Papers

In the generalization of token graphs (or supertoken graphs), the vertices of a $k$-token graph of a graph $G$ correspond to $k$ distinguishable or indistinguishable tokens placed on $k$ different or equal vertices of $G$. In several cases, we provide the whole spectrum of a supertoken graph of a star by analyzing its regular partitions.

The analysis of large data panels is important in econometrics and beyond. Prediction and inference methods for such data typically rely on simplifying model assumptions for the covariance structure of errors. One convenient assumption is what we call groupwise sphericity: that errors are uncorrelated across individuals and have constant variance within certain groups. While theoretically useful, in large panels groupwise sphericity is often too restrictive to apply in practice. We therefore develop new quantitative inference tools to test whether deviations from this model assumption are practically relevant. Our approach covers both large-dimensional data matrices and regression panels, in a regime where the cross-sectional dimension is proportional to the sample size. The theory is based on the analysis of extreme eigenvalues of the empirical covariance matrix and uses recent advances in random matrix theory. Numerical experiments demonstrate accurate size control and good power in finite samples.

We give a complete classification of Schur positivity for three-legged spiders. Every spider with at least two even legs is Schur positive, whereas every spider with three odd legs has a negative Schur coefficient. If $a$ is even and $b,c$ are odd, then the spider $S(a,b,c)$ is Schur positive if and only if $a\le 5b+5c+2$. Whenever Schur positivity fails, there is exactly one negative Schur coefficient, whose value we determine explicitly. These results extend those of Thibon and Wang for the families $S(a,2,1)$ and $S(a,4,1)$, and those of Wang and Wang for $S(a,b,2)$. The proof combines known Schur-positivity results for clique-spiders with Pieri's rule and simultaneous induction.

For a matrix rank-metric code $\mathcal{C}$ and an integer $i$, the $i$-th subcode distance of $\mathcal{C}$ is the largest minimum rank distance of an $i$-dimensional subcode of $\mathcal{C}$. We study how subcode distances change under the Add-and-Remove construction, in which a subcode of codimension $\ell_s$ is kept and a space of dimension $\ell_a$ is added; this construction is used, with Gabidulin codes, in the $\mathsf{MIRANDA}$ signature scheme. For arbitrary matrix codes, we give inequalities between the subcode distances of the original and of the new code, and we show that Delsarte duality interchanges $\ell_s$ and $\ell_a$. For matrix Gabidulin codes, whose subcode distances are given by an explicit formula, we show that, if $\ell_s<\max\{m,n\}$, then within each block of indices $(a-1)\max\{m,n\}+1,\ldots,a\max\{m,n\}$ not exceeding $\dim_{\mathbb{F}_q}(\mathcal{C}_s)$, all but the last $\ell_s$ subcode distances are the same as for the Gabidulin code, and the remaining ones decrease by at most one; a small example shows that such a decrease can occur. We also express the minimum distance of the new code in terms of the rank distance between the added space and the common subcode. For $\mathsf{MIRANDA}$, the lower bound on the minimum distance of the dual public code used by the known distinguisher when $\ell_s=0$ is a special case of these results; for the proposed parameters, where $\ell_s>0$, the same bound holds for a subcode of codimension $\ell_s$ of the dual public code.

This paper establishes uniform local large-deviation asymptotics for the entrance law $\underline n_t(dx)$ of excursions of a Lévy process with negative drift and locally regularly varying Lévy measure $\nu$, reflected at its running infimum. In the fixed-time regime, we prove that as $x\to\infty$, $$ \underline n_t\big((x,x+\delta]\big) \sim \underline n(\zeta\wedge t)\, \nu\big((x,x+\delta]\big), $$ uniformly in $\delta\in[\delta_0,\infty]$ for every $\delta_0>0$. Our principal result concerns the large-time regime and shows that, as $t\to\infty$, $$ \underline n_t\big((x,x+\delta]\big)\sim \int_x^{x+\delta} V(z) \,\nu(\beta t+dz), $$ uniformly in $x\geq0$ and $\delta\in[\delta_0,\infty]$, where $V$ denotes the renewal function. Using a fluctuation-theoretic decomposition of the killed semigroup, we further derive corresponding uniform local large-deviation asymptotics for the Lévy process killed upon its first passage below zero.

Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini–Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini–Study distances generated by local operations. We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and, independently, to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.

