Despite the strong performance of large language models (LLMs) across diverse tasks, their susceptibility to adversarial attacks and unsafe content generation remains a significant obstacle to deployment, particularly in high-stakes settings. Addressing this challenge requires safety mechanisms that are both practically effective and theoretically grounded. In this paper, we introduce BarrierSteer, a novel inference-time framework that improves response safety by embedding learned nonlinear safety constraints directly into the model's latent representation space. BarrierSteer treats hidden-state safety classifiers as Control Barrier Functions (CBFs), enabling constraint-guided steering of unsafe latent trajectories during generation. By composing multiple safety constraints through efficient constraint merging without modifying the underlying LLM parameters, BarrierSteer preserves model utility. We provide theoretical results showing that applying CBFs in the latent space yields a principled, modular, and computationally efficient approach for steering with respect to learned safety constraints, with guarantees conditional on the learned barriers capturing the intended safety prope
Self-concordant barriers are essential for interior-point algorithms in conic programming. To speed up the convergence it is of interest to find a barrier with the lowest possible parameter for a given cone. The barrier parameter is a non-convex function on the set of self-concordant barriers on a given cone, and finding an optimal barrier amounts to solving a non-convex infinite-dimensional optimization problem. In this work we study the degradation of the optimal value of the problem when the problem is convexified, and provide an estimate of the accuracy of the convex relaxation. The amount of degradation can be computed by comparing a 1-parameter family of non-convex bodies in $R^3$ with their convex hulls. Our study provides insight into the degree of non-convexity of the problem and opens up the possibility of constructing suboptimal barriers by solving the convex relaxation
We introduce a generalized Langevin model system for different non-Markovian effects in the well and barrier regions of a potential, and use it to numerically study the dependence of the barrier-crossing time. In the appropriate limits, our model interpolates between the theoretical barrier-crossing-time predictions by Grote and Hynes (GH), as well as by Pollak et al., which for a single barrier memory time can differ by several orders of magnitude. Our model furthermore allows to test an analytic rate theory for space-inhomogeneous memory, which disagrees with our numerical results in the long well-memory regime. In this regime, we find that short barrier memory decreases the barrier-crossing time as compared to long barrier memory. This is in contrast with the short well-memory regime, where both our numerical results and GH theory predict an acceleration of the barrier crossing time with increasing barrier memory time. Both effects, the `Markovian-barrier acceleration' and GH `non-Markovian-barrier acceleration' can be understood from a committor analysis. Our model combines finite relaxation times of orthogonal degrees of freedom with a space-inhomogeneous coupling to such degr
Considering energy dependence of the barrier radius in heavy-ion fusion reactions, a modified Siwek-Wilczyński (MSW) fusion cross section formula is proposed. With the MSW formula, the fusion barrier parameters for 367 reaction systems are systematically extracted, based on 443 datasets of measured cross sections. We find that the fusion excitation functions for about $60\%$ reaction systems can be better described by introducing the energy dependence of the barrier radius which is due to the dynamical effects at energies near and below the barrier. Considering both the influence of the geometry radii and that of the reduced de Broglie wavelength of the colliding nuclei, the barrier heights are well reproduced with only one model parameter. The extracted barrier radius parameters linearly decrease with the effective fissility parameter, and the width of the barrier distribution relates to the barrier height and as well as the reduced de Broglie wavelength at energies around the Coulomb barrier.
