Papers

We embed the Jacobi-like and Laguerre-like systems for ordinary type-I/type-II multiple orthogonality introduced by Wolfs into a single Askey-type confluence scheme. Applying the Bernstein transform simultaneously to the $q$ Jacobi-like weights produces a positive finite-lattice Hahn-like ancestor, represented by multiple beta integrals, reducing to the classical Hahn weight for $q=1$, and converging back to the continuous system. Parameter and scaling limits then yield Kravchuk-like, two Meixner-like, two Charlier-like, Jacobi-like, two Laguerre-like, and Hermite-like families. We realize every arrow through explicit hypergeometric orthogonality data. For each near-diagonal multi-index $1\leq |\mathbf m|\leq N$, the Hahn-like type-II polynomial is a terminating ${}_{q+2}F_{q+1}(1)$ series satisfying an exact inverse Bernstein identity. We construct the normalized type-I form and, under explicit separation and nonvanishing assumptions, recover its individual polynomial components by finite sums of terminating hypergeometric functions and prove uniqueness. Finite-pole terms regroup into terminating Kampé de Fériet blocks. Their limits are obtained by coefficient extraction and reconstruction at infinity on the unreflected Kravchuk-like and Meixner-II-like branches, and by finite Lauricella–Horn sector sums on the reflected branches. This yields sectorwise component confluence along the Kravchuk–Charlier, Meixner–Charlier, Meixner–Laguerre, and Charlier–Hermite arrows. For $q=1$ all formulas reduce to the classical families; for $q>1$ the two Meixner-like systems, and likewise the two Charlier-like systems, are inequivalent under permutation of normalized rows.

For a triangle of area \(T\), a planar corner-area characterization states that an interior point \(M\) lies on the Steiner inellipse precisely when the three corner triangles cut off by the lines through \(M\) parallel to the sides have areas \(T_1,T_2,T_3\) satisfying $$ T_1+T_2+T_3=\frac12 T. $$ We give the corresponding statement for a simplex in arbitrary dimension. If \(S\) is a nondegenerate \(n\)-simplex of volume \(V\) and \(V_1(M),\ldots,V_{n+1}(M)\) are the volumes of the facet-parallel corner simplices determined by \(M\), then $$ M\in\partial E_J(S) \quad\Longleftrightarrow\quad \sum_{i=1}^{n+1}V_i(M)^{2/n}=\frac1n V^{2/n}, $$ where \(E_J(S)\) is the John ellipsoid of \(S\). We also identify the entire corner-volume functional with the central second-moment quadratic of the uniform simplex. The novelty claimed here is limited to the corner-volume formulations and their connections with the planar Steiner-inellipse result; the underlying barycentric, covariance, and John-ellipsoid facts are classical.

Settling a central problem in set-theoretic topology, and many related problems, we prove that there is, consistently, a regular Lindelöf space of cardinality $\aleph_1$ that is not a D-space. The existence of such a space is independent of ZFC; we establish its equivalence to a weak form of the Continuum Hypothesis. The proof of the main theorem is adjustable to produce consistent examples with a variety of stronger properties.

We present GridapGeosciences$.$jl, a new parallel distributed-memory Julia package for the numerical approximation of partial differential equations on general manifolds. Our abstract approach to generating discrete manifold domains relies on mesh refinement in general, and is implemented within the Gridap finite element library, whose high-level interface enables seamless definition of both intrinsic and extrinsic variational formulations. GridapGeosciences$.$jl features the cubed sphere manifold, which is used in this study to showcase the functionality of GridapGeosciences$.$jl in different applications. The first application involves scalar transport where we develop intrinsic formulations of common stabilisation terms, which are shown to be equivalent to well-known extrinsic stabilisation formalisations. Such stabilisation terms are used to develop an intrinsic formulation of the two-dimensional thermal shallow water equations that includes upwinded numerical fluxes. The final application is a linearisation of the Boussinesq equations where we demonstrate the capacity of GridapGeosciences$.$jl to capture both surface and vertical dynamics on a three-dimensional manifold.

