Papers

A subsemiring $S$ of the nonnegative cone of the real line containing $1$ is called a positive semiring. The Bi-UF Positive Conjecture, posed by Baeth, Chapman, and Gotti in 2021, states that the prototypical semiring $\mathbb{N}_0$ is the only positive semiring whose additive monoid $(S,+)$ and nonzero multiplicative monoid $(S^\bullet,\cdot)$, where $S^\bullet=S\setminus\{0\}$, are both factorial. In this paper, we prove their conjecture.

A real symmetric matrix is called $d$-locally positive semidefinite if all of its $d \times d$ principal submatrices are positive semidefinite. We investigate the spectral geometry of $d$-locally positive semidefinite matrices. The set of vectors of eigenvalues of $d$-locally positive semidefinite matrices of size $n \times n$ is fully understood and known to be convex when $d \in \{1,n-1,n\}$ \cite{blekherman2022hyperbolic}. In the smallest remaining case $n=4, d=2$, non-convexity of the set of vectors of eigenvalues was proved in \cite{kozhasov2023eigenvalues}, but even in this case the full description was unknown. We provide a basic semialgebraic description of the set of vectors of eigenvalues for $n=4, d=2$ by establishing a Fischer-type inequality for $2$-locally positive semidefinite matrices of size $4 \times 4$ and prove non-convexity for $n \geq 4$ and $d \in \{2, n-2\}$. Non-convexity is established via solving certain non-smooth and non-convex min-max point configuration problems in the complex plane, which could be interesting in themselves. Similar problems were considered in \cite{nesterenko2024submatrices,sengupta2026submatrices} in the context of matrix decomposition and approximation.

We construct explicit unirational del Pezzo surfaces of degree $1$ with arithmetic Picard rank one over $\mathbb{Q}$, $\mathbb{F}_5$, and $\mathbb{C}(t)$. Moreover, we prove that every smooth real geometrically rational surface is unirational over $\mathbb{R}$ if and only if it has a real point.

In this note, we prove a recent conjecture of Hung, Martínez and Navarro on the sums of character degrees of finite groups.

We present a proof of a conjecture we made about primitive polynomials, for $q$ sufficiently large. The argument follows a standard route, however a deeper result than usual with a character sum bound is required. This is a theorem of Fu and Wan. The proof is AI assisted.

Let $\Lambda$ be the Leech lattice and let $\mathrm{Co}_0=\operatorname{Aut}(\Lambda)$. Sun and Wang proved that the space of $\mathrm{Co}_0$-invariant holomorphic Jacobi forms of singular weight $12$ and index $4$ satisfies \[ 4\leq \dim J^{\mathrm{Co}_0}_{12,\Lambda,4}\leq 9, \] and left its exact dimension open. We prove \[ \dim J^{\mathrm{Co}_0}_{12,\Lambda,4}=6. \] At singular weight, theta decomposition identifies this space with the simultaneous $\mathrm{Co}_0$- and Weil-invariant subspace of $\mathbb{C}[\Lambda/4\Lambda]$. Conway symmetry and $T$-invariance reduce the problem to a twelve-dimensional space of isotropic orbit sums. On this space the projected Weil $S$-operator satisfies the universal relation \[ S\left(S+\frac{1}{2}I\right)(S-I)=0, \] obtained from the level-$4$ Hecke algebra. Equivalently, the associated integral character matrix $K$ satisfies \[ K(K+2^{23}I)(K-2^{24}I)=0. \] Combining this relation with known index-$4$ forms, reduction modulo $2$, and character data obtained from the $A_3^8$ deep hole reduces the remaining possibilities to a finite exact calculation. A final torsion evaluation of the known index-$3$ form $\Phi_{12,3}$ determines the last required character value, and exact elimination leaves a unique admissible branch, of dimension $6$. We also construct two Conway-averaged theta forms from explicit markings of the Niemeier lattices with root systems $D_6^4$ and $D_4^6$. Together with the four forms previously exhibited by Sun and Wang, they give a natural basis of $J^{\mathrm{Co}_0}_{12,\Lambda,4}$.

