Papers

Large Language Models (LLMs) have created an urgent need for reliable watermarking methods that enable attribution of generated text while remaining robust to editing and paraphrasing. We propose a novel synchronization-free watermarking scheme in which every watermark consists of a single binary congruence generated from a pair of neighbouring tokens. For each token pair, a cryptographic hash determines an evaluation point of a Reed–Solomon polynomial representing the secret identity, while the parity of the polynomial evaluation determines the watermark bit embedded into the second token of the pair. Since each congruence is self-contained and depends only on the local token pair, the proposed construction is naturally resistant to insertions, deletions, and token reordering. We analyse the recovery problem from an algebraic perspective, discuss several decoding algorithms suitable for different identity sizes, and model watermark corruption as a Binary Symmetric Channel. The analysis shows that reliable recovery requires only a small redundancy even for relatively high token corruption rates. Unlike existing block-based watermarking schemes, the proposed method avoids synchronization problems while providing a flexible framework for embedding both short and long secret identities.

Principal component regression (PCR) regularizes high-dimensional prediction by choosing a spectral cutoff, but rank selection cannot correct systematic inflation of the retained empirical eigenvalues. We study clean Gaussian random designs in which the aggregate covariance tail creates a nearly scalar sample-space floor comparable to the predictive head scale. De-floored principal component regression (dPCR) retains the cutoff and subtracts an estimated floor from the retained denominators. We prove an ordinary-PCR prediction-risk lower bound uniform over all ranks and a high-probability dPCR upper bound. When the floor is sharp and inexpensive to remove in population prediction risk, the conditional risk of dPCR is asymptotically negligible relative to that of the best ordinary PCR rank. An exact risk decomposition explains the separation: denominator inflation is governed by first spectral mass, whereas the clean prediction cost of correction is governed by squared spectral mass. A same-sample trimmed-mean floor estimate attains the oracle dPCR upper-bound rate at a prespecified rank, and the separation persists under approximate predictive alignment when the tail prediction-energy fraction vanishes. Separate pointwise fixed-aspect formulas show that the risk-optimal positive scalar correction improves rank-$1$ PCR, whereas mean-floor subtraction is generally not optimal for a broad Marchenko–Pastur bulk.

The numerical solution of the simplified Ericksen–Leslie model for nematic liquid crystals is challenging because the flow and director equations are strongly coupled and because incompressibility and the unit-length condition must be enforced simultaneously. A Lagrange multiplier formulation avoids a small Ginzburg–Landau parameter, but the Newton systems have a double saddle-point structure. We develop an augmented Lagrangian block preconditioner in which both constraints are augmented while their discrete enforcement remains multiplier based. After finite element discretization and backward Euler time integration, the Newton increments are grouped into velocity–director and pressure-multiplier variables. A block-diagonal approximation of the coupled velocity-director block then leads to separate, physically scaled approximations of the pressure and director-multiplier Schur complements. Manufactured-solution tests show the expected spatial accuracy and first-order temporal convergence for the primary variables; the multiplier error reaches a spatial-error floor on the fixed mesh used in the temporal study. In the reported parameter ranges, the outer FGMRES iteration counts are nearly mesh independent, remain stable under time-step and viscosity variation, and improve as the augmentation parameters increase. A smooth benchmark also exhibits monotone decay of the computed total energy.

We construct Sugawara operators for the quantum affine superalgebra $\mathrm{U}_q(\widehat{\mathfrak{gl}}_{M|N})$ in an explicit form which generalizes the results given in \cite{JLM} for the quantum affine algebras. We also calculate the Harish-Chandra images of the Sugawara operators.

In the earlier paper arXiv:2603.11533, the authors gave an explicit tensor product formula, modulo projectives, for modules over the group algebra $F\mathfrak{S}_p$. In this paper, we use this formula to study tensor powers of such modules. In certain cases, we give combinatorial descriptions of the multiplicities of the indecomposable summands appearing in their direct-sum decompositions. We also study the asymptotic behaviour of these multiplicities as the underlying prime tends to infinity.

