Papers

We generalize Zeckendorf's theorem to second-order linear recurrences of the form $G_{k+2} = gG_{k+1} + hG_{k}$ with arbitrary, coprime initial values $(G_1, G_2)$. We analyze the asymptotic behavior of the count $\#R_G(X)$ of positive integers less than or equal to $X$ that have an expansion under the standard rule of expansions $\mathcal{E}$ associated with the recurrence. Additionally, we provide sufficient conditions under which $G$ has the unique expansion property under the rule of expansions. Finally, we completely characterize the $g$-golden ratio recurrence family when $G_1 = 1$ or $G_2 = 1$. Our approach directly investigates the algebraic structure of the expansions, simplifying and completing previous partial results.

This short article resolves a question about Arthur parameters highlighted in previous work of Balodis, Cunningham, Fiori and Zhang, in the case of $\mathrm{GL}_n/F$ where $F/\mathbb{Q}_p$ is a $p$-adic field. Namely, we prove that every Arthur parameter $\psi$ of $\mathrm{GL}_n/F$, can be factored into Arthur parameters $\psi\cong \psi_1\oplus\cdots \oplus\psi_r$ such that the orbit $C_\psi=C_{\psi_1}\times\cdots\times C_{\psi_r}\subseteq V_{\lambda_{\psi_1}}\times \cdots \times V_{\lambda_{\psi_r}}$ in the associated Vogan variety has smooth closure if and only if each $\bar{C}_{\psi_i}$ is smooth for $i=1, \ldots, r$, and that $C_{\psi_i}$ has smooth closure if and only if it is open or closed.

In secure distributed matrix multiplication, a user wishes to compute the product of two matrices with the assistance of servers, in such a way that any $T$ colluding servers learn nothing about either matrix. Degree tables are a combinatorial tool for constructing polynomial codes for this problem and underlie several state-of-the-art schemes, including $\mathsf{GASP}_r$, $\mathsf{GASP}_{r,s}$, and $\mathsf{DOG}_{r,s}$. We introduce a periodic-gap framework for constructing degree tables that contains these three families as special cases and leads to two new constructions, $\mathsf{SHIFT}_{r,s}$ and $\mathsf{COVER}_r$. We determine their exact recovery thresholds and show that, in many cases, they outperform the current state of the art. We also prove a new lower bound on the recovery threshold of ordinary integer degree tables. In the balanced case, in which the partitioning parameters and the security parameter are equal, we sharpen this bound to match $\mathsf{SHIFT}_{r,s}$ up to lower-order terms, showing that it is asymptotically optimal among degree-table constructions.

We establish a uniform product-difference theorem for dense subsets of $\mathbb Z^2$, which gives an affirmative answer to Problem~2 of Fish and, as consequences, to both parts of his Problem~1. More precisely, we prove that for every $\delta>0$ there exists an integer $K(\delta)\geq 1$ such that every set $E\subseteq\mathbb Z^2$ with upper Banach density $d^\star(E)\geq\delta$ satisfies \[ K(\delta)\mathbb Z \subseteq \{ab:(a,b)\in E-E\}. \] As consequences, we obtain affirmative answers to both parts of Fish's Problem~1: for positive-density sets $E_1,E_2\subseteq\mathbb Z$ and $E\subseteq\mathbb Z^2$, respectively, the sets \[ (E_1-E_1)^2-(E_2-E_2)^2 \quad\text{and}\quad \{x^2-y^2:(x,y)\in E-E\} \] contain nontrivial ideals of $\mathbb Z$, with generators depending only on the corresponding density thresholds. In particular, the latter result also settles a conjecture of Davies concerning differences of the indefinite quadratic form $x^2-y^2$ in dense subsets of $\mathbb Z^2$.

Let $s^+(G)$ denote the sum of the squares of the positive adjacency eigenvalues of a graph $G$. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan, proved by Liu, Tang, and Zhang, gives a lower bound of $n-1$ for any connected graph of order $n$. We strengthen this bound to $s^+(G)\ge n$ for every connected graph $G$ of order $n$ with minimum degree at least two, unless $G$ is a cycle.

