The relevant limit for such systems is the block capacity $C_N(d)$, the largest mutual information between a length-$N$ input and the output of a binary deletion channel with deletion probability $d$. Because the boundaries are known, $C_N(d)/N$ is never smaller than the classical capacity $C(d)$. Computing $C_N(d)$ is hard because the input takes $2^N$ values. For $0\le d<1$, we show that the optimal input is unique, gives positive probability to every string, and is unchanged by complementing or reversing the strings. We then introduce the optimized run distribution (ORD), which assigns one probability to each number of runs; it is optimal for $N\le3$ and is solved with a numerical optimality certificate up to $N=16$. For longer strands, an exact recursion for the output distribution gives unbiased rate estimates, and an empirical Bernstein inequality turns them into confidence intervals. Within the statistical precision, a one-parameter Markov input performs as well as the best inputs found. For strands of 100-200 bits, known boundaries increase the rate by up to 0.026 bits per symbol beyond the best certified upper bound on the capacity of the unsegmented channel. Via Fano's inequality, the block capacity also bounds the rate of any code that uses a single strand; in the tabulated case, this bound is tighter than the best known finite-length bound for strands longer than about 90 bits. Conversely, block rates yield lower bounds on $C(d)$; at $d=0.1$, the bound is within $1.1\times10^{-4}$ bits/use of the best certified lower bound. Finally, InfoNCE estimates, even with the optimal critic, empirically lose most of their ability to rank inputs as the strand length grows.

The constraints for the effective local gauge groups stemming from the causal action principle for causal fermion systems are reviewed. The constraint which are quadratic in the gauge potentials (coming from the so-called bilinear logarithmic terms) are shown to make a connection between the structures of Lie algebras and Clifford algebras. This connection is worked out systematically starting from general chiral potentials corresponding to the gauge group $\mathrm{U}(N) \times \mathrm{U}(N)$. The general results are illustrated in various examples.

We establish an exact composition principle for spherical immersions coupled through coefficient profiles and norm-preserving bilinear maps. Under an explicit mixed-orthogonality condition, the spherical mean-curvature vector splits into orthogonal input and profile contributions. The composition is minimal if and only if every input is minimal and the profile is $W$-minimal for an explicit monomial weight. Using Hsiang–Lawson reduction and Kapouleas–McGrath gluing, we construct closed embedded $W$-minimal profiles, including families with unbounded intermediate Betti numbers. For sufficiently large comparable input dimensions, a Gaussian transfer developed in this paper produces interior profiles from closed embedded Gaussian seeds that are nondegenerate modulo rotations.

We prove that a smooth toric Fano fibration admits a complete gradient shrinking Kähler-Ricci soliton.

We prove a structure theorem for numerical semigroups H that are almost symmetric and almost complete intersections. We show that a row-factorization (RF) matrix of H must possess a highly regular structure, which we call a cascade matrix. Consequently, the defining ideal I_H of the semigroup ring k[H] also exhibits a highly regular structure, derived from this cascade matrix. Both the RF-matrix and the binomial minimal generating set of I_H are unique. Conversely, this structure completely characterizes almost symmetric almost complete intersection numerical semigroups: to every cascade matrix M we associate a monoid H and, whenever this is a numerical semigroup, we prove that H is pseudo-symmetric, almost complete intersection, and has M as RF-matrix. As a consequence of our study, we obtain several key results. 1) Rigidty: if an almost complete intersection semigroup is almost symmetric, then it is forced to have odd embedding dimension and to be pseudo-symmetric. This result is a “next step” after Kunz's theorem, which states that an almost complete intersection semigroup is never symmetric. 2) Huneke-Ulrich structure: the toric ideal can be realized as the ideal of entries of an alternating matrix multiplied by a vector of variables, and this yields the licci property. 3) Cascade polynomials: for each odd positive integer e, we construct a multivariate squarefree polynomial P_e with integer coefficients, arising from a cascade matrix of variables. We provide an enumerative interpretation of its coefficients, thereby proving their non-negativity. 4) Herzog–Watanabe question: we prove that every minimal relation of an arbitrary numerical semigroup H can be obtained by subtracting two rows in some RF-matrix of H, affirmatively answering a 2019 question by Herzog and Watanabe.