Connectors are ubiquitous in technical systems and frequently combine mechanical contact with electric current transfer and heat generation. To capture these interacting processes, an electro-thermo-mechanical contact formulation based on the third-medium concept is proposed. Mechanical contact, electric current flow, and heat conduction are represented within a unified finite element framework without explicit contact-surface tracking. Deformation-dependent switching functions govern the onset of electrical and thermal transport during interface closure. Electrical conduction is coupled to the transient thermal problem through Joule heating, while the temperature dependence of the electrical conductivity provides the corresponding feedback on the current flow. The coupled displacement, electric potential, and temperature fields are solved monolithically and combined with an independently interpolated deformation-gradient-like field for robust third-medium regularization under severe compression. Numerical examples demonstrate contact-induced current transfer, subsequent Joule heating and transient temperature evolution, as well as localized current paths during progressive closure of rough interfaces.

The thesis develops symbolic representations of dynamical systems in finite-dimensional commutative real algebras. The algebraic framework (orthogonal idempotents, zero divisors, and the structure of finite-order algebras) supports algebra-valued Fourier and Laplace transforms and Walsh-function-based system identification. The methods yield exact, coordinate-free vectorial solutions for motion in non-inertial reference frames, including a generalized Larmor theorem, the Foucault pendulum, motion in central positional-force fields, and the two-body problem in arbitrarily rotating reference frames. This English edition is an audited LaTeX reconstruction of the 1995 original, with an editorial note documenting provenance.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and, for a dominant integral weight $\lambda$, let $\Theta_\lambda$ be the multiplicity-free sum of the weights of the irreducible module $V(\lambda)$. The $\Theta_\mu$ form a $\mathbb{Z}$-basis of the $W$-invariants, so there are unique integers $a(\mu,\lambda)$, the atomic numbers, with $\mathrm{ch}\,V(\lambda) = \sum_\mu a(\mu,\lambda)\Theta_\mu$ over dominant $\mu$. We ask when they are nonnegative. They are the Moebius transform of the weight-multiplicity function on the dominant-weight poset, which the crosscut theorem turns into an alternating sum of at most $2^{\mathrm{rank}\,\Phi}$ multiplicities indexed by subsets of the covers of $\mu$ in $[\mu,\lambda]$. We prove that $a(\mu,\lambda)\ge 0$ whenever every connected component of the Dynkin diagram is a path, and identify the coefficient with the dimension of an explicit weight space: it is cut out of $V(\lambda)$ by alternately taking kernels of raising operators and cokernels of lowering operators, one per cover, in order along the path. Type $D_4$ shows the hypothesis is necessary. Put $\beta = \alpha_1 + 2\alpha_2 + 2\alpha_3 + 2\alpha_4$; for every dominant $\mu$ with $\langle\mu,\alpha_2^\vee\rangle = 1$ and $\mu+\beta$ dominant, $a(\mu,\mu+\beta)$ is $-2$ if $\langle\mu,\alpha_1^\vee\rangle = 0$ and $-1$ otherwise. Restriction to the support of $\lambda-\mu$ carries this family into every irreducible type with a trivalent node. Thus an irreducible finite root system has all atomic numbers nonnegative exactly when its Dynkin diagram is a path, namely in types $A_n$, $B_n$, $C_n$, $F_4$, $G_2$. Deep in the dominant chamber, in an explicit range, $a(\mu,\lambda)$ is the Kostant partition number of $\lambda-\mu$ for the nonsimple positive roots; negative atomic numbers are therefore confined to boundary slabs.

This paper studies lower-level-constrained bilevel optimization in a response-oracle setting, where lower-level model information is unavailable and the induced response mapping is locally Lipschitz but potentially nonsmooth. In this setting, the classical response Jacobian and reduced hypergradient may fail to exist. We propose Bilevel Zeroth-Order Learning (Bi-ZOL), a structure-guided zeroth-order method for finding stationary points of the nonsmooth reduced problem. Instead of estimating the gradient of a fully smoothed reduced hyperobjective, Bi-ZOL separates the bilevel chain-rule structure: it keeps the exact upper-level partial gradients at the queried response and uses zeroth-order sampling only to estimate the response Jacobian. This construction yields an approximate hypergradient that is more directly aligned with the Clarke chain-rule subdifferential. We show that the Bi-ZOL direction admits a partial-smoothing interpretation, quantify its pointwise structural bias, and prove finite-time convergence to a $(\delta,\epsilon)$-Bi-ZOL Frank–Wolfe stationary point. The bias is $O(\delta)$ for piecewise $C^{1,1}$ responses under local regularity and vanishes for piecewise affine responses on active-cell neighborhoods. Experiments on incentive-based tracking problems show that Bi-ZOL achieves smaller stationarity gaps and lower hyperobjective values than vanilla zeroth-order smoothing under comparable response-oracle budgets.

