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We study selective refusal editing as a three-way control problem: induce non-refusal on designated edit prompts while preserving benign behavior and harmful refusals outside the edit set. We introduce Residual Paving, a routed residual editing method for frozen instruction-tuned transformers that separates route selectivity, whether to intervene, from residual-edit capacity, what edit to apply. An early-layer router predicts a scalar gate and expert mixture; when active, prompt-conditioned bottleneck residual experts apply later-layer residual updates while leaving the backbone unchanged. This decomposition supports an oracle-routing diagnostic where only the learned scalar gate is replaced with the held-out edit/keep label, leaving the residual editor and frozen backbone fixed. On the primary Gemma-3-4B-IT held-out split, learned Residual Paving reduces edit refusal from 88.6% to 4.0%, with 95.5% benign distribution preservation and 87.3% harmful distribution preservation. Same-protocol one-direction steering controls are much weaker on edit success, leaving edit refusal at 86.8% for Edit-target ActAdd and 78.9% for DIM-style refusal steering. The remaining failure is off-target

Using Postnikov's Le-diagrams, decorated permutations, and Grassmann necklaces, we classify which positroids are sparse paving matroids. This allows us to enumerate sparse paving positroids, making connections to a known sequence involving the golden ratio and to the Lucas numbers.

For a matroid of rank $r$ and a non-negative integer $k$, an element is called $k$-loose if every circuit containing it has size greater than $r-k$. Zaslavsky and the author characterized all binary matroids with a $1$-loose element. In this paper, we establish a sharp linear bound on the size of a binary matroid, in terms of its rank, that contains a $k$-loose element. A matroid is called $k$-paving if all its elements are $k$-loose. Rajpal showed that for a prime power $q$, the rank of a $GF(q)$-matroid that is $k$-paving is bounded. We provide a bound on the rank of $GF(q)$-matroids that are cosimple and have two $k$-loose elements. Consequently, we deduce a bound on the rank of $GF(q)$-matroids that are $k$-paving. Additionally, we provide a bound on the size of binary matroids that are $k$-paving.

We introduce the new problems of quantum packing, quantum covering, and quantum paving. These problems arise naturally when considering an algebra of non-commutative operators that is deeply rooted in quantum physics as well as in Gabor analysis. Quantum packing and quantum covering show similarities with energy minimization and the dual problem of polarization. Quantum paving, in turn, aims to simultaneously optimize both quantum packing and quantum covering. Classical sphere packing and covering hint the optimal configurations for our new problems. We present solutions in certain cases, state several conjectures related to quantum paving and discuss some applications.

We consider the discrete anisotropic Maxwell operator DaH0 on a bounded paving $Ω$ $\subset$ Z3 , where H0 denotes discrete isotropic Maxwell operator and Da a diagonal operator of multiplication containing information about the anisotropy of the medium inside $Ω$. Letting a complex number $λ$ __ = 0 such the Dirichlet-to-Neumann operator $Λ$(Da) associated with the system DaH0 u = $λ$u on $Ω$ admits a unique solution, we show that knowing $Λ$(Da) is sufficient to determine Da by a reconstruction procedure for Da.

Gao and Xie (2021) conjectured that the inverse Kazhdan-Lusztig polynomial of any matroid is log-concave. Although the inverse Kazhdan-Lusztig polynomial may not always have only real roots, we conjecture that the Hadamard product of an inverse Kazhdan-Lusztig polynomial of degree $n$ and $(1+t)^n$ has only real roots. Using interlacing polynomials and multiplier sequences, we confirm this conjecture for paving matroids. This result allows us to confirm the log-concavity conjecture for these matroids by applying Newton's inequalities.

White's conjecture asserts that any two tuples of matroid bases that have the same multi-set union can be transformed from one to another by symmetric exchanges; it also implies that the toric ideals of matroids are generated by the binomials encoding these exchanges. We prove White's conjecture for the class of paving matroids. Our strategy is to generalize the inductive argument using circuit-hyperplane relaxations in the recent work of Han et al. to stressed hyperplane relaxations.

We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of these varieties across any dimension. Additionally, we explain the relationship between polynomials constructed using these different methods. We then compute a comprehensive and finite set of defining equations for matroid varieties associated with specific classes of paving matroids. Finally, we focus on the class of paving matroids of rank $3$, known as point-line configurations, which essentially contain simple matroids of rank $3$. Furthermore, we provide a decomposition for the associated circuit variety of point-line configurations, where all points have a degree less than $3$. Lastly, we present several examples applying our results and compare them with the known cases in the literature.

In this paper, we study positroids and its overlap with two classes of matroids: transversal and paving matroids. We exhibit a new class of fundamental transversal matroids and classify the Le-diagram for rank two transversal positroids. We also establish a combinatorial description for paving positroids in terms of Le-diagrams.