Bimatrix games may have an exponential number of mixed Nash equilibria if both dimensions of the game are allowed to grow. Bounds on their maximal number give structural insights that have been used to construct hard-to-solve games. We show new sharp or asymptotically sharp bounds on the (polynomial) number of equilibria for generic games where one dimension of the game is fixed and the number of strategies of the other player grows. These results go beyond the hitherto studied square games. Our methods employ combinatorial properties of polytopes, and recent obstructions that relate to the graph of those polytopes. For $n\ge5$, we construct $3\times n$ games that have all $2n+1$ vertices of the best-response polytope as equilibrium strategies, proved using a simple case of the 4-color theorem for planar graphs. Generic $4\times 5$ games are shown to have at most 17 equilibria, using computer calculations with existing datasets for all combinatorial types of the relevant polytopes. For $d\times n$ games, we construct games where all but a fraction of $O(1/n)$ of the maximum number of vertices are equilibrium strategies.

High-order differential operators and nonlinear coupling make it challenging to construct conservative and energy-stable schemes for coupled gradient-flow systems. We present GradAgent, a knowledge-guided multi-agent system that coordinates three agents across model analysis, algorithm design and proofs, and numerical implementation and validation. Independent audits strengthen reliability by uncovering mathematical errors and proof gaps, guiding revisions, and maintaining consistency across stages. In Reconstruction Mode, GradAgent reconstructs 20 published studies and organizes audited knowledge in an extensible knowledge graph (KG), GradAgent-KG, linking model structures, discretization strategies and proofs, implementations, and numerical evidence. In Design Mode, the agents assess the applicability of retrieved knowledge and develop new schemes informed by relevant discretization strategies. Applied to the fully coupled multicomponent vesicle phase-field-fluid model, GradAgent yields three first-order and three second-order schemes across three algorithmic families, including four linear, decoupled schemes. Under stated assumptions, all six schemes conserve membrane component mass and vesicle volume and dissipate their respective temporally discrete energies unconditionally. Comparisons with and without GradAgent-KG show that it promotes diversity in structure-preserving scheme design for this target model. Numerical tests confirm second-order spatial accuracy, the expected temporal orders, conservation, and temporally discrete energy dissipation, while three-dimensional shear-flow simulations agree qualitatively with experiments. These results demonstrate GradAgent's ability to combine reusable knowledge, coordinated reasoning, and independent auditing to develop and validate structure-preserving algorithms for complex coupled systems.

We study the asymptotic behavior of Bergman kernels for high tensor powers of semipositive line bundles over Hermitian manifolds. At points where the curvature degenerates, the classical asymptotic expansion may fail. In this paper, we establish a full local asymptotic expansion and rapid off-diagonal decay near degenerate points at which the metric admits a local decoupled model. More generally, we make the following two spectral hypotheses: a localized mild spectral gap for the Kodaira Laplacian and a spectral gap for the rescaled local model. Under these assumptions, we prove a localization property and the rapid off-diagonal decay for the Bergman kernel. Furthermore, if the metric has a local quasi-homogeneous structure, we obtain a full local asymptotic expansion in the $C^\infty$-topology. As an application, we study pull-backs of positive line bundles under branched coverings. Near a smooth ramification hypersurface, the resulting asymptotic expansion reflects the branching order. Finally, for certain non-quasi-homogeneous models, we still obtain localization and leading-order asymptotics.

How do the zeros of a polynomial evolve under the holomorphic heat flow or repeated differentiation? In this work, we develop a unified probabilistic proof to three conjectures on the evolution of polynomial zeros under holomorphic heat flow and repeated differentiation. We study these two evolutions for random polynomials with i.i.d.~roots $z_1,\dots,z_n\sim\mu_0$, where $\mu_0$ on $\mathbb C$ satisfies suitable $2+\delta$ logarithmic moment conditions. For the heat flow of small time $t>0$ and Lipschitz continuous Stieltjes transform of $\mu_0$, we identify the limit distribution $\mu_t$ as explicit push-forward of $\mu_0$ under a transport map. For rotationally invariant $\mu_0$, we characterize the limit after $\lfloor tn\rfloor$ differentiations through its radial quantiles. For instance, the heat flow evolves the circular law into the elliptic and semicircle law, while differentiation moves surviving mass towards the origin. We also show that $o(n)$ derivatives preserve the initial distribution $\mu_0$. In fact, all convergences hold almost surely. The common proof strategy crucially relies on recursion identities from leaving out a root, and concentration inequalities, which lead to self-consistent equations for Stieltjes transforms. This brings a method familiar from random matrix theory to polynomial evolutions. Moreover, it allows for generalizations beyond rotational symmetric distributions and quantitative stability estimates for $o(n)$ differentiations and vanishing heat-flow times.