Temporary plastic film barriers are widely used to separate occupied rooms from exterior renovation zones, yet their effect on indoor particulate exposure is poorly quantified. We monitored PM$_{2.5}$ in a Tampa, Florida, apartment for 48 days with a low-cost optical sensor (Temtop LKC-1000S+), spanning pre-barrier, barrier-on, and post-barrier periods. A quadratic baseline was fitted to "background" minutes devoid of identifiable indoor sources, allowing excess concentrations ($Δ$PM) to be partitioned into facade work, cooking, and passive accumulation without outdoor co-monitoring. The barrier prevented large construction spikes indoors but curtailed natural ventilation, doubling the mean baseline from 1.9 to 3.9 $μ$g m$^{-3}$. During this stage, passive build-up accounted for $45\,\%$ of the daily excess dose, with facade work and cooking contributing $31\,\%$ and $24\,\%$, respectively. Once the new window was installed and evening airing resumed, the baseline fell to 0.8 $μ$g m$^{-3}$, the lowest of the campaign. Our findings highlight the trade-off between dust shielding and background elevation and demonstrate that simple polynomial fitting bolsters low-cost IAQ diagnostics
This paper generalizes the control barrier function framework by replacing scalar-valued functions with matrix-valued ones. Specifically, we develop barrier conditions for safe sets defined by matrix inequalities -- both semidefinite and indefinite. Matrix inequalities can be used to describe a richer class of safe sets, including nonsmooth ones. The safety filters constructed from our proposed matrix control barrier functions via semidefinite programming (CBF-SDP) are shown to be continuous. Our matrix formulation naturally provides a continuous safety filter for Boolean-based control barrier functions, notably for disjunctions (OR), without relaxing the safe set. We illustrate the effectiveness of the proposed framework with applications in drone network connectivity maintenance and nonsmooth obstacle avoidance, both in simulations and hardware experiments.
We study path components of ropelength sublevel spaces for knot and link types. For an ordered oriented link type $\mathcal L$, let $Y_Λ(\mathcal L)$ be the moduli space of its $C^{1,1}$ representatives with standard thickness at least $1$ and total length at most $Λ$. The first nonempty level is the ropelength, and its minimizer locus is the ideal stratum $I(\mathcal L)$. The first main result is a compactness theorem: if the graph of nonzero pairwise linking numbers of $\mathcal L$ is connected, then every $Y_Λ(\mathcal L)$ is compact in the quotient constant-speed $C^1$ topology. We combine this with a general minimax principle showing that distinct connected components of a compact minimizer locus are separated by a strictly positive energy barrier. Applied to the Gordian pair of Kusner and Kusner, this gives a rigorous nontrivial lower bound for the number of ideal deformation components: their two minimizing configurations lie in distinct ideal components and have strictly positive ropelength merge distance. We also prove a general sink-to-birth principle for sublevel persistence. A non-global sink of an energy functional produces a path component born strictly above the glob
Barrier billiards are simple examples of pseudo-integrable models which form an appealing but poorly investigated subclass of dynamical systems. The paper examines the semiclassical limit of the exact quantum transfer operator for barrier billiards constructed in [J. Phys. A: Math. Theor. \textbf{55}, 024001 (2022)]. The obtained asymptotic expressions are used to provide analytical arguments to support the conjecture that spectral statistical properties of barrier billiards are described by the semi-Poisson distribution and to derive the trace formulas for such billiards which in the transfer operator approach is not automatic.
This work develops a robust adaptive control strategy for discrete-time systems using Control Barrier Functions (CBFs) to ensure safety under parametric model uncertainty and disturbances. A key contribution of this work is establishing a barrier function certificate in discrete time for general online parameter estimation algorithms. This barrier function certificate guarantees positive invariance of the safe set despite disturbances and parametric uncertainty without access to the true system parameters. In addition, real-time implementation and inherent robustness guarantees are provided. The proposed robust adaptive safe control framework demonstrates that the parameter estimation module can be designed separately from the CBF-based safety filter, simplifying the development of safe adaptive controllers for discrete-time systems. The resulting safe control approach guarantees that the system remains within the safe set while adapting to model uncertainties, making it a promising strategy for discrete-time safety-critical systems.
