Papers

We prove a conditional upper bound for moments of the analytic rank of elliptic curves over number fields. We also prove conditional bounds for the density of elliptic curves with analytic rank larger than a given bound.

We study rationality of conic bundles over finite fields defined over the line. In particular, for a fixed number $n$ of degenerate fibers, what is the proportion of good conic bundles that are rational over a finite field of order $q$? The Galois action on their Picard groups factors through the Weyl group $W(D_n)$, reducing rationality to a condition on signed permutation types. Using work of Colliot-Thélène and Yang together with Chebotarev density, we compute the asymptotic proportion of such conic bundles that are rational as $q$ goes to infinity.

A 140-GHz transmitter with direct raised-cosine-like envelope shaping and wide-range phase tuning is presented in 90-nm SiGe BiCMOS. The proposed architecture relaxes conventional baseband pulse shaping and high-speed digital-to-analog converters by directly synthesizing 3-level and 5-level RF envelope states that approximate a raised-cosine waveform, enabled by a 23-dB modulation dynamic range. Measured results demonstrate data rate up to 8 Gbpers with 34-dB sidelobe suppression. To support scalable phased-array applications, an injection-locked-oscillator phase-tuning path is also integrated, providing 22.5deg digital phase resolution together with continuous analog tuning over more than 360 deg. The transmitter achieves up to 2 dBm output power and combines direct RF envelope shaping with wide-range fine phase control for sub-THz wireless links

We extend to abelian length categories of finite rank a characterisation of brick-finiteness known for finite-dimensional algebras. We prove that such a category is brick-finite if and only if every torsion class is generated by a wide subcategory and every torsionfree class is cogenerated by a wide subcategory. The proof is based on an exchange relation for brick labels across wide intervals which is a shadow of the mutation of simple minded collections. As a corollary,we extend the validity of the first Brick Brauer-Thrall conjecture to this setting.

The Hurwitz bound on the order of the $\mathbb K$-automorphism group ${\rm{Aut}}({\mathcal{X}})$ of an algebraic curve ${\mathcal{X}}$ of genus $g(\mathcal{X})\ge 2$ defined over a field $\mathbb K$ of zero characteristic states that $|{\rm{Aut}}({\mathcal{X}})|\le 84(g(\mathcal{X})-1)$. Improved bounds are available for the order of certain types of subgroups within automorphism groups. For instance, if a subgroup $H$ of ${\rm{Aut}}({\mathcal{X}})$ is dihedral, then in the complex case, $|H| \leq 4g(\mathcal{X}) + 4$. More recently it has been shown that a tighter bound holds for $H$ a generalized quasi-dihedral group. In this paper we explore the more general setting of a curve defined over a field of any characteristic, and $H$ a group admitting a cyclic subgroup of index two. We show that the same upper bound for the size of a dihedral group of automorphisms holds for curves defined over an algebraically closed field of characteristic $p\ne 2$. Then we provide some classification results about (non-dihedral) groups of size larger than $4g(\mathcal{X})+4$ admitting a cyclic subgroup of index $2$.

We variationally characterize the bottom of the spectrum of the buckling problem in an infinite cylinder $A\times\mathbb{R}^{n-d}$, where $A$ is an open bounded subset of $\mathbb{R}^d$, and compute it explicitly when $A$ is a ball.

We study the last iterate of the projected subGradient Method with a constant stepsize for convex $L$-Lipschitz objectives in $\mathbb{R}^d$, minimized over closed convex sets. Assuming that the initial iterate $x_1$ has distance at most $R$ from a minimizer, we establish matching upper and lower bounds on the minimax optimization error at $x_n$, where the infimum is taken over constant stepsizes. Specifically, this error is of order $L\cdot R\cdot\sqrt{\min\{d,\log n\}/n}$. Notably, when the dimension is fixed, the minimax error decays as $1/\sqrt{n}$, without the extra logarithmic factor intrinsic to dimension-free analyses. Our analysis exploits a combinatorial argument based on linear algebra and novel geometric constructions.

