Papers

The main computational challenges of solving the Vlasov-Maxwell (VM) system include the high dimensionality of the phase space, nonlinearity, inherent conservation properties, among others. In this paper, we develop a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for the VM system, as a continuation of our previous work (arXiv:2207.00518). The method takes advantage of the tensor friendly structure of the Vlasov equation and employs the low rank hierarchical Tucker decomposition to approximate the Vlasov solution in high dimensions. Hence, the curse of dimensionality can be mitigated. Furthermore, to realize the LoMaC property, the algorithm simultaneously evolves the conservation laws of mass, momentum and energy alongside the Vlasov equation using a high order conservative method with the kinetic flux vector splitting. By a conservative orthogonal projection, the low rank solution is guaranteed to have the same macroscopic observables updated from the conservation laws. A collection of numerical tests on the VM system are presented to demonstrate the efficiency and efficacy of the proposed algorithm.

For linear, time-invariant stochastic delay-differential equations with a single constant delay and both multiplicative and additive noise, this paper derives optimal semi-analytic algebraic equality conditions that can be used to identify second-moment stability boundaries without the use of problem discretization. Successful validation against Monte Carlo simulations and published results for several low-dimensional models clarifies limitations of stability conditions proposed in the literature and demonstrates considerable savings in computational effort relative to discretization-based approaches. In particular, using the theory derived in this paper, second-moment stability boundaries are shown to be computable using parameter continuation techniques applied to discretization-free equality conditions that scale only with the square of the problem dimension. For the case of one-dimensional stochastic delay-differential equations, in particular, the analysis is entirely closed form with a stability condition expressed entirely in terms of elementary functions. These results are enabled by the derivation of an advection-type boundary-value problem with non-local boundary conditions for a three-variable correlation function followed by a reduction to a delay-differential boundary-value problem for a two-variable correlation function. For the former problem, observations regarding the spectral abscissa of the discretization of the corresponding infinitesimal generator, particularly that second-moment stability is lost when a real eigenvalue passes through the origin, motivate identification of second-moment stability boundaries with a loss of uniqueness of stationary solutions to the latter problem.

We introduce $L_{\infty}$-Kuranishi spaces by associating, to each chart, $L_{\infty}[1]$-algebras defined on open neighborhoods of points in the zero locus of the Kuranishi section. We show that these objects collectively form a category into which the category of smooth manifolds naturally embeds. Some notions in \cite{FOOO1} are modified to achieve the desired categorical structures; for instance, the tangent bundle condition for chart embeddings is replaced by a quasi-isomorphism condition for the $L_{\infty}[1]$-structures.

Graphon estimation requires structural assumptions to address its intrinsic non-identifiability. A standard approach is degree-based identifiability, where the degree function is assumed to be strictly monotonic. This assumption is rather restrictive and fails for graphons with constant or non-injective degree function, even when distinct latent positions have different connectivity profiles. In this paper, we introduce rooted motif signatures as higher-order node-level representations for graphons. They extend the degree function by recording, at each latent position, the densities of rooted motifs such as triangles, cycles, paths, and other local subgraph patterns. We study the extent to which these signatures can distinguish latent positions beyond degree information. For generic finite-rank graphons, we prove that suitable rooted motif signatures determine the connectivity profiles of latent positions. We also explain why such a property cannot hold for arbitrary graphons without additional assumptions, since different latent positions may have identical rooted motif signatures. On the statistical side, we define empirical rooted motif signatures from a single observed graph and prove uniform concentration bounds for these estimators. Simulation experiments illustrate that rooted motif signatures can reveal latent structure in settings where degree-based representations are uninformative, including graphons with constant or non-injective degree functions and stochastic block models with equal block degrees.

We prove Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. We also identify the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, giving an extension of Markman's tautological generation theorem from primitive to arbitrary Mukai vectors. The main structure input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S. The coproduct is obtained, by dimensional reduction, from a factorization coproduct for 3d cohomological Hall algebras, and gives rise to a global BPS Lie algebra attached to the stack of coherent sheaves on S. The link between this structure and the applications to chi-independence and tautological generation is provided by Hecke operators on BPS cohomology, which modify one-dimensional sheaves by zero-dimensional quotients. To make this construction work, we prove that there is an identification between the affinized BPS cohomology of the semistable locus and the primitive part of the coproduct on the entire moduli stack

We prove that a complete Berwaldian manifold $\left(M,F\right)$ admitting a nontrivial conformal circle preserving transformation (\cpt for short) must be Riemannian, provided that it has a dense subset on which no flag curvature vanishes (in particular, if $(M,F)$ has positive or negative flag curvature).

