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This paper introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional on compact Riemannian manifolds. Instead of considering arbitrary symmetric variations of the metric, we restrict the variational problem to an infinite-dimensional subspace determined by the Chen-Nagano gauge constraint. We derive the corresponding restricted Euler-Lagrange equations and obtain a novel structural characterization of critical metrics. The resulting criticality condition is expressed by the equation $E_g = B_g^{*}(θ) + c\,g$ which may be regarded as a restricted counterpart to the classical Einstein equation. Furthermore, we demonstrate that this variational framework naturally leads to generalized Ricci almost soliton structures and, in the gradient case, to gradient Ricci almost solitons. Several global rigidity consequences of this restricted principle are also established.

The Fourier restriction problem asks when it is meaningful to restrict the Fourier transform of a function to a given set. Many of the key examples are smooth co-dimension 1 manifolds, although there is increasing interest in fractal sets. Here we propose a natural intermediary problem where one considers the fractal surface generated by the graph of the additive Brownian sheet in $\mathbb{R}^k$. We obtain the first non-trivial estimates in this direction, giving both a sufficient condition on the range of $q\in[1,2]$ for the Fourier transform to be $L^{q}(\mathbb{R}^{k+1})\to L^2(G(W))$ bounded and a necessary condition for it to be $L^{q}(\mathbb{R}^{k+1})\to L^p(G(W))$ bounded. The sufficient condition is obtained via the Fourier spectrum, which is a family of dimensions that interpolate between the Fourier and Hausdorff dimensions. Our main technical result, which is of interest in its own right, gives a precise formula for the Fourier spectrum of the natural measure on the graph of the additive Brownian sheet, and we apply this result to the Fourier restriction problem. Our restriction estimate is stronger than the estimate obtained from the well-known Stein--Tomas restriction

Large language models rely on web-scraped text for training; concurrently, content creators are increasingly blocking AI crawlers to retain control over their data. We analyze crawler restrictions across the top one million most-visited websites since 2023 and examine their potential downstream effects on training data composition. Our analysis reveals growing restrictions, with blocking patterns varying by website popularity and content type. A quarter of the top thousand websites restrict AI crawlers, decreasing to one-tenth across the broader top million. Content type matters significantly: 34.2% of news outlets disallow OpenAI's GPTBot, rising to 55% for outlets with high factual reporting. Additionally, outlets with neutral political positions impose the strongest restrictions (58%), whereas hyperpartisan websites and those with low factual reporting impose fewer restrictions -only 4.1% of right-leaning outlets block access to OpenAI. Our findings suggest that heterogeneous blocking patterns may skew training datasets toward low-quality or polarized content, potentially affecting the capabilities of models served by prominent AI-as-a-Service providers.

The length $\mathsf{is}(π)$ of a longest increasing subsequence in a permutation $π$ has been extensively studied. An increasing subsequence is one that has no descents. We study generalizations of this statistic by finding longest subsequences with other descent restrictions. We first consider the statistic which encodes the longest length of a subsequence with a given number of descents. We then generalize this to restrict the descent set of the subsequence. Extending the classical result for $\mathsf{is}(π)$, we show how these statistics can be obtained using the RSK correspondence and the Schützenberger involution. In particular, these statistics only depend on the recording tableau of the permutation.

The Fourier and Fourier-Stieltjes algebras over locally compact groupoids have been defined in a way that parallels their construction for groups. In this article, we extend the results on surjectivity or lack of surjectivity of the restriction map on the Fourier and Fourier-Stieltjes algebras of groups to the groupoid setting. In particular, we consider the maps that restrict the domain of these functions in the Fourier or Fourier-Stieltjes algebra of a groupoid to an isotropy subgroup. These maps are continuous contractive algebra homomorphisms. When the groupoid is étale, we show that the restriction map on the Fourier algebra is surjective. The restriction map on the Fourier-Stieltjes algebra is not surjective in general. We prove that for a transitive groupoid with a continuous section or a group bundle with discrete unit space, the restriction map on the Fourier-Stieltjes algebra is surjective. We further discuss the example of an HLS groupoid, and obtain a necessary condition for surjectivity of the restriction map in terms of property FD for groups, introduced by Lubotzky and Shalom. As a result, we present examples where the restriction map for the Fourier-Stieltjes algebr

We study three natural types of restrictions on Fubini rankings and unit interval parking functions, which are motivated by their correspondence with ordered set partitions. For each restriction type, we define the corresponding subset of Fubini rankings and unit interval parking functions, establish enumerative results, and provide bijections between the restricted families. We also obtain exponential generating functions and combinatorial interpretations, including connections with exceedances in permutations and with the absence of cyclical adjacencies in set partitions.