We provide some examples of homotopy coherent smooth finite group actions on spheres which do not arise from strict actions.

We settle the Rajchman problem for canonical scalar dyadic Mandelbrot cascades at the minimal Kahane–Peyrière integrability threshold. If $\mu$ is the cascade on $[0,1]$, then $\widehat{\mu}(\xi)\to 0$ as $|\xi|\to\infty$, almost surely on non-extinction. For every fixed nondegenerate $C^2$ embedded arc $\gamma:[0,1]\to\mathbb{R}^2$, the pushforward $\gamma_\#\mu$ is likewise Rajchman almost surely on non-extinction. The analogous conclusion holds for the scalar cascade on the parameter circle pushed forward by any fixed nondegenerate $C^2$ Jordan curve. No moment condition of order strictly greater than one is imposed; in particular, the results include the regime $\mathbb{E}[W^q]=\infty$ for every $q>1$. The proof combines a spine-based lower-deviation principle, adaptive terminal approximation, and predictable capping to obtain almost-sure estimates uniform over large frequency annuli without higher moments. For curved pushforwards, an endpoint-safe phase decomposition controls direction-dependent stationary regions, including those meeting the endpoints of an arc, and couples the geometric and probabilistic arguments through a common dyadic kernel. Combined with the exact Fourier-dimension formulas for the corresponding models, the theorems show that Rajchman decay persists at zero Fourier dimension.

Motivated by Sarnak's conjecture in topological dynamics for the Möbius function $\mu$, we study, for a power-bounded $T$ on a Banach space $E$, the weak convergence $$ (*) \qquad \qquad \frac1N\sum_{n=1}^N \mu(n)T^nv \to 0 \text{ weakly } \forall v\in E. $$ For that, we introduce a notion of dynamical entropy for operators, which we denote $h^*_{top}(T)$, and show that if Sarnak's conjecture is true, then $h^*_{top}(T)=0$ implies the desired convergence (*). We conclude an equivalent operator formulation of Sarnak's conjecture. For several classes of operators we prove that (*) holds, and that $h^*_{top}(T)=0$.

We extend the proof-theoretic treatment of uniform bound extraction from normed structures axiomatized in positive bounded logic [Advances in Mathematics, 290:503-551, 2016] (as developed for the model theory of Banach spaces) to the more general setting of abstract metric structures, including discrete structures viewed as classical first-order models. In particular, we establish uniform bound extraction theorems for our generalized framework for $\forall\exists$-sentences whose matrix is the negation of (an embedding of) a formula in positive bounded logic, whose proofs use saturation. In this way, we provide a formal explanation for the successes in the extraction of uniform bounds from nonstandard proofs given in [Advances in Mathematics, 343:567-623, 2019], which had informally followed the perspective of the monotone functional interpretation. As an application of the formal framework we develop, we provide novel explicit bounds for a structural theorem for stable subsets of groups given in [Mathematical Proceedings of the Cambridge Philosophical Society, 168(2):405-413, 2020].

We consider the Dirichlet problem for the mean curvature operator in Minkowski space, \[ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda u + \mu h(x,u) \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega, \] in a bounded domain $\Omega \subset \mathbb{R}^N$, where $\lambda, \mu$ are real parameters, and the nonlinearity $h$ is superlinear at $u = 0$. In particular, we study the combined effect of the parameters $\lambda,\,\mu$ on the multiplicity of solutions. In the general setting, following Szulkin's approach for nonsmooth functionals, we prove the existence, for $\lambda$ not belonging to the spectrum of the Dirichlet Laplacian and $\mu$ sufficiently large, of a global minimizing solution (with negative action level) and of a min-max solution (with positive action level). Moreover, we characterize the limiting profiles of these solutions as $\mu \to +\infty$. More precisely, when the global minimizer is positive, its limit profile is $\mathrm{dist}(\cdot,\partial\Omega)$, thus saturating, in the limit, the geometric constraint $|\nabla u|\le1$, while min-max solutions collapse uniformly to zero as $\mu\to+\infty$. A nonexistence criterion is also given for suitable values of $\lambda$ and $\mu$. Finally, when the domain $\Omega$ is a ball, using a shooting approach, we establish the existence of arbitrarily many nodal radial solutions for every $\lambda \ge 0$ and for $\mu$ sufficiently large.