In recent years, owing to the high accuracy in characterizing non-separable channels prevalent in next-generation wireless applications, the classical Weichselberger channel model has gained widespread adoption in multiple-input multiple-output (MIMO) systems. However, its non-separable structure also introduces severe analytical complexity, leading to a lack of tractable mathematical frameworks in the literature and thus raises an urgent need for further research. To address the aforementioned analytical complexity, we first derive the nearest separable (double-correlated Rayleigh) fading model to the Weichselberger model under the Kullback-Leibler divergence (KLD), a problem equivalent to rank-1 nonnegative matrix factorization under the Itakura-Saito (IS) distance criterion. The results of our asymptotic analysis in the high-SNR regime reveal that the KLD-enabled approximation achieves a tighter capacity estimate than the conventional Kronecker model, especially in sparse and non-regular scattering environments. Yet, a key limitation of the KLD-enabled model is its tendency to mischaracterize the channel capacity in the low-SNR regime due to its inability to preserve total channel power. As a more robust alternative, we introduce a novel moment matching method (MMM) aimed at mapping the exact channel statistics to those of a Wishart distribution. Both the KLD-enabled and MMM-enabled separable channel directly enable the use of exact closed-form expressions for the ergodic capacity. Numerical results demonstrate that the MMM-enabled model consistently improves upon the capacity accuracy of the conventional Kronecker model across all SNR regimes.

Evolving crack fields in structural health monitoring and fracture assessment must often be inferred from sparse mechanical measurements rather than dense full-field observations. We develop CNN2D–ConvGRU, a convolutional-recurrent framework for measurement-conditioned reconstruction of time-dependent phase-field brittle fracture. At each load step, the model maps a fixed-length history of phase and displacement fields and sparse current-step displacement measurements to the current full-field state. New measurements are assimilated during sequential deployment, making the framework a state-inference surrogate rather than an autonomous time integrator. It reproduces crack paths, damage evolution, and bulk displacement response, with the largest errors near propagating crack tips and steep displacement gradients and some drift at late stages. Without retraining, weights learned on a $256 \times 256$ raster are evaluated on a $512 \times 512$ raster of the same physical domain and finite-element discretization using a proportionally refined measurement grid. This empirical raster-and-sensing transfer preserves the principal damage topology and global damage evolution, although fine-scale displacement errors increase near crack tips. Comparisons with alternative spatial and temporal architectures show that CNN2D–ConvGRU offers a favorable balance between reconstruction accuracy and computational cost. Relative to repeated finite-element solutions, sequential reconstruction achieves mean speedups of $175\times$ on CPU and $253\times$ on GPU. These results demonstrate efficient full-field fracture-state reconstruction from sparse observations while retaining the spatial structure and history dependence of phase-field fracture.

In this paper we study Borel graphs generated by finitely many commuting Borel functions. We give a geometric analysis of the free part of such graphs based on marker sets and marker regions. Assuming the existence of $r$-forward-independent hitting sets with bounded syndeticity, we obtain marker decompositions of the free part into rootless and rooted regions with controlled geometry. As applications, we derive finite Borel asymptotic dimension and hyperfiniteness, and obtain upper bounds for Borel edge chromatic numbers which improve previously known results. For the case in which each of the commuting Borel functions is bounded-to-one, we verify the existence of $r$-forward-independent hitting sets with syndeticity $Cr$ for some constant $C$. This gives another proof of a recent theorem of Shinko-Weilacher-Yu, and is used to show that if one of the commuting Borel functions is injective and another one is bounded-to-one and exactly even-to-one, then the graph has a Borel perfect matching.