We study continuous-time mean-variance portfolio selection with no-shorting constraints and unknown investment opportunity sets from a reinforcement learning (RL) perspective. The problem is a constrained stochastic linear – quadratic control problem for which the entropy-regularized exploratory formulation of Wang et al. (2020) leads to difficulty in theoretical analysis, because enforcing the constraint on the support of randomized policies nullifies the tractable Gaussian exploration. To tackle this challenge, we introduce an auxiliary exploratory problem without entropy in which exploratory policies are still Gaussian whose samples may violate the no-shorting requirement but their means satisfy it. We then prove that, for a suitable choice of exploration variance, the mean of the optimal Gaussian policy of the auxiliary problem coincides with the optimal policy of the original problem. Motivated by this theoretical result, we develop a model-free RL algorithm that learns the optimal policy of the auxiliary (and hence the original) problem directly from trajectory data without estimating the investment opportunity set. A numerical example demonstrates the performance of the proposed algorithm.

Problem definition: The paper studies an integrated end-to-end planning problem in large-scale paper manufacturing, where production scheduling, trimming decisions, vehicle loading, and multi-period fulfillment of make-to-order and make-to-stock demand must be coordinated over time. In practice, these decisions are often optimized sequentially, leading to material waste, inefficient transportation, and degraded service levels. Solving the fully integrated problem at industrial scale remains computationally challenging due to its combinatorial structure. Methodology/results: A key structural feature of the problem is that downstream fulfillment decisions depend on upstream production and logistics choices only through aggregate supply availability over time. By exploiting this structure, the paper develops an exact mathematical formulation and proposes a two-phase hybrid framework (BDCG-DP) that integrates column generation (CG) using exact dynamic-programming (DP) for supply-side decisions with Benders decomposition (BD) for downstream fulfillment. Computational experiments on proprietary instances from a major North American paper manufacturer show that BDCG-DP lowers total costs by 24.4% compared to a traditional CG-DP on challenging eight-week planning problems. Median runtime for four-week planning problems decreases from over five hours using CG-DP to under one hour using BDCG-DP. Managerial implications: This paper provides the first exact model that integrates production, trimming, load planning, and multi-period fulfillment at an industrial scale. The proposed approach returns integer-feasible plans within 2.3 to 6 hours for the most complex planning problems, enabling planners to access high-quality implementable schedules within hours, a capability that was previously unavailable in practice.

For a set $A\subseteq\mathbb{N}_0$, let $r_1(A,n)$ denote the number of solutions of the equation $a+a^{\prime}=n$ with $a,a^{\prime}\in A$, and let $r_2(A,n)$ denote the number of such solutions subject to $a\le a^{\prime}$. These functions are called additive representation functions (as first considered by Erdős, Sárközy and Sós). In this paper, we resolve two problems posed by Sárközy and Sós in 1997. First, if $A$ is infinite and $r_2(A,2m+1)\ge r_2(A,2m)$ for every sufficiently large $m$, then the complement of $A$ is finite. This gives a negative answer to Problem 3.1 in~\cite{SarkozySos1997}. Secondly, there exist an arithmetic function $f$ satisfying $f(n) \to \infty$, $f(n+1) \ge f(n)$ for $n > n_0$, and $f(n) = o\left(\frac{n}{(\log n)^2}\right)$, and a set $A$ such that \( |r_1(A,n) - f(n)| = o((f(n))^{1/2}) \) holds on a sequence of integers $n$ whose density is $1$. This gives a positive answer to Problem 3.3 in~\cite{SarkozySos1997}.

