Papers

We prove that the integral cycle class map for $\SP^d(\PP^n)$ is split injective, with finite cokernel annihilated by $d!$,and we determine the ranks of Chow groups for $\mathrm{SP}^d(\mathbb{P}^n)$. Compatible integral transfer retractions show that $\CH_q(\SP^d(\PP^n))$ stabilizes for $d\geq q$. An explicit nonalgebraic order two class persists under addition of points, producing a split cokernel summand even in the range $q\leq d$ and in the infinite symmetric product. For the smooth configuration open set, we identify the codimension two cokernel with third unramified cohomology and detect its ramification on a rational resolution. In contrast to addition, even multiplicities annihilate a nonzero Chow class.

We introduce the parity module of a matroid with coefficients in a field $\mathbb{k}$. When $\mathbb{k}$ has characteristic zero, we show that the parity module of a matroid is isomorphic to its intersection cohomology module, yielding a new intrinsic characterization of the latter. The structure and dimension of the parity module can vary depending on the characteristic, giving rise to characteristic-dependent generalizations of the Kazhdan-Lusztig and Z-polynomials of matroids. When the matroid is realizable over the complex numbers, the parity module is isomorphic to the hypercohomology of the parity sheaf supported on the arrangement Schubert variety of any such realization. We also introduce the notion of pure modules over any field $\mathbb{k}$ and show that they satisfy a matroid analogue of the Beilinson-Bernstein-Deligne-Gabber decomposition theorem: Every pure module over the graded Möbius algebra decomposes as a direct sum of shifts of the parity modules of matroids obtained by restricting to flats.

Spatial reception under directed molecule transport requires balancing useful molecule capture against intersymbol interference (ISI). To characterize this balance, we study a fully absorbing spherical receiver that counts absorption events only within a prescribed cap. Using the known joint first-hitting law under aligned drift, we derive exact cap impulse, cumulative, and eventual responses. We perform stable cumulative evaluation. We then assess the derived responses through particle simulations and time-step comparisons, and demonstrate that our conditional-timing analysis explains when opposite drifts of equal magnitude can favor the same wider aperture. We also show that for the studied geometry and a symbol duration of $0.15$ s, the aperture that maximizes the signal-to-interference difference (SID) increases from $29.13^\circ$ at zero drift to $61.86^\circ$ at a drift magnitude of $40$ $\mu$m/s. These results (i) connect joint arrival-time and location statistics to spatial reception design, (ii) characterize the tradeoff between useful molecule capture and ISI, (iii) support SID-based aperture selection for a given drift and symbol duration, and (iv) quantify the SID loss incurred by retaining the zero-drift optimal aperture for the same symbol duration.

We classify, up to equivalence, all left-invariant complex structures on the three-dimensional Euclidean rigid motion group $SE(3).$ Our approach is based on the correspondence between complex structures on a real Lie algebra $\mathfrak{e}(3)$ of the motion group and complex Lie subalgebras of the complexification of $\mathfrak{e}(3).$ The resulting structures form a one-parameter family $J_\phi.$ We further establish a connection with the classical Calabi-Eckmann complex structures on $S^3\times S^3.$ More precisely, we construct a one-parameter deformation of Lie brackets connecting $\mathfrak{so}(3)\oplus \mathfrak{so}(3)$ to $\mathfrak{e}(3),$ for which each endomorphism $J_\phi$ remains integrable. Consequently, every left-invariant complex structure on $SE(3)$ arises as a contraction limit of Calabi-Eckmann structures. The same family of endomorphisms also defines complex structures on $\mathfrak{so}(3,1).$

