Papers

Philip L. Roe, the renowned British scientist, passed away on April 26, 2026, at the age of 87, while Professor Emeritus at the University of Michigan. My association with Phil Roe began in 1984, when he joined the Cranfield Institute of Technology, UK, as Professor of Computational Fluid Dynamics, while I was a postdoctoral research fellow at the same institution. For more than four decades, his incisive ideas, physical principles and distinctive approach to research on numerical methods for solving hyperbolic partial differential equations have had a lasting influence on my own approach to research in the field. Here I recount some personal memories and anecdotes emerging from the rewarding experience of interacting with Phil during the Cranfield years and beyond. I also review, quite succinctly in places, some of the main research topics that he pursued throughout much of his academic career. These notes are intended as a modest homage to Roe's impact on the subject, as well as to his influence on the careers of generations of scientists in both academia and industry.

We study Gaussian maxima arising from isotropic families of unit vectors and show that their excess expected maximum is controlled by the distance from an orthonormal basis. We apply this stability result to correlation matrices with equal nonzero eigenvalues, obtaining an exact minimum when the matrix size is divisible by the rank and a quantitative gap otherwise. In dimension two, we determine the leading asymptotic term and describe the structure of near-optimal configurations. We also obtain covariance-block recovery results and a finite-dimensional lower bound for the higher-rank limiting problem.

We study a model of a random centrally symmetric polytope and ask about the existence of points in the polytope with non-zero, integer coordinates. Our main results are the location of a sharp threshold for the existence of these integer points, a characterization of the typical empirical distribution of their coordinates, and showing that near the feasibility threshold these integer points are almost always totally isolated — at linear Hamming distance to the nearest other integer point in the polytope. The behavior of these random integer programs is related to that of the symmetric binary perceptron (the case when coordinates of points are $\pm 1$-valued). The lack of symmetry in the set of potential solutions leads to new questions and phenomena, such as the distribution of the coordinate sizes in typical solutions, and requires the development of new information-theoretic techniques to deal with higher-dimensional optimization problems arising in the method of moments.

Let p(n,k) denote the number of partitions of n into exactly k parts. In 1942 Auluck, Chowla and Gupta conjectured that for every n the sequence p(n,1), p(n,2), ..., p(n,n) is unimodal. Szekeres proved this for all sufficiently large n, but with a threshold that was never made explicit. We prove the conjecture for every n >= 1. For n >= 10^5 we give an analytic argument with fully explicit constants: an exact probabilistic representation of the ratio p(n,k+1)/p(n,k), a second-order saddle-point main term with a rigorous uniform error bound in the transition window, and separate elementary or saddle-point estimates in the two tails. Every numerical constant in this argument is certified in ball arithmetic. For n <= 2*10^5 the conjecture is verified exhaustively by two independent programs, one of them using exact integer arithmetic only. Since log-concavity of k -> p(n,k) fails for every n >= 4, the argument works with first differences and never uses log-concavity.

We study roots of Bernstein-Sato polynomials of ideals. Our first main result gives a general partial differential equation criterion for detecting roots of the Bernstein-Sato polynomial of an ideal. For homogeneous case, we recast this criterion as an analytic condition, connecting the algebraic problem of locating roots with analytic properties of the defining data. As an application, we give a series of roots for the general determinantal ideal prove the strong monodromy conjecture for them. Keywords. Bernstein-Sato polynomial, determinantal variety, monodromy conjecture.

Exact permutation covariance does not itself supply a sparse metric geometry. For permutation-covariant, full-support $k$-body interaction orbits, the incident interaction norm scales as $\binom{N-1}{k-1}|J_N|\|h\|$; such orbits cannot have uniformly finite range in metrics of diverging diameter. A permutation-invariant Hamiltonian diagonal in the graph basis has ground-state multiplicity at least $N!/|\mathrm{Aut}(G)|$ whenever $G$ is a ground graph. We give a quantitative version for approximately symmetric energies, together with fixed-radius connectivity and unitary-isometry obstructions. The trace-distance asymmetry of an arbitrary state also bounds its probability of selecting one labeled graph. We then construct two exactly solvable finite relational history Hamiltonians. A cyclic clock with a reversible incidence update has a unique ground state; gauge reduction yields a cycle Laplacian with a rank-one pin and the exact gap $\Delta_{N,L}=E_* (1-\cos\theta_L)$, where $\tan(L\theta_L/2)=1/\sin\theta_L$ and $0<\theta_L<\pi/L$. In particular, $\Delta_{N,L}/E_*=\pi^2/(2L^2)-2\pi^2/L^3+O(L^{-4})$, independently of the number of vertices. In an alternative order-free architecture, one clock qubit per unordered pair gives commuting four-local terms, $2^N$ ground states and a size-independent gap $E_*(1-1/\sqrt2)$. A fixed-weight control penalty selects $\binom Nq$ ground states without reducing the gap. These exact finite results expose clock and locality resource tradeoffs; they do not establish an emergent geometric phase.