We derive general families of Ramanujan-type congruences for the function $\overline{p}_{\ell}(n)$, which counts the $2$-color overpartitions of $n$ in which one of the colors appears only in parts that are multiples of $\ell$. For example, for all $n\geq0$ we prove that $$\overline{p}_{4}(32n+28)\equiv 0 \pmod{512}.$$

We show that the finite group reduction of the Dubrovin-Zhang/Double Ramification hierarchy associated with a semisimple CohFT preserves its bihamiltonian structure and its tau structure. By applying this result to the orbifold Gromov-Witten theory with the target $\mathbb{P}^1_{2,2,2,2}$, we obtain two non-equivalent integrable hierarchies which can be associated with the Frobenius manifold structure on the Hurwitz space $M_{1;1}$, and we show that one of them is equivalent to the genus one topological recursion.

This paper investigates second-order asymptotic expansions for the conditional higher moment (CoHM) coherent risk measure under a Farlie-Gumbel-Morgenstern (FGM) dependence structure, capturing a weak contagion between a primary loss risk and a reference risk. Assuming that the primary risk belongs to the Fréchet, Weibull, or Gumbel maximum domain of attraction, we systematically derive second-order asymptotic expansions using extreme value theory and second-order regular variation theory. Compared with existing first-order results, our refined approximations capture higher-order tail behavior and dependence effects more accurately. Numerical simulations confirm that the second-order asymptotics substantially reduce approximation errors, especially at extreme confidence levels. Empirical applications to insurance claim data further illustrate the practical superiority of the second-order approach.

A variable-rate lossy joint source-channel coding scheme for burst-noise communication channels with noiseless feedback is introduced. The scheme comprises a multi-stage channel optimized vector quantization system that dynamically allocates bits via a greedy algorithm among the variable-rate residual quantizers at each stage, based on the channel output sequence received at the encoder through the feedback link, thereby generalizing a prior fixed-rate scheme. Simulations over an $M$-th order Markov noise Polya contagion channel demonstrate that the proposed variable-rate scheme consistently outperforms the fixed-rate scheme, regardless of the channel bit error rate and noise correlation, achieving signal-to-noise ratio gains of up to about 4.5 dB.

A conucleus $\delta$ on a partially ordered monoid $\mathbf{A}$ is an interior operator that satisfies $\delta(a) \cdot \delta(b) \leq \delta(a \cdot b)$ and $\delta(a) \cdot \delta(1) = \delta(a)$ for all $a,b \in A$. A conucleus is multiplicative if the equality $\delta(a \cdot b) = \delta(a) \cdot \delta(b)$ holds for all $a,b \in A$. In this article we focus on the study of conuclei on hoops, structures which generalize well-known classes of algebras, such as the class of MV-algebras and BL-algebras. Among several results, we provide a Glivenko-type theorem for conuclei. Special emphasis is given to term definable conuclei. The main result of this article is an explicit description of all terms that define a multiplicative conucleus on every structure of an arbitrary proper variety of Wajsberg hoops. We also show that the problem of finding terms that define (multiplicative) conuclei on a variety of basic hoops or BL-algebras is equivalent to finding such terms on some variety or some pair of varieties of Wajsberg hoops. We provide nontrivial interesting examples.

The rapid digitalisation of elite sport has created new opportunities for integrating artificial intelligence (AI), performance analytics, and decision-support systems into athlete development and competition management. However, existing solutions remain fragmented, typically addressing isolated tasks such as performance analysis, athlete monitoring, or referee support. This paper presents \textbf{FST$\cdot$ai~2.5}, an explainable, uncertainty-aware, and secure AI framework for Olympic and Para-Taekwondo. \textbf{FST$\cdot$ai~2.5} introduces a unified digital ecosystem integrating athlete intelligence, competition analytics, federation-scale data management, AI-assisted decision support, athlete and event digital twins, explainable performance indicators, and adaptive training recommendations. The framework supports World Taekwondo (WT), Member National Associations (MNAs), coaches, referees, analysts, and athletes through transparent, secure, and federation-aware governance. By combining multi-source competition data, athlete-performance information, and contextual evidence, \textbf{FST$\cdot$ai~2.5} provides tactical diagnostics, longitudinal athlete monitoring, performance forecasting, personalised development planning, and federation-wide benchmarking using explainable and uncertainty-aware AI. Prototype deployments demonstrate the feasibility of the proposed framework. Although developed for Olympic and Para-Taekwondo, the methodology is broadly applicable to explainable AI, digital twins, and trustworthy decision support in combat sports and other high-performance sporting environments.