We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let $\psi:\N\to(0,\infty)$ be a non-increasing approximation function and let $\tau_\psi$ denote its lower order. For $m,n\ge1$ and $\boldsymbol\theta\in\mathbb R^m$, denote $\Exact_{m,n}(\psi,\boldsymbol\theta)$ the set of $m\times n$ matrices that are $\psi$-approximable with shift $\boldsymbol\theta$, but are not $c\psi$-approximable for any $0<c<1$. It is shown that \[ \hdim \Exact_{m,n}(\psi,\boldsymbol\theta)=m(n-1)+\frac{m+n}{1+\tau_\psi},\ \ \ {\text{for}}\ \tau_\psi>\frac{n}{m}. \] We adopt a probabilistic viewpoint, i.e. by an averaged counting argument to quantify the distribution of rational vectors and subspaces $$\left\{\frac{\mathbf p+\boldsymbol\theta}{q}: (\mathbf p, q)\in \mathbb{Z}^{m+1}\right\},\ \ \ \bigg\{\big\{A\in \Mat_{m, n}(\R): A\mathbf q-\mathbf p-\boldsymbol\theta=0\big\}: (\mathbf p, \mathbf q)\in \mathbb{Z}^{m+n}\bigg\}.$$

We extend the partition pairing theory of Andrews and Dastidar by replacing pairs with groups of $k$ equal parts. Two weight-preserving bijections give combinatorial interpretations of the joint pairing index–width distribution and the negative-rank enumerations. The first combines conjugation with the Stockhofe–Keith correspondence and sends the $k$-pairing index and width to the number of parts and the largest part, respectively. Consequently, their joint distribution is independent of $k$ and is given by a Gaussian polynomial. We also obtain finite refinements that record the residual statistics. The second bijection sends simply $k$-paired partitions of negative $k$-pairing rank to $k$-regular partitions with marked internal gaps. In a fixed nonzero residue class, sign cancellation leaves rectangular partitions, while gap markings correspond to overlining choices with the smallest part not overlined. This gives direct combinatorial explanations of the divisor counts and the factor $1/2$ in the overpartition enumeration, extending the odd-divisor and odd-overpartition results of Andrews and Dastidar. Finally, motivated by the diagonal pairing of Andrews and Dastidar, we extend the construction to ordered tuples of $k$ Young diagrams. Applying the two-wing transfer operation to pairs of component diagrams defines an equivalence relation on these tuples. We show that two tuples are equivalent if and only if they have the same cell-multiplicity function, and that every equivalence class contains a unique representative whose component diagrams are nested. We also determine the cardinalities of the equivalence classes and identify the nested representatives with plane partitions of rectangular shape.

The susceptibility tensor of a homogeneous collisionless magnetized plasma possesses parity-time (PT) symmetry. For a stationary gyrotropic equilibrium, with the wave vector $\mathbf{k}$ and the equilibrium magnetic field $\mathbf{B}_0$ in the $x$–$z$ plane, the kinetic response satisfies $\chi(\omega^*,\mathbf{k})=R\chi(\omega,\mathbf{k})^*R$, where $R=\operatorname{diag}(1,-1,1)$. We show that the susceptibility PT-symmetry is a reduced-response-space manifestation of PT symmetry of the underpinning kinetic system. We identify parity $P$ and time reversal $T$ on the electromagnetic kinetic eigenmode space and show that the Hamiltonian operator $\mathcal{H}_{\mathrm{VM}}$ of the linearized Vlasov–Maxwell system is PT-symmetric, satisfying $PT\mathcal{H}_{\mathrm{VM}}(PT)^{-1}=\mathcal{H}_{\mathrm{VM}}$. The PT symmetry of the homogeneous magnetized Vlasov–Poisson eigenproblem is also established. Many kinetic instabilities associated with nonmonotonic distributions, rings, loss cones, and resonant energetic-particle populations are fundamentally spontaneous PT-symmetry breaking, which is achieved through and only through Krein collisions. Three examples demonstrate distinct utilities of this analysis. Definite-sign cyclotron residues protect thermal Bernstein modes from spontaneous PT-symmetry breaking. The nonmonotonic Dory–Guest–Harris distribution permits sign-indefinite residues and Krein collisions, for which we derive an analytical instability threshold and growth rate. The recently discovered mode-pole resonance instability provides a distinct realization of spontaneous PT-symmetry breaking and illustrates how PT symmetry can guide the discovery of kinetic instabilities with applications to fusion energy. These results establish PT symmetry as a structural principle of plasma waves and instabilities and open new directions for studying spectral topology in kinetic plasmas.