We study the coordinate-plane Kac walk on $\mathrm{SO}(n)$ with independent uniform rotation angles. For the unnormalized Hilbert-Schmidt Riemannian distance, we prove that the Wasserstein-$2$ Lipschitz coefficient of a block of $2\binom n2$ steps is at most $2n^{-1/200}$ for sufficiently large $n$. This gives a fixed-accuracy $O(n^2)$ Wasserstein mixing bound and inverse-polynomial accuracy in a constant number of blocks. We also obtain an $O(n^2)$ total-variation upper bound. The coupling averages each component of a regularized-inverse angle correction over its own angle, so its exact flow preserves the joint angle law. A deterministic decreasing regularization parameter cancels the main scalar and matrix-valued resolvent drifts. Its stability factor is polynomial in the inverse regularization, and the matrix estimates close on two partial traces. The same scalar estimate bounds the fraction of deficient covariance directions; independent cores and a cutoff integration-by-parts argument give the total-variation transfer. We provide the parameter estimates, conditioning arguments, and finite-time singular-mass treatment explicitly. The results imply total-variation pre-cutoff on the quadratic scale, but do not establish either the presence or the absence of cutoff.

We study the Bernoulli property for skew products of the form $f(x,y)=(g(x),A(x)y)$ on $M\times PSL_2(\mathbb{R})/\Gamma$, where $g$ is a smooth, measure-preserving, exponentially mixing diffeomorphism, $\Gamma<PSL_2(\mathbb{R})$ is a cocompact lattice, and $A\colon M\to \operatorname{PSL}_2(\mathbb{R})$ is a $C^{1+\alpha}$ cocycle. We prove that if $A$ has a nonzero Lyapunov exponent, then $f$ is Bernoulli. The proof does not require that $f$ be exponentially mixing. This provides a new method for establishing the Bernoulli property, which applies to a large class of skew products.

Let $r \ge 2$ and $t \ge 1$. It is shown that if $T$ is an $r$-uniform tight tree with $t$ edges and $H$ is a $T$-free $r$-uniform hypergraph, then $|E(H)|\le (t-1)|\sh H|/r$, where $\sh H$ is the $(r-1)$-shadow of $H$. \iffalse Equality holds only for $(n,t + r - 2,r)$-designs.\fi The bound is tight infinitely often, and establishes Kalai's Conjecture, whose $r=2$ case is the Erd\H os-Sós Conjecture. The proof was found by GPT-6 Astra, extending its method of proof for the Erd\H os-Sós conjecture to the hypergraph setting. It is noteworthy that previous proofs of special cases of the Erd\H os-Sós conjecture do not extend to give tights bounds in the hypergraph setting. A strengthening of the Erd\H os-Sós conjecture due to the authors about tight lower bounds on the number of copies of a tree in a graph with average degree $d \ge t-1\ge 0$ remains open.

We prove that the realization graph of every graphical degree sequence is maximally Hamiltonian: it is Hamilton-laceable when bipartite on more than one vertex, and Hamilton-connected otherwise. This answers Problem P59 of Mütze's survey of combinatorial Gray codes, and the Hamiltonicity question recorded as open by Barrus, in the strongest form either admits. The argument is an induction on the number of ground vertices, cutting the realization graph at a single ground vertex into fibers and the quotient they lie over. The proof is formalized in Lean 4 and checked by its kernel, with seven results cited from the literature and nothing else assumed. Its engine is a classification. The realizable neighborhoods of a ground vertex form a shifted family – one closed under replacing an element by a smaller one – and the quotient is the Johnson graph of that family. Such a Johnson graph can fail to be Hamilton-connected, and we determine exactly when: the failures are one explicit family of examples, the Y-families, and each of them fails between a single pair of its members. A shifted family with a greatest member never fails, and those families are exactly the shifted matroids, where the conclusion already follows from the theorem of Naddef and Pulleyblank on the graphs of 0/1-polytopes. The obstruction lives entirely outside the matroid case, which is why it has not been met before.