Ferroelectric tunnel junctions (FTJs) have attracted increasing research interest as a promising candidate for non-volatile memories. Recently, significant enhancements of tunneling electroresistance (TER) have been realized through modifications of electrode materials. However, direct control of the FTJ performance through modifying the tunneling barrier has not been adequately explored. Here, adding a new direction to FTJ research, we fabricated FTJs with BaTiO3 single barriers (SB-FTJs) and BaTiO3/SrTiO3 composite barriers (CB-FTJs), and reported a systematic study of FTJ performances by varying the barrier thicknesses and compositions. For the SB-FTJs, the TER is limited by pronounced leakage current for ultrathin barriers and extremely small tunneling current for thick barriers. For the CB-FTJs, the extra SrTiO3 barrier provides an additional degree of freedom to modulate the barrier potential and tunneling behavior. The resultant high tunability can be utilized to overcome the barrier thickness limits and enhance the overall CB-FTJ performances beyond those of SB-FTJ. Our results reveal a new paradigm to manipulate the FTJs through designing multilayer tunneling barriers with
A macroscopic energy barrier is a necessary condition for self-correcting quantum memory. In this paper, we prove tight bounds on the energy barrier applicable to any quantum code obtained from the hypergraph product of two classical codes. If the underlying classical codes are low-density parity-check codes (LDPC), the energy barrier of the quantum code is shown to be the minimum energy barrier of the underlying classical codes (and their transposes) up to an additive $O(1)$ constant.
A barrier certificate can separate the state space of a con- sidered hybrid system (HS) into safe and unsafe parts ac- cording to the safety property to be verified. Therefore this notion has been widely used in the verification of HSs. A stronger condition on barrier certificates means that less expressive barrier certificates can be synthesized. On the other hand, synthesizing more expressive barrier certificates often means high complexity. In [9], Kong et al consid- ered how to relax the condition of barrier certificates while still keeping their convexity so that one can synthesize more expressive barrier certificates efficiently using semi-definite programming (SDP). In this paper, we first discuss how to relax the condition of barrier certificates in a general way, while still keeping their convexity. Particularly, one can then utilize different weaker conditions flexibly to synthesize dif- ferent kinds of barrier certificates with more expressiveness efficiently using SDP. These barriers give more opportuni- ties to verify the considered system. We also show how to combine two functions together to form a combined barrier certificate in order to prove a safety property unde
Robots operating alongside people, particularly in sensitive scenarios such as aiding the elderly with daily tasks or collaborating with workers in manufacturing, must guarantee safety and cultivate user trust. Continuum soft manipulators promise safety through material compliance, but as designs evolve for greater precision, payload capacity, and speed, and increasingly incorporate rigid elements, their injury risk resurfaces. In this letter, we introduce a comprehensive High-Order Control Barrier Function (HOCBF) + High-Order Control Lyapunov Function (HOCLF) framework that enforces strict contact force limits across the entire soft-robot body during environmental interactions. Our approach combines a differentiable Piecewise Cosserat-Segment (PCS) dynamics model with a convex-polygon distance approximation metric, named Differentiable Conservative Separating Axis Theorem (DCSAT), based on the soft robot geometry to enable real-time, whole-body collision detection, resolution, and enforcement of the safety constraints. By embedding HOCBFs into our optimization routine, we guarantee safety, allowing, for instance, safe navigation in operational space under HOCLF-driven motion obje
This paper presents a novel and direct approach to price boundary and final-value problems, corresponding to barrier options, using forward deep learning to solve forward-backward stochastic differential equations (FBSDEs). Barrier instruments are instruments that expire or transform into another instrument if a barrier condition is satisfied before maturity; otherwise they perform like the instrument without the barrier condition. In the PDE formulation, this corresponds to adding boundary conditions to the final value problem. The deep BSDE methods developed so far have not addressed barrier/boundary conditions directly. We extend the forward deep BSDE to the barrier condition case by adding nodes to the computational graph to explicitly monitor the barrier conditions for each realization of the dynamics as well as nodes that preserve the time, state variables, and trading strategy value at barrier breach or at maturity otherwise. Given these additional nodes in the computational graph, the forward loss function quantifies the replication of the barrier or final payoff according to a chosen risk measure such as squared sum of differences. The proposed method can handle any barrie
Transport of a moving V-shaped barrier exposed to a bath of chiral active particles is investigated in a two-dimensional channel. Due to the chirality of active particles and the transversal asymmetry of the barrier position, active particles can power and steer the directed transport of the barrier in the longitudinal direction. The transport of the barrier is determined by the chirality of active particles. The moving barrier and active particles move in the opposite directions. The average velocity of the barrier is much larger than that of active particles. There exist optimal parameters (the chirality, the self-propulsion speed, the packing fraction, and the channel width) at which the average velocity of the barrier takes its maximal value. In particular, tailoring the geometry of the barrier and the active concentration provides novel strategies to control the transport properties of micro-objects or cargoes in an active medium.