For every positive integer $d$ divisible by $4$, we prove the following new upper bound for the Bourgain-Clozel-Kahane sign uncertainty constant: \[ \mathrm{A}_+(d) \le \sqrt{2 \left\lfloor \frac{d}{16} \right\rfloor + 2}. \] It recovers the optimal bound $\mathrm{A}_+(12) \le \sqrt{2}$ in dimension $12$ and improves the previously best known bound $\sqrt{(d+2)/(2\pi)}$ for all $d \ge 52$ divisible by $4$. The proof uses Fourier eigenfunctions and associated quasimodular forms constructed by Feigenbaum, Grabner, and Hardin.

While recent advances in minimax dual control have led to exact solutions for uncertain general linear time-invariant systems as well as (sub)optimal dual controllers, corresponding results for linear positive systems are still lacking. This paper aims to fill this gap and thereby pave the way toward scalable dual control algorithms. We study the general minimax optimal dual control problem for positive linear systems with unknown dynamics and reformulate it as a standard zero-sum dynamic game. By allowing randomized control inputs, we solve the corresponding Bellman equation exactly for the scalar case with sign uncertainty in the input. This yields an implicit dual control policy that is optimal both in terms of cost and $\ell_1$-gain. The optimal dual policy uses exploration in a specific region of the hyperstate space to conduct optimal probing. Outside this exploration regime, the controller reduces to a deterministic certainty equivalence policy, indicating that sufficient information has been obtained to identify the correct input direction. In addition, these results allow us to analyze fundamental limitations of minimax dual control for positive systems and provide a foundation for more general dual control problems for positive systems for future work.

Towards a question of Voiculescu, two notions of $1$-bounded entropy, $h$ and $h^\text{top}$, are defined for $C^*$-algebras. The quantity $h$ involves the tracial completion with respect to all traces while the quantity $h^\text{top}$ depends on operator norm microstates. It is demonstrated that $h^\text{top}(\mathscr{A})$ does not exceed $h(\mathscr{A})$ for all $C^*$-algebras $\mathscr{A}$. Moreover, a variational principle enabling the computation of $h$ is introduced. The quantity $h$ is computed in various examples, and the computation is used to justify structural conclusions such as $C^*$-primeness, crossed product indecomposability, and free indecomposability. These notions of entropy are generalized to operator systems, where it is demonstrated that both may be computed on a basis.

Let $F_n$ be a free group of rank $n\geq3$, freely generated by $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ denote the Nielsen automorphism of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $d_{i,j}(x_k)=x_k$ for $k\neq i$, and let $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$. We determine the lower central series of $D_n$. We first prove that, for $1\leq r<i\leq n$, $d_{i,r}\in \gamma_{i-r}(D_n)\setminus\gamma_{i-r+1}(D_n)$. For each $i=2,\ldots,n$, this calculation leads to a filtration $\{W_{i,c}\}_{c\geq1}$ of $U_i=\langle d_{i,1},\ldots,d_{i,i-1}\rangle$. For every $c\geq1$, we obtain an explicit iterated semidirect-product decomposition of $\gamma_c(D_n)$ in terms of the subgroups $W_{i,c}$, and prove that $U_i\cap\gamma_c(D_n)=W_{i,c}$ for $i=2,\ldots,n$. The construction also gives an explicit basis for each quotient $\gamma_c(D_n)/\gamma_{c+1}(D_n)$ in terms of basic commutators and determines the exact lower-central depth of every such commutator. Consequently, $D_n$ is a Magnus group.

In his seminal paper, Kac famously asked whether "one can hear the shape of a drum" - that is whether the isometry class of a bounded domain in a Euclidean space is uniquely determined by the spectrum of its Laplace spectrum. The Laplace spectrum is closely related to the length spectrum of the associated billiard. For a convex bounded planar domain, to each billiard periodic orbit one can associate not only its length but also its rotation number. The set of pairs of length and rotation number of each periodic orbit is called the marked length spectrum. Using the marked length spectrum of the domain one can associate with it its minimal action function also known as Mathers $\beta$-function denoted by $\beta_\Omega$. Via unique quasianalytic continuation, we prove the following rigidity problem: knowing that Mather's $\beta$-functions of two billiards in two analytic planar domains coincide on a set of positive measure of KAM diophantine numbers implies that they coincide on a set of all KAM diophantine numbers. In particular, knowing that Mather's $\beta$-functions of two domains coincide on a positive measure set of KAM diophantine numbers implies that the Marvizi-Melrose invariants of these domains coincide.