We propose a stochastic behavioral modeling framework, termed Gaussian behaviors, which augments a deterministic linear time-invariant (LTI) behavior with a Gaussian noise component. We show that this notion is a tractable subclass of stochastic behaviors and encompasses classical parametric stochastic LTI state-space system models as special cases. Analogously to deterministic LTI behaviors, the framework enables simple and tractable stochastic data-driven control methods. To this end, we obtain a method for prediction by conditioning the Gaussian behavior on the known part of the trajectory, which is identified directly from the sample covariance of trajectory data. Building on this method, we develop predictive control formulations that optimize over feedforward or disturbance affine feedback policies. The resulting formulations are shown to be convex. We further derive a finite-sample confidence bound on the prediction accounting for both aleatoric and epistemic uncertainty, and incorporate it into a robust control method, for which a tractable convex upper bound is obtained. Within this framework, subspace predictive control is recovered when only the mean prediction is used, while data-enabled predictive control is shown to account for the prediction uncertainty in an optimistic fashion. Numerical case studies illustrate the benefits of the proposed methods.

This work develops a statistical test for the maximal rank of the deterministic instantaneous (or spot) covariance matrix of a continuous-time $\mathbb{R}^d$-valued Itô semi-martingale $X(t)$ using high-frequency observations with a particular focus on the impact of an adapted drift. We explicitly account for the presence of an adapted drift process, which, as our results demonstrate, cannot be neglected, by introducing a re-centred covariance estimator instead of relying solely on a second moment estimator. Building on this estimator, we test the null hypothesis that the rank of the spot covariance matrix is at most $r<d$ for all $t$ against local alternatives in which the $(r+1)$th eigenvalue is greater than some vanishing signal detection rate. Critical values are derived in a non-asymptotic framework and can be significantly affected by a potential drift. However, the power analysis establishes asymptotic consistency for separation rates, which depend on the Hölder regularity of both the drift and the spot covariance matrix, as well as on a potential spectral gap $\underline{\lambda}_r \geq 0$ under the null hypothesis. Simulation results indicate that the covariance-based test achieves higher power across a wider range of alternatives compared to classical second moment-based procedures.

For a graph $G$, let $ \lambda_1(G)\ge \lambda_2(G)\ge \cdots \ge \lambda_n(G)$ denote the adjacency eigenvalues of $G$. We prove that for all looped graphs $G$ on $n$ vertices, \[\lambda_1(G) + \lambda_2(\overline{G}) \le \frac87 n. \] Our method also gives a short new proof of the Nordhaus-Gaddum result for the spectral radius proved by Terpai that $\lambda_1(G) + \lambda_1(\overline{G}) \le \frac43n - 1$. We investigate the asymptotic maximum of \[ \lambda_i(G)+\lambda_j(\overline G) \] for fixed $i$ and $j$. We prove general bounds on $\lambda_i(G) + \lambda_{j}(\overline{G})$ for all pairs $(i, j)$ and also give general bounds on the related problem of minimizing $\lambda_{n-i+1}(G) + \lambda_{n-j+1}(\overline{G})$ for fixed $i$ and $j$. We also show the close relation of these Nordhaus-Gaddum type problems to recent work on the maximum spectral gaps of graphs by Brooks, Linz and Lu.