We improve an earlier characterization of the Sobolev space $\dot W^{1,p}(\mathbb R^d)$, $1<p<\infty$, in terms of the oscillations of a function on balls of varying centers and radii. In contrast to earlier work, finiteness of the supremum is replaced by the formally weaker assumption of finiteness of the liminf, thereby answering a question by Brezis.

We study the QWEP property of twisted crossed products and its relation to Fourier multiplier transference on noncommutative $L_p$-spaces. We show that QWEP is not preserved by crossed products of QWEP von Neumann algebras by hyperlinear groups. Our examples include trace-preserving actions of a torsion-free residually finite Kazhdan group on a diffuse abelian algebra and on the hyperfinite type II$_1$ factor. We also obtain non-QWEP crossed products with nontrivial scalar cocycles. For scalar twists, we prove that a finite trace-preserving crossed product is QWEP if and only if its ordinary crossed product and its twisted group von Neumann algebra are QWEP. In the positive direction, we establish a restriction theorem for scalar twists of regular wreath products. These results provide QWEP inputs for Fourier multiplier transference and identify a completely bounded multiplier norm obstruction to QWEP. Finally, operator-valued cocycles give finite QWEP crossed products whose acting groups are not hyperlinear.

In this paper we consider groups of complex hyperbolic isometries generated by three complex involutions. Our first main result states that if the group is discrete and at least two of the mirrors of complex reflections intersect, then the group may be put into a normal form involving the ordering of traces of short, even length words. Our second main result refines a conjecture of Schwartz (proved by Grossi) that determines the type of the group in terms of the angles between the mirrors. That is, in the parameter space of complex triangles with given angles, it determines which of two given words becomes elliptic first.

Bent partitions were introduced as a generalization of partial spread construction of bent functions and they became a hot research topic recently. In this paper, we prove two general constructions of normal bent partitions that are related to vectorial dual-bent functions. They cover many of the currently known bent partitions of this type as partial cases. These constructions also provide a large number of new bent partitions. Relevant properties of bent functions obtained from such partitions are proven.

We show that the set $\mathbb{P}-1$ of shifted primes contains infinitely many sum-product patterns of the form $\{x,x+y,xy\}$ with $x,y$ arbitrarily large distinct integers. More strongly, we can also show that, for any $k\geq 1$, the set $\mathbb{P}-1$ contains longer patterns of the form $\{x,x+y,\ldots, x+ky,xy\}$ with $x,y$ arbitrarily large distinct integers, a statement that contains the Green–Tao theorem as a special case.

We study the persistence of non-planarity in boundaries of hyperbolic groups under quotients, focusing on quotients by large enough powers of infinite order elements. We prove that the non-planarity of the boundary of a one-ended hyperbolic group is preserved under such quotients. As a consequence, we obtain an analogous result about the persistence of Menger curve boundaries.

We give a detailed and complete algebraic proof of Simons' Riemannian holonomy theorem by a single strong induction on the dimension of the linear span of the curvature orbit. The proof uses the construction of flats and root centralizers, but replaces the detailed analysis of the common zero-weight space, the most technical core of previous algebraic proofs, by a trace-vanishing lemma. The lemma applies to curvature operators contained in an ideal that annihilates the curvature module. Restriction to a common geodesic subspace produces a proper invariant kernel. Induction makes this kernel fixed by the holonomy action, after which the trace lemma shows that the root centralizers span the ambient space. Their intersection recovers the maximal flat, and irreducibility completes the proof. For the completeness we also include the detailed derivation of the local symmetry of the underlying Riemannian manifold from the algebraic theorem.

Let $(\mathcal{M},\tau)$ be a semi-finite von Neumann algebra. We prove that $L_1(\mathcal{M})$ has Talagrand type $(1,\psi_{1,1})$, where $\psi_{1,1}(t)=t/\log(e+t)$. As an consequence, the Schatten trace class $S_1$ has Talagrand type $(1,\psi_{1,1})$, thus settled an open problem of Question 2 in Cordero-Erausquin and Eskenazis (2023). Our proof is based on the noncommutative Mazur maps initiated by Ricard.

