We develop a rigorous theoretical framework for principal manifold estimation that recovers a latent low-dimensional manifold from a point cloud observed in a high-dimensional ambient space. Our framework accommodates manifolds with general, potentially non-Euclidean topology, which can be inferred using tools from topological data analysis. Using the theory of Sobolev spaces on Riemannian manifolds, we establish that the proposed principal manifolds are well defined, prove convergence of the iterative algorithm used to compute them, and show consistency of the finite-sample estimator. Furthermore, we introduce a novel method for selecting the complexity level of a fitted manifold, which addresses the shortcomings of the classical fitting-error criterion. We also provide a detailed geometric interpretation of the penalty term in our framework. In addition to the theoretical developments, we present extensive numerical experiments supporting our results. This article provides theoretical foundations for approaches that have been used in applications such as robotics. More importantly, it extends these approaches to general topological settings with potential applications across a broad range of disciplines, including neuroimaging and shape data analysis.

Several families of sharp Bernstein inequalities are established on the weighted $L^2$ space over parabolic domains, which include bounded or unbounded rotational paraboloids and parabolic surfaces. The main tool is a second-order differential operator satisfied by a specific basis of orthogonal polynomials in weighted $L^2$ space.

Distributed Port-Hamiltonian (dPHS) theory provides a powerful framework for modeling physical systems governed by partial differential equations and has enabled a broad class of boundary control methodologies. Their effectiveness, however, relies heavily on the availability of accurate system models, which may be difficult to obtain in the presence of nonlinear and partially unknown dynamics. To address this challenge, we combine Gaussian Process distributed Port-Hamiltonian system (GP-dPHS) learning with boundary control by interconnection. The GP-dPHS model is used to infer the unknown Hamiltonian structure from data, while its posterior uncertainty is incorporated into an energy-based robustness analysis. This yields probabilistic conditions under which the closed-loop trajectories remain bounded despite model mismatch. The method is illustrated on a simulated shallow water system.

We present a comprehensive analysis of Round-Delayed Amnesiac Flooding (RDAF), a variant of Amnesiac Flooding that introduces round-based asynchrony through adversarial delays. We establish fundamental properties of RDAF, including termination characteristics for different graph types and decidability results under various adversarial models. Our key contributions include: (1) a formal model of RDAF incorporating round-based asynchrony, (2) a proof that flooding always terminates on acyclic graphs despite adversarial delays, (3) a construction showing non-termination is possible on any cyclic graph, (4) a demonstration that termination is undecidable with arbitrary computable adversaries, and (5) the introduction of Eventually Periodic Adversaries (EPA) under which termination becomes decidable. These results enhance our understanding of flooding in communication-delay settings and provide insights for designing robust distributed protocols.

We study the finite-rank frame operators generated by cylinder indicator functions for the Bernoulli Cantor measure $μ_{p}$. In the symmetric case $p=\frac{1}{2}$, the natural Haar differences diagonalize these operators. For general $0<p<1$, we show that the weighted Haar basis still yields a sparse tree-banded matrix form, although diagonalization is lost. We also prove a filtration representation in terms of conditional expectations and level-wise mass operators. This leads to a norm convergent limit operator $K_{\infty}$, which is compact, positive, and self-adjoint. Finally, we show that $K_{\infty}$ is characterized by a self-similar operator identity induced by the first-level Cantor decomposition, and we derive corresponding block and scalar resolvent renormalization formulas.

We study the long-time asymptotic behavior of small-data solutions to the three-dimensional Vlasov--Riesz system with the inverse power-law potential $λ|x|^{-α}$ in the strictly long-range regime ($0 < α< 1$). By introducing finite- and infinite-time modified wave operators for the characteristic flows, we describe the asymptotic dynamics via convergence to an effective profile along a suitably modified reference flow, and establish modified scattering of solutions. Our proof relies mainly on ODE techniques for the characteristic flows, while also using PDE methods for weighted $W^{1,\infty}$-bounds. Compared with the earlier result (of Huang and Kwon), our Lagrangian approach extends modified scattering to the broader regime $\frac{1}{2}<α<1$ and provides a distinct and more robust argument.