We establish a joint stable limit theorem for integrated discretization errors of finite-dimensional càdlàg semimartingales whose predictable characteristics are absolutely continuous in time and satisfy a pathwise integrability condition. The normalized integrated error, evaluated on the discretization grid, converges stably jointly with the discretized semimartingale in the vector-valued Skorokhod $J_1$-topology. No martingale representation property is assumed. We apply this theorem to the Euler-Maruyama scheme for stochastic differential equations with additive semimartingale noise. For a $C^1$ drift with locally Lipschitz derivative and a nonexplosive solution, the grid-point error normalized by $n$ converges stably jointly with the discretized driver. The limiting error solves a linear integral equation with an explicit inhomogeneous term involving the driver, an auxiliary Brownian motion, and independent uniform marks attached to the jumps. We also obtain the same limit under two alternative drift regularity conditions based on small-jump integrability or Lévy semigroup smoothing. For continuous drivers, the continuous-time error converges stably in the uniform topology.

We define genus character $L$-functions of orders in quadratic function fields of odd characteristic. We express each as a product of two quadratic $L$-functions and explicit local factors at primes dividing the conductor, with a possible shift of the variable in one of the two quadratic $L$-functions. This product formula yields meromorphic continuation and functional equations. In the real case, we evaluate special values in terms of continued fractions and derive formulas for the product of class numbers of two imaginary quadratic function fields. These formulas are function field analogues of the Kaneko-Mizuno-Zagier formula.

We prove that every stable atomic (twisted) vector bundle on a projective hyper-Kähler manifold is 1-obstructed. In the appendix, we construct a simple atomic object that is not 1-obstructed, providing a counterexample to Beckmann's conjecture.

A well-known conjecture of Toda's asserts that the BPS invariants associated to a Calabi–Yau threefold are independent of the chosen Euler characteristic. We answer the question in the affirmative for a certain class of local surfaces, which we call warped local curves. Along the way we show that the moduli stack of sheaves on a warped local curve can be written as the global critical locus of a function. We also show that the moduli stack of sheaves on a curve fibration over $\mathbb{A}^1$ is the global critical locus of a function.

We construct a rank-9 uniform non-homogeneous vector bundle on ${\mathbb P}^5$. This yields the first counterexample to the longstanding open conjecture that every uniform vector bundle of rank $r<2n$ on ${\mathbb P}^n$ is homogeneous.

Let $\Omega=\Omega^+\subset \mathbb{R}^{n+1}$ be a bounded open set with $n$-Ahlfors regular boundary and let $\Omega^-=\mathbb{R}^{n+1}\setminus\overline{\Omega^+}$. For $x\in\partial\Omega$, $r>0$, define \begin{equation*} \varepsilon_{n,\Omega}(x,r):=\frac{1}{r^n} \inf_{H^+}\mathcal{H}^n\left(((\partial B(x,r)\cap H^+)\setminus \Omega^+) \cup ((\partial B(x,r)\cap H^-)\setminus \Omega^-)\right), \end{equation*} where $\mathcal{H}^n$ stands for $n$-dimensional Hausdorff measure and the infimum is taken over all open affine half-spaces $H^+$ such that $x\in \partial H^+$. Here $H^-=\mathbb{R}^{n+1}\setminus \overline{H^+}$. In this work we prove that $\Omega$ is a $C^{1,\alpha}$ open set, with $0<\alpha<1$, if and only if there exists some constant $C>0$ such that $\varepsilon_{n,\Omega}(x,r)\leq C\,r^\alpha$ for all $x\in\partial\Omega$ and all $0<r\leq \mathrm{diam}(\partial\Omega)$.

We develop novel methods for the fast construction of spectral approximations to smooth linear and weakly singular Volterra integral operators and for the fast evaluation of smooth nonlinear Volterra integrals. The proposed methods employ the Duffy mapping, which transforms the original indefinite integral into a definite integral and maps the original causal triangular domain to a square, thereby avoiding any artificial extension of the kernel function. For smooth linear and weakly singular Volterra operators, the resulting approximations are almost-banded and lower-banded, respectively. The $N\times N$ principal finite section of these approximations can be constructed in $O(N)$ and $O(N^2)$ flops, respectively, via recurrence relations. For Volterra integrals with a smooth kernel that depends nonlinearly on a given function, recurrence relations lead to fast evaluation with a complexity of $O(N^2\log N)$ when the function is approximated by a degree-$N$ Chebyshev approximant. These spectral approximations and evaluation procedures give rise to fast and stable spectral methods for solving Volterra integral equations, as well as a broad class of problems involving Volterra integral operators. Extensive numerical experiments show that these spectral methods significantly outperform existing methods in terms of stability, speed, accuracy, and ease of implementation.