We investigate the zeros of Leonardo polynomials and their distribution in the complex plane. We show that the structure of the zeros differs between even and odd degrees. For even degrees, all zeros are simple and purely imaginary, with imaginary parts lying in \((-2,2)\). For odd degrees, there is exactly one positive real zero, whereas no negative real or purely imaginary zeros occur. We obtain two-sided bounds for the unique positive real zero, prove that it decreases strictly to zero as the degree increases through odd values, and determine its asymptotic behavior. Using the relation between Leonardo and Fibonacci polynomials together with a complex trigonometric parametrization, we transform the zero equation into a family of fixed-point problems. Suitable invariant disks are constructed, and the corresponding mappings are shown to be contractions. This yields a constructive description of the complete zero set, proves the simplicity of all zeros, and provides geometrically convergent iterations for their numerical approximation. The method also provides explicit control over the approximation error. Finally, we establish a degree-dependent vertical strip containing all zeros and show that its width tends to zero as the degree increases. Consequently, the zeros become uniformly concentrated near the imaginary axis. Numerical examples illustrate the convergence of the proposed method, the associated error estimates, and the resulting zero distributions.

An LZ77 codec bounds every match by one sliding window, trading cheap offsets for short matches against reach for long ones. Match-length-dependent windows refine this trade-off: each match length has its own window, and the decoder infers the offset range from the decoded length. We present two such codecs: WLZ4, a byte-aligned LZ4-class format with one-byte and flagged offsets, and WZIP, an entropy-coded Zstandard-class format. Each block begins with its decoded length, from which WZIP derives its widest window and a decoder sizes its output. LZ4 blocks may expand incompressible input by 0.4%; WLZ4 adds at most 15 bytes, WZIP 2. We describe the formats, threaded match finders, decoders and parser. On Silesia, WLZ4 reaches a ratio up to 16% above LZ4HC's, decoding more slowly, and WZIP exceeds Zstandard 22's ratio while its trusted decoder runs at 91% of Zstandard's speed (87% checked; on enwik9, about as fast).

We implement (in a concrete example) a procedure of Dubrovin which gives a braid group action on spaces of solutions of ordinary differential equations such as the quantum differential equation of complex projective space. Such actions originated in physics for the tt* (topological-antitopological fusion) equations. We make the procedure systematic by utilizing an algebraic description of braid group actions on distinguished bases. We also generalize to the case of (arbitrary) solutions of the tt*-Toda equations, one of which corresponds to the quantum cohomology of complex projective space.

We study the length of one round of Uno under the official rules, for N players who each start with M cards. The fastest possible round takes at least M turns with two players and at least 2M-1 turns with three or more, and both bounds are attained for small M. An exact search over random deals shows that real deals rarely allow them. Under the official rules a round has no maximum length. Under forced play the probability that a round lasts more than t turns decays exponentially in t at a rate that does not depend on M. Our main result concerns very large hands. If every card a player ever holds is eventually played, a round lasts (27/19)NM turns to first order, and the constant depends only on the composition of the deck. The official rule that a Wild Draw Four may only be played by a player who holds no card of the current colour breaks this law. Every large hand hoards Wild Draw Fours, M/23 of them per player, and the round ends with a dumping endgame. For an even number of players this gives (53/46)NM turns. For an odd number of players the dumping cannot alternate round the table, and the endgame settles into a seating pattern whose turn rate we compute numerically. Simulations with up to 32,000 cards per hand support these limits. Finally we solve a small strategy game: with two players the Nash equilibrium over a family of 48 strategies is a single pure strategy that wins 62.8% of rounds against uniformly random play.