Let \(\Lambda\) be a finite-dimensional Koszul algebra. Using a description of the linear part of minimal projective resolutions in terms of Koszul duality, we prove that \(\Lambda\) is absolutely Koszul if and only if its Koszul dual \(\Lambda^{!}\) is graded left co-coherent. We further show that an absolutely Koszul algebra of finite global linearity defect has finite graded finitistic dimension. Finally, we refine Koszul duality for finite-dimensional absolutely Koszul algebras. As a further application, we answer a question of Green et al.~\cite{12} by means of the Koszul dual algebra.

We prove a Frobenius-dilation inequality for Hilbert functions of ideals generated by powers of fixed homogeneous forms in characteristic zero. Via Emsalem–Iarrobino inverse system duality, it yields quantitative fat-point postulation estimates and a new proof of the recent theorem of Ha–Sivakumar resolving Demailly's conjecture for arbitrary finite point sets. A fixed-characteristic version controls tangent spaces to varieties of powers and, together with plane interpolation and secant-theoretic results, proves complete nondefectivity of $V^k_{2,d}$ for all $k\ge3$ and $d\ge2$; in particular, it settles the ternary Waring problem for generic $k$-rank conjectured by Lundqvist–Oneto–Reznick–Shapiro (suggested by Ottaviani). We also show that, for a prime power exponent $q$, any failure of Nicklasson's conjecture in three variables is confined to $2q-2$ consecutive degrees.

Hyperbolic geometry exhibits geometric phenomena, such as fast area growth, that are difficult to visualize faithfully in Euclidean space, and which standard models like the Poincaré disk can obscure. To bring hyperbolic geometry to life, we embed hyperbolic surfaces in Euclidean space by discretizing the surfaces into meshes, and minimizing a distortion energy so that the edge lengths in the embeddings match those in the hyperbolic plane. The resulting surfaces buckle and ruffle to accommodate the extra area, making visible what flat models hide. We present exemplary illustrations, such as embedded disks, equidistant strips, diverging geodesics, and also artistic organic-like renders. We discuss our use of these models, as renders and 3D prints, in research talks, public engagement, outreach, and education.

We establish the uniform doubling property for $G$-invariant metrics on a wide class of compact Riemannian homogeneous spaces $(G/K,g)$, describing explicitly the volume growth of the metric balls $B_g(p,r)$. This property was previously known only for specific cases, including abelian Lie groups, the Lie group $SU(2)$ and quotients of $SU(2)\times \mathbb R^n$. Building on the approach of Eldredge, Gordina and Saloff-Coste for $SU(2)$, we refine and develop geometric and Lie-theoretic tools that allow us to replace the explicit identities of the Milnor basis in $SU(2)$ by a general structural assumption on the metric eigenspace decomposition. In particular, we show that the uniform doubling of $(G/K,g)$ is a consequence of the cyclic bracket condition $[\mathfrak{m}_i,\mathfrak{m}_j]=\mathfrak{m}_k$, for $i,j,k$ pairwise distinct, a property that naturally generalizes the Milnor structure of $SU(2)$. We apply our results to $\mathbb Z_2\times \mathbb Z_2$-symmetric spaces, establishing the uniform doubling property for the complete family of $G$-invariant metrics on several classes of generalized Wallach spaces. For compact homogeneous spaces, we also derive a global Poincaré inequality which holds uniformly for the spaces $(G/K,g)$ under consideration, with a constant controlled by the volume doubling constant of the space.