We establish new restrictions on the virtual cohomological dimension of word hyperbolic groups admitting Anosov representations into $\operatorname{PO}(p, q)$ ($p > q \geq 2$) or the exceptional Lie group $\operatorname{G}_2'$. Our main result states that, if $P_k$ is the parabolic subgroup of $\operatorname{PO}(p, q)$ stabilizing an isotropic $k$-plane in $\mathbb{R}^{p, q}$, a group admitting a $\{P_1,P_q\}$-Anosov representation into $\operatorname{PO}(p, q)$ has virtual cohomological dimension strictly less than $p$. As a result, a $\mathbf{H}^{p, q-1}$-convex cocompact subgroup of $\operatorname{PO}(p, q)$ cannot act properly discontinuously and cocompactly on $\mathbf{H}^{p, q-1}$. The main theorem also implies that a group admitting a $P_{\alpha}$-Anosov representation into the exceptional Lie group $\operatorname{G}_2'$ has virtual cohomological dimension strictly less than $4$, where $\alpha$ is a short restricted root of $\operatorname{G}_2'$. As an application, we prove that the homogeneous space $\operatorname{G}_2'/ \operatorname{SU}(2, 1)$ does not admit any compact quotient.

We identify a normalization defect in a determinantal construction of fourth-order Hahn difference equations and compare its scalar equation with Foupouagnigni's 1998 elimination and its published 2001 counterpart. On an explicit positive orthogonal family, all five prescribed coefficients vanish: the formula gives \(0=0\) and imposes no condition. Along an explicit regular perturbation, division by \(\varepsilon^2\) before specialization recovers a nonzero equation on that family, the same scalar equation as the correctly executed old shift-and-eliminate procedure. On the common nondegenerate domain specified below, both constructions give the same scalar operator up to a nonzero rational factor. The first-order degree bounds follow directly from valid identities of 1998; preserving cancellations gives a bound that can be arbitrarily sharper on explicit admissible inputs, together with exact top coefficients and cancellation criteria. Explicit coefficient formulas do not by themselves yield a new scalar equation on this domain. Our comparison does not exclude corrective or expository contributions; additional computational gain or exceptional coverage must be demonstrated.

Tracking changing comparators is a central problem in online learning. For exp-concave losses, previous work established the $T^{1/3}P_T^{2/3}$ dependence on the horizon $T$ and Euclidean path length $P_T$ for fixed dimension, while the optimal cost of dimension remained unresolved. We resolve this question by establishing a sharp $d^{1/3}$ curved dimension factor, improving the $d^{2/3}$ factor of recent switching and interval reductions [1, 2]. To achieve this improvement, we develop a proper, full-information learner on compact convex subsets of $\mathbb{R}^d$ that adapts to the unknown path length and, under fixed dimension-independent diameter, gradient, and curvature parameters, guarantees $$\operatorname{Reg}_T(u_{1:T}) \le \widetilde{O}\left(d+\min\left\{d^{1/3}T^{1/3}P_T^{2/3},\sqrt{TP_T},T\right\}\right).$$ A matching lower bound on a diameter-one Euclidean ball establishes the minimax frontier up to logarithmic factors when $T\gtrsim d^2$, including the transition from the curved rate to larger path budgets. The lower bound already holds for losses that depend on only one coordinate per round, isolating an unavoidable cost of learning across directions. Our upper bound follows from square-root Gaussian KL control of comparator movement and a cancellation argument that pays for history restarts. The curved learner maintains $O(\log^2 T)$ Gaussian states, uses one gradient per round, and runs in time near-linear in $T$ under a Mahalanobis-projection oracle.