In this article, we mainly investigate when the binary operation $T^{[n]}:[0, 1]^2\rightarrow [0, 1]$ satisfies the associative law and thus becomes a triangular norm where \begin{equation*} T^{[n]}(x,y)=\left\{\begin{array}{ll} f_{n}(T(f_{n}(x),f_{n}(y))), &\text{if}\,\, (x,y)\in{[0,1)^{2}}, \\ \min\{x,y\}, &\text{otherwise,} \end{array}\right. \end{equation*} $f_{n}:[0,1]\rightarrow\left\{0, \frac{1}{n}, \frac{2}{n}, \dots, \frac{n-1}{n}, 1\right\}$ given by $f_{n}(x)=\frac{1}{n}{\lfloor nx \rfloor}$ with $n$ a positive integer and $T:[0, 1]^2\rightarrow [0, 1]$ is a triangular norm. We present some sufficient conditions for $T^{[n]}$ to be triangular norms, and for $T^{[n]}$ to be never associative when $T$ is strictly monotonic.

We construct compact, embedded solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ that arise from various non-compact initial data and exist for a finite time.

For $\kappa > 1$, we study the reciprocal LCM-power function \[ \sigma^{[\kappa]}_{-1}(n) := \sum_{d \mid n}\mu(d)\,\sigma_{-1}(n / d)^\kappa. \] We determine its average and extremal orders. A natural Robin-scale normalization is strictly increasing in $\kappa$ and tends to $\sigma_{-1}$, yielding an RH-equivalent criterion. Unconditionally, the corresponding Robin inequality holds eventually on every bounded parameter interval. Directed interval arithmetic proves the inequality for each $n > 5040$ and $1 < \kappa \le 100{,}000$.

We prove a large deviation principle for additive functionals of local Lipschitz functions of a one-dimensional symmetric zero-range process on the diffusive scale. The rate function is obtained from the dynamical large deviation principle for the empirical density and is characterized by a variational constraint involving the density at a fixed spatial point. On the microscopic side, a super-exponential replacement is needed to replace the local observable with a function of the empirical density. On the macroscopic side, the spatial regularity enjoyed by finite-rate trajectories is needed to pass from spatial averages to point evaluation.

In this paper, we study the transition threshold problem for the 2D Navier-Stokes Boussinesq equations around the Couette flow with a constant background temperature in two domains: $\mathbb{T} \times [-1, 1]$ and $\mathbb{R} \times [-1, 1]$. It was proved that if the sizes of initial perturbations of the vorticity and the temperature in Sobolev spaces are $\nu^{\fr13}$ and $\nu^{\fr56}$, respectively, then this system admits a globally stable solution. Besides, we derive the enhanced dissipation estimate for the vorticity and temperature, as well as the inviscid damping estimate for the velocity. The coupling in the equations poses some difficulties to the analysis. We overcome these challenges and establish the results via the quasi-linear method and energy estimates.

We provide a presentation of the braid group in which the generating set is the set of all positive half-twists, the relations are balanced positive relations, and every relation is conjugate to one in a finite list. We then prove that for any such presentation, its positive monoid is not isomorphic to the quasipositive braid monoid. Finally, we show that the quasipositive braid monoid has no proper Garside family. These results settle a problem of Calegari from the K3 problems list.

We study strictly positive multivariate binary distributions by separating one-body marginals from dependence. Local positive reweighting defines a dependence nucleus, completely characterized by the higher-order Walsh log-linear coefficients $J_S$, $|S|\ge2$. Any such nucleus and any strictly positive one-body marginals determine a unique joint distribution. In finite-spin Gibbs systems, these coefficients are the many-body interaction couplings.

Assuming that $\zeta(s)$ has no nontrivial zero with $\mathrm{Re}\, s > 7/8$, as asserted in a recent unrefereed preprint of OpenAI, we obtain $\Lambda \le 13963/100000 = 0.13963$ for the de Bruijn-Newman constant. A published Polymath boundary computation supplies a finite-region strip of half-width $1/5$. We derive an approximation on $3 \le \mathrm{Im}\, z \le 6$ from its general Riemann-Siegel coefficient bounds. At height $61/20$, a fixed cutoff permits two differentiations, and comparison with $\zeta$ bounds both the zero-density kernel and its vertical derivative. A new conjugate-pair inequality and a twenty-four-step strip comparison give the stated bound. The finite scalar and polynomial certificates are included in the paper; earlier comparisons giving $0.17973$, $0.17172$, and $0.16408$ are retained to show the source of the gain. The argument uses neither Gordon's analytic inputs nor Gomila's new computations. The seven-eighths hypothesis remains an assumption, and the published numerical boundary input is retained.