We determine the exact probability that the closed hexagonal polygon obtained by cyclically joining six independent points uniformly distributed on the unit sphere is knotted. The only possible nontrivial knot is a trefoil. Almost surely, the convex hull of the six points has one of two simplicial combinatorial types: a combinatorially regular octahedron and a combinatorially non-regular octahedron. We show that the straight-line complete graph on the six vertices of the hull of the combinatorially regular octahedral type admits exactly one unoriented trefoil Hamiltonian cycle. On the other hand, no cycle connecting vertices of the irregular octahedron can produce a trefoil. We then use stereographic projection to transform the probability of the regular hull type to a Sylvester-type problem for the planar beta-prime probability measure $d\mu(x,y)=\frac{dx dy}{\pi\ (1+x^2+y^2)^2}$. Applying Stokes' theorem and the Blaschke-Petkantschin formula, we compute the expected squared $\mu$-content of a random triangle. This yields a regular octahedral probability of $\frac{15}{4\pi^2}$ and a trefoil probability of $\frac{1}{16\pi^2}$.

Let $\Omega\subset\mathbb C^n$($n\geq 3$) be a pseudoconvex domain with a smooth strongly pseudoconvex boundary point. We prove that if the Bergman metric of $\Omega$ has constant scalar curvature, then it is necessarily Kähler–Einstein.

A description of the endomorphisms of Zappa-Szép products of two groups as a group of $2\times 2$ matrices of maps is discussed in \cite{detsem}. Using this description, we have studied the concept of determinant for the endomorphisms of Zappa-Szép products of two groups. A characterization of the invertible endomorphisms is given with the help of the tools developed using the determinants. The determinants are also computed for the alternating group $A_{5}$.

We study the finite-population approximation of a heterogeneous linear–quadratic mean-field control problem with common noise and random coefficients. Starting from the centralized social planner problem for $N$ non-exchangeable agents, we derive its exact finite-dimensional stochastic Riccati system and identify the scaling of its diagonal, off-diagonal, affine, and scalar components. We then compare this system with the Hilbert-space-valued Riccati system of the continuum model. Our main estimates give quantitative convergence of the complete backward system, including its common-noise martingale integrands. The error separates coefficient and kernel consistency, common-noise approximation, interface effects, exact diagonal corrections, and the mass of the limiting interaction kernel on the discrete diagonal band. We propagate these estimates to the feedback gains and, by a cellwise coupling, to the optimal states, controls, and initial social values. Finally, a cellwise projection of the limiting representative-agent feedback yields a decentralized finite-population strategy whose optimality gap vanishes. Under the stated regularity assumptions, the error bounds are expressed through explicit approximation and concentration moduli. Quantitative bounds on these moduli yield algebraic rates, including an $N^{-1/2}$ rate for the backward system in finite-type regimes; the rates for optimal trajectories additionally reflect the available conditional moment bounds.

How many distinct symbols should a password use? If its length is fixed at $n$ and an observer learns only which symbols appear, the number of compatible passwords is maximized asymptotically when $k/n\to1/(2\log2)$. We ask what changes when, in addition to the length, aggregate information about the repetition pattern is revealed. We model this by fixing a second additive profile $V_k=\sum_i v(J_i)$ at scale $V_k/n\approx\rho$. For $v(j)=\log(j!)$, the profile records the reduction in the logarithm of the number of compatible words caused by repetitions; we show that once the normalized profile $\rho$ exceeds $0.507834\ldots$, the limiting optimal fraction is pinned at $1/2$. For a typical multiplicity profile at the optimal support above this threshold, the excess in $V_k$ is carried by a vanishing fraction of used symbols. We interpret this as a form of non-equivalence of ensembles and extend the mechanism to other profiles and non-uniform allocation models.

$R\Pi\Sigma^*$-extensions form a rich class of difference rings that provide a unified algebraic framework for modeling indefinite nested sums, transcendental products, and nested products over roots of unity structures that frequently appear in combinatorics, number theory, and particle physics. For a large subclass of these extensions whose ring of constants is a field, we introduce a complete reduction approach to resolve the telescoping problem without solving any difference equations. More precisely, we explicitly construct a complement to the subspace of differences over the constant field and develop an algorithm that decomposes any element of the extension into the sum of a difference and a component lying in this complement. Consequently, summability holds if and only if this complementary component is zero. This structural approach yields significant speed-ups for parameterized telescoping and, notably, creative telescoping for deriving linear recurrences of definite sums. Finally, we compute an explicit idempotent representation that extends existing telescoping algorithms and our complete reduction framework to the general class of $R\Pi\Sigma^*$-extensions, opening up previously untreatable classes of sums and products.