This paper extends Macdonald's duality theory and Cuong-Nam's local homology theory – which is in some sense dual to the local cohomology theory of A. Grothendieck [Gro1] – to semilocal noetherian rings equipped with the $J$-adic topology. By overcoming the key structural obstructions of the semilocal setting via Zöschinger's structure theory and semilocal Matlis duality, this work establishes topological limit duality isomorphisms and links local homology with local cohomology. Ultimately, this framework provides the foundational machinery required to extend the classical results of Macdonald [Mac] and Cuong-Nam [C-N1] to the semilocal setting.

Shifted convolution sums of $\mathrm{GL}(3)\times\mathrm{GL}(2)$ type have been extensively studied in several important settings, while a number of natural cases remain open. In this paper, we first revisit the shifted convolution sums involving $\lambda_\pi(1,n)$ and the divisor function $d(n)$, improving the recent power saving of $1/42$ obtained by Blomer and Li to $1/22$. We also treat the corresponding sums involving $\lambda_\pi(1,n)$ and $r(n,\mathcal{Q})$, settling a previously open case and obtaining a power saving of $1/22$. Here $\lambda_\pi(1,n)$ denotes the normalized Fourier coefficients of a Hecke–Maass cusp form $\pi$ on $\mathrm{SL}(3,\mathbb{Z})$, and $r(n,\mathcal{Q})$ denotes the number of representations of a positive integer $n$ by a positive definite integral binary quadratic form $\mathcal{Q}$. The proof combines two circle methods with a different organization of the major-arc analysis. A key point is that the nonzero Poisson frequencies are kept together in a single coefficient vector, rather than estimated separately, which allows them to be controlled by a finite matrix norm. The zero Poisson frequency is treated separately: its contribution is independent of the reduced residue class, and its arithmetic mean cancels the constant term in the complementary Jutila weight before absolute values are taken. We make this cancellation explicit, including the relevant normalization and local factors. The same mechanism extends to the binary quadratic-form setting, while the remaining Jutila contribution is handled by the available correlation estimates.

We study Boolean entropy and Fisher information in information inequalities and the Boolean central limit theorem. We prove a fractional Stam inequality for the Boolean Fisher information associated with arbitrary fractional partitions of subset sums, and derive an entropy power inequality for sums of Boolean independent random variables and its weighted version. We then establish a Boolean analogue of Barron's entropy convergence result along the Boolean central limit theorem: Boolean entropy converges to the entropy of the Rademacher law if and only if the initial entropy is finite. Furthermore, under a finite fourth moment, in the Boolean central limit theorem, we obtain \[ \Gamma(\mu_n\mid\mathbf b)= \frac{m_4(\mu)-1-m_3(\mu)^2}{2}\frac{\log n}{n} +o\!\left(\frac{\log n}{n}\right), \] with an explicit example showing sharpness. For symmetric laws with finite fourth moment and finite Boolean Fisher information, \[ \Psi(\mu_n\mid\mathbf b) =2\bigl(m_4(\mu)-1\bigr)\frac{\log n}{n} +o\left(\frac{\log n}{n}\right). \] Our approach is based on the Nevanlinna representation of Cauchy transforms.