Alexander's conjecture states that for every two finite triangulations of the same topological space, if they have a common subdivision, then they have a common stellar subdivision. We generalize the recent result of Adiprasito and Pak, who resolved Alexander's conjecture for finite simplicial complexes, to infinite simplicial complexes.

We study constrained bilinear optimal control for nonlocal age–space structured population equations with renewal boundary conditions and endogenous surveillance feedback. The control acts as a coefficient in a mixed transport–diffusion equation, while a scalar observable generated by the state enters both the interior dynamics and the renewal law. This produces a nonlinear closed-loop control-to-state map and a feedback-dependent adjoint system. Using a characteristic mild formulation rather than a standard Lions–Magenes argument, we establish closed-loop well-posedness and Fréchet differentiability. We then derive the reduced and feedback-corrected adjoint equations. The feedback derivative is identified as a low-rank perturbation \(\ell_{\bar y,\bar u}(p)(t)\chi(a,x)\); in the Volterra-kernel regime, the associated transfer operator is quasinilpotent, yielding an explicit resolvent representation of the adjoint. Finally, we prove first-order optimality conditions and decompose the switching function into reduced and feedback-induced components.

This paper presents a new hybrid MC/deterministic method for solving the one-group steady-state Boltzmann transport equation based on decomposition of solution in macro and micro components. The macro component captures the large-scale structure of the solution. It is represented by angular moments of the high-order transport solution. The $P_1$ approximation is applied to define the macro component. The first two angular moments are obtained as a solution of hybrid low-order moment equations with exact closures. The equation for the micro component is solved using a MC simulation. The hybrid two-level system of equations for macro and micro components is solved by fixed-point iteration scheme. Numerical results are presented to demonstrate variance reduction of stochastic numerical solution and improvement in computational efficiency.

We investigate a coupled system of partial differential equations modeling the interaction between Brinkman flow, biofilm evolution, and nutrient transport in a porous medium. The model captures the mutual influence between the fluid velocity and the biofilm through drag and diffusion coefficients that depend on the local biofilm volume fraction. A hard constraint on the admissible range of the biofilm fraction is incorporated through the subdifferential of an indicator functional, which leads naturally to an evolution variational inequality formulation for the biofilm dynamics. Assuming standard coercivity, ellipticity, and growth conditions on the model coefficients and reaction terms, we prove the global-in-time existence of weak solutions. The analysis relies on a decomposition of the system into three interconnected subproblems: the Brinkman equation with a fixed biofilm profile, the constrained biofilm evolution treated through maximal monotone operator theory, and the nutrient equation viewed as a semilinear parabolic problem. These components are then coupled through a Leray–Schauder type fixed-point argument, with the passage to the limit justified by Aubin–Lions and Simon compactness results. We further establish the nonnegativity of the nutrient concentration under a natural quasi-positivity assumption on the reaction term. Finally, we provide conditional uniqueness results for weak solutions in two spatial dimensions under additional smallness assumptions.

As Coleman famously argued, the apparent breakdown of partition-function factorization in quantum gravity associated with Euclidean wormholes is a red herring, arising from a hidden average over an ensemble of theories. We present a direct analog of Coleman's argument for the apparent breakdown of Hilbert-space factorization associated with spatial wormholes, i.e., Einstein–Rosen bridges. Our main result is the following reconstruction theorem for quantum field theories: unitary QFTs are determined, up to unitary isomorphism, by their closed-manifold partition functions; every reflection-positive partition function arises from a unitary quantum field theory; and the states prepared by manifolds span the space of invariant states under the reconstructed theory's symmetry group. Interpreting the result gravitationally, we conclude that any apparent breakdown of Hilbert-space factorization is a red herring, arising from restricting to an incomplete spectrum of charged states.