We study the initial boundary value problem (IBVP) for the higher order Korteweg–de Vries type equation \[\partial_tu+(-1)^{j+1}\partial_x^{2j+1}u+\frac12\partial_x(u^2)=0,\qquad j\in\mathbb{N},\] on the right half line, subject to the Robin boundary conditions \[(\partial_x+\gamma)\partial_x^{\ell-1}u(t,0)=\varphi_\ell(t),\qquad 1\le\ell\le j,\] where $\gamma\in\mathbb{R}$ is common to all boundary conditions. We prove local well posedness for \[u_0\in H^s(\mathbb{R}^+),\qquad \varphi_\ell\in H^{\frac{s+j-\ell}{2j+1}}(0,T),\qquad -j+\frac14<s<\frac32.\] In particular, the regularity range extends the low regularity theory for the higher order Dirichlet IBVP to Robin boundary conditions. The main ingredient is an explicit unified transform representation for the higher order IBVP with Robin boundary conditions. The Robin hierarchy couples adjacent boundary traces and leads, in each spectral sector, to a nontrivial system for the unknown boundary transforms. Rather than computing the inverse of this system, we resolve precisely the linear combination required by the unified transform representation through Lagrange interpolation at the rotated spectral points. This yields an exact factorization in which the dependence on the Robin parameter is concentrated in the single factor $(k-i\gamma)^{-1}$. Consequently, the only possible Robin pole is $k=i\gamma$, which contributes the residue mode \[e^{-\gamma x+\gamma^{2j+1}t}\] exactly when $j$ is odd and $\gamma>0$. Combining this representation with linear estimates in modified Fourier restriction spaces and higher order KdV bilinear estimates yields the low regularity well posedness result. The representation also recovers the classical KdV formulas with Robin and Neumann boundary data when $j=1$ and is formally consistent with the higher order Dirichlet representation under the reciprocal Robin limit.

For every nonempty Hassett divisor $\mathcal C_d$ parameterizing special cubic fourfolds, we consider the quaternion class $\beta_d=(d/2,-3)$ in the two-torsion Brauer group. We prove that if a Hodge-general member of ${C}_d$ is rational then $\beta_d=0$ – or, equivalently, that Huybrechts' twisted-K3 condition $(**')$ holds. Consequently, a very general member of $\mathcal C_d$ is irrational if $\beta_d\ne0$. Thus, a very general cubic fourfold containing a smooth cubic scroll or a Veronese surface is irrational, and hence so is a very general Küchle fourfold of type $(\mathrm{c7})$. We also obtain an analogous obstruction for Hodge–special Gushel–Mukai fourfolds: a very general member in the discriminant–$d$ locus can be rational only if $d$ is a sum of two squares. Consequently, a very general Gushel–Mukai fourfold containing a cubic scroll is irrational.

Without a constraint qualification, second-order cone programs can have a duality gap or an unattained dual value. For every such program with arbitrary real coefficients, we construct an exact dual using only affine equations and memberships in products of Lorentz cones. Every feasible dual point gives a valid bound. Whenever the primal is feasible with finite value, the dual attains that value. This answers a question raised by Pólik and Terlaky and reiterated in subsequent work: whether an exact dual for second-order cone programming can use only Lorentz-cone constraints. The same construction produces an affine second-order cone system that is feasible exactly when a given affine second-order cone system is infeasible, including in the weakly infeasible case. Both formulations have polynomial size and are obtained uniformly from the input coefficients using only fixed rational constants. These are formulation-size results over exact real data; they do not imply polynomial-time solvability or polynomial bounds on the bit length of rational certificates.

We prove that if H and K are subgroups of a free group F of arbitrary rank and $H'=K'\ne 1$, then $H=K$. As a consequence, if R is a noncyclic subgroup of a free group F and $R'$ is normal in F, then R is normal in F. This gives a positive answer to a problem of Shpilrain, originally posed as Problem 11.124 in the Kourovka Notebook and later appearing as Problem F15 in Kapovich-Myasnikov-Shpilrain.