We give a deterministic polynomial-time list-decoding algorithm for Reed-Solomon codes over prime fields that approaches list-decoding capacity for every evaluation set and every constant rate.

Extending the classical $K$-means and $K$-medians models, this paper introduces an $\ell_{p,q}$ mixed-norm clustering model where the centroid updates and cluster assignments are under the $\ell_p$ and $\ell_q$ norms, respectively. The model is formulated as a mixed-integer program (MIP) with Heaviside composite constraints that describe the nearest-center assignments. The framework recovers $K$-means and $K$-medians when $p=q=2$ and $p=q=1$, respectively, and yields new models when $p\ne q$. To address the computational challenges, we develop a progressive integer programming (PIP) method that adaptively fixes confident assignments and solves restricted mixed-integer subproblems. For $q=1$, we develop a convex inner approximation of the difference-of-convex constraints in the restricted subproblems, for which a global solution can be computed. Importantly, we establish the connection between the local minimizer and the strong center-local minimizer of the mixed-norm clustering problem and the global optimal solution of the restricted subproblems under certain assumptions. This connection provides a practical certificate of a local minimizer of the nonconvex mixed-norm clustering model. We further develop techniques for constructing adaptive fixing sets and working sets for $q=1$. Extensive numerical experiments demonstrate the superior performance of the mixed-norm clustering model and the efficiency of PIP for solving the MIP model, which may be intractable otherwise. In particular, the $\ell_{2,1}$ mixed-norm clustering model is effective under coordinate-sparse, mean-balanced contamination, whereas the $\ell_{1,2}$ model is preferred under dense coordinatewise Cauchy contamination. The numerical results also show that PIP can escape poor alternating solutions and obtain substantially better feasible clustering, while preserving strong warm starts when no improvement is found.

For a Coxeter system $(W,S)$ of finite rank and an involutive automorphism $\ast:W\to W$ which preserves $S$, we prove that the length-generating function $\sum_{z\in\mathbf{I}_{W,\ast}}q^{\ell(z)}$ of the set of $\ast$-twisted involutions is rational. We derive recurrence relations analogous to the well-known recurrence relations expressing the growth series $W(q):=\sum_{w\in W}q^{\ell(w)}$ in terms of the growth series $W_I(q)$ of the parabolic subgroups of $W$. Our result extends to any subset $\mathcal{C}$ of $\ast$-twisted involutions that is closed under $\ast$-twisted conjugation. We use these recurrence relations to give an alternative proof of a power-series identity for twisted involutions due to Lusztig.

We study the control of Markov decision processes in which the quality of a policy is evaluated by a dynamic, time-consistent Markov risk measure rather than by an expected discounted cost. The main obstacle to combining such measures with reinforcement learning is that a transition risk mapping depends on the transition kernel in a nonlinear way, and therefore cannot be estimated from a single observed transition. We remove this obstacle by employing mini-batch transition risk mappings: the mapping is applied to the empirical measure of $N$ independent next-state samples, and the result is averaged. The resulting mapping is again coherent. However, as an expected value of a function of $N$ next-state values, it admits an unbiased one-sample estimator. We embed this mapping into a double deep Q-network, analyze the two sources of estimation bias that arise, and obtain a risk-averse Q-learning method applicable to state spaces far beyond the reach of tabular schemes. The method is applied to an underwater robot navigation problem, in which a vehicle must visit collection points, gather stochastic information payloads, and deliver them at transmission points, while exposed at each step to the risk of destruction. A hierarchical decomposition delegates path execution to an exact graph search and confines learning to the high-level “collect or transmit” decision. A low-dimensional feature map, invariant under the symmetries of the problem, replaces the raw state–configuration encoding. In experiments on $300$ held-out environments, the resulting policies transfer to instance sizes never seen in training, and already $N=2$ reduces the upper semideviation of the outcome distribution while simultaneously improving its mean whenever the simulator is misspecified—an empirical counterpart of the duality between coherent risk measures and distributional robustness.