Tunneling is an iconic concept that captures the peculiarity of quantum dynamics but, despite its ubiquity, questions remain. We focus on strong-field tunneling, which is vital to all attosecond science. We find an unexpected optical tunneling event that happens when the instantaneous electric field vanishes and there is no barrier. This event arises from a color switchover in a strongly polychromatic field. The tunneling without a barrier reveals the disconnect between the standard intuition built on the picture of a quasistatic barrier and the nonadiabatic nature of the process.
We discuss the pricing methodology for Bonus Certificates and Barrier Reverse-Convertible Structured Products. Pricing for a European barrier condition is straightforward for products of both types and depends on an efficient interpolation of observed market option pricing. Pricing products We discuss the pricing methodology for Bonus Certificates and Barrier Reverse-Convertible Structured Products. Pricing for a European barrier condition is straightforward for products of both types and depends on an efficient interpolation of observed market option pricing. Pricing products with an American barrier condition requires stochastic modelling. We show that for typical market parameters, this stochastic pricing problem can be systematically reduced to evaluating only one fairly simple stochastic parameter being the asymmetry of hitting the barrier. Eventually, pricing Bonus Certificates and Barrier Reverse Convertibles with an American barrier condition, shows to be dependent on stochastic modelling only within a range of $\pm\frac{2}{3}$ of accuracy - e.g. within this accuracy limitation we can price these products without stochastic modelling. We show that the remaining price compon
A theoretical study of the transport properties of zigzag and armchair graphene nanoribbons with a magnetic barrier on top is presented. The magnetic barrier modifies the energy spectrum of the nanoribbons locally, which results in an energy shift of the conductance steps towards higher energies. The magnetic barrier also induces Fabry-Perot type oscillations, provided the edges of the barrier are sufficiently sharp. The lowest propagating state present in zigzag and metallic armchair nanoribbons prevent confinement of the charge carriers by the magnetic barrier. Disordered edges in nanoribbons tend to localize the lowest propagating state, which get delocalized in the magnetic barrier region. Thus, in sharp contrast to the case of two-dimensional graphene, the charge carriers in graphene nanoribbons cannot be confined by magnetic barriers. We also present a novel method based on the Green's function technique for the calculation of the magnetosubband structure, Bloch states and magnetoconductance of the graphene nanoribbons in a perpendicular magnetic field. Utilization of this method greatly facilitates the conductance calculations, because, in contrast to excising methods, the p
This paper examines a class of barrier options-multi-step barrier options, which can have any finite number of barriers of any level. We obtain a general, explicit expression of option prices of this type under the Black-Scholes model. Multi-step barrier options are not only useful in that they can handle barriers of different levels and time steps, but can also approximate options with arbitrary barriers. Moreover, they can be embedded in financial products such as deposit insurances based on jump models with simple barriers. Along the way, we derive multi-step reflection principle, which generalizes the reflection principle of Brownian motion.
Epitaxial Fe/magnesium gallium spinel oxide (MgGa2O4)/Fe(001) magnetic tunnel junctions (MTJs) were fabricated by magnetron sputtering. Tunnel magnetoresistance (TMR) ratio up to 121% at room temperature (196% at 4 K) was observed, suggesting a TMR enhancement by the coherent tunneling effect in the MgGa2O4 barrier. The MgGa2O4 layer had a spinel structure and it showed good lattice matching with the Fe layers owing to slight tetragonal lattice distortion of MgGa2O4. Barrier thickness dependence of the tunneling resistance and current-voltage characteristics revealed that the barrier height of the MgGa2O4 barrier is much lower than that in an MgAl2O4 barrier. This study demonstrates the potential of Ga-based spinel oxides for MTJ barriers having a large TMR ratio at a low resistance area product.