For a random polynomial, the number of real zeros $N_{\mathbb R}$ is a highly nonlinear function of its coefficients, and its statistical properties have been studied extensively. A natural question is whether $N_{\mathbb R}$ satisfies a central limit theorem. For various ensembles with iid standard Gaussian coefficients, such central limit theorems have been established; see, for instance, arXiv:1911.12182, arXiv:1801.06331, arXiv:1401.5745; Azais and Leon, Electron. J. Probab. 18 (2013), no. 68; arXiv:1504.05355, arXiv:2111.09015, arXiv:1707.09276, arXiv:1005.4113. These results rely on a rich range of tools, including Kac-Rice formulas, moment methods, and Wiener chaos decompositions. In the non-Gaussian setting, however, many of these tools are unavailable. To the best of our knowledge, prior central limit theorems beyond the Gaussian setting were limited to Kac-type polynomials, including hyperbolic polynomials; see the works of Maslova (1974), O. Nguyen and Vu (arXiv:1904.04347), and, more recently, Do, N. Nguyen, and O'Rourke (arXiv:2605.26402). In this paper, we prove a central limit theorem for the total number of real zeros of Weyl polynomials whose coefficients are iid copies of a symmetric, mean-zero, variance-one subgaussian random variable $\xi$. This substantially extends one of the main results of Do and Vu (arXiv:1707.09276) to a broad class of non-Gaussian distributions, including the Rademacher distribution. Without the symmetry assumption, we prove central limit theorems for the number of real zeros for positive bulk intervals, as well as for $[0,\infty)$. Our proof combines the uniform one-point anti-concentration estimates from our recent work (arXiv:2511.07735) with the localization of Weyl polynomials around the coefficient index $i\approx x^2$. While our proofs use comparison to compute the variances, the CLT deduction is rather direct.

We study bounded derived categories of semisimple tensor categories and their semisimple module categories. These categories may have infinitely many simple isomorphism classes. We classify bounded monoidal $t$-structures by integer-valued characters of the universal grading group and an integer deviation. We also classify compatible module $t$-structures using stabilizer subgroups of the module components. The corresponding hearts are tensor and module equivalent to the original categories. It follows that a monoidal derived equivalence and a coherent derived module equivalence recover both the tensor category and its module category.

This paper presents a study of the Weber-Schafheitlin integral and its connections to random walk theory. The Weber-Schafheitlin integral, defined as the improper integral of a product of Bessel functions, arises naturally in the evaluation of probability distributions for multidimensional random walks. This work bridges pure special function theory with applied probability.

For a connected regular graph G and a vertex a, we study the algebra generated by the adjacency and degree matrices of G-a and its cyclic module P_a generated by the all-ones vector. Our main theorem determines dim P_a for Cartesian products whose factors have equitable distance partitions at the chosen roots. A normalized logarithmic derivative of the local spectral generating function partitions the factors into boundary classes. We identify the boundary-return space exactly and express dim P_a as a sum of affine ranks on additive spectral fibres. For a distance-regular factor with distinct spectrum \Theta, this gives dim P_a(F^{\square m}) = |m\Theta| - 1. For products of powers of two distinct complete graphs, we evaluate the fibre formula in closed form. We also determine the full punctured algebras of all Hamming graphs: equality with the compressed Terwilliger algebra holds precisely in dimensions at most four for the hypercube and at most two for larger alphabets. For distance-regular graphs, adjacency moments alone determine the intersection array, with an explicit finite reconstruction. Finally, Cartesian stabilizer formulas separate metric loss from orbit splitting; on Doob graphs their distance-graded defect recovers the number of Shrikhande factors.

In this paper, we prove a global-in-time uniform bound for solutions to a thermo-diffusive system in the whole space with bounded nonnegative initial data. The result allows exponential reaction rates and a range of superexponential rates determined by the diffusivities. This extends the recent results of La, Roquejoffre, and Ryzhik \cite{LRR24} and of Leontev and Ryzhik \cite{LR26} beyond their strictly subexponential range.