We characterize the sharp structure-agnostic minimax risk for coefficient estimation in the partial linear model when the outcome and treatment nuisances are learned by two distinct black-box learners, which resolves the open problem in double machine learning posed by Gu (2025). For each nuisance \(q\in\{\mu,\pi\}\), we characterize the available learner by an approximation-error budget \(a_q\) and a stochastic-error budget \(s_q\), with the latter controlled through localized Rademacher complexity. Writing \(\mathcal E_n\) for the minimax mean-squared error, we show that \[\mathcal E_n\asymp1\wedge\left\{\frac1n+\left(a_\mu a_\pi+\min\left\{a_\pi s_\mu+s_\pi^2,\,a_\mu s_\pi+s_\mu^2\right\}\right)^2\right\}.\] The main new ingredient is a novel lower bound for the general two-learner problem. Our proof constructs four finite-mixture testing experiments using orthogonal code functions. Across these experiments, the hidden perturbations are placed outside both learner classes, outside only the treatment learner class, outside only the outcome learner class, or inside both learner classes. These four configurations capture, respectively, the interaction between the two approximation errors, the two asymmetric interactions between one learner's approximation error and the other learner's learning error, and the joint estimation difficulty of learning both nuisances. Combining the four resulting lower bounds yields the displayed rate, which matches the latest upper bound in Gu (2026). Our result shows that standard double machine learning can overstate the intrinsic difficulty of target estimation and provides a target-specific principle for learner selection: approximation error and stochastic complexity must be jointly balanced across the two nuisance learners rather than optimized separately.

We construct a counterexample to Köthe's conjecture and use this to also answer a question due to Rowen.

We develop a constructive method for solving semilinear elliptic equations $\Delta u(x)=f(x,u(x))$ on locally finite, connected infinite graphs with layered structure. Using Eidelheit's theorem, we establish coupling criteria ensuring that arbitrary initial-layer data extend to global solutions for every $f$. We apply combinatorial criteria to prove solvability on leafless infinite trees, integer lattices, the triangular and hexagonal lattices, and a Cayley graph of the discrete Heisenberg group. We further establish solvability for a broad class of Cayley graphs of semidirect products $G\cong\mathbb Z\ltimes_\theta H$. In particular, $\Delta u=e^u$ has infinitely many solutions on $\mathbb Z^2$, but none of finite energy. We also extend the method to the bi-Laplacian under two-step coupling conditions, to the $p$-Laplacian under a unique-neighbor condition, and to magnetic Laplacians.

Finite-range alignment can end either in a single flock or in several noninteracting clusters, yet convergence results rarely determine which outcome follows from a given finite-particle state. We study a one-dimensional normalized alignment model with a hard interaction cut-off and obtain exact predictions in an expansive regime. Velocity order is invariant, so pair separations are nondecreasing and the communication graph evolves through finitely many irreversible edge deletions. On each fixed graph, the group inverse of the random-walk Laplacian gives the total relative displacement remaining before relaxation. Comparing this displacement with the available interaction slack selects the next deletion and yields a finite recursion for the complete switching sequence, terminal partition, limiting cluster velocities, and internal geometry. For path configurations, an explicit Green kernel gives a necessary-and-sufficient fragmentation criterion and a sharp critical alignment rate, including asymptotic boundary contact at criticality. A spectral-geometric condition extends the theory to an open set of initially nonordered velocities, while a common self-weight extension covers both self-excluding and self-including local averages. Numerical computations reproduce the thresholds and multi-event cascades. The results provide an exact finite-size theory of fragmentation and terminal state selection for a class of finite-range interacting particle systems.

The local clique cover number $lcc(G)$ is the minimum valency of an edge-clique cover of $G$. We prove the conjectured inequality $lcc(G)+\chi(G)\le |V(G)|+1$ for every finite simple graph. The proof gives an independent set improvement of an endpoint-cover estimate and applies the resulting construction to an induced four-vertex path. We further prove that the cover can be chosen so that every nonuniversal vertex has valency at most $|V(G)|-\chi(G)$. Consequently, every graph attaining equality has a universal vertex. The stronger statement follows by reducing a counterexample of minimum order to a prime double-critical graph and refining the induced path extension. We also show that an induced matching of size $m\ge2$ yields $lcc(G)+\chi(G)\le |V(G)|+3-m$, which improves the general bound when $m\ge3$, and determine the restrictions imposed by equality on deletion of an induced $2K_2$.