The generalized pancake graph $P(m,n)$ is the Cayley graph of the group of colored permutations $\mathbb{Z}_m\wr S_n=(\mathbb{Z}_m)^n\rtimes S_n$ generated by generalized prefix reversals. In this paper, we establish that, for all $m,n\geq2$, the spectral gap $\gamma(P(m,n))$ of the normalized Laplacian satisfies $\alpha_m/n\leq\gamma(P(m,n))\leq1/n$, where $\alpha_m$ is a positive constant that depends only on $m$. As a consequence, for every fixed $m\geq2$, $\gamma(P(m,n))$ is $\Theta_m(1/n)$ as $n\to\infty$. The proof combines Cesi's semi-recursive spectral-gap inequality with a Fourier decomposition of the appropriate operators associated with a coset Schreier graph of color-position pairs. For fixed $n\geq2$, we also establish that $\gamma(P(m,n))$ is $\Theta_n(m^{-2})$ as $m\to\infty$. This disproves a conjecture of Blanco and Buehrle asserting that, for fixed $n$, the corresponding undirected generalized pancake graphs form an expander family. Additionally, we present a counterexample to a recent conjecture of Greaves and Zhu concerning equality between the spectral gaps of the full Cayley graph and the associated coset Schreier graph.

Colour-kinematics duality is, even at tree level, not manifest from the standard Lagrangian of Yang-Mills theory in that the usual Feynman-diagram expansion does not follow the kinematic Jacobi identities. For a Lagrangian manifesting colour-kinematics duality, the kinematic Jacobi identities of the Feynman-diagram expansion follow from the existence of a second-order differential operator $\mathsf{b}$ acting on the algebra of colour-stripped fields that forms part of the data of a BV$^\square$-algebra. We argue that the existence of differential operators of order greater than two implies weaker but nontrivial fragments of kinematic Jacobi identities. We further show that a superspace formulation of the first-order Yang-Mills action in the Batalin-Vilkovisky formalism admits a differential operator of order six in every spacetime dimension. Therefore, the Feynman-diagram expansion of tree and loop scattering amplitudes of the first-order formulation of Yang-Mills theory automatically enjoy a weak form of colour-kinematics duality.

We characterize all symmetric, embedded surfaces of revolution with constant mean curvature that meet the unit sphere orthogonally. It is well known that there exists a unique free boundary catenoid inside of the unit ball. By analogy, we prove the existence of a unique free boundary nodoid inside of the unit ball with constant mean curvature equal to $-1$. Moreover, there exists a unique compact free boundary nodoid outside of the unit ball with constant mean curvature equal to $-1$. However, there exists no symmetric surface of revolution with constant mean curvature $1$ that meets the unit sphere orthogonally.

We show that if M is a full II_infinity factor such that the fundamental group of a finite corner is R and Out(M) is a locally compact second countable group, then M admits either a unique continuous trace scaling flow or a continuum of pairwise distinct ones, up to cocycle conjugacy. Equivalently, there is either a unique III_1 factor or a continuum of pairwise non-isomorphic III_1 factors with continuous core M. We apply this dichotomy result to factors constructed in [Dep12] and [Cha25] where Out(M) is a Lie group. We also give two independent constructions of full factors admitting a continuum of pairwise distinct trace-scaling flows where the outer automorphism group is unknown.