Approximating the effective anisotropic hyperelastic response of a representative volume element (RVE) requires constitutive expressiveness, mechanical consistency, and computational efficiency. This work develops a polyconvex physics-augmented neural network (PANN) construction based on paired directional features whose orientations, stretch powers, and determinant exponents adapt to the RVE response. The features are composed with convex, componentwise nondecreasing cores, realized as an input-convex neural network (ICNN) or an input-convex Kolmogorov-Arnold network (ICKAN). The complete energies satisfy objectivity, an exactly stress-free reference, two-dimensional polyconvexity, and prescribed growth conditions by construction, and nonnegative energy is certified separately for the selected models. On a multicavity and a single-cavity periodic cell, these energies are compared with an Unconstrained energy, trained identically without the curvature and growth constraints, and with three hyperreduced projection-based reduced-order models (HPROMs), which serve as physics-based references. On the first cell, learning the features lowers the stress error sevenfold, and six features suffice on both. The constrained energies remain below one percent in stress error and in the displacement error of a tensile-coupon FE$^2$ application with a full-order reference, whereas the Unconstrained energy and the HPROMs are more accurate inside the data. Beyond the data, however, the Unconstrained energy loses rank-one convexity, and under confined compression it stops below half of the 500 MPa that the constrained energies reach. These results quantify what the guarantees cost in accuracy and what they secure beyond the data.

We establish a Carathéodory-type extension theorem for premeasures with values in general separated abelian net convergence groups. Starting from hereditary exhaustivity on the source ring and intrinsic conditions for combining null errors, we construct a countably additive extension on a regulated Boolean sigma-algebra, with full upper and lower regularity in the original convergence. Completeness is required only on the limit-stable hull of the source range. Under closed summation for arbitrary directed rows and finite-response reflection, we identify the regulated domain with the splitting class of Sion's inner and outer processes. For a Dedekind-complete real vector lattice, we show that sequential closed summation on a bounded interval is equivalent to weak sigma-distributivity of the generated band, giving a converge-theoretic interpretation of the concept. Source representations allow us to recover the partially ordered-group extension of Volauf and the finite-variation extension of Kusraev and Malyugin from the general theorem. We then treat finite semivariation by adjoining dominated coordinate variations to an additive lift of the premeasure. The resulting extension is defined on the full semivariation regulator domain and preserves semivariation exactly, without requiring decomposability or the countable sup property. It includes measures of infinite vector variation.

A normalized sequence $(y_n)$ in the dual $E^{*}$ of a Banach space $E$ is called normal if it is weak-star null, and strongly normal if [ G(y_n)=\left{x\in E:\sum_{n=1}^{\infty}|y_n(x)|<\infty\right} ] is dense in $E$. We prove that, within the class of Grothendieck Banach spaces, reflexivity is characterized by the subsequence behavior of normal sequences: a Grothendieck space $E$ is reflexive if and only if every normal sequence in $E^{*}$ contains a strongly normal subsequence. If $E$ is nonreflexive and Grothendieck, then $E^{*}$ contains a normalized weakly null sequence none of whose subsequences is strongly normal; more strongly, it contains a closed hereditarily $\ell_{1}$ subspace $Y$ with a normalized weakly null Schauder basis such that no sequence in $S(Y)$ is strongly normal. We also obtain an exact quotient criterion valid for arbitrary Banach spaces: a normal sequence has a strongly normal subsequence precisely when, for some subsequence $z=(z_n)$, the quotient $E/\bigcap_n\ker z_n$ is separable. The proofs combine a weak-star basic-sequence argument with results of Haydon, Johnson–Maurey–Schechtman and Popov. Concrete realizations are given in $(L_{\infty}(\mu))^{*}$ for nonatomic probability spaces (and, more generally, for $\sigma$-finite spaces with a nonatomic part) and in $\ell_{\infty}^{*}$.