Quantum entanglement plays a leading role in modern understanding of physical systems, from quantum phases of matter to quantum gravity. In quantum information theory, one seeks operationally meaningful quantifiers of entanglement, which for large quantum systems are notoriously difficult to evaluate due to the lack of computationally efficient algorithms. In this Letter, we show that for large random induced mixed states the logarithmic negativity, an efficiently computable entanglement measure, generically coincides with the exact entanglement cost under positive-partial-transpose-preserving operations, thereby acquiring a precise operational interpretation. Our results establish logarithmic negativity as an exact characterization of entanglement in generic many-body states and provide a tractable route for quantifying entanglement in complex quantum systems.

Maker-Breaker subgraph games are among the most famous combinatorial games. For $n,q\in\mathbb{N}$ and a fixed subgraph $C$ of the complete graph $K_n$, the two players, called Maker and Breaker, alternately claim edges of $K_n$. Maker claims one unclaimed edge per round and Breaker may claim up to $q$ edges per round. If Maker is able to claim all edges of a copy of $C$, he wins the game. Otherwise Breaker wins. Bednarska and Łuczak (2000) determined in a landmark work the asymptotics of the treshold bias as $\Theta(n^{1/m(C)})$ where $m(C)$ is the 2-density of $C$, analysing random strategies. Since then it has been a major open problem to determine the treshhold bias, if it exists, with corresponding strategies, leading to sharp constants in the $\Theta$-notion. A famous case is the triangle game ($C=C_3$), studied by Chvatal and Erd"os (1978), who showed Maker wins if $q\le \sqrt{2n}$ and Breaker wins if $q\ge2\sqrt{n}$. Glazik and Srivastav (2022) improved this via a potential method, showing Breaker wins already for $q\ge\sqrt{8/3}\sqrt{n}$. Spencer (2019) conjectured generalizability to arbitrary subgraphs $C$. We confirm this conjecture, presenting a general winning strategy for Breaker if the potential function fullfils conditions depending on $C$. With this result we give the first constructive (polynomial-time) strategies for Breaker in the $k$-cycle Maker-Breaker game for arbitrary, but fixed $k \geq 4$: Breaker wins if $q>\sqrt[k-1]{(k-1)\big(\frac{2(k-1)}{k}\big)^{k-2}n^{k-2}}$. By Bednarska and Łuczak (2000) our bound is asymptotically optimal. However, our constants are better than those arising from their random strategies. More recently, Sowa and Srivastav (2025) gave the first constructive Maker strategy for $C_4$. Our work may motivate study of Maker strategies for $C_k, k \ge 5$, narrowing the gap towards the Breaker bounds presented.

We consider a general bond percolation on an infinite locally finite tree, where the edge retention probabilities $p_e$ are replaced by $\min\{1,q_{|e|}p_e\}$, where $\{q_n\}_{n\ge 1}$ is a sequence of positive perturbation factors and $|e|$ denotes the distance between the edge $e$ and the root. If the original percolation model admits infinite clusters, it is of interest to investigate under which perturbations $0<q_n\le 1$ this connectivity property is preserved. Conversely, if the original percolation does not admit infinite clusters, we are led to study the stability of such a property under perturbations satisfying $q_n\ge 1$. In both cases, under minimal assumptions on the original model, we show that the percolative behaviour is stable against certain quantitative non-trivial perturbations. We also discuss an application of our results to the Erdős similarity conjecture for Cantor sets.

The canonical formulation of physical theories with irregular nomic structure is as constrained Hamiltonian theories within which ill-posedness of the equations of motion is connected to a pernicious form of surplus representational capacity. Such theories can be converted into theories with regular nomic structure and a well-posed initial value problem via the process of symplectic reduction. We analyse, synthesise, and contrast different approaches to the presentation and analysis of constrained Hamiltonian theories, drawing upon recent work on formalisation of nomic structure on model spaces (Gryb and Thébault 2024) and comparisons of theoretical structure and representational capacity via category theory (Bradley and Weatherall 2020; Bradley 2025b). We suggest that the case of irregular nomic structure is most naturally suited to a category theoretic presentation in which state spaces are arrows and symplectic reduction is arrow composition (Landsman 2005). Under this approach one obtains the natural results that theories with isomorphic state spaces are equivalent and theories whose reduced state spaces are isomorphic are equivalent at the level of the regular representations of their nomic structure. This analysis provides a suitable foundation for the case of quantization of theories with irregular nomic structure, which will be in a companion paper.