We study the process $e^+e^- \to W^+W^-$ with the aim of estimating the prospects for observing heavy neutrinos contributions at future $e^+e^-$-colliders. In this work, we consider two implementations of heavy-light neutrino mixing: a linearized mixing approximation applied in popular models and an exact unitary mixing scheme. We conclude that the approximate realization leads to physically incorrect results for this process, while exact unitary mixing provides some signatures that can be experimentally checked.

Let $A$ be a unital $C^*$-algebra and $H$ a Hilbert space. The cone $\CP(A,B(H))$ of completely positive maps carries the Bures metric $β$, closely related to the cb-norm. We introduce a family of Bures--Kuratowski (BK) metrics on $\CB(A,B(H))$ that extend $β$ exactly on $\CP(A,B(H))$. The construction combines a Kuratowski embedding of the Bures cone, based at an anchor $θ\in\CP(A,B(H))$, with a regular-representation Hausdorff coordinate arising from universal regular models. Each BK metric admits an $\ell^p$-wedge decomposition, splitting $\CB(A,B(H))$ into the Bures cone and a non-CP component attached at $θ$. We then study Vietoris--Rips and Čech complexes of BK metric spaces. The wedge formula yields explicit criteria for mixed simplices, a join-type description of the mixed Rips complex, and ball-intersection criteria for mixed Čech simplices. For finite point clouds, this makes the mixed simplicial geometry computable from the two component metrics and reveals new homological features arising from the interaction between the CP and non-CP sectors.

This paper presents a smooth parameterization of continuous-time Signal Temporal Logic (CT-STL) specifications for nonconvex trajectory optimization that is sound and complete up to the accuracy of the underlying numerical integration scheme. CT-STL provides a natural framework for encoding rich temporal and logical task requirements, but existing trajectory-optimization formulations typically enforce such specifications only at discrete sampling nodes. In contrast, the proposed method evaluates specifications in dense time, thereby guaranteeing continuous-time satisfaction of always predicates, which is critical for path constraints such as obstacle avoidance, while eliminating the node-induced conservatism of eventually predicates by allowing satisfaction at any time within the prescribed interval. These two dense-time constructions also serve as the main building blocks for handling more general CT-STL formulas, including complex until specifications. Furthermore, the proposed parameterization resolves the locality and gradient-masking issues inherent in standard quantitative semantics, yielding a more favorable landscape for gradient-based solvers. Although dense-time evaluation introduces additional function evaluations during discretization, it also permits substantially coarser temporal grids without sacrificing safety or logical fidelity. This, in turn, reduces the dimension of the resulting nonconvex program, which is often the dominant factor in trajectory-generation cost. The numerical effectiveness and semantic exactness of the proposed framework are demonstrated on an agile quadrotor flight problem subject to a complex continuous-time until specification. The implementation is available at https:// github.com/UW-ACL/TrajOpt_CT-STL

Physics-based simulation involves trade-offs between performance and accuracy. In collision detection, one trade-off is the granularity of collider geometry. Primitive-based colliders such as bounding boxes are efficient, while using the original mesh is more accurate but often computationally expensive. Approximate Convex Decomposition (ACD) methods strive for a balance of efficiency and accuracy. Prior works can produce high-quality decompositions but require large numbers of convex parts and are sensitive to the orientation of the input mesh. We address these weaknesses with VisACD, a visibility-based, rotation-equivariant, and intersection-free ACD algorithm with GPU acceleration. Our approach produces high-quality decompositions with fewer convex parts, is not sensitive to shape orientation, and is more efficient than prior work.

Federated learning (FL) over heterogeneous IoT edge devices faces coupled system-modality-data heterogeneity: the lower-cost device carries both fewer sensors and less computational power, so the slowest device (straggler) produces the most incomplete gradient signals. Naively averaging their updates dilutes rare-modality information and wastes computation on absent-sensor parameters, whereas existing methods handle the triple heterogeneity (system, modality, data) in isolation and none addresses their coupling. To resolve this issue, we propose RELIEF, a framework that partitions the fusion-layer Low-Rank Adaptation (LoRA) projection matrix into modality-aligned column blocks and uses this partition as a unified interface for aggregation, elastic training, and communication. Each block is aggregated only within the cohort of devices possessing that modality, which eliminates cross-modal gradient interference; the server then allocates personalized training budgets by prioritizing blocks with the highest cohort-internal divergence, so that resource-constrained devices train fewer but more impactful parameters. We prove that cohort-wise aggregation removes interference from the convergence bound and that the divergence-guided allocation achieves sublinear regret. Experiments on two IoT sensor datasets (PAMAP2, MHEALTH) under both full-parameter (CNN) and parameter-efficient (LoRA) training show that RELIEF achieves up to 9.41x speedup and 37% energy reduction over FedAvg with up to 15.3 pp rare-modality F1 gains, and real-device validation on a two-Jetson AGX Orin testbed confirms these results.