We construct closed Levi non-degenerate CR manifolds with non-vanishing Chern monomials. For every $n \geq 2$ and $\lceil n / 4 \rceil \leq m \leq \lfloor n / 2 \rfloor$, there exists a $(2 n + 1)$-dimensional example of signature $(\lfloor n / 2 \rfloor, \lceil n / 2 \rceil)$ on which all Chern monomials with positive indices at most $m$ and total weight at most $n - m$ are non-zero in cohomology with real coefficients. Taking $m = \lceil n / 4 \rceil$ gives examples with $c_{1}^{\lfloor 3 n / 4 \rfloor} \neq 0$. For $n = 4$ or $n \geq 6$, the latter examples show that the Chern-class vanishing theorem for strictly pseudoconvex CR manifolds does not extend to the Levi non-degenerate case. In these dimensions, their underlying manifolds admit no strictly pseudoconvex CR structures. The construction uses circle bundles over products of integral pseudo-Kähler manifolds.

We propose a low-complexity SVD-based transceiver for zero-padded memory-$1$ ISI channels. Appending a single zero-padding symbol to an $n$-symbol block yields an $n\times(n-1)$ channel matrix with an exact symmetric tridiagonal Toeplitz Gram matrix. This structure provides closed-form singular values and vectors, enabling exact SVD precoding and equalization via FFTs without explicitly forming dense SVD matrices. The resulting implementation has $O(n\log n)$ complexity and incurs only a $1/n$ rate penalty. The resulting parallel scalar subchannels enable joint adaptive PAM and power allocation for SER minimization, followed by exact soft-information computation for LDPC decoding. At comparable code rates near $0.945$, simulations show that our proposed scheme outperforms a conventional TX-FFE/RX-DFE system with hard-decision Reed-Solomon (RS) decoding by approximately $5$ dB at FER $10^{-5}$. At high SNR, it achieves nearly the same FER performance as a SOVA-based receiver with LDPC decoding, while replacing trellis-based detection with fast SVD precoding and exact scalar soft-output computation.

We introduce coset lifted product (CLP) codes, a family of quantum LDPC codes that combine the protograph structure of lifted product codes with the coset action of Aydin, Tamo, and Barg \cite{AydinTamoBarg2026}. Given a finite group $G$, a subgroup $H \le G$ that need not be normal, a protograph $A$ over $\mathbb F_2[G]$, and a protograph $B$ over $\mathbb F_2[N_G(H)]$, the two protographs act on the cosets $G/H$ from the left and from the right to produce quantum CSS codes. Further, for groups of odd order, we give an explicit formula for the dimension as a sum over the irreducible representations $\rho$ of $G$ of hypergraph product dimensions. A search over $14,843$ pairs $(G,H)$ with $H$ restricted to be non-normal and $120 \le n \le 400$ yields certified codes that exceed the best published value of $Q=kd^2/n$ at check weights $6,7$ and $9$, i.e., we obtain $[[360,20,20]]$ at $w=6$, a $[[360,24,20]]$ code at $w=7$, and codes $[[252,16,18]]$ and $[[288,16,20]]$ at $w=9$. Our search also produce high-distance codes with parameters $[[168,10,14]], [[240,8,20]]$ at $w=6$ and $[[216,8,18]]$ at $w=7$.

Let $A_n=\mathbb{k}[x_1^{\pm1},\ldots,x_n^{\pm1}]$ be a Laurent polynomial algebra over a field $\mathbb{k}$ of characteristic zero. We determine explicitly the automorphisms commuting with each locally finite derivation \[ D_\mu=\sum_{i=1}^n\mu_i x_i\frac{\partial}{\partial x_i}, \qquad \mu=(\mu_1,\ldots,\mu_n)^T\in\Bbbk^n. \] These are precisely the automorphisms $\rho(x_i)=\lambda_i x^{m_i}$, where $\lambda_i\in\mathbb{k}^\ast$, the exponent vectors $m_i$ are the columns of a matrix $M_\rho\in\operatorname{GL}_n(\mathbb Z)$, and $\mu^TM_\rho=\mu^T$. We also give an explicit parametrization in dimension two and relate these automorphisms to symmetries of diagonal flows on algebraic tori.