We study a dynamic strategic communication problem in which a sender controls the timing of truthful updates from binary continuous-time Markov sources. The receiver chooses between a zero-order-hold estimator that follows the sender's updates and a prior-only default estimator, aiming to maximize a weighted correct-estimation utility. In contrast, the sender seeks to persuade the receiver to estimate the state as 1, regardless of the true state. This misalignment leads to a Stackelberg game in which the sender, as the leader, commits to state-dependent Poisson update rates, and the receiver, as the follower, decides whether to follow the sender's messages. The sender maximizes the long-term average time that the receiver's estimate equals 1, subject to a conditional intensity budget and a participation constraint (PC) ensuring that following the sender's messages does not degrade the receiver's average utility relative to its prior information. For a single source, we show that the sender's optimal policy allocates a minimum state-0 update intensity to the undesired state-0, just enough to satisfy the PC, and the remaining budget to the desired state-1. For multiple sources with heterogeneous minimum state-0 update intensities, we develop a branch-and-bound algorithm that typically avoids exhaustive search. Finally, we extend the solution to multiple receivers over dedicated channels. Our results show that controlling timeliness alone enables the sender to persuade the receiver and increase its utility.

We propose a projected drift-randomized Milstein (PRM) method for stochastic differ ential equations with non-differentiable and super-linearly growing drift coefficients. The method extends the randomized Milstein approach beyond the globally Lipschitz setting by incorporating a drift projection into the randomized quadrature approximation. Moreover, unlike existing first-order Milstein-type methods for SDEs with super-linearly growing drift coefficients, the proposed method does not require spatial differentiability of the drift coefficient. Under suitable polynomial Lipschitz and one-sided Lipschitz conditions on the drift, together with standard regularity assumptions on the diffusion, we establish a one-step mean-square stability estimate and derive the required local residual bounds. These esti mates yield first-order strong convergence of the PRM method in the L2-sense. Numerical experiments confirm the theoretical convergence rate and demonstrate the applicability of the method to SDEs with non-differentiable and super-linearly growing drifts.

This paper presents a practical guide to building lightweight neural-network surrogates for the plastic return-mapping process in concurrent multiscale (FE2) simulations. Rather than proposing a new architecture, we show how a deliberately simple feed-forward network, structured to mirror the classical return-mapping update, can replace the prohibitively expensive nested fine-scale solves that dominate the cost of conventional FE2 schemes based on FFT homogenization at the meso-scale. We walk through the full workflow: generating training data from incremental homogenization analyses, constructing a compact yet sufficient dataset, embedding material symmetries directly into the mapping, and deploying the trained network as a user-defined material subroutine (UMAT) in a standard finite-element solver – enabling widespread use. A sensitivity study examines the model's robustness to data density, increment size, and mesh refinement, and we characterize the regimes in which the surrogate holds and where it breaks down. For the macroscopically isotropic, two-dimensional plane-stress setting considered here, the surrogate reproduces the reference response while reducing the per-analysis cost from hours to seconds with speed-ups up to 30,000 over standard FE2. The approach extends naturally to three dimensions and to weaker symmetry assumptions, given an appropriate sampling strategy and dataset.

The approach of squeezing more transistors in the same area in order to speed up computing is no longer effective. Currently, researchers and engineers are searching for novel solutions that offer faster computing. One of these solutions is to compute where you store, known as in-memory computing. Resistive random access memories (ReRAMs), which are based on memristor crossbar arrays, enable in-memory computing. Moreover, ReRAMs offer large storage capacity associated with energy efficiency. In this work, we focus on storing digital data in memristor crossbar arrays. A critical challenge here is the sneak-path problem, occurring when there is a rectangle on the array with three low and one high resistances at the corners. The electric current in this case is prone to sneaking through the low-resistance path upon reading, which results in the high resistance data becoming erroneous. In this paper, we propose effective constrained coding solutions to the sneak-path problem after finding the expected number of sneak paths over a two-dimensional array given their circumferences. In particular, we adopt a literature model where $b$ rows on the crossbar array are read simultaneously while the others are grounded, and we design capacity-achieving non-binary constrained codes for the cases of $b=2$ and $b=3$. We focus more on the sneak paths with shorter circumferences as they are more detrimental. Here, GF refers to Galois field. Our GF$(4)$ codes, for $b=2$, and GF$(8)$ codes, for $b=3$, are a class of lexicographically-ordered constrained (LOCO) codes, and we call them resistive-LOCO (RES-LOCO) codes. RES-LOCO codes operate horizontally, and we also suggest a run-length-limited scheme for coding data on the crossbar array vertically to mitigate the sneak-path problem for $b=4$. We experimentally demonstrate the effectiveness of our RES-LOCO codes for various array setups.