We study strategic location choice by customers and sellers, termed the Bakers and Millers Game in the literature. In our generalized setting, each miller can freely choose any location for setting up a mill, while each baker is restricted in the choice of location for setting up a bakery. For optimal bargaining power, a baker would like to select a location with many millers to buy flour from and with little competition from other bakers. Likewise, a miller aims for a location with many bakers and few competing millers. Thus, both types of agents choose locations to optimize the ratio of agents of opposite type divided by agents of the same type at their chosen location. Originally raised in the context of Fractional Hedonic Games, the Bakers and Millers Game has applications that range from commerce to product design. We study the impact of location restrictions on the properties of the game. While pure Nash equilibria trivially exist in the setting without location restrictions, we show via a sophisticated, efficient algorithm that even the more challenging restricted setting admits equilibria. Moreover, the computed equilibrium approximates the optimal social welfare by a facto

The formalism of general probabilistic theories provides a universal paradigm that is suitable for describing various physical systems including classical and quantum ones as particular cases. Contrary to the usual no-restriction hypothesis, the set of accessible meters within a given theory can be limited for different reasons, and this raises a question of what restrictions on meters are operationally relevant. We argue that all operational restrictions must be closed under simulation, where the simulation scheme involves mixing and classical post-processing of meters. We distinguish three classes of such operational restrictions: restrictions on meters originating from restrictions on effects; restrictions on meters that do not restrict the set of effects in any way; and all other restrictions. We fully characterize the first class of restrictions and discuss its connection to convex effect subalgebras. We show that the restrictions belonging to the second class can impose severe physical limitations despite the fact that all effects are accessible, which takes place, e.g., in the unambiguous discrimination of pure quantum states via effectively dichotomic meters. We further dem

In this work we explore ways to restrict the ability to call Solidity smart contract functions for a specified duration. We describe methods to restrict functions from being called twice in the same transaction, block, or time period. This is related to the notion of non-reentrant functions, which are functions that can be called within a previous execution. These methods can be used to restrict interactions with entire sets of functions of smart contracts. We are motivated to revisit this topic for two reasons. First, we note that sixteen real-world smart contracts exploits in 2023 resulting in over $136M USD lost or stolen that could have been prevented by restricting function calls. As part of this survey, we dissect a new class of exploit that involves so-called read-only reentrancy: exploits that re-enter read-only functions to make smart contract state inconsistent in order to enable their exploitation. Second, while some of these approaches are simple, they may not always behave the same across different blockchains that support Solidity.

In this paper, we use Hasse diagrams and generating functions to count alternating permutations with restricted prefix and suffix of lengths 3 and 4. In other words, for an alternating permutation $σ=σ_1σ_2σ_3\cdotsσ_{n}\in S_{n}$, we restrict length-3 prefixes $σ_1σ_2σ_3$ to follow certain patterns, such as $231$ and $132$, or follow certain restrictions such as $σ_2 \geq \max\{σ_1,σ_3\}+2$, similarly for prefixes of length 4. We also study the enumeration of alternating permutations with restrictions on both prefix and suffix.

In this paper, a restricted transverse ray transform acting on vector and symmetric $m$-tensor fields is studied. We developed inversion algorithms using restricted transverse ray transform data to recover symmetric $m$-tensor fields in $\mathbb{R}^3$ and vector fields in $\mathbb{R}^n$. We restrict the transverse ray transform to all lines going through a fixed curve $γ$ that satisfies the Kirillov-Tuy condition. We show that the known restricted data can be used to reconstruct a specific weighted Radon transform of the unknown vector/tensor field's components, which we then use to explicitly recover the unknown field.

Counting logics with a bounded number of variables form one of the central concepts in descriptive complexity theory. Although they restrict the number of variables that a formula can contain, the variables can be nested within scopes of quantified occurrences of themselves. In other words, the variables can be requantified. We study the fragments obtained from counting logics by restricting requantification for some but not necessarily all the variables. Similar to the logics without limitation on requantification, we develop tools to investigate the restricted variants. Specifically, we introduce a bijective pebble game in which certain pebbles can only be placed once and for all, and a corresponding two-parametric family of Weisfeiler-Leman algorithms. We show close correspondences between the three concepts. By using a suitable cops-and-robber game and adaptations of the Cai-Fürer-Immerman construction, we completely clarify the relative expressive power of the new logics. We show that the restriction of requantification has beneficial algorithmic implications in terms of graph identification. Indeed, we argue that with regard to space complexity, non-requantifiable variables o