In this paper, we study Zhang twist of Leavitt path algebra $L_K(E)$ using a graded automorphism $\sigma$ induced by a graph automorphism such that the twisted algebra is a Leavitt path algebra over another graph $E_{\sigma}$ which is a twisted graph obtained from the original graph $E$. We establish combinatorial connection between the graphs $E$ and $E_{\sigma}$ and as a consequence, we show that many ring-theoretic properties are invariant for Leavitt path algebras under the twist by $\sigma$. We also define the notion of noncommutative projective scheme for $\mathbb Z$-graded algebras that coincides with the notion of noncommutative projective scheme for connected $\mathbb N$-graded algebras defined by Artin and Zhang and study it in the context of twists of Leavitt path algebras.

We study the smooth stabilization number of a locally flat embedded surface in a smooth $4$-manifold, defined as the minimum number of connected sums with $S^2\times S^2$ needed to make the surface topologically isotopic to a smooth embedding. We prove that a nonsmoothable locally flat surface in a closed, simply connected smooth $4$-manifold becomes smoothable after a single stabilization if its homology class is primitive and noncharacteristic and its complement has abelian fundamental group. This yields infinitely many examples of nonsmoothable surfaces with smooth stabilization number one. We also exhibit examples for which a single stabilization produces infinitely many pairwise smoothly nonisotopic representatives in the same topological isotopy class. In contrast, we construct a locally flat genus-eight surface representing $6h$ in $\mathbb{CP}^2$ whose smooth stabilization number is exactly two. More generally, we show that smooth stabilization numbers of locally flat embedded surfaces in $\mathbb{CP}^2$ are unbounded.

We prove global well-posedness and scattering for the three-dimensional defocusing cubic nonlinear Schrödinger equation with arbitrary initial data in $H^s(\R^3)$, $s>\frac{1}{2}$, without radial symmetry. The proof combines the $I$-method with long-time $L^2_{t,x}$ bounds for products of frequency-localized components of the nonlinear solution. The key high–low estimate follows from a directional interaction identity pairing high-frequency mass with low-frequency energy, together with an induction on frequency.

We consider balanced doubly intermittent full-branch interval maps with two neutral fixed points. Although the ordinary empirical measures fail to converge for Lebesgue-almost every initial point, we show that this non-statistical behavior persists under a broad class of deterministic observation schemes: if the sampling sequence is block-dominating, then both endpoint Dirac masses are accumulation points of the sparse empirical measures for Lebesgue-almost every orbit. This class includes sampling sequences of positive lower density, arithmetic and polynomial sequences, and finite-index regularly varying sequences. We also prove that every map in the balanced class has the strong natural-measure property, including at the boundary exponent $\beta=1$: the pushforwards of every absolutely continuous initial probability converge to a distinguished endpoint mixture $\nu_*$. Abstract consequences of this one-time limit include annealed convergence along every deterministic sampling sequence, the existence of arbitrarily sparse sequences with almost-sure convergence, and an almost-sure convergence criterion based on a power-saving variance bound. For the symmetric quadratic map $g_1\in\mathfrak F_*$, we verify the dyadic covariance condition on a countable uniformly dense endpoint-flat family: the double covariance sum is $O_\phi(n\log(n+2))$. Consequently its dyadic empirical measures converge for Lebesgue-almost every initial point to $\tfrac12(\delta_{-1}+\delta_1)$.

A domain derivative for time-dependent electromagnetic scattering from perfect conductors is established. By proceeding through the Laplace domain, frequency-dependent bounds on solutions to Maxwell's equations are derived. These bounds are used both for establishing time regularity properties of the domain derivative and for proving convergence of the proposed Runge–Kutta convolution quadrature semi-discretization in time. A full convergence analysis is also carried out for pointwise evaluations of the domain derivative, when this time discretization is combined with a Galerkin method in space. Numerical experiments validate the convergence orders in time and space. Eventually, the domain derivative is applied in an iterative shape reconstruction algorithm, in which the electric near field is measured at some receiver positions away from the perfect conductor. Numerical examples show the feasibility of this algorithm and in particular highlight its robustness, when additional noise is applied to the data.