A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. It is called $\epsilon$-positive if the ratio of the restrictions of the imaginary and real parts of the holomorphic volume form is bounded by $\frac{1}{\epsilon}.$ An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We prove that the Riemannian metric restricted to the subset of $\epsilon$-positive Lagrangians induces a path metric when the relative Kähler class of the ambient Calabi-Yau manifold lies on a line of rational slope. The proof relies on the construction of a sequence of quantization maps from a circle bundle over the space of positive Lagrangians to the dual of the space of holomorphic sections of powers of a prequantum line bundle. The rationality condition for the relative Kähler class allows us to reduce to the case of Bohr-Sommerfeld Lagrangians. Building on ideas of Tian and Nirenberg-Wells, Hörmander estimates are used to show that functions on a Bohr-Sommerfeld Lagrangian can be approximated by holomorphic sections of a sufficiently high power of the prequantum line bundle. Consequently, the quantization maps eventually separate fibers of the circle bundle.

For fixed positive degrees $n,m$, alternating monic reciprocal transformations define a rational discrete dynamical system on the finite-dimensional coefficient space of polynomial pairs. This system arises from congruence jumping in polynomial divisibility problems, where a quotient move changes the polynomial state, but its coefficient dynamics can be studied independently. We prove that every state with nonzero constant terms returns after four alternating updates to its rational gauge classes and classify when this return orbit is realized over $\mathbb Z$. The local obstruction is encoded by endpoint $p$-adic valuation defects. After the first integral block, constant-term valuations evolve by an explicit matrix $M_{n,m}\in\operatorname{SL}_2(\mathbb Z)$ whose trace depends only on $s=nm$. This yields a finite-order, elliptic-type regime for $s<4$, a parabolic dilation regime for $s=4$, and a hyperbolic regime for $s>4$. In the hyperbolic case, invariant thresholds, irrational eigencones, and discrete valley formulas give finite criteria for forward and bi-infinite integral orbits; nevertheless, failure can be delayed arbitrarily long. At the pointwise level, separated auxiliary-relation transport factors through the jumped-variable projection. One jump preserves the visible support rank, while two alternating jumps collapse it to dimension at most one; saturation cannot restore rank. In the quadratic specialization, a rank-two $p$-adic relation lattice gives an exact local cancellation-depth formula and a finite criterion on periodic Vieta orbits. We classify the fixed auxiliary relations that saturate indefinitely on an irreducible six-cycle and construct arbitrarily long persistence on parabolic and fixed hyperbolic orbits.

We study continuous-time q-learning for entropy-regularized diffusion control without assuming classical differentiability of the value function. Rewards and admissible policies need only be Borel measurable, which allows discontinuous rewards. The diffusion coefficient $\sigma$ is action independent and locally uniformly elliptic, and the drift is $b_0 + \sigma\psi$, with $b_0$ action independent and $\psi$ bounded. In this setting the q-function involves the value only through its gradient. For each admissible policy, we represent its value by a backward stochastic differential equation (BSDE) and prove that this value is continuous before the horizon, although the data are only measurable. Replacing the gradient with a measurable representation of the martingale integrand, we define the q-function up to Lebesgue-null sets and recover the classical definition when the value function is smooth. We then prove policy improvement, when the Gibbs update is admissible, and martingale characterizations of the value and the q-function from on-policy and off-policy observations. Here the candidate value need only be continuous before the horizon, and actions are sampled on a deterministic finite grid and held constant between sampling times. Under an integrability condition on the reward, we show that a second BSDE gives an upper bound for the optimal value, attained when its Gibbs policy is admissible, and we characterize its solution in the same way.

A recent graph-theoretic criterion for a Hamiltonian to exhibit a free-parafermion spectrum is that its frustration graph be an indifference graph. We show that an arbitrary finite indifference graph can be realized as a frustration graph of the quantized six-vertex model at odd roots of unity, by a suitable choice of the defining wiring diagram and parameters. This provides a concrete realization of the criterion through the quantum Frobenius property of the quantized six-vertex model, recently established from its three-dimensional integrable structure.