Ribet's 1976 proof of the converse to Herbrand's theorem created a new paradigm in algebraic number theory by illustrating that unramified abelian Galois extensions of number fields can be constructed using Galois representations associated to Eisenstein congruences, i.e., cuspidal eigenforms that are congruent to Eisenstein series. In recent years, further progress has been made in investigating the quantity of such Eisenstein congruences, showing that when there are many such congruences, additional and finer information can be gleaned about the arithmetic structure of the related extensions. In this vein, we use new results on the existence of Eisenstein congruences in tame families to explain a computational observation about splitting behavior of primes in a certain metabelian number field.

This article is devoted to the mathematical analysis of a new Navier–Stokes-$\alpha$ model involving a nonlinear filter equation. The resulting model is governed by a doubly nonlinear parabolic–elliptic coupled system. In our previous work [M.~F.~Cortez and O.~Jarrín, Mathematical study of a new Navier–Stokes-alpha model with nonlinear filter equation – Part I, J. Math. Fluid Mech. 28, no.~12 (2026)], we established the global well-posedness of weak Leray-type solutions and the existence of a global attractor. In the present work, under natural assumptions on $A(\cdot)$, we investigate several additional properties of this model, with particular emphasis on higher-order regularity and its consequences for the long-time dynamics. The regularity analysis constitutes a central and delicate issue, since the nonlinear structure of the elliptic filter and its coupling with the evolution equation preclude a direct application of the standard regularity arguments available for linearly filtered Navier–Stokes-$\alpha$ models. Overcoming this difficulty requires new higher-order estimates for the nonlinear elliptic filter equation, which may also be of independent interest. Among the results obtained, two constitute the main contributions of the article. First, we establish uniform higher-order regularity for the global attractor. This result is then used as a fundamental ingredient in proving an exact determining-modes property for complete trajectories on the attractor: if the projections of two complete trajectories onto a sufficiently large finite-dimensional space of Stokes modes coincide at every time, then the two trajectories coincide identically.

To an oriented quiver with boundary $\Gamma$ and a compact connected Lie group $G$ we associate a Nahm-Kirchhoff moduli space $\mathcal{M}_{\mathbb{H}}(\Gamma)$: solutions of Nahm's equations on the edges with Kirchhoff matching at interior vertices, modulo boundary-trivial gauge transformations. For connected $\Gamma$, $\mathcal{M}_{\mathbb{H}}(\Gamma)$ is a smooth manifold of dimension $4(|E|-|\Gamma_{\mathrm{int}}|)\dim G$ with a hyperkähler structure arising as an infinite-dimensional hyperkähler quotient. Via a Kempf-Ness argument on the Cartan-Hadamard manifold $G_{\mathbb{C}}/G$, we prove a quiver analogue of Donaldson's theorem identifying it with the complex-symplectic quotient $\mathcal{M}_{\mathbb{C}}(\Gamma)$ of the complex Nahm equations by the complexified gauge group. This quotient is independent of edge lengths and isomorphic to a finite-dimensional complex-symplectic quotient of $(T^*G_{\mathbb{C}})^E$, while the hyperkähler metric depends on them: uniform rescaling acts by a homothety, and the transported metrics genuinely vary. By cutting and gluing, $\Gamma\mapsto\mathcal{M}_{\mathbb{C}}(\Gamma)$ is a 2D TQFT valued in complex Hamiltonian manifolds, à la Moore-Tachikawa, whereas the hyperkähler refinement is not functorial on $\mathbf{Cob}_2$: spines of a cylinder with different lengths give non-isomorphic spaces. Retaining lengths, these spaces form a metric field theory, a functor of non-unital monoidal categories from metric cobordisms to hyperkähler manifolds composed by hyperkähler reduction; over each cone of $M_{g,n}^{\mathrm{trop}}$ they give a family with fixed complex-symplectic fibre and varying metric. Finally, harmonic gauge representatives converge as an internal edge collapses, and on a long edge with regular semisimple Higgs field the Kempf-Ness minimizer is exponentially confined near its centralizer; the limiting metrics are left open.