In this article, we consider discontinuous martingales on tangent bundles over submanifolds of Euclidean space. First, we introduce a connection rule on tangent bundles and establish the Itô calculus for discontinuous semimartingales on tangent bundles. Then we focus on harmonic maps with respect to non-local Dirichlet forms and show that the derivative of harmonic maps along infinitesimal symmetries induces discontinuous martingales on tangent bundles. This process may be viewed as a stochastic Jacobi field along the image martingale. We also introduce the stochastic parallel transport of tangent vectors along càdlàg semimartingales on Riemannian submanifolds with projected jumps. Using the parallel transport, we obtain the mean-value property for the differential of harmonic maps involving a jump part expressed through the second fundamental form. We also obtain a derivative formula for harmonic maps for isotropic Lévy processes with a Brownian component on compact Riemannian manifolds.

We show that the stable norm of flat metrics on $H_{d}(T^n,\mathbb{R})$ is locally rigid if $1\leq d<n-1$ and locally rigid among metrics of the same volume if $d=n-1$. We also show that the stable norm on $H_{2}(T^3,\mathbb{R})$ is not locally rigid. As applications, we answer negatively a question raised by Bangert in his ICM address, prove local rigidity of the marked $k$-area spectrum of flat metrics for $1\leq k\leq n-2$, and prove a local rigidity result for the volume spectrum of flat metrics on $T^n$.

This paper develops Admissibility-Preserving Control (APC), a realization-centered safety-critical control framework for strict-feedback systems subject to asymmetric actuator limits, time-varying output constraints, and actuator-rate limitations. APC denotes the overall control architecture, whereas an Admissibility-Preserving Input Realization (APIR) denotes its constraint-realization module. Therein, the APIR dynamically generates the physical plant input while rendering its prescribed asymmetric actuator set forward invariant. In contrast to algebraic clipping and post-design saturation compensation, the actuator limits are embedded directly in a continuously differentiable dynamic realization with user-selectable regularity and interpretable tuning parameters. The APIR is integrated with recursive backstepping by treating the realized plant input as an additional state. The resulting design does not require an input-to-state stability assumption on the uncontrolled plant. Instead, the nonlinear drift terms are compensated recursively, subject to an explicit compatibility condition between the desired motion, the available control authority, and the APIR interior gain. The framework is further extended to time-varying output-safe tracking through a smooth asymmetric logarithmic barrier coordinate and its associated Lyapunov function and to simultaneous actuator-magnitude and rate constraints through a cascaded APIR. Rigorous Lyapunov and invariance analyses establish regional asymptotic tracking, forward invariance of the compatible admissible sets, and boundedness of all closed-loop signals. Numerical studies illustrate asymmetric actuator utilization, output-safety preservation, and magnitude-rate constraint enforcement.

Uniform asymptotic expansions are derived as $\kappa \to \infty$ for the Whittaker functions $W_{\kappa,\mu}(z)$, $M_{\kappa,\mu}(z)$, as well as related functions including generalized Laguerre polynomials. The results are uniformly valid for $0\leq\mu\leq(1-\delta_0)\kappa<\kappa$, where $\delta_0\in(0,1)$ is arbitrary and fixed. The analysis is based on the associated differential equation, which has a double pole and two turning points. At one of the turning points, which may coalesce with the double pole, a recently developed asymptotic theory is applied to obtain expansions involving Bessel functions. At the second turning point, expansions involving Airy functions are obtained. In both cases, the coefficients are readily computable, in contrast to those occurring in earlier results. The expansions, when taken together, uniformly cover the entire complex $z$-plane on the principal branch. Standard analytic continuation and connection formulas extend the results to all branches of $z$ and to negative $\mu$.

Inverse physics-informed neural networks (PINNs) can reconstruct a field accurately while returning an incorrect physical parameter. We introduce a two-axis post-training diagnosis that separates finite-sample resolution under a specified observation-and-estimation protocol from the signed parameter preference encoded by the final learned field and residual metric. The first axis repeatedly fits noisy observations with a matched forward estimator. At known synthetic truth, the second freezes the field and residual view and computes a local score displacement toward a nearby residual-profile minimum. Endpoint consistency then tests whether joint training delivers that preference under the same final view. Across three synthetic one-dimensional, scalar-parameter PDEs, matched-forward mean absolute relative error ranges from 2.34 percent to 17.46 percent. The displacement tracks frozen-profile minima across locked seeds, architectures, and fresh-noise retraining (r from .945 to .982), and it tracks delivered signed log-error in 240 fresh-noise RBA runs (r = .994; 237/240 correct directions). A coupled two-parameter Darcy check validates the full matrix calculation. The axes are complementary diagnostic coordinates, not additive error components or a deployable oracle-free estimator. Together, they route follow-up work toward observations, residual evidence, or endpoint delivery.