Suppose we want to assign a certainty equivalent–one number–to a multivariate risk. Which such assignments are law-invariant, monotone with respect to vector stochastic dominance, and invariant to independent background risk? I show that every such certainty equivalent is a positive mixture of scalar entropic certainty equivalents applied to positive projections of the vector risk. The same representation yields a robust-order characterization: unanimity across such certainty equivalents is equivalent, up to closure, to dominance after adding independent multidimensional background risk. In a social-welfare specialization, the corresponding shadow valuations are welfare weights.

We consider closed contact manifolds $(M,\xi)$ with periodic positive equivariant symplectic homology. This is a very large class of contact manifolds and, to the best of our knowledge, includes all currently known examples admitting Reeb flows with finitely many closed orbits for which this homology is well defined. Under weak and homologically natural index assumptions on a non-degenerate contact form $\alpha$ on $M$, we establish a sharp lower bound $r_M$ for the number of simple closed Reeb orbits of $\alpha$. The quantity $r_M$ is completely defined in terms of the positive equivariant symplectic homology of $M$. Moreover, we show that this bound is attained if and only if $\alpha$ is lacunary, i.e., the Conley-Zehnder indices of all closed orbits have the same parity. Consequently, the invariant $r_M$ admits a clean dynamical characterization: whenever $M$ admits a non-degenerate lacunary contact form, $r_M$ equals the number of its simple closed Reeb orbits and is therefore independent of the choice of such a form. In particular, in the lacunary case the number of such orbits is a contact invariant completely determined by Floer theory. We compute $r_M$ for a broad class of examples, including several prequantizations of symplectic orbifolds, and show that in this case $r_M=\dim H_*(M/S^1;\mathbb{Q})$, thereby giving a topological characterization of the invariant. Motivated by these results, we conjecture that any contact form with finitely many closed Reeb orbits is necessarily non-degenerate and lacunary, and that the underlying contact manifold is a prequantization of this type.

The optimal control of three-phase permanent-magnet synchronous motors (PMSMs) is challenging due to their nonlinearity and the discrete nature of the control set. Existing approaches either rely on mixed-integer trajectory optimization or require computationally intensive value-iteration procedures. This paper proposes a Linear Matrix Inequality (LMI)-based method for approximating the infinite-horizon value function using a quadratic parameterization and iterated Bellman inequalities, yielding a tractable convex program. The computed function can be obtained efficiently offline and used online as a tail cost in a horizon-one optimal control law. Simulation results show that the proposed approach achieves a favorable trade-off between switching effort and current ripple, with performance comparable to that of finite-control-set MPC but with a significantly lower computational cost.

In this paper, we prove a sign phenomenon first observed by Andrews for certain $q$-series from Ramanujan's Lost Notebook. For three of the series considered by Andrews, namely $v_2(q)$, $v_3(q)$, and $v_4(q)$, we show that the coefficients are alternating in sign, with only a density-zero set of exceptions. Our approach yields precise asymptotic formulas for the coefficients via an adapted circle method, inspired by the work of Folsom-Males-Rolen-Storzer on the $q$-series $v_1(q)$, revealing an interplay between exponential growth and oscillatory behaviour. This interaction produces a dominant alternating sign factor, which governs the sign regularity observed numerically by Andrews. More broadly, we establish the same sign behaviour for explicit infinite families of $q$-hypergeometric series encompassing these examples, and show that it arises systematically from oscillatory asymptotics of these $q$-series near roots of unity. We introduce an additional family whose coefficients appear to exhibit similar sign regularity, suggesting that this phenomenon is widespread and may point towards a deeper underlying theory.