We investigate the physical mechanisms governing the continuous transition from the droplet-dominated to the ion-dominated regime in electrosprays of highly conducting liquids. We characterize electrosprays of four ionic liquids using time-of-flight spectrometry and direct flow rate measurements. In the droplet regime, the jet breakup process exhibits self-similar lognormal mass-to-charge distributions with a constant coefficient of variation. In the mixed and ionic regimes, the average solvation state of the emitted ions decreases with decreasing flow rate, consistent with a shift of the primary ion emission zone toward the cooler cone-jet neck. Modeling ion evaporation from the post-breakup droplet population yields an estimate for the ion solvation energy of $ΔG_0 \gtrsim 1.9$~eV, a value difficult to reconcile with jet-less ion emission from a Taylor cone tip. Furthermore, we identify two fundamental limits on the performance of highly conducting electrosprays near minimum flow rate: substantial neutral mass losses driven by the evaporation of small droplets, and a dissociation limit imposed by the finite fraction of free ions in the bulk liquid. The dissociation limit yields an analytical expression for the maximum specific impulse of electrospray thrusters, showing excellent agreement with experimental data across multiple propellants and electrospray sources.

Generating performant executables from high level languages is critical to software performance across a wide range of domains. Modern compilers perform this task by passing code through a series of well-studied optimizations at progressively lower levels of abstraction, but may miss optimization opportunities that require high-level reasoning about a program's purpose. Recent work has proposed using LLMs to fill this gap. While LLMs can achieve large speedups on some programs, they frequently generate code that is incorrect. In this work, we propose a method to balance the correctness of conventional compiler optimizations with the ``creativity'' of LLM-based code generation: compiler-LLM cooperation. Our approach integrates existing compiler optimization passes with LLM-based code generation at multiple levels of abstraction, retaining the best features of both types of code optimization. We realize our approach with a multi-agent system that includes (1) LLM-based optimization agents for each level of abstraction, (2) individual compiler constituents as tools, (3) an LLM-based test generation agent that probes the correctness and performance of generated code, and (4) a guiding LLM that orchestrates the other components. The strategy enables LLM-based optimization of input programs at multiple levels of abstraction and introduces a method for distributing computational budget between levels. Our extensive evaluation shows that compiler-LLM cooperation outperforms both existing compiler optimizations and level-specific LLM-based baselines, producing speedups up to 1.25x.

Coarse-Grained Reconfigurable Architectures (CGRAs) are a promising and versatile accelerator platform, offering a balance between the performance and efficiency of specialized accelerators and the software programmability. However, their full potential is severely hindered by control flow in accelerated kernels, as the control flow (e.g., loops, branches) is fundamentally incompatible with the parallel, data-driven CGRA fabric. Prior strategies to resolve this mismatch in CGRA kernel acceleration are either inefficient, sacrificing performance for generality, or lack generality due to the difficulty of adapting them across different execution models. Thus, a general and unified solution for efficient CGRA kernel acceleration remains elusive. This paper introduces NEURA, a unified and retargetable compilation framework that systematically resolves the control-dataflow mismatch in CGRAs. NEURA's core innovation is a novel, pure dataflow intermediate representation (IR) built on a predicated type system. In this IR, control contexts are embedded as a predicate within each data, making control an intrinsic property of data. This mechanism enables NEURA to systematically flatten complex control flow into a single unified dataflow graph. This unified representation decouples kernel representation from hardware, empowering NEURA to retarget diverse CGRAs with different execution models and microarchitectural features. When targeted to a high-performance spatio-temporal CGRA, NEURA delivers a 2.20x speedup on kernel benchmarks and up to 2.71x geometric mean speedup on real-world applications over state-of-the-art (SOTA) high-performance baselines. It also provides a competitive solution against the SOTA low-power CGRA when retargeted to a spatial-only CGRA. NEURA is open-source and available at https://github.com/coredac/neura.