Control barrier functions (CBFs) provide a systematic framework for enforcing safety constraints in nonlinear control systems. However, their implementation typically relies on accurate system models, which can limit their applicability in the presence of significant modeling uncertainties or unknown dynamics. This paper proposes a model-free framework for the computation of recursive control barrier functions based on the ultra-local model approach that leverages online estimation of the unknown system dynamics to construct CBF constraints. This approach does not require an explicit model of the system dynamics and enhances robustness with respect to disturbances and model mismatch. The resulting control architecture enables the enforcement as well as the anticipation of safety constraints for systems with higher relative degree. The effectiveness of the proposed approach is illustrated on the adaptive cruise control benchmark.

Let $K$ be a finite pure $k$-dimensional simplicial complex, with $k\ge1$, on the vertex set $[n]$ and with facet family $K_k$. Let $\lambda_1(K)\ge\lambda_2(K)\ge\cdots>0$ be the nonzero eigenvalues of its $(k-1)$-dimensional up-Laplacian, and, after ordering the vertices so that $\deg_K(1)\ge\cdots\ge\deg_K(n)$, let $\dvT{r}(K)$ be the number of vertices contained in at least $r$ facets. A complex is shifted if replacing a vertex of a face by a smaller vertex outside the face always yields another face. We prove that there is a shifted family $\HH$ of $(k+1)$-element subsets of $[n]$, with the same number of members as $K_k$, such that \[ \tfrac12\bigl|K_k\,\triangle\,\HH\bigr| \;\le\; \tfrac12\left[\sum_{r\ge1}\bigl(\dvT{r}(K)\bigr)^{2}-\sum_{r}\lambda_r(K)^{2}\right]. \] The left-hand side counts the facets that have to be exchanged to reach $\HH$; thus one half of the gap between the second power sums of the two sequences bounds the distance of $K_k$ to a shifted family. The characterization $\lambda(K)=\dv(K)^{\mathsf T}\iff K$ is isomorphic to a shifted complex was established in \cite{Gupta} from the identity that this gap equals twice the number of failed elementary shifts. The present paper converts that identity into a quantitative stability statement and recovers the equality characterization at zero defect. For $k=1$ this bounds the number of edge exchanges needed to reach a threshold graph with the same number of edges.

Trefftz functions satisfy the differential equation locally and exactly; quasi-Trefftz functions do so approximately to prescribed high order. This paper considers (quasi-)Trefftz approximations for two-dimensional scalar electromagnetic problems. Established discretizations include the Flexible Local Approximation MEthod (FLAME), Trefftz elements ($T$-elements), and Trefftz discontinuous Galerkin (Trefftz-DG) methods. New developments are gradient-enriched FLAME (GEFLAME), conforming Trefftz–FLAME finite elements (TFF), and a full-field Bloch-wavevector-vs-frequency solution with Schur–DtN reduction. Applications cover electrostatics, scattering, singular fields, and Bloch waves in periodic structures. These methods make different compromises between conformity and flexibility. FLAME and GEFLAME incorporate Trefftz functions directly into local difference schemes; TFF uses elementwise FLAME schemes to lift polynomial traces into element interiors; $T$-elements match elementwise Trefftz spaces weakly to such traces; and Trefftz-DG couples broken Trefftz spaces through fluxes and penalties. In the reported wave-scattering examples, GEFLAME gives field and gradient errors several orders of magnitude below those of the quadratic finite-element discretization at comparable algebraic cost. Localized singular-function enrichment removes the dominant reentrant-corner error. Conforming TFF and quasi-conforming $T$-elements admit standard finite-element assembly but expose polynomial edge traces as an accuracy bottleneck. For periodic media, the bilinear Trefftz-DG formulation gives an analytic polynomial wavenumber-vs-frequency eigenproblem without restricting the Bloch multiplier to the unit circle. Schur–DtN reduction yields compact boundary-response problems.