We study the boundedness of the Fourier transform on variable Lebesgue spaces. We obtain necessary conditions and, independently, sufficient conditions on the exponents $p(\cdot)$ and $q(\cdot)$ under which the inequality $$\|\hat{f}\|_{q(\cdot)}\leq C\|f\|_{p(\cdot)}$$ holds for some constant $C>0$ and all $f\in L^{p(\cdot)}(\mathbb{R})$. As a byproduct, we improve some recent results of Saucedo and Tikhonov. Moreover, when $p(x)\to 1$ and $q(x)\to +\infty$ as $|x|\to +\infty$, we characterize the exponents for which the Fourier transform is bounded.

We classify skew braces with additive and multiplicative groups both isomorphic to the dihedral group $ D_{n} $. As a consequence, we obtain an alternative proof of the classification of Hopf-Galois structures of dihedral type on a dihedral Galois extension.

This work is concerned with the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with non-diagonal diffusion terms of type \begin{equation} u_{t}-\sum_{i,j=1}^{d} D^{2}_{ij}(a_{ij}(x)\beta(x,u))+ \text{div}(b(x,u)u)=0 \quad \text{in}\; (0, \infty) \times \mathbb{R}^{d} ,\notag \end{equation} with initial condition $u(0,x)\equiv u_{0}(x)$, where $a_{ij}$, $\beta$, and $b$ are suitable functions. Under suitable assumptions, this equation generates a continuous contraction semigroup $S(t): L^{1}(\mathbb{R}^{d}) \rightarrow L^{1}(\mathbb{R}^{d})$, and $u(t)=S(t)u_{0}$ is a mild solution to the equation. Our main contribution is to prove that this mild solution is unique in the much larger class of distributional solutions. This extends previous uniqueness results for the diagonal (also called isotropic) diffusion case $a_{ij} \equiv \delta_{ij}$. Another key analytical result of this paper is the uniqueness for distributional solutions of the associated linearized equation. As a main application, we prove weak uniqueness for the corresponding McKean-Vlasov SDEs. Moreover, we prove that, the probabilistically weak solution to the McKean-Vlasov SDEs is also the unique probabilistically strong solution. Furthermore, we establish a new $L^{\infty}$ estimate for mild solutions starting from data in $L^{1}\cap L^{\infty}$ and this estimate is used in the construction of nonlinear Markov processes. Finally, we prove that the path laws of the solutions to the McKean-Vlasov SDEs form a nonlinear Markov process in the sense of McKean.

In the present work, we investigate the hypothesis of the ultrametric organization of abstract energy landscapes defined on random Erdős–Rényi graphs. Within the proposed model, each vertex of the graph is assigned a random free energy uniformly distributed on a given interval, and the kinetics of the system is described by a Markov process with a Kramers transition rate matrix. Based on the spectral decomposition of this matrix, a kinetic metric between basins of attraction is constructed. Computational experiments for graphs with $V=5000$ vertices and $E=5000$ edges (average degree $\langle k\rangle=2.0$) demonstrate that this metric reveals a high degree of nontrivial ultrametricity, which monotonically increases from $\approx42\%$ at an energy interval width of $\Delta F=10$ kJ/mol to $\approx96\%$ at $\Delta F=1000$ kJ/mol. For a rigorous mathematical justification of this phenomenon, a limit theorem is formulated and proved, establishing that as the width of this interval tends to infinity, the logarithmic asymptotics of the proposed metric converges pointwise to the classical single-linkage ultrametric. The analysis of computational experiments shows that for finite values of the interval width, suboptimal paths and thermal fluctuations introduce corrections that violate the strong triangle inequality; however, in the regime of large energy fluctuations, their contribution is exponentially suppressed, and consequently the metric asymptotically acquires ultrametric properties. The obtained results indicate that the ultrametric organization of energy landscapes represents a universal asymptotic property of networks of arbitrary topology with a strongly rugged relief, which for sparse, locally tree-like graphs manifests itself already at moderate energy spreads.