Smart agriculture platforms usually bundle AI diagnosis, IoT sensing and decision push into a single package, so the green benefit attributable to each component remains unclear and resource-allocation decisions lack quantitative evidence. Building on a previous platform-level Monte Carlo assessment, this paper makes the components explicit and runs two controlled simulation experiments. Experiment 1 follows the chain from AI capability to farmer behavior to agrochemical input reduction, modeling pesticide/fertilizer reduction as avoidable blind-application share times prescription effectiveness times decision-touch coverage times adoption rate, and compares an experienced-extension mode with the AI mode: the probability of reaching 20% pesticide reduction is essentially zero in the extension mode but 20.7% at baseline, up to 49% with diagnosis accuracy 0.95 and adoption 0.85 under AI; the probability of 15% fertilizer reduction rises from near zero to 52.0%. Experiment 2 compares current practice (P0), IoT engineering retrofit (P1), and P1 plus AI irrigation scheduling (P2): median aggregate water saving rises from 7.8% (P0) to 11.0% (P1) and 16.0% (P2), with AI adding 5.0 percentage points beyond engineering; paddy CH4 reduction reaches 30.5% under AI scheduling versus 19.8% under manual operation, and the rice irrigation-methane subsystem carbon intensity declines 27.9%. Sensitivity analyses of both experiments consistently indicate that the primary bottleneck for meeting green targets is farmer adoption rather than algorithm accuracy, and that AI data fusion is robust to soil-moisture sensing errors. This work provides a reproducible simulation framework for component-level green-value evaluation and promotion-strategy optimization of smart agriculture platforms.

New biholomorphic invariants of a germ of a real-analytic set at a singular point (either an RC-singular point or a singular point of a smooth structure) are found: the Bloom-Graham singular type and the singular l-type. Their construction is based on the Bloom-Graham type and l-type constructions, respectively. A series of examples is considered.

We study how much observable finite-dimensional information is sufficient to identify a finite-state generator $\epsilon$-machine. The observable summaries are labeled tables of conditional probabilities given finite histories and contiguous block laws. For stationary ergodic finite-alphabet processes, we first prove that a table with past horizon $L$ and future horizon $R$ is equivalent to the block law of length $L+R$. We then recall the classical linear-realization bound. Two hidden Markov presentations with $r$ and $s$ states agree on the entire process law if they agree on words through length $r+s-1$. Our main result gives a structural synchronization bound adapted to exact finite-state generator $\epsilon$-machines. If their target synchronization radii are $a$ and $b$, and their maximum predictive separation radius is $d$, then equality of one block law of length $a+b+d+1$ implies isomorphism. The proof aligns the states of both machines with words that synchronize the two generators simultaneously. For the full-support binary context subfamily of order $m$ with pairwise distinct parameters, the structural synchronization bound is $2m+2$, whereas the state-count and rank-dimension forms of the classical linear-realization bound are $2^{m+1}-1$. A construction shows that a horizon of order $m$ is necessary. When the order-$m$ context structure is known in advance, the sharp horizon is $m+1$ symbols. Thus the raw state-count and rank-dimension formulas can be exponentially pessimistic on this subfamily. The comparison is between general sufficient bounds, not with an optimal realization horizon.

We study Maxwell-Stefan diffusion with additive friction coefficients $f_{ij}=g_i+g_j$. In mass fractions, the system isolates the constrained pair-friction block; in mole fractions, it is the classical ideal isothermal/isobaric Maxwell-Stefan system at constant total molar concentration. Additivity makes the constrained pair-friction dissipation species-diagonal; conversely, species-diagonality on one interior barycentric constraint space forces a pair-sum representation. At operator level, the positive constrained relaxation operator is a scalar shift of a compression of $G=diag[g_1,\ldots,g_N]$. Its scalar resolvent yields both an explicit constrained inverse and interlacing spectral roots, which form global real-analytic coordinates on the open simplex and whose differentials are left eigen-covectors. In root coordinates the principal part is diagonal, no self-square gradient term occurs, and scalar comparison yields invariant rectangles and separation from the simplex boundary. For regularity we introduce entropy-stabilized one-sided multi-EPD truncations: Euler-Poisson-Darboux entropies cancel mixed quadratic production, while for $N\ge4$ a truncation-weighted mixing-entropy correction supplies transverse coercivity. Caccioppoli and logarithmic estimates, shrinking, and critical mass yield Hölder continuity up to the Neumann boundary. The mixing entropy also symmetrizes the moment system; frozen conormal estimates give spatial Lipschitz bounds. Together with time Hölder control and short-interval maximal regularity, this yields global strong solvability on bounded $C^{2+\alpha}$ domains with $0<\alpha<1$, for each $N\ge2$, $d\ge2$, and $p>d+2$, for all uniformly positive, compatible initial concentrations in the natural trace class. Solutions become classical for positive times and converge exponentially to equilibrium in relative entropy, $L^2$, and $C^1(\bar\Omega)$.