We develop a differential-geometric approach to contractive projections on complex Banach spaces based on Levi forms of plurisubharmonic functions. We show that if a projection $P$ contracts a $\mathrm{C}^2$ plurisubharmonic function $\Phi$, then, at any point of its range, the Levi form of $\Phi$ defines a tangent Hilbertian geometry for which $P$ becomes an orthogonal projection. When the range is finite-dimensional and the relevant Levi forms are nondegenerate, comparing the Levi geometries associated with different points leads naturally to positive relative operators and associated one-parameter groups. We apply this framework to tracial noncommutative $\mathrm{L}^4$-spaces. Exploiting the cubic form of the duality mapping, we characterize contractive projections by a Jordan triple orthogonality property and prove that a projection on $\mathrm{L}^4(\mathcal{M})$ is $2$-contractive if and only if it is completely contractive. When the ambient von Neumann algebra is finite-dimensional, we identify the relative operator associated with the left and right tangent forms on the range with a restriction of an inverse square root of a modular operator on a linking von Neumann algebra. This entails that the range of any $2$-contractive projection is completely isometric to a rectangular noncommutative $\mathrm{L}^4$-space associated with a $\mathrm{W}^*$-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every $1 < p < \infty$ with $p \neq 2$, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative $\mathrm{L}^p$-space is completely contractive. For $p > 2$, we obtain a weighted ternary description of its range in terms of a finite-dimensional $\mathrm{W}^*$-$\mathrm{TRO}$.

We prove that metric measure spaces arising from $2$-manifolds equipped with Riemannian metrics $g$ such that $g, g^{-1} \in L^\infty_{\rm loc} \cap W^{1,p}_{\rm loc}$ are infinitesimally Hilbertian. Along the way, we establish uniform lower $1$-capacity bounds of several $1$-dimensional subsets of $\R^2$.

The Eisenbud-Goto conjecture stated that for a homogeneous prime ideal in a polynomial ring over an algebraically closed field, the Castelnuovo-Mumford regularity is bounded above by the multiplicity minus height plus one. In 2017, McCullough and Peeva showed that the regularity of an arbitrary homogeneous prime cannot be bounded by any polynomial function of the multiplicity. For ideals generated by polynomials with at most $c$ terms, we present a bound on the regularity which is less than triple exponential in $c$ and the multiplicity. Furthermore, for toric ideals we get a bound that is smaller than double exponential in terms of multiplicity.

For the coupled three-dimensional double well $H=\tfrac12|p|^2+2\sum_i(q_i^2-1)^2-\sum_{i<j}q_i^2q_j^2$ we exhibit, at the first excited energy level (and also at the ground state), a linear functional on polynomials of degree $\le 6$ that satisfies every constraint of the standard level-3 eigenstate bootstrap, has a strictly positive definite moment matrix, and nevertheless assigns a negative value to a nonnegative polynomial. It is therefore an interior point of the level-3 relaxation that is realized by no quantum state. The construction reflects a fact that holds at every level: the eigenstate rows never reach the top-degree momentum moments. At level 3, where strictly feasible points exist, this means that the corresponding block of the moment matrix carries no information about which energies admit one. Every algebraic ingredient is an exact rational certificate, and the only numerical ingredients (spectral enclosures of an eigenfunction) are carried out in rigorous ball arithmetic.

For a closed orientable surface of genus at least 2, we study the large-scale geometry of the (symmetrized) Lipschitz metric on its space of projective filling geodesic currents. We show that the rank and, hence, asymptotic dimension of the space of filling geodesic currents is infinite. This gives the first example of a natural proper space associated to a closed surface, with a metrically proper isometric action of the mapping class group, that has infinite asymptotic dimension. In contrast, we show that any finite simplex of geodesic currents has rank uniformly bounded above by $8g-9$. We then show a sufficient criterion for its rank and asymptotic dimension to be equal to $8g-9$ and construct examples satisfying it, thus showing the bound is sharp. We further prove that Teichmüller space equipped with the Lipschitz metric has rank equal to $3g-3$. In particular, the space of projective filling geodesic currents and the Teichmüller space are not quasi-isometric.