We study the joint diffusive-mean field limit for a system of weakly interacting kinetic Langevin dynamics, extending the results of Delgadino, Gvalani and Pavliotis (Arch. Ration. Mech. Anal., 2021) to the hypoelliptic/hypocoercive case. We show that, in the absence of phase transitions, the two limits commute, and we calculate the covariance matrix of the limiting Brownian motion using the Green-Kubo/Kipnis-Varadhan formula. However, at low temperatures, and in the presence of phase transitions, the two limits may not commute. We demonstrate our findings by providing a detailed analysis of the diffusive-mean field limit for the $O(2)$ model in a magnetic field. Our analysis is based on the systematic use of recently developed hypocoercivity techniques, together with an appropriate linearization of the mean field McKean-Vlasov-Fokker-Planck partial differential equation.

In \cite{BD} Bourgain and Demeter found sharp upper bounds for the number of $k$-th powers inside arbitrary arithmetic progressions whose step has $O(1)$ many divisors. We make the easy observation that the same arguments are still valid if the step does not grow too rapidly in relation to the length of the progression. Furthermore, we give sharp bounds for the number of $k$-th powers among the first $N$ terms for $N$ large enough. Both results should be known. Nevertheless, we add to the literature.

Let $T(n,k)$ be the minimum total surface area of $k$ axis-aligned boxes with integer sides and pairwise distinct dimension multisets whose union is the cube $[0,n]^3$. We determine the column $k=6$ completely: $T(n,6)=8n^2+2n+12$ for $5\le n\le 9$, and $T(n,6)=8n^2+2n+6$ for all $n\ge 10$, together with the exceptional values $T(3,6)=100$ and $T(4,6)=148$. The threshold $n=10$ equals $1+2+3+4$, the least possible sum of four distinct stick lengths, and the general law holds: for every $k\ge 4$ and every $n\ge (k-2)(k-1)/2$, $T(n,k)=8n^2+2n+2(k-3)$, with thresholds at the triangular numbers. Three structural results support and extend these values. First, a fault-free law: the minimum internal interface of a partition of the cube into six boxes with no fault plane is exactly $2n^2+n$ for all $n\ge 3$ (OEIS A014105), proved by an exact accounting of spanning pieces, floating pieces and cube corners. Second, a reduction theorem: within an explicit range, the three-dimensional problem collapses to a two-dimensional one, $I(n,k)=n^2+W^*(n,k-1)$, where $W^*(n,m)$ is the minimum internal wall of a tiling of the $n\times n$ square by $m$ rectangles of pairwise distinct dimensions; the key ingredient is an unconditional slab lemma. Third, a doubling law in the middle regime of the 2D problem: $W^*(n,4)=n+4$ for $n=4,5$ and $W^*(n,5)=n+6$ for $4\le n\le 9$, proved by finite case trees; via the reduction theorem this gives computer-free proofs of the middle regimes of the columns $k=5$ and $k=6$. The lower bound for the main family does not use the distinctness of the pieces.