We introduce the twisted super Yangian $Y^{\mathfrak{s}}_{\imath}$ of quasi-split type A in the Drinfeld current presentation for an arbitrary symmetric parity sequence $\mathfrak{s}$. We prove via Gauss decomposition that $Y^{\mathfrak{s}}_{\imath}$ is isomorphic to the special twisted super Yangian $\mathscr{SY}^{\mathfrak{s}}$, a subalgebra of the twisted super Yangian $\mathscr{Y}^{\mathfrak{s}}$ previously defined via the R-matrix presentation. As a corollary, we show that the algebras $Y^{\mathfrak{s}}_{\imath}$ corresponding to different symmetric parity sequences with the same $\mathfrak{m}$ and $\mathfrak{n}$ are isomorphic. We also establish a PBW theorem for $Y^{\mathfrak{s}}_{\imath}$ and describe the center of $\mathscr{Y}^{\mathfrak{s}}$ in terms of Gaussian generators, thereby generalizing known results for the nonsuper quasi-split type A case.

Partial-mastery (PM) cognitive diagnostic models (CDMs) extend traditional CDMs by replacing binary latent attribute mastery indicators with continuous mastery scores for multiple latent attributes. In PM-CDMs, each subject is characterized by a fixed continuous latent mastery vector, from which item-specific binary attribute profiles are independently generated. This formulation provides a bridge between classical CDMs and continuous latent variable models. Despite growing interest in PM-CDMs, their identifiability properties remain unexplored. In this work, we establish the first identifiability results for PM-CDMs. We derive sufficient conditions for identifiability that are direct analogues of established conditions for traditional CDMs. To develop the main argument, we use symbolic computation on a minimal example with five items and two latent attributes to show that the Jacobian of the model parameterization is generically nonzero. Combining tools from real analysis and algebraic statistics, we prove that this local property implies generic finite-to-one identifiability of the item parameters and the marginal distributions of the relevant latent attributes. We further show that if the $Q$-matrix contains such identifiable local structures for all attribute pairs, identifiability extends to the full PM-CDM. These findings provide a rigorous theoretical foundation for estimation and inference in partial-mastery cognitive diagnostic models.

We develop a machinery for deriving ideals on a countable set from a partition of $\omega$ indexed by $\omega^{<\omega}$. A derivative operator on trace trees, parametrized by an auxiliary ideal $\mathcal J$, yields a strict transfinite hierarchy $\mathcal H^{\mathcal J}_\alpha$ of proper ideals, tall from level one onward and independent of the chosen partition. Its finite levels are exactly the Fubini powers, $\mathcal H_n\cong\mathrm{Fin}^{\otimes(n+1)}$, while $\mathcal H_\omega=\bigcup_n\mathcal H_n$ amalgamates all finite powers. We establish presentation independence, local homogeneity, Fubini recursion, and $\Pi^1_1$-completeness of the full hierarchy. Let $\mathcal F_\omega$ be Kwela's canonical inductive limit and $\mathcal F'_\omega$ his independent-partitions limit. We prove $\mathcal F_\omega\not\leq_K\mathcal H_\omega$, although $\mathcal F'_\omega\sqsubseteq\mathcal H_\omega$ and every finite coherent fragment of a putative reduction is realizable over $\mathcal H_\omega$. The proof introduces essential depth, an invariant monotone along Katetov reductions of $\mathrm{Fin}\otimes\mathrm{Fin}$, and gives the sharp non-extension bound $N(m)=m+2$. Thus $\mathcal H_\omega$ contains no isomorphic copy of $\mathcal F_\omega$, and $\mathcal F_\omega\not\leq_K\mathcal F'_\omega$. Applications include chromatic ideals $\mathcal G_k\in\mathcal H_2\setminus\mathcal H_1$ whose inclusion order records divisibility and whose Katetov order records arithmetic. We also prove the orthogonality of Cantor–Bendixson and derivative ranks. Finally, $\mathcal P(\omega)/\mathcal H$ is a $\sigma$-closed reduced power $\mathbb B\cong\mathbb B^\omega/\mathrm{Fin}$, contains $\mathcal P(\omega)/\mathrm{Fin}$ regularly, and under CH is forcing-equivalent to $(\mathcal P(\omega)/\mathrm{Fin})^+$.