Given a Riemannian manifold $M$ and an $L^2$-normalized Laplacian eigenfunction $ψ$ on $M$ with eigenvalue $λ^2$, a general problem in analysis is to understand how the mass of $ψ$ distributes around $M$. There are different ways to attack this problem. One of them is to analyze the $L^p$-norm of $ψ$ restricted to a submanifold of $M$. Here, we concentrate on the case $M=S^2$, $p=2$, and we restrict to geodesics of the sphere. Burq, Gérard, and Tzvetkov showed, for $γ$ a geodesic of $S^2$ (and indeed for more general surfaces), that $||ψ|_γ||_{L^2} \ll λ^{1/4}$ and that this bound is optimal in general. In this paper, we specialize to the case in which $ψ$ is an eigenfunction of all the Hecke operators on the sphere and consider the set of geodesics $\mathcal{C}_{D}$ of $S^2$ associated to fundamental discriminants $D<0$. By combining approaches of Ali and Magee, we improve the previous upper bound to $||ψ|_{\mathcal{C}_{D}}||_{L^2} \ll_{D,\varepsilon} λ^{\varepsilon}$ for any $\varepsilon>0$, which is essentially sharp.

Restricted Heisenberg Lie superalgebras are studied over an algebraically closed field F of characteristic p > 2. We use the ordinary 1- and 2-cohomology spaces with trivial coefficients to compute the restricted 2-cohomology spaces. As an application, the corresponding restricted one-dimensional central extensions are described.

Many identification results in instrumental variables (IV) models hold without requiring any restrictions on the distribution of potential outcomes, or how those outcomes are correlated with selection behavior. This enables IV models to allow for arbitrary heterogeneity in treatment effects and the possibility of selection on gains in the outcome. I provide a necessary and sufficient condition for treatment effects to be point identified in a manner that does not restrict outcomes, when the instruments take a finite number of values. The condition generalizes the well-known LATE monotonicity assumption, and unifies a wide variety of other known IV identification results. The result also yields a brute-force approach to reveal all selection models that allow for point identification of treatment effects without restricting outcomes, and then enumerate all of the identified parameters within each such selection model. The search uncovers new selection models that yield identification, provides impossibility results for others, and offers opportunities to relax assumptions on selection used in existing literature. An application considers the identification of complementarities betwee

Wormhole solutions, bridges that connect different parts of spacetime, were proposed early in the history of General Relativity. Soon after, it was shown that all wormholes violate classical energy conditions, which are non-negativity constraints on contractions of the stress-energy tensor. Since these conditions are violated by quantum fields, it was believed that wormholes can be constructed in the context of semiclassical gravity. But negative energies in quantum field theory are not without restriction: quantum energy inequalities (QEIs) control renormalized negative energies averaged over a geodesic. Thus, QEIs provide restrictions on the construction of wormholes. This work is a review of the relevant literature, thus focusing on results where QEIs restrict traversable wormholes. Both 'short' and 'long' (without causality violations) wormhole solutions in the context of semiclassical gravity are examined. A new result is presented on constraints on the Maldacena, Milekhin, and Popov 'long' wormhole from the recently derived doubled smeared null energy condition.

This paper mainly studies the ResLieDer pair in characteristic 2, that is, a restricted Lie algebra with a restricted derivation. We define the restricted representation of a ResLieDer pair and the corresponding cohomology complex. We show that a ResLieDer pair is rigid if the second cohomology group is trivial and a deformation of order $n$ is extensible if and only if its obstruction class is trivial. Moreover, we prove that the central extensions of a ResLieDer pair are classified by the second cohomology group. Finally, we show that a pair of restricted derivations is extensible if and only if its obstruction class is trivial.

The naive combination of polymorphic effects and polymorphic type assignment has been well known to break type safety. Existing approaches to this problem are classified into two groups: one for restricting how effects are triggered and the other for restricting how they are implemented. This work explores a new approach to ensuring the safety of polymorphic effects in polymorphic type assignment. A novelty of our work lies in finding a restriction on effect interfaces. To formalize our idea, we employ algebraic effects and handlers, where an effect interface is given by a set of operations coupled with type signatures. We propose signature restriction, a new notion to restrict the type signatures of operations, and show that signature restriction is sufficient to ensure type safety of an effectful language equipped with unrestricted polymorphic type assignment. We also develop a type-and-effect system to enable the use of both operations that satisfy and do not satisfy the signature restriction in a single program.

In this paper we prove that in prime characteristic there is a functor $-_{p-Leib}$ from the category of diassociative algebras to the category of restricted Leibniz algebras, generalizing the functor from associative algebras to restricted Lie algebras. Moreover we define the notion of restricted universal enveloping diassociative algebra $Ud_{p}(g)$ of a restricted Leibniz algebra $g$ and we show that $Ud_{p}$ is left adjoint to the functor $-_{p-Leib}$. We also construct the restricted enveloping algebra, which classifies the restricted Leibniz modules. In the last section we put a restricted pre-Lie structure on the tensor product of a Leibniz algebra by a Zinbiel algebra.