The Hilbert specialization property of a field, whereby irreducibility of polynomials is preserved under specialization of some of the variables, admits an integral counterpart where specialization takes place inside a ring and irreducibility is required over the ring. It is known that the ring $\mathcal{O}_K$ of integers of any global field $K$ satisfies this integrally Hilbertian property. Using a different approach, we reprove this result and strengthen it by showing that the corresponding integral Hilbert subsets of $\mathcal{O}_K^k$ are not only Zariski-dense, as required by integral hilbertianity, but also of positive density. A central ingredient is a generalization to the ring $\mathcal{O}_K$ of the Ekedahl–Poonen formula originally proved for $\mathbb{Z}$ and $\mathbb{F}_q[t]$. This formula, of independent interest, gives the density of tuples in $\mathcal{O}_K^k$ for which coprime polynomials in $k$ variables and with coefficients in ${\mathcal O}_K$ take coprime values. Our generalization requires estimates for the number of points of bounded height in an arithmetic progression. We obtain such estimates as our third main result, thereby contributing to the theory of heights.

We revisit the effective radial Schrödinger equation for a charged spinless particle in collinear static electric and magnetic fields that was formulated in our earlier work within the tridiagonal representation approach (TRA). In that model the angular factor of the electric coupling is replaced by the orientation parameter $C=\cos\theta$, the linear Zeeman term is diagonal in the azimuthal quantum number, and the effective field-modified central potential takes the form $V(r)=\Vt(r)+qECr$. Here we show that the same equation can be solved exactly without the TRA, by the Nikiforov–Uvarov (NU) method, once the potential component $\Vt(r)$ is prescribed. Three examples are worked out in closed form: a Kratzer-type component, which yields a linear-plus-Coulomb-plus-inverse-square central potential; a pseudoharmonic-type component, which yields a linear-plus-harmonic-plus-inverse-square potential; and a Hulthén-type component with a repulsive core (Manning–Rosen form), which yields a screened confining potential with a finite number of $s$-wave bound states. For each case we obtain the energy spectrum, the normalized eigenfunctions in terms of Laguerre or Jacobi polynomials, the location of the potential minimum, and closed-form radial moments. The spectra are independent of the orientation $\theta$ and of the electric field, in agreement with the structure found by the TRA, while the field enters through the shape of $V(r)$, the radial moments, and the expectation value $\langle V\rangle_k$. All analytical results are confirmed by an independent finite-difference diagonalization.

Generalized complex structures give rise to generalized Dolbeault, Bott–Chern and Aeppli cohomologies. We study these cohomologies for invariant generalized complex structures on almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0\ltimes_A\mathfrak h$, where $\mathfrak h$ is an abelian ideal of codimension one, and on almost abelian solvmanifolds $\Gamma\backslash G_A$. We first show that the invariant generalized Dolbeault cohomology can be identified with the Chevalley–Eilenberg cohomology of the complex Lie algebra $\mathcal L$, the $+{\mathbf i}$-eigenspace of the structure, with coefficients in a character determined by the canonical line. We prove that $\mathcal L$ is itself an almost abelian Lie algebra; thus the cohomology reduces to the kernel and cokernel of an explicit derivation. Assuming that $A$ is diagonalizable over $\mathbb R$, we obtain a combinatorial formula for the cohomology dimensions and compare with compact quotients. Other comparison results are discussed, noting that the cohomology of invariant forms always injects into that of $\Gamma\backslash G_A$.

A graph $H$ is called common if the number of monochromatic copies of $H$ in a $2$-edge-coloring of a large complete graph is asymptotically minimized by the random coloring with an equal proportion of each color. The house graph and $K_4$ with one edge subdivided once are known to be common by the flag algebra method. Grzesik, Lee, Lidický, and Volec asked for “human-friendly” proofs of the commonality of these two graphs. In this short note, we give flag-algebra-free proofs that both graphs are common.

Let $W$ be a complex reflection group and $X$ the De Concini-Procesi wonderful model of its reflection arrangement. Inside Etingof's sheaf of Cherednik algebras on $X$ we define a subsheaf generated by the Dunkl operators that extend the standard ones on the hyperplane complement. We conjecture that at $t=0$ the image of the logarithmic vector fields in a natural module over this subsheaf is the sheaf of Gaudin algebras on $X$ defined by Aguirre, extending a construction of Bonnafé and Rouquier from the hyperplane complement to $X$. We prove the conjecture for $W=S_{3}$.