This paper investigates the structural properties, operator representations, and algebraic duality of Hom-Novikov superalgebras and to introduce Hom-anti-pre-Novikov superalgebras. We establish a formal equivalence between Hom-Novikov superalgebras and admissible Hom-Novikov superalgebras via $q$-superalgebra extensions, proving that a Hom-pre-Lie superalgebra admits a Hom-Novikov structure if and only if its associated $2$-superalgebra is admissible. Introducing $q$-admissible pairs $(\mathcal P, \mathcal Q)$ on super-commutative Hom-associative superalgebras, we detail systematic constructions of Hom-Novikov superalgebras and their admissibility classes. Furthermore, we characterize anti-super-$\mathcal{O}$-operators on Hom-Lie superalgebras, showing that invertible operators induce compatible Hom-anti-pre-Lie structures and extend directly to admissible setups. Extending these frameworks to Poisson-type structures, we demonstrate that the sub-adjacent Hom-Lie superalgebra of a Hom-anti-pre-Lie Poisson superalgebra yields a transposed Hom-Poisson superalgebra. Finally, we establish the fundamental connection between Hom-anti-pre-Novikov superalgebras and invertible anti-super-$\mathcal{O}$-operators.

Arbitrary-order directional derivatives along multiple (possibly distinct) directions are computed through a generalized dual-number formulation for both scalar- and vector-valued functions. The proposed framework combines generalized dual evaluations with an inclusion–exclusion reconstruction of symmetric multilinear forms, allowing general multidirectional derivatives to be reconstructed from repeated-direction evaluations without explicitly constructing higher-order derivative tensors. Mixed directional and mixed partial derivatives arise naturally as particular cases of the formulation. The methodology further enables the systematic computation of arbitrary-order kinematic quantities and the construction of Taylor-series methods for systems of ordinary differential equations through automatically generated time derivatives. Numerical examples include the computation of high-order directional derivatives in high-dimensional functions, mixed partial derivatives, arbitrary-order kinematic quantities, and Taylor-series integration methods. The implementation is developed in modern Fortran within an open-source framework compatible with the Fortran Package Manager ecosystem.

While Harris recurrent Markov chains (in discrete time) automatically exhibit wide-sense regenerative structure, we construct a Harris recurrent Markov process (in continuous time) that is not wide-sense regenerative, thereby giving a negative answer to the open problem first raised in the 1990s and later posed in Glynn(2011). The counterexample exhibits the following rigidity property: every almost surely finite random time that is independent of the state observed at that time must be almost surely constant. A Cantor set linearly independent over the rationals plays a key role in the construction, turning calendar time into an algebraic record of the path already traversed.

We study the wave kinetic equation (WKE) derived by Shavit–Bühler–Shatah from the two-dimensional Boussinesq system of internal waves. The equation is anisotropic and vector-valued, with a two-branch sign-changing dispersion relation, coupled propagation branches, and a sign-indefinite pseudo-momentum invariant. The resonant manifold has angular degeneracies that obstruct the standard analytic treatment of the collision operator. We introduce a natural angular cut-off adapted to these degeneracies, prove boundedness of the cut-off collision operator, and establish local well-posedness in weighted $L^\infty$ spaces. We also show that, even with the zero-frequency cut-off retained, removing the cut-off near $|\cos\theta|=\tfrac12$ makes the collision operator unbounded on these weighted spaces. This provides, to our knowledge, the first rigorous analytic framework for an anisotropic vector-valued WKE arising from Boussinesq dynamics.