Safety filters based on control barrier functions (CBFs) and high-order control barrier functions (HOCBFs) are often implemented through quadratic programs (QPs). In general, especially in the presence of multiple constraints, feasibility is difficult to certify before solving the QP and may be lost as the state evolves. This paper addresses this issue for linear time-invariant (LTI) systems with affine safety constraints. Exploiting the resulting geometry of the constraint normals, and considering both unbounded and bounded inputs, we characterize feasibility for several structured classes of constraints. For certain such cases, we also derive closed-form safety filters. These explicit filters avoid online optimization and provide a simple alternative to QP-based implementations. Numerical examples illustrate the results.

Control barrier function (CBF)-QP safety filters enforce safety by minimally modifying a nominal controller. While prior work has mainly addressed robustness of safety under uncertainty, robustness of the resulting closed-loop \emph{stability} is much less understood. This issue is important because once the safety filter becomes active, it modifies the nominal dynamics and can reduce stability margins or even destabilize the system, despite preserving safety. For linear systems with a single affine safety constraint, we show that the active-mode dynamics admit an exact scalar loop representation, leading to a classical robust-control interpretation in terms of gain, phase, and delay margins. This viewpoint yields exact stability-margin characterizations and tractable linear matrix inequality (LMI)-based certificates and synthesis conditions for controllers with certified robustness guarantees. Numerical examples illustrate the proposed analysis and the enlargement of certified stability margins for safety-filtered systems.

We present a unified C++ implementation of the functional renormalization group and the parquet equations within the single-boson exchange formalism for several paradigmatic tight-binding and impurity models at equilibrium. The implementation computes the full dynamic vertex and self-energies, with momentum dependence treated using a truncated unity framework. We implement multiple self-energy flow equations, cutoff schemes, and extensions ranging from the dynamical functional renormalization group to multiloop flow equations that incorporate cutoffs in both the propagator and the interaction. The codebase serves as a reference for recent developments in the fRG and parquet methods for correlated electron systems and provides a flexible foundation for developing new many-body approaches and extensions.

We study robust regression under a contamination model in which covariates are clean while the responses may be corrupted in an adaptive manner. Unlike the classical Huber's contamination model, where both covariates and responses may be contaminated and consistent estimation is impossible when the contamination proportion is a non-vanishing constant, it turns out that the clean-covariate setting admits strictly improved statistical guarantees. Specifically, we show that the additional information in the clean covariates can be carefully exploited to construct an estimator that achieves a better estimation rate than that attainable under Huber contamination. In contrast to the Huber model, this improved rate implies consistency even when the contamination is a constant. A matching minimax lower bound is established using Fano's inequality together with the construction of contamination processes that match $m> 2$ distributions simultaneously, extending the previous two-point lower bound argument in Huber's setting. Despite the improvement over the Huber model from an information-theoretic perspective, we provide formal evidence -- in the form of Statistical Query and Low-Degree Polynomial lower bounds -- that the problem exhibits strong information-computation gaps. Our results strongly suggest that the information-theoretic improvements cannot be achieved by polynomial-time algorithms, revealing a fundamental gap between information-theoretic and computational limits in robust regression with clean covariates.

We propose an overview of optimal transport theory and its applications to econometric methodology. This review is specifically designed for practitioners, be they econometric theorists or applied econometricians. The review of applications of optimal transport to econometrics is organized around the particular aspects of the mathematical theory of optimal transport they rely on.

We establish an equivalence between categories of 'formally nilpotent' Lie algebras and exponential groups in characteristic zero. It extends the equivalences of Mal'cev, Lazard, Quillen and Warfield, and applies to groups under composition of generalized formal series or automorphisms of algebras of generalized formal series. We obtain first-order transfer results from finite dimensional nilpotent objects to formally nilpotent ones. We give applications to solving equations over groups, to the theory of nilpotent exponential groups as per Miasnikov-Remeslennikov, and to definability problems in certain groups of formal series.