Classical Block Withholding (BWH) attacks have been extensively studied in block-dependent reward schemes, where pool members are compensated upon a block discovery within the pool. However, most contemporary mining pools operate under share-based schemes, wherein participants are paid immediately upon submission of valid shares. In this paper, we analyze BWH under Pay-Per-Share (PPS) for Nakamoto-style blockchains and prove that these mechanisms are not incentive compatible, contrary to conclusions drawn under fixed difficulty in prior literature. Under PPS, the optimal strategy for a BWH attacker is the All-out Attack (AoA): the adversary allocates its entire hashpower toward the victim pool, submitting only partial Proof-of-Work shares (pPoW) while withholding all valid blocks, i.e., full Proof-of-Work (fPoW). Prior to the first difficulty adjustment, the adversary incurs negligible loss from withheld fPoWs. After the adjustment reduces block difficulty, the adversary either generates more pPoWs per unit time when the pPoW difficulty is reduced accordingly or, if the pPoW difficulty is held fixed, earns a higher reward per share. In both cases, it achieves a post-adjustment reward rate of $\frac{\alpha}{1-\alpha}$ per target epoch, compared with the honest baseline rate of $\alpha$. Remarkably, this gain matches the theoretical upper bound achieved by optimal selfish mining in Nakamoto consensus under perfect network influence. The results further indicate that BWH is substantially more profitable under PPS than under mainstream block-dependent payout schemes, even when compared with advanced BWH variants. Honest miners benefit at the same rate as the adversary per unit hashpower, while the victim pool operator bears all losses, paying out-of-pocket for pPoW submissions without receiving fPoW compensation in return.

The greedy Tamari poset, inspired by the well-studied Tamari lattice, was recently defined by Dermenjian in the more general setting of greedy $\nu$-Tamari posets. Bousquet-Mélou and Chapoton counted intervals of the greedy $m$-Tamari poset in 2024 by solving a functional equation, and found that they are equi-enumerous to planar $(m+1)$-constellations. In this work, we give a combinatorial proof of this fact for the case $m = 1$, which also gives the refined enumeration conjectured by Bousquet-Mélou and Chapoton. This is done by establishing a recursive decomposition of greedy Tamari intervals isomorphic to that of bipartite planar maps. We also propose a more general and refined conjecture for the case of general $m$.

This paper studies real-time robust optimal control for uncertain nonlinear systems, where linear time-varying (LTV) approximations make planning tractable but require sound linearization error bounds (LEBs) to guarantee robust constraint satisfaction. We develop tight, differentiable, GPU-parallel LEBs for LTV approximations of nonlinear and neural network (NN) dynamics. For analytic dynamics, we introduce path-based Hessian bounds that are tighter than standard interval methods. For NN dynamics, we derive certified LEBs using NN verifier-generated affine relaxations and local Jacobian corrections. We adapt a GPU-parallel system-level synthesis LTV-based robust control solver to be compatible with these LEBs by extending it to handle right-invertible disturbance matrices and non-zero-centered disturbance sets for tight zonotopic uncertainty propagation. Our method, GPUSLS-LEO, enables online optimization of robust feedback policies that account for linearization error, producing tight, formally verified reachable tubes. On complex nonlinear and NN dynamics up to 168 state dimensions, our method can compute robust control policies on the GPU at rates up to 67 Hz, reducing solve times and conservativeness relative to baselines while preserving formal guarantees and real-time performance.

Towards developing the tools of geometric group theory for non-locally compact topological groups, we give one of the first complete classifications of a family of such groups up to coarse equivalence, and when possible, up to quasi-isometry. In a previous paper, we placed the homeomorphism groups of countable Stone spaces into three classes: coarsely bounded, unbounded yet generated by a coarsely bounded set, and unbounded but not generated by any coarsely bounded set. Now we show that these are the coarse equivalence classes: Any two groups within one of these classes are in fact coarsely equivalent. Furthermore, we show that groups in the second class are quasi-isometric to the Hamming cube, the space comprising infinite binary sequences with finitely many nonzero entries equipped with the Hamming distance, and that groups in the third class are coarsely equivalent to the set of leaves of the regular one-ended tree of countably infinite valence. As part of the proof, we show that infinite Hamming graphs over finite alphabets are all bi-Lipschitz equivalent and prove a coarse geometric classification result for topological groups admitting exhaustions by proper, open, coarsely bounded subgroups.