We propose and analyze a projection-based model reduction method for the rapid and reliable solution of time-harmonic acoustic wave propagation problems over a frequency band, with particular emphasis on seismic applications. The high-fidelity model is based on a hybridizable discontinuous Galerkin (HDG) discretization, formulated as a generalized saddle-point system coupling the volume and trace unknowns. The basis construction combines residual-based greedy sampling of the frequency parameter, enrichment with frequency derivatives of the HDG solution, and proper orthogonal decomposition (POD) compression. We establish the well posedness of both the high-fidelity and reduced problems and characterize their stability with inf-sup constants. We investigate the interaction between static condensation, inherent to the HDG discretization, and projection onto the reduced spaces, characterizing its implications for the reduced formulation and approximation errors. Since these operations do not commute in general, we derive a priori error estimates for both the trace and volume variables to quantify the gap between the two operations. We further identify a lifting condition on the volume reduced space that eliminates these gaps and yields a trace error estimate independent of the volume approximation error. Numerical experiments on the two-dimensional Marmousi model and a three-dimensional heterogeneous medium, with more than $10^{7}$ unknowns, illustrate the theoretical estimates and show that the reduced model achieves speedups of several orders of magnitude compared to the high-fidelity solver.

We give a randomized polynomial-time $0.48$-approximation for maximizing a nonnegative non-monotone $k$-submodular function under a matroid constraint, for every $k\ge2$. The algorithm combines a top-two label rule with a nested-chain potential and maximum-weight matroid bases. Its analysis reduces the approximation guarantee to a one-dimensional time allocation: a uniform local lower bound leads to a linear differential equation, concavity closes the remaining interval, and two elementary budgets complete the proof. A strict saving in the time budget absorbs sampling and discretization errors, yielding the exact $0.48$ guarantee in the value-oracle model.

In this paper, we give a family of the arithmetic progressions in which the numbers of odd unimodal and odd strongly unimodal sequences are divisible by 2 and by 4. In particular, we answer a question of Bringmann and Lovejoy on the existence of such arithmetic progressions for odd strongly unimodal sequences modulo 4.

We present eMOOD, an entropy-guided evolution of the Multidimensional Optimal Order Detection (MOOD) framework for high-order finite-volume methods on unstructured meshes. A novel troubled-cell indicator based on numerical entropy production replaces the classical relaxed discrete maximum principle. Using entropy generation as a physically motivated measure of non-smoothness, the detector targets shocks, contact discontinuities and under-resolved flow features with a compact stencil. Compared with conventional MOOD detection, eMOOD reduces flagged cells by up to 75%, improving efficiency and reducing memory-access overhead. The detector is used within both the classical MOOD cascade of progressively reduced-order schemes and a novel MOOD-CWENO hybrid, which retains inexpensive linear reconstruction in smooth regions and activates nonlinear high-order reconstruction only where needed. Implemented in the open-source parallel solver UCNS3D, the methodology is assessed on the Shu-Osher shock-entropy-wave interaction, double Mach reflection, supersonic forward-facing step, shock-helium-bubble interaction and triple material point problems. These benchmarks encompass strong shock interactions, shock-induced instabilities, multi-material interfaces and vortex-dominated flows, testing robustness, shock capturing and accuracy on unstructured meshes. Results show that eMOOD preserves the accuracy and robustness of reference CWENO and MOOD formulations while achieving speed-ups of up to 20% without loss of solution quality. Its compact detection stencil, reduced memory footprint and improved localisation offer a more scalable and efficient framework for massively parallel high-order compressible-flow simulations. UK Ministry of Defence (copyright) Crown owned copyright 2026 AWE.