We directly show that the local ratio of the shear viscosity to the entropy density for Unruh radiation at a finite distance from the horizon is universal and satisfies the relation $ η/s = 1/(4πc_s^2) $, which involves the speed of sound $ c_s $. Since $ c_s^2 \leq 1 $ by causality, this establishes the close connection between the famous Kovtun-Son-Starinets bound and causality. Moreover, we show that the ratio of bulk to shear viscosity saturates another well-known bound for the bulk viscosity, predicted within holographic approach. We also show that the condition of isotropy of thermal radiation in the Rindler space leads to a novel sum rule relating the $ c^{(0)}(μ) $ and $ c^{(2)}(μ) $ spectral densities, and we explicitly demonstrate its validity for conformal field theory and free massive Dirac fields in any number of dimensions. The sum rule provides the validity of Pascal law and bears some similarity with Burkhardt-Cottingham sum rule for spin-dependent parton distributions. Our result suggests a new perspective on dissipative transport phenomena in media undergoing extreme acceleration, such as quark-gluon plasma created in relativistic heavy-ion collisions.

We present a theoretical framework demonstrating a deterministic initialization mechanism for Warm Inflation via classical conformal boundary conditions. A persistent challenge in dissipative inflationary models is the "cold start" paradox: initializing the requisite thermal bath to generate the dissipative friction that subsequently sustains radiation production. Postulating an idealized, asymptotically scale invariant pre-inflationary phase, we mathematically prove that a conformal Weyl mapping to the emergent metric furnishes a finite, analytically derived initial radiation density. Implementing a spontaneous conformal symmetry-breaking ansatz, an emergent inflaton field is subjected to this inherited thermal bath. We analytically derive the initial kinematics of this framework, demonstrating that for strict sub-Planckian temperatures, the universe naturally initializes in the weak dissipative regime (Q << 1). The initial Hubble friction provided by the boundary radiation enables a smooth, deterministic kinematic handoff to the warm slow-roll steady-state attractor. As a mathematical proof-of-concept, this mechanism provides a fully realized framework to bypass the bootstrap problem of warm inflation.

We investigate the origin of the broad exponential wings observed in a significant fraction of the Halpha profiles of JWST-discovered little red dots (LRDs) and little blue dots (LBDs). Recent studies have shown that exponential broad-line profiles are not a prerogative of LRDs, are often also present in LBDs, and need not imply that electron scattering is the dominant broadening mechanism in every source. Motivated by our unification picture in which LRDs are the dust-reddened, high-inclination counterparts of compact blue broad-line AGNs, we model the broad Balmer emission with a virialized, radially stratified broad-line region (BLR). In this framework, the observed profile is the luminosity-weighted superposition of clouds spanning a range of radii and therefore a range of characteristic virial velocities. We show that such a stratified BLR can reproduce the extended exponential-like wings observed in three representative LRDs, without requiring electron scattering to be the primary origin of the broad wings. Our results support a picture in which the broad wings and the line cores encode different physics: the wings arise primarily from virial BLR stratification, whereas the cores retain additional imprints of absorption and radiative transfer in dense gas. The successful fits further suggest that the cloud radial distribution peaks near the dust sublimation radius, while the exponential wings are shaped by the line-emitting inner BLR shells where the higher virial velocities produce the high-velocity tails. This offers a simple physical explanation for the exponential wings of little dots, without invoking exotic new components or scenarios.

This paper introduces a novel joint communication and proactive monitoring (JCAM) system that simultaneously monitors multiple untrusted links and serves multiple legitimate users. The system leverages a cell-free massive multiple-input multiple-output (CF-mMIMO) architecture, where one subset of access points (APs) is dedicated to receiving signals from untrusted links, while another subset transmits data to legitimate users and jamming signals into the untrusted links. This dual functionality not only ensures reliable communication for legitimate users but also degrades the performance of untrusted links, thereby enhancing monitoring effectiveness. Closed-form expressions for the spectral efficiency (SE) of legitimate users and the monitoring success probability (MSP) are derived under partial zero-forcing (PZF) precoding/combining schemes with imperfect channel state information. Leveraging these expressions, we develop a simple yet effective AP mode assignment strategy that determines which APs perform downlink transmission and jamming, and which APs are dedicated to receiving signals from untrusted links. The objective is to maximize the MSP while satisfying predefined quality-of-service (QoS) requirements for all legitimate users. Numerical results show that the proposed mode assignment strategy significantly outperforms the benchmark, achieving up to a $32\%$ improvement in monitoring performance, while maintaining low computational complexity. Moreover, our proposed JCAM framework provides nearly a six-fold improvement in the minimum MSP over the co-located massive MIMO baseline.