The solvability of two-point boundary value problems for constant-coefficient differential-algebraic equations is investigated. Unlike previous studies, which often assume that the matrix pair associated with the equation is regular, we consider the general singular case. Using the Kronecker canonical form, we decompose the problem into simpler subsystems, enabling a systematic analysis of solvability. The method of parameterization reduces the boundary value problem to a system of algebraic equations. We derive criteria for the existence and uniqueness of solutions and provide a comprehensive framework for solving such boundary value problems. Several illustrative examples are presented and show the applicability of the results.

We prove that the magnitude of the tope graph of a simple oriented matroid is the specialization at $x=-1$ of the motivic zeta function of its underlying matroid. This refines the Las Vergnas–Zaslavsky theorem and defines magnitude for arbitrary matroids. For simple orientable matroids, we compute the order of the pole at $q=-1$ from chains of flats and identify it with the Varchenko–Gelfand degree of the tope parity function. Comparing pole orders, we construct a rank-six real arrangement $\mathcal A$ such that $\operatorname{Mag}(\mathcal A;-t)$ has infinitely many negative coefficients, disproving Koizumi–Liu's eventual sign alternation conjecture. We then construct a canonical multiplicative Varchenko–Gelfand filtration on the mod-$2$ magnitude cohomology of a simple oriented matroid and show that its graded dimensions recover the full motivic zeta function.

Bringing multiscale geometric analysis directly to irregular point clouds remains difficult: quantities such as local dimension, anisotropy, density variation, and geometric transitions are typically estimated through explicit neighborhood, manifold, or graph constructions, or left for neural networks to infer from coordinates. We introduce Heat Field Signatures (HFS), which lift a point cloud to a multiscale family of smooth ambient heat fields, providing a direct interface from discrete samples to geometric analysis. From this field, HFS computes closed-form global and local signatures directly from pairwise distances, capturing heat concentration, intrinsic dimension, anisotropy, and scale transitions. We further introduce the Heat Dimension Spectrum (HDS), a compact summary of multiscale geometric composition. HFS can be used as a closed-form descriptor, a lightweight learned representation, or a geometric feature channel for neural point-cloud models. Across synthetic and real-world benchmarks spanning subcellular, neuronal, tree, and protein data, HFS outperforms strong point-cloud and multiparameter-persistence baselines while substantially reducing end-to-end cost. On SCOP protein-fold classification, HFS improves over the strongest deep baseline by nearly $24$ percentage points using coordinates alone, while standalone HFS representations are exactly rotation-invariant by construction. More broadly, HFS turns a classical heat field into a practical interface for multiscale geometric analysis in modern point-cloud learning.

Let $(M,\tau)$ be a semifinite von Neumann algebra, let $J$ be a trace-preserving Jordan isomorphism, and let $(n_k)_{k\geq 1}$ be a random increasing sequence of integers obtained by selecting each integer $n\geq 1$ independently with probability $n^{-\alpha}$, where $0<\alpha<\frac12$. We show that, almost surely, for every $x\in L^1(M)$, $ \frac1m\sum_{k=1}^m J^{n_k}(x)$ converges bilaterally almost uniformly. This extends LaVictoire's classical random $L^1$ ergodic theorem to the non-commutative setting.