Every polynomial with real non-negative coefficients yields a finite probability distribution after normalization. The Ehrhart $h^*$-polynomial of a lattice polytope $P$ is a non-negative integer polynomial that encodes the integer-point counts for positive integer dilations of $P$. We study the corresponding finite distributions, which we call $h^*$-distributions. We determine the mean and variance of these distributions, establish a connection between higher moments and Ehrhart polynomial coefficients, and study their cluster points in the $d$-dimensional probability simplex. We consider the special case of real-rooted $h^*$-distributions, applying existing tail bounds to obtain new linear inequalities for the coefficients of real-rooted $h^*$-polynomials arising from reflexive polytopes. We conclude by establishing sufficient conditions under which a sequence of real-rooted $h^*$-distributions is asymptotically normal, and we apply our results to various families of polytopes, including zonotopes and Pitman-Stanley polytopes.

Extremal hyperbolic surfaces are finite-type hyperbolic surfaces that contain an embedded disc of the largest possible radius. We introduce a new general approach to these objects, based on the uniformization by NEC groups, obtaining a structural description of the Dirichlet domains associated to extremal disc centers for surfaces with cusps and/or geodesic boundary. We also provide a number of applications of our new description of extremal surfaces, as an effective counting formula for extremal disc configurations, the determination of every such configuration with Euler characteristic -1 and -2, the location of hidden extremal discs in several families of extremal surfaces, and the full description of their automorphism groups.

We study the class $\#\mathsf{L}$ of functions counting accepting paths of non-deterministic log-space Turing machines and construct methods to prove containment in $\#\mathsf{L}$. We prove that a large number of classical combinatorial and number theoretic functions belong to this class: classical functions from enumerative combinatorics (multinomial coefficients, Catalan numbers, linear extensions of trees, Stirling numbers, etc), algebraic combinatorics (number of standard Young tableaux, etc), discrete geometry, number theoretic functions, representation theoretic multiplicities in a large class of cases. We show that $\mathrm{GL}_2$-plethysm coefficients of bounded length outer partition can be counted by log$^2$-space polytime verifiers. We pose numerous questions and conjectures on $\#\mathsf{L}$ containment and its generalizations, that suggest venues for conditionally disproving $\#\mathsf{P}$-completeness. While studying which combinatorial functions are in $\#\mathsf{P}$ provides a formal way of (dis)proving the existence of combinatorial interpretations, the lower class $\#\mathsf{L}$ serves as an analogue for functions computable in polynomial time.

We consider the Cauchy problem for the incompressible Euler-Vlasov-Fokker-Planck (Euler-VFP) system in the whole space \(\mathbb R^3\) near the global Maxwellian equilibrium. The Fokker-Planck operator and the particle-fluid drag dissipate the relative momentum but do not separately control the common particle-fluid momentum; in Fourier variables, this degeneracy occurs in the transverse momentum components. To recover the missing coercivity, we augment the classical four-moment compensator with a finite-rank skew-adjoint correction constructed from second-order Hermite modes. Combined with the cancellation between the kinetic and fluid drag terms and the incompressibility constraint, the resulting compensated Fourier energy yields a unique global classical solution for sufficiently small initial data $(u_0,f_0)\in H^N\times L_v^2(H^N)$, with $N\geq 4$. The high-order energy argument involves only spatial derivatives of the kinetic perturbation and requires no mixed \(x\)-\(v\) derivative estimates. We further construct a positive-order Lyapunov functional and establish the decay rate \((1+t)^{-1/2}\) for all positive-order spatial derivatives in the \(L^2\)-norm and for the corresponding pointwise-in-space norms, without any additional \(L^1\) integrability or low-frequency assumption on the initial data. Although no uniform algebraic decay rate is asserted for the zero-order energy of \((u,f)\), the directly dissipative variables \(u-J(f)\) and \(\{\mathbf I-\mathbf P_0\}f\) decay in the \(L^2\)-norm at the same rate, where $ J(f)=\int_{\mathbb R^3}v\sqrt M f\,{\rm d}v$ denotes the particle momentum and \(\mathbf P_0\) is the orthogonal projection onto \(\operatorname{span}\{\sqrt M,v_1\sqrt M,v_2\sqrt M,v_3\sqrt M\}\). To the best of our knowledge, these positive-order and zero-order decay estimates have not previously been established for the incompressible Euler-VFP system.