Automatic differentiation (AD) is a cornerstone of modern machine learning. Since all AD frameworks implicitly fix a basis of Euclidean space to define gradients and Jacobians, they natively process mathematical objects that are arrays of a fixed set of coordinates. This makes it awkward to express objects such as rotations, volume forms, and curvatures that arise, for example, in applications of AI to science. Bespoke workarounds lack reusability. We show that (i) a single, basis-independent derivative operator can represent many geometric differential operations, and (ii) this operator can be expressed entirely in terms of existing AD primitives. This reusable transformation can be composed with JAX batching, compilation, and hardware acceleration. Geometry enters as algebraic data. We experimentally demonstrate its reuse across different manifold operators, its composition through a spacetime inverse-problem pipeline, and its use in optimization over different rotor subgroups.

Autonomous small-body proximity operations require continuous-thrust guidance under uncertain gravity, solar-radiation pressure (SRP), and stochastic disturbances. We combine iterative covariance steering (iCS) with $L_1$ distributionally robust adaptive control ($L_1$-DRAC) to provide risk-aware collision avoidance despite mismatch between the nominal planning model and the true dynamics. Iterative covariance steering plans a nominal feedback policy satisfying distributionally robust conditional value-at-risk keep-out constraints, while $L_1$-DRAC compensates for dynamics uncertainty and bounds the residual distributional mismatch through a Wasserstein radius used by the planner. In $\mathbf{1000}$ paired Monte Carlo trials with gravity, SRP, and diffusion mismatch, $L_1$-DRAC adaptive control reduces the keep-out violation rate from $\mathbf{32.7\%}$ to $\mathbf{0.2\%}$ and the physical collision rate from $\mathbf{16.4\%}$ to zero, without actuator saturation.

This paper studies continuous covariant kernel families over hulls of Delone sets of finite local complexity using graded Wiener-algebra methods. The norm given by the least common weighted summable off-diagonal profile is identified isometrically with a graded $\ell^1$-norm. The resulting Banach $*$-algebras are symmetric, spectrally invariant in the reduced groupoid C*-algebra, and stable under holomorphic functional calculus. Exact physical weights give preservation of polynomial and stretched-exponential classes, including a fixed stretched-exponential rate. A configuration with dense orbit has a faithful regular representation. Invertibility on that configuration therefore gives an inverse family with one profile valid throughout the hull; for repetitive sets every configuration has this property. For polynomial weights of order $s>0$, return-vector counting yields a differential estimate with parameter $\kappa/(\kappa+s)$, where $\kappa$ is a growth exponent of the return vectors and may be taken equal to $d$ for Meyer sets. Combined with established norm-controlled inversion criteria, this gives explicit bounds in the common-profile norm. Explicit bounds are also obtained for stretched-exponential physical weights and for gapped Fermi projections.

We propose constructions of spherical $5$- and $7$-designs in odd dimensions by lifting balanced halves of even-dimensional designs to several layers of suitable latitudes. For every odd dimension, we obtain an explicit spherical $5$-design with at most $60(d^2-1)$ points, attaining the optimal quadratic order in the dimension. Assuming Zauner's conjecture, this bound improves to $7.5(d-1)^2$. For every odd $d \geq 9$, we also construct a spherical $7$-design with $110M$ points from a complex projective $3$-design with $M$ points in $\mathbb{CP}^{(d-3)/2}$.

We study testing a random geometric graph with a fixed kernel against an Erdős-Rényi graph with the same edge density. Under the null hypothesis, the graph has $n$ vertices and independent edges with probability $p$. Under the alternative hypothesis, latent points $X_1,\ldots,X_n$ are independent and uniform on $\mathbb{S}^{d-1}$, and edges are conditionally independent with probabilities $K(\langle X_i,X_j\rangle)$, where $p=\mathbb{E}[K(\langle X_1,X_2\rangle)]$. The goal is to determine a critical dimension $d_\star = d_\star(n,K)$: the total variation distance between the two graph laws tends to one for $d\ll d_\star$ and to zero for $d\gg d_\star$. For dense graphs, [BDER16] established $d_\star\asymp n^3$ for hard threshold kernels, whereas [MWX26] proved $d_\star\asymp n^{3/4}$ for smooth kernels with $K'(0)\ne0$. We connect these results by characterizing the detection thresholds for kernels that need not be differentiable at zero. We also determine the thresholds for smooth kernels whose first nonzero derivative at zero has finite order greater than one. Throughout these classes, signed triangle or quadrilateral counts attain the optimal detection threshold. For certain nondifferentiable kernels, quadrilateral counts are optimal while triangle counts are suboptimal.