Dinh and Nguyên asked whether the determinant of a Griffiths positive matrix of $(1,1)$-forms belongs to the Hodge–Riemann cone. For every $n\geqslant4$ and $2\leqslant k\leqslant n-2$, we present counterexamples constructed by AI: Nakano positive $k\times k$ matrices of constant $(1,1)$-forms on $\mathbb{C}^n$ whose determinants have singular Lefschetz maps in bidegree $(1,n-k-1)$. Under the simultaneous diagonalizability (SD) condition, we prove the Hodge–Riemann property in every bidegree $(p,q)$ with $p+q=n-k$ and $\min(p,q)\leqslant1$. The proof uses the generalized Alexandrov–Fenchel inequality of Ross, Süß, and Wannerer. For every $n\geqslant6$ and $2\leqslant k\leqslant n-4$, we also present SD examples whose determinants have singular Lefschetz maps in bidegree $(2,n-k-2)$. Their underlying idea comes from Ross–Toma's construction. The remaining cases $n=k\geqslant4$ and $n=k+1\geqslant5$ are open for general Griffiths positive matrices. They reduce to the top-Chern case of Griffiths' positivity question, equivalently to Finski's double mixed discriminant problem. We also ask whether a $2\times2$ counterexample on $\mathbb{C}^4$ can be both Nakano positive and dual Nakano positive.

Fourier-Motzkin elimination computes an inequality description of the projection of a polyhedron onto a subset of its coordinates by eliminating one variable at a time. It is used in several areas of optimisation and computer science, and it is a standard way of obtaining the entropic constraints of a causal structure, where the marginalisation over the latent variables produces such a projection. Its limitation is the growth of the intermediate systems of inequalities, which can be doubly exponential in the number of eliminated variables even though the projection itself grows only as a single exponential. In practice the computational overload of the method therefore depends on two choices: how the redundant inequalities are removed after each step, and the order in which the variables are eliminated. We consider both. We first show, by an explicit example, that Imbert's redundancy test cannot be interleaved with redundancy removal by linear programming. We show that the two methods, however, can be combined soundly if the derivation records used by Imbert's test are re-initialised after every step at which linear programming is used. We then propose a rule for choosing the elimination order of the variables that gives a significant computational advantage, however, at the cost of increased resource usage. We demonstrate this advantage on some random polytopes, where the rule reduces the running time by factors of between 6 and 25 compared with the same elimination under a fixed order. For entropic descriptions of causal structures, with more than 250 inequalities and more than 100 variables to eliminate, our rule keeps the number of inequalities handled at each step one to two orders of magnitude lower than a fixed order.

We prove Malliavin nondegeneracy for the third-order Hermite process. The key step is to show that, for every nonzero test direction $h\in C_c^\infty(0,1)$, the directional Malliavin derivative of a third-order Hermite random variable is an infinite-rank Gaussian quadratic form. Using arbitrarily large orthonormal families of such directions, we derive a self-contained Fourier bound for their joint characteristic function and obtain polynomial small-ball estimates of arbitrary order for the Malliavin norm. This yields negative moments of every order. Combining the one-time estimate with determinant factorization and Malliavin strong local nondeterminism, we obtain negative moments of all orders for finite-dimensional Malliavin determinants. Consequently, all finite-dimensional distributions, as well as arbitrary vectors of non-overlapping increments, admit Schwartz densities. We further establish grid-uniform Sobolev bounds for inverse Malliavin determinants of normalized increment vectors and derive stretched-exponential estimates for all partial derivatives of their densities. The decay exponent is $2/3$, reflecting the third Wiener chaos. This settles the next non-Gaussian Hermite order after the Rosenblatt case and provides a quantitative counterpart to the order-two smoothness theory.

Let $d \geqslant 1$ be an integer, $\mathscr{M}$ a suitable $d$-abelian category. There is a notion of higher torsion classes in $\mathscr{M}$, also known as $d$-torsion classes. We generalise a classic theorem of Hoshino by providing criteria for a $d$-torsion class $\mathscr{U}$ to be splitting, notably that $\mathscr{U}$ is stable under $\tau_d^-$, the inverse $d$-Auslander–Reiten translation. We apply the criteria to show that under additional assumptions, which are satisfied in Dynkin type $A$, the elements of the spine of the lattice of $d$-torsion classes are precisely the splitting $d$-torsion classes. The spine consists of the elements which belong to a chain of maximal length.

We analyze the poetico-comedic criteria by which Tom Lehrer arranged the chemical elements in his 1959 hit "The Elements," a parody to the tune of Gilbert and Sullivan's "The Modern Major-General" (1879). We compose a new version to the same tune incorporating all 118 currently known elements and fulfilling, so far as possible, the same artistic aims.