The notion of $n$-stationary subsets of $\mathcal{P}_\kappa \lambda$ for $n < \omega$ were introduced and studied by Cody, Lambie-Hanson & Zhang \cite{CLHZ} and Torres \cite{T}. They proved that if $\kappa$ is supercompact, then $\mathcal{P}_\kappa \lambda$ is $n$-stationary in itself for all cardinals $\lambda \geq \kappa$ and all $n < \omega$. In this paper we prove that strong compactness of $\kappa$ does not imply the $2$-stationarity of $\mathcal{P}_\kappa \lambda$ for cardinals $\lambda > \kappa$. We also discuss Menas' Theorem for $1$-stationary and $2$-stationary subsets of $\mathcal{P}_\kappa \lambda$.

Whether exact scalar feedback intrinsically incurs the additional dimension $d$ paid by known zeroth-order methods remains open even for Lipschitz convex optimization. For a universal Lipschitz scale, the value only bound $O(d^2\log(d+1)\log(1/\epsilon))$ and two-point bound $O(d\epsilon^{-2})$ yield the upper bound $\widetilde O\left(d\min\{d,\epsilon^{-2}\}\right)$. By contrast, prior lower bounds for arbitrary randomized algorithms give only $\Omega(\min\{d,\epsilon^{-2}\})$, leaving a factor $d$ unexplained. We close this gap, up to logarithmic factors, for arbitrary adaptive randomized algorithms minimizing a convex objective with a universal Lipschitz scale over the $d$-dimensional Euclidean unit ball, where each query returns only the exact scalar value. Let $T_\epsilon$ denote the minimum number of queries required to return an $\epsilon$-suboptimal point with probability at least $1/2$, uniformly over the function class. We prove that \[T_\epsilon\ge c\,\frac{d\min\{d,\epsilon^{-2}\}}{\log\!\bigl(\min\{d,\epsilon^{-2}\}\bigr)},\] for $d\ge d_0$ and $0<\epsilon\le\epsilon_0$, where $c,\epsilon_0>0$ and $d_0\in\mathbb N$ are universal constants. This gives $\Omega\left(\frac{d}{\epsilon^2\log(1/\epsilon)}\right)$ in the low-accuracy regime $\epsilon\ge d^{-1/2}$ and $\Omega\left(\frac{d^2}{\log d}\right)$ in the high-accuracy regime $\epsilon\le d^{-1/2}$ with the latter independent of $\epsilon$. These bounds match the corresponding upper bound up to logarithmic factors. To our knowledge, this is the first near-optimal lower bound for arbitrary adaptive randomized algorithms throughout both accuracy regimes of exact value Lipschitz convex optimization. The proof uses a random support function hard family and develops a posterior mean energy method for adaptive exact max observations, in place of first-order zero chain constructions and noise based transcript inequalities.