We consider a paving property for a maximal abelian *-subalgebra (MASA) $A$ in a von Neumann algebra $M$, that we call so-paving, involving approximation in the so-topology, rather than in norm (as in classical Kadison-Singer paving). If $A$ is the range of a normal conditional expectation, then so-paving is equivalent to norm paving in the ultrapower inclusion $A^ω\subset M^ω$. We conjecture that any MASA in any von Neumann algebra satisfies so-paving. We use [MSS13] to check this for all MASAs in $\mathcal B(\ell^2 \mathbb N)$, all Cartan subalgebras in amenable von Neumann algebras and in group measure space II$_1$ factors arising from profinite actions. By [P13], the conjecture also holds true for singular MASAs in II$_1$ factors, and we obtain here an improved paving size $C\varepsilon^{-2}$, which we show to be sharp.

We prove that every paving matroid that is an excluded minor of interval positroids can be reduced to one of three fundamental families of excluded minors of interval positroids by relaxing dependent hyperplanes. Using this result, we classify all non-positroid excluded minors of interval positroids that are paving matroids. Additionally, we provide a criterion that characterizes all excluded minors of interval positroids that are paving positroids.

The Chow class of the closure of the torus orbit of a point in a Grassmannian only depends on the matroid associated to the point. The Chow class can be extended to a matroid invariant of arbitrary matroids. We call the coefficients appearing in the expansion of the Chow class in the Schubert basis the Schubert coefficients of the matroid. These Schubert coefficients are conjectured by Berget and Fink to be non-negative. We compute the Schubert coefficients of a disconnected matroid in terms of the Schubert coefficients of its connected components. And we compute the Schubert coefficients for all sparse paving matroids, and confirm their non-negativity.

Panhandle matroids are a specific family of lattice-path matroids corresponding to panhandle-shaped Ferrers diagrams. Their matroid polytopes are the subpolytopes carved from a hypersimplex to form matroid polytopes of paving matroids. It has been an active area of research to determine which families of matroid polytopes are Ehrhart positive. We prove Ehrhart positivity for panhandle matroid polytopes, thus confirming a conjecture of Hanely, Martin, McGinnis, Miyata, Nasr, Vindas-Meléndez, and Yin (2023). Another standing conjecture posed by Ferroni (2022) asserts that the coefficients of the Ehrhart polynomial of a connected matroid are bounded above by those of the corresponding uniform matroid. We prove Ferroni's conjecture for paving matroids -- a class conjectured to asymptotically contain all matroids. These results follow from purely enumerative statements, the main one being conjectured by Hanely et. al concerning the enumeration of a certain class of ordered chain forests.

In this paper, we pave the way to six-generation (6G) by investigating the outage probability (OP) of fluid antenna system (FAS)-active reconfigurable intelligent surface (ARIS) communication systems. We consider a FAS-ARIS setup consisting of a base station (BS) with a single fixed-position antenna and a receiver equipped with a fluid antenna (FA). Utilizing the block-correlation model, we derive a closed-form expression for the OP. Our analysis, supported by numerical results, confirms the accuracy and effectiveness of the derivation. Furthermore, the results demonstrate that the FAS-ARIS system significantly outperforms other configurations in terms of OP, highlighting its potential to enhance communication performance and reliability in future 6G networks.

A matroid M is cyclically orderable if there is a cyclic permutation of the elements of M such that any r consecutive elements form a basis in M. An old conjecture of Kajitani, Miyano, and Ueno states that a matroid M is cyclically orderable if and only if for all nonempty subsets X in E(M), |X|/r(M) is less than or equal to |E(M)|/r(M). In this paper, we verify this conjecture for all paving matroids.

We show that the base polytope $P_M$ of any paving matroid $M$ can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of $P_M$, starting with Katzman's formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni's work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr, and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show t

This article proves, in the case of split groups over arbitrary fields, that all fibers of convolution morphisms attached to parahoric affine flag varieties are paved by products of affine lines and affine lines minus a point. This applies in particular to the affine Grassmannian and to the convolution morphisms in the context of the geometric Satake correspondence. The second part of the article extends these results over $\mathbb Z$. Those in turn relate to the recent work of Cass-van den Hove-Scholbach on the geometric Satake equivalence for integral motives, and provide some alternative proofs for some of their results.

In this work we present an algorithm to construct sparse-paving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove in a simple way an asymptotic relation between the class of Sparse-paving matroids and the whole class of matroids. Additionally we introduce a matrix based method which render an explicit partition of the $r$-subsets of $S$, $\binom{S}{r}=\sqcup_{i=1}^{γ}\mathcal{U}_{i}$ such that each $\mathcal{U}_{i}$ defines a sparse-paving matroid of rank $r$.