We introduce and analyze a discrete-time network process in which each node holds a (weak) preference ordering over a finite set of alternatives and updates by local Borda aggregation. At each step, a node forms a weighted average (row-stochastic random-walk normalization) of its neighbors' Borda score vectors and projects the aggregated score back to a weak order. Updates are bounded: in each round, a node advances by at most one step along a shortest path in the fixed graph of preference orderings, following the direction prescribed by its neighbors' Borda-aggregated preferences. Our emphasis is dynamical: we develop sufficient conditions, stated directly in terms of graph topology, weights, and the bounded step rule, for (i) self-sustained oscillations in the absence of persistent sources, and (ii) forced oscillations under contrarian persistent camps. We also record robustness (structural stability) away from score-tie hyperplanes and contrast synchronous (Variant S) and asynchronous (Variant A) updating.

In this paper, we investigate the behavior of the family of graph-based splitting algorithms specialized to the problem of finding a point in the intersection of linear subspaces. The algorithms in this family, which encompasses several classical methods such as the Douglas-Rachford algorithm, are defined by a connected graph and a subgraph. Our main result establishes that when the graph and subgraph coincide, the optimal relaxation parameter is exactly $1$, thereby extending known results for the Douglas-Rachford algorithm to a much broader class of methods. Our analysis hinges on some properties of iso-averaged linear operators, which are defined as the average of an isometry and the identity, and are characterized by a specific symmetry of the norm of their relaxation.

Unitary $k$-designs are central to quantum information and quantum many-body physics as efficient proxies for Haar-random dynamics. We study how chaotic Hamiltonian evolution can generate unitary $k$-designs. Standard approaches typically rely on many independent Hamiltonian realizations or fine-tuning evolution times. Here we show that unitary designs can instead arise from a quenched temporal ensemble, where Hamiltonians are sampled once and held fixed, while randomness enters only through the evolution times. We analyze a two-step protocol (2SP), applying $H_1$ for time $t_1$ and $H_2$ for time $t_2$, and a three-step protocol (3SP) with an additional quench, with all times randomly drawn from a prescribed distribution. Time averaging imposes energy-index matching in the frame potential (FP), which quantifies the distance to Haar random. Analytically and numerically, we show that 2SP cannot realize a general unitary $k$-design, whereas 3SP can do so for arbitrary $k$. The advantage of 3SP is that the additional random phases impose stronger constraints, eliminating independent permutation degrees of freedom in the FP. For Gaussian unitary ensemble Hamiltonians, we prove these results rigorously and show that under imperfect time averaging, 3SP achieves the same accuracy as 2SP with a parametrically narrower time window.

We predict ultrafast switching in a chiral anti-ferromagnet that occurs at femtosecond times, nearly 5 orders of magnitude faster than the torque induced nanosecond switching previously observed. The physical mechanism, quite different from that which drives slow switching, involves the creation of massive effective magnetic fields by ultrafast spin current injection. Identifying these fields as key to femtosecond rotation, we establish simple practical rules for their maximisation with wide applicability to all magnetised materials. Employing state-of-the-art time-dependent density-functional theory and using the example of chiral magnet, Mn$_3$Sn, we induce ultrafast rotation enough to drive the switching of magnetic order between the six possible non-collinear ground states. We further demonstrate the possibility of undoing this switching by subsequent injection of oppositely polarized spin current. Our findings place chiral anti-ferromagnets as a materials platform for femtosecond Néel-vector switching, opening a route towards the manipulation of magnetic matter at ultrafast times.

We study the geometry of $\mathbb{R}^3$ equipped with a rotationally invariant Carnot-Carthéodory metric obtained by weighting motion in the $z$-direction by a function $f(r)$ of the cylindrical radius. When $f$ vanishes only at $r=0$, the space exhibits a Grushin--type singularity along the vertical axis. We provide sufficient conditions on $f$ ensuring a Grushin--like structure and describe the full optimal synthesis at singular points. For Riemannian points, we propose a candidate cut time determined by a discrete symmetry of the Hamiltonian flow. In the integrable case $f(r)=r$, we prove that this candidate coincides with the true cut time and give an explicit description of the cut locus.

In this work, we construct a persistent version of the well-known Leray spectral sequence. More precisely, we construct a spectral sequence that computes the persistent cohomology of a space from the persistent cohomology in each open set and its intersections with a covering that is the pre-image under a function of a covering of a known space.