We introduce a Many-Body Expansion framework for scaling black-box solvers for discrete optimization. The method decomposes a large problem into fragments, repeatedly improves each fragment with the rest of the assignment held fixed, and then applies targeted corrections on unions of fragments. The solver itself is unchanged: it only receives a restricted subproblem and returns a candidate assignment. The framework separates decomposition from solver design. Moebius inversion gives an exact fragment expansion of the objective, and for pairwise objectives such as graph coloring, MAX-CUT, and QUBO the expansion closes at second order. The promoted solver is monotone by construction, terminates in finitely many productive sweeps for non-negative integer-valued objectives, and connects to junction-tree dynamic programming when the fragment interaction structure is acyclic and full response functions are propagated. On graph coloring benchmarks, the promoted versions of TabuSearch and simulated annealing (SA) generally outperform flat applications of the same solvers, often finding better solutions in less CPU time, and the gain proves insensitive to the choice of graph partition. A facility-location study delimits the claim: there the partition must respect the objective's interaction locality, and the promoted solver reaches within 0.3-1.4% of GLPK-certified optima at every tested size. The same wrapper interface is used in each solver-swap case, and the facility-location check reuses the same monotone guard with a sign-adjusted interaction diagnostic; the improvements therefore come from the MBE promotion mechanism rather than solver-specific tuning.

The Erdős–Rado sunflower conjecture asserts that, for every fixed integer $r\ge 3$, there is a constant $K(r)$ such that every $\ell$-uniform family with more than $K(r)^{\ell}$ members contains an $r$-sunflower. We prove this conjecture for families of bounded VC-dimension.

Let $K$ be an uncountable compact metrizable space, and $E$ be a separable Banach space with the uniform primary factorisation property (UPFP) and containing no isomorphic copy of $c_0$. We prove that $C(K, E)$ also has the UPFP. As a consequence, we obtain the primariness of the bi-parameter spaces $C([0,1],\ell_p)$ and $C([0,1], L_p)$ for $1 \leq p<\infty$.

Witsenhausen's counterexample is a canonical problem in decentralized control: a two-stage linear-quadratic-Gaussian team problem in which nonlinear policies strictly outperform every affine one. This paper develops a variational and computational framework for the scalar counterexample based on optimal transport. Building on the transport formulation of Wu and Verdú, we absorb the first controller into a transport map and recast the problem as $J^\star=\inf_Q\{k^2W_2^2(P,Q)+\mathrm{mmse}(Q)\}$, which trades a quadratic Wasserstein transport cost against a minimum mean-square estimation error. The optimal first controller is the monotone rearrangement that pushes the prior $P$ to a minimizer $Q^\star$. We establish the existence and absolute continuity of minimizers, derive an Euler–Lagrange optimality condition, and obtain a semi-closed-form Gaussian benchmark with an explicit second-variation stability threshold. We then show that restricting to finitely supported laws yields an MMSE-regularized optimal quantizer: its optimal values decrease to $J^\star$, its weak limit points are minimizers of the continuous problem, and its stationarity conditions pair centroid levels with power cells, recovering Lloyd–Max quantization as control becomes expensive. This structure leads to an estimation-aware Lloyd algorithm with signalling-based initialization. As $k\to\infty$, the optimal controllers converge to the classical affine pair; as $k\to0$, $J^\star(k)=\sigma^2k^2+o(k^2)$ and the optimal laws collapse onto $\delta_0$, so signalling constellations are a moderate-$k$ phenomenon. Numerical experiments corroborate the theory, and the framework extends to the vector counterexample, where the finite-level program becomes an MMSE-regularized vector quantizer.

This paper addresses the problem of whether or not a vector-valued multivariate functions can be expressed as a sum or a product of vector-valued functions in disjoint sets of variables through a proper invertible linear change of variables. The crux is an invariant algebra, the so-called center, that we introduce for a set of multivariate functions with second order partial derivatives. We thus provide simple criteria and algorithms for simultaneous additive and multiplicative decompositions of any set of multivariate functions with minor analytic conditions. This is applied to the factorization problem of multivariate homogeneous polynomials, in particular those that are products of linear forms.

That the inclusion $Q(F_3(X))\subseteq \mathsf{Pa}^-_3$ is proper and Mod$(\mathbf{qb}_3)$ is not generated by free $p$-algebras is proved, answering two open questions from Kowalski-Słomczyńska \cite{KowSlom}.