For every finite $P\subset\mathbb{R}^2$ we prove $|\{p\cdot q: p,q\in P\}|\gg |P|^{199/295}$, with an absolute constant and no logarithmic loss, where $199/295 = 2/3+7/885$. This improves the bound $2/3+7/1425$ of Kokkinos (arXiv:2502.12727), which in turn had improved the first superthreshold bound of Hanson, Roche-Newton, and Senger. The main new ingredient is the inequality $|F^{(2)}G^{(2)}/(F^{(2)}G)| \le |AB|(|AF|/|A|)^4(|BG|/|B|)^3$: each radial profile retains its own Petridis growth-minimizing subset, and the product of the two subsets serves as a common base for both growth operators; a prime-power construction shows this is sharp under its hypotheses. The second ingredient is a weighted-median replacement for the dyadic selection in the squeezing argument of Roche-Newton and Wong, upgrading their seven-factor expander to a logarithm-free form; an appendix gives the complete proof from the Solymosi-Zahl incidence theorem. We complement the lower bound with a third-moment structure theorem for near-extremal configurations and a construction showing that the pinned problem admits no linear lower bound: $\max_{p\in P}|p\cdot P| = O(|P|/\sqrt{\log |P|})$ is attainable. The algebraic core, the median lemma, the surface identities, and the pinned construction are formally verified in Lean 4 and included as ancillary files.

We study trend estimation in state-space models in which the trend has a fractional stochastic difference of order $d>0$ and the observation errors form a short-range-dependent stationary process. Using finite-sequence fractional summation and differencing operators, we analyze the penalized least-squares estimator obtained by shrinking the fractional differences of the trend. We derive asymptotic mean squared error (MSE) approximations for all $d>0$ and identify a sharp phase transition at $d=1/2$. When $d>1/2$, the estimator is consistent and its optimally balanced MSE has order $n^{-(2d-1)/(2d)}$. At the boundary $d=1/2$, we obtain a refined finite-sample approximation and show that the MSE decreases at the slower order $\log\log n/\log n$. When $0<d<1/2$, the MSE converges to an explicit positive limit, so consistent recovery of the trend is impossible under the considered scaling. We also describe a practical criterion for choosing the penalty parameter and differencing order, and numerical experiments illustrate the MSE approximations and the behavior of the selection procedure.

We study cyclic orderings of all edges of a complete $k$-uniform hypergraph on $n$ vertices in which the binary incidence sequences of the vertices are cyclic shifts of a common word. The shifts are chosen independently, with no prescribed action on the vertices. For $2\leq k<n$, coprimality $\gcd(n,k)=1$ is known to suffice even when consecutive edges must differ by a single vertex exchange. We recall a short orbit construction and prove the converse for the first two nontrivial uniformities without any adjacency requirement. For $k=2$, an ordering exists exactly when $n=2$ or $n$ is odd; for $k=3$, exactly when $n=3$ or $3\nmid n$. The necessity proofs use reflected convolution identities and pair-intersection counts to constrain the vertex shifts to a torsion coset. For triples, multiplicity-preserving dilation and conditional prime-power capacity bounds complete the argument.