We present a Hopf algebraic generalization of the Abe-Kanno theorem on a correspondence between subgroups of an algebraic group and invariant subfields of the field of rational functions. It applies to residually finite-dimensional Hopf algebras admitting an artinian classical quotient ring and is used in the paper to derive some general properties of such Hopf algebras.

We establish sharp lower bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r^{0.499}$, and $0 \leq q \leq 1$ is real. These match the better than squareroot cancellation upper bounds obtained in previous work of the author. We prove the same sharp lower bounds for the moments $\frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{it}|^{2q} dt$ of zeta sums, and more generally for moments of character sums $\sum_{n \leq x} h(n) \chi(n)$ with suitably bounded multiplicative twist $h(n)$. The proofs are based on a comparison of the sizes of $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} (\sum_{n \leq x} \chi(n)) \overline{I(\chi)}$, $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |I(\chi)|^2$ and $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |I(\chi)|^4$, where $I(\chi)$ is a certain “barrier adjusted” Perron integral inspired by the analogous results for random multiplicative functions. In a companion paper, we extend these arguments to the full interesting range $x \leq 0.99r$ for the unweighted character sum moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$. This leads to a positive proportion non-vanishing result for Dirichlet theta functions $\theta(1,\chi)$.

In this work, we introduce a classification scheme for topological phases of matter based on the topology of the space of pure states of a model $C^*$-algebra. Under it, topological phases are described by homotopy classes of sections of certain fiber bundles of (pure) states. Applying this classification procedure on states of the Weyl $C^*$-algebra that are invariant under translations by a lattice, we recover the $K$-theoretic classification of gapped spectral projectors for topological insulators of types A and AI, thus essentially generalizing this notion.

V. Drinfeld and E. Lau introduced a “decompletion” of the ring of $p$-typical Witt vectors, following earlier work of T. Zink. The goal of this paper is to offer an exposition of this construction, which we call the sheared Witt vectors, on the category of rings $R$ whose reduction is a perfect $\mathbb{F}_p$-algebra.

We bound the dimension of the Prym-Brill-Noether variety for an open subset of the moduli space of étale double covers of k-elliptic curves. We also obtain new bounds on the dimension of the Prym-Brill-Noether variety for general étale double covers of k-gonal curves, disproving a conjecture of Creech, Len, Ritter, and Wu, and provide a new conjecture for its dimension. To do this, we completely describe the Prym-Brill-Noether variety of a double cover of a certain tropical curve known as the loop of loops. We use the combinatorics of Coxeter groups to establish several topological properties of these tropical Prym-Brill-Noether varieties, and prove a lifting result when the edge lengths are generic.

Let $F(N)$ denote the largest cardinality of a Sidon subset of $\{0,1,\ldots,N-1\}$. We prove \[ F(N)\le N^{1/2}+\gamma_0N^{1/4}+O(1), \qquad \gamma_0=0.94349\ldots<0.9435. \] This improves the previously published coefficient $0.98183$. Our argument develops a vector-valued smoothing method that combines several discrete smoothing kernels, each accompanied by boundary weights that compensate for endpoint effects, so that their weighted combination satisfies the required finite covering inequalities. We also show that averaging systems that satisfy these inequalities individually cannot improve upon the best constituent. Numerical optimization is used to find an eight-component candidate system, which is then certified by exact rational arithmetic.

We introduce the notion of one-sided median porosity for subsets $E$ of $\mathbb{R}$. We prove that this condition is necessary and sufficient for the distance weight $d_E^{-\alpha}$ to belong to a one-sided Muckenhoupt $A_p$ class for some $\alpha>0$ and $1<p<\infty$. As part of the proof, we obtain new characterizations of one-sided $A_p$ weights and one-sided $\mathrm{BMO}$ functions, in terms of medians. It was recently shown that $d_E^{-\alpha}$ is a one-sided Muckenhoupt $A_1$ weight for some $\alpha>0$ if and only if $E$ is one-sided weakly porous. In this paper, we find the precise range of exponents $\alpha>0$ such that $d_E^{-\alpha}$ belongs to a one-sided $A_p$ class, both for $p=1$ and for $1<p<\infty$. In addition, we show that $E$ is median porous if and only if it is both left and right median porous, and we give an example of a one-sided median porous set which is neither median porous nor one-sided weakly porous.