We present various discretisation techniques that mark a departure from the classical algorithms of numerical analysis, in the aim of preserving the physical behaviour of the solutions to the differential systems we are discretising, even for discretisation steps that are not very small. These techniques are inspired by the principles laid down by Mickens and are close to those proposed by Hirota and Kahan. Detailed applications of these methods are presented for the logistic equation and for the Lotka-Volterra system. We discuss the importance of preserving positivity whenever the latter is an expected property of the solution and give a practical rule for achieving this. Possible problems arising from the discretisations we propose are also discussed. A section is devoted to discretisations of integrable systems aimed at preserving integrability. Finally we conclude this review with a discussion of the possible relevance of a discrete space-time.

We study nonlinear wave equations with dynamic boundary conditions on smooth bounded domains and analyze a fully discrete approximation in the low-regularity regime. The method combines isoparametric bulk–surface finite elements of degree $k$ with an exponential integrator in time. Assuming only bounded energy of the exact solution, we prove convergence of the displacement–velocity pair in the weak norm $L^2(\Omega;\Gamma)\times H^{-1}(\Omega;\Gamma)$. The scheme achieves first-order convergence in time and spatial convergence of order $h^{2/3}$ for $k=1$ and $h^{(k+2)/(k+3)}$ for $k\ge 2$. In particular, these rates show that higher-order finite elements retain a provable asymptotic advantage even at low regularity. A central difficulty is that the continuous and discrete bulk–surface problems are posed on different geometries and must therefore be compared directly in weak norms. To address this, we develop a weak-norm framework for non-conforming geometries based on lift and adjoint-lift operators, combined with a frequency-decomposition argument. To the best of our knowledge, this is the first fully discrete low-regularity convergence result for nonlinear wave equations with dynamic boundary conditions in a non-conforming bulk–surface finite element setting. Numerical experiments confirm the predicted rates and illustrate the improved efficiency of higher-order methods.

Let $a(n)$ be the sequence A028342 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\sum_{n\ge0} a(n)x^n/n! = \prod_{i\ge1}(1-x^i)^{-1/i}$. Equivalently, $a(n)$ counts permutations of an $n$-element labeled set in which every cycle is assigned one divisor of its length, where a cycle of length $m$ has $d(m)$ choices, $d(m)$ being the number of positive divisors of $m$. We prove a family of congruences for $a$, conjectured by Peter Bala. They state that $k \mid a(n+k)+a(n)$ for odd $k$, that $k \mid a(n+k)-a(n)$ for $k\equiv 0,2,6 \pmod 8$, and that $k \mid 2(a(n+k)-a(n))$ for $k\equiv 4\pmod 8$. The proof first establishes a product congruence $a(n+k)\equiv a(n)a(k)\pmod k$, and then computes $a(p^r)\bmod p^r$ for each prime power by counting the colored permutations fixed by a subgroup of order $p$.

We introduce a novel discretization of the incompressible Euler equations based on their interpretation as geodesic equations on the Lie group of volume-preserving diffeomorphisms. It is well known that encoding diffeomorphisms and their infinitesimal generators through a discretized Koopman representation places a nonholonomic constraint on discrete velocities, for which there is no consensus on a variational treatment. We show that taking the vakonomic perspective, as opposed to the usual perspective of Lagrange–d'Alembert, yields discrete fluid trajectories that remain geodesics on a (sub-)Riemannian manifold. In particular, the resulting vakonomic dynamics are Lie–Poisson and their solutions admit a discrete relabeling symmetry, leading to machine-precision satisfaction of Casimir invariants along with a discrete analogue of Kelvin's Circulation Theorem. Using an efficient momentum map representation based on low-rank Clebsch variables, we show that these vakonomic fluids behave stably and consistently even at low grid resolutions, leading to increased robustness and physical realism in the long term.

Let $L(t)$ denote the largest number of edges induced by $t$ vertices from two vertex layers of a hypercube. We show that $$\frac14 t\log_2 t+\frac18 t\log_2\log_2 t-O(t) \leq L(t) \leq \frac14 t\log_2 t+ (1+o(1))t\log_2\log_2 t.$$