We study homogeneous forms with invariant Jacobian under group actions. For finite-dimensional irreducible complex representations, we show that the canonical decomposition defined by the center of the form is either trivial or gives a system of imprimitivity, which determines the module structure of Jacobian $J(f)$. We apply the theory to symmetric groups and classify homogeneous forms with invariant Jacobian on the natural permutation module.

Let $a(n,k)$ be the number of $1324$-avoiding permutations of length $n$ with $k$ inversions. Linusson and Verkama proved $a(n,k)\le a(n+1,k)$ for $k\le2n-7$. We study the obstruction beyond that line: the residuals $\mathcal R_{\delta,n}$, namely the indecomposable, non-almost-decomposable avoiders at defect $\delta=k-2n+7$. Contracting maximal increasing consecutive runs reduces residuality to a quadratic equation on a finite family of skeletons. It follows that, for every fixed $\delta$, the eventual count has the form $|\mathcal R_{\delta,n}|=A_\delta n^2+B_\delta n+C_\delta$. Our central uniform result determines the quadratic coefficient at every defect: with $P(q)=\prod_{j\ge1}(1-q^j)^{-1}$, $\sum_{\delta\ge0}A_\delta q^\delta=4q^3(1+q)P(q)^2/(1-q)^2$. This is a formula for the leading coefficient of the residual count, not for the full count. The same structural estimates give computer-assisted proofs of $a(n,k)\le a(n+1,k)$ for every $n\ge1$ and $k\le2n+6$, and of regions whose width grows with $n$: for $n\ge2^{16},2^{18},2^{20}$ the defect may be as large as $\lfloor\sqrt n/4\rfloor$, $\lfloor\sqrt n/3\rfloor$, $\lfloor\sqrt n/2\rfloor$, respectively. More generally, every fixed $c<\sqrt2\log(5)/\log(68)$ is admissible for all sufficiently large $n$. The three added fixed defects $11,12,13$ use complete catalogue and rational-sum certificates supplied in the accompanying archival supplement. The leading-coefficient theorem is obtained from a complete finite classification of marked rank-three cores and all-parameter extension lemmas. We also determine the exact rank-three stabilization onset, while keeping it separate from the still unknown onset of the complete residual count. The unrestricted Claesson–Jel'inek–Steingr'imsson conjecture, the full residual polynomials, and the sharp global base-length bound remain open.

Based on the conjectured (and checked for tree-type quivers) quiver Yangian/Coulomb branch algebra correspondence, we give a quiver Yangian interpretation of the monopole formulas for 3D $\mathcal N=4$ good $A$-type quiver gauge theories, also known as the $T_\rho(SU(N))$ theories. Using the algebra action on the $\frac12$-BPS vortices, we reorganize the generators of the truncated shifted quiver Yangian into boundary-adapted generators, and obtain the classical relations in terms of $U(N)$ Casimirs on the Slodowy slice $\mathcal S^{\mathfrak{gl}_N}_\rho$, whose coordinates are given by these boundary-adapted generators. We also show that the boundary-adapted generators and the Casimir relations have the correct fugacities as predicted by the Hall-Littlewood expression of the monopole formula for $T_\rho(SU(N))$, hence reconstructing it as the Hilbert series of the truncated shifted quiver Yangian.

We classify all filtered deformations of polynomial rings in three variables. We use our characterization to answer a special case of a question of Etingof by showing that if $F$ is an algebraically closed field of positive characteristic and $A$ is an $F$-algebra with a filtration by finite dimensional subspaces whose associated graded ring is a polynomial ring in three variables then $A$ satisfies a polynomial identity.

The Lusternik–Schnirelmann category of a closed symplectic manifold admits various estimates. Locally conformally symplectic (LCS) geometry is a generalization of symplectic geometry involving a closed one-form, called the Lee form. In this paper, we introduce Farber's Lusternik–Schnirelmann category for closed one-forms into LCS geometry by applying it to the Lee class. In particular, this provides a new topological approach to LCS structures of the second kind. We then obtain lower bounds for this invariant under LCS blow-ups.