The canonical Ramsey theorem establishes that for every positive integer $t$, there is a sufficiently large $n$ such that every edge-colouring of a complete graph $K_n$ contains a copy of $K_t$ that is canonically coloured, i.e. monochromatic, rainbow, or lexicographic. In this paper we investigate two variations of this theorem. First, we study canonical Ramsey numbers in the multipartite hypergraph setting. We prove that in every edge-colouring of the 3-uniform complete hypergraph on at least $t^{151t^2}$ vertices, there always exists a canonically coloured copy of $K_{t, t, t}^{(3)}$. This estimate is sharp up to the constant in the exponent. Second, we explore off-diagonal canonical Ramsey numbers. Let $ER(a, b, c)$ denote the minimum number of vertices $n$ required to guarantee a monochromatic $K_a$, a lexicographic $K_b$, or a rainbow $K_c$ in every edge-colouring of the complete graph $K_n$. We establish sharp bounds for these numbers across different parameter regimes. Specifically, we prove that $ER(a, b, c)\geq c^{\Omega(ab)}$ when $c$ is sufficiently large relative to $a$, and that $ER(a, b, c)\leq 2^{O_b(a)}$ when $b\ge 4$ is a fixed constant and $c\le a$. Finally, we analyse the behaviour of the function when avoiding lexicographic triangles (i.e. $b=3$), showing that $ER(a, 3, c)\le (a-1)(c-2)+O(c^6)$.

Dynkin classified the maximal subalgebras of the simple finite-dimensional Lie algebras. The super version of his problem is solved, for the matrix Lie superalgebras, by Shchepochkina, and, for the simple vectorial ones with polynomial coefficients taken with their Weisfeiler gradings, in the companion paper by Leites and Shchepochkina — with three exceptions left open: the three simple finite-dimensional Lie superalgebras $\mathfrak{osp}_a(4|2)$, $\mathfrak{ag}(2)$ and $\mathfrak{ab}(3)$, which are vectorial in more than one way and are not covered by the general argument. Here we settle these three cases for graded subalgebras. For each of the three Lie superalgebras, we list all the $\mathbb{Z}$-gradings to which the method of that paper applies — two for $\mathfrak{osp}_a(4|2)$, five for $\mathfrak{ag}(2)$, eight for $\mathfrak{ab}(3)$ — and describe, for every one of them, all the maximal graded subalgebras, the semi-simple ones explicitly. A simple criterion upgrades most of the answers from “maximal graded\rq\rq\ to “maximal\rq\rq. Every answer is computed over $\mathbb{Q}$. Two by-products: the classical list of five cases into which the graded subalgebras of a vectorial Lie superalgebra split is incomplete for ambients of depth $>1$ — a sixth case has to be added, and it is non-empty precisely here; and the parameter $a$ of the copies of $\mathfrak{osp}_a(4|2)$ sitting inside $\mathfrak{ag}(2)$, resp. $\mathfrak{ab}(3)$, is computed intrinsically.

We study a constrained basis-selection problem for an ordered block matrix $B=[ B_1\mid\cdots\mid B_n ].$ The successive tail matrices determine a nested family of column spaces, and we consider bases of representatives which are compatible with this induced flag. Our main linear-algebraic result shows that, among bases of representatives, compatibility with the entire flag is characterized by the intrinsic block quotas $r_k=w_k-w_{k+1}$, where $w_k$ is the rank of the $k$-th tail matrix. We then show that the same structure arises naturally in finite-horizon linear control. For the reachability matrix $[A^{N-1}G\mid \cdots\mid G]$, requiring every actuator to be used exactly once turns an actuator schedule into a representative system. Preservation of the complete reachability profile is then equivalent to compatibility with the tail-space flag. This correspondence leads to an exact depth-first search with a future-rank pruning criterion. For an explicit structured family and a fixed search ordering, the baseline search visits $3\cdot2^m+m-2$ nodes before the first solution, whereas future-rank pruning reduces this number to $3m+1$. Exact-arithmetic experiments reproduce these formulas for all tested instances.

Inspired by work of Gaster, Greene, and Vlamis on the chromatic number of curve graphs, we prove that every endomorphism of the total Kneser graph is an automorphism. We also show that the automorphism group of the total Kneser graph is the product of a pair of permutation groups.

We consider the interaction-round-a-face version of the nineteen-vertex model and derive a new quantum spin-$1$ chain. In the isotropic limit, the associated spin chain can be seen as three coupled Babujian-Taktahjan spin chains with different boundary twists. We compute the ground state short-distance correlation functions of the isotropic model in the thermodynamic limit by exploiting the shared integrable structure between the models.