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Network verification (NWV), broadly defined as the verification of properties of distributed protocols used in network systems, cannot be efficiently solved on classical hardware via brute force. Prior work has developed a variety of methods that scale by observing a structure in the search space and then evaluating classes within the search space instead of individual instances. However, even these classification mechanisms have their limitations. In this paper, we consider a radically different approach: applying quantum computing to more efficiently solve NWV problems. We provide an overview of how to map variants of NWV problems into unstructured search problems that can be solved via quantum computing with quadratic speedup, making the approach feasible in theory to problems that are double in size (of the input). Emerging quantum systems cannot yet tackle problems of practical interest, but rapid advances in hardware and algorithm development make now a great time to start thinking about their application. With this in mind, we explore the limits of scale of the problem for which quantum computing can solve NWV problems as unstructured search.

Biopharmaceutical products, particularly monoclonal antibodies (mAbs), have gained prominence in the pharmaceutical market due to their high specificity and efficacy. As these products are projected to constitute a substantial portion of global pharmaceutical sales, the application of machine learning models in mAb development and manufacturing is gaining momentum. This paper addresses the critical need for uncertainty quantification in machine learning predictions, particularly in scenarios with limited training data. Leveraging ensemble learning and Monte Carlo simulations, our proposed method generates additional input samples to enhance the robustness of the model in small training datasets. We evaluate the efficacy of our approach through two case studies: predicting antibody concentrations in advance and real-time monitoring of glucose concentrations during bioreactor runs using Raman spectra data. Our findings demonstrate the effectiveness of the proposed method in estimating the uncertainty levels associated with process performance predictions and facilitating real-time decision-making in biopharmaceutical manufacturing. This contribution not only introduces a novel approach for uncertainty quantification but also provides insights into overcoming challenges posed by small training datasets in bioprocess development. The evaluation demonstrates the effectiveness of our method in addressing key challenges related to uncertainty estimation within upstream cell cultivation, illustrating its potential impact on enhancing process control and product quality in the dynamic field of biopharmaceuticals.

The stability of complex networks, from power grids to biological systems, is crucial for their proper functioning. It is thus important to control such systems to maintain or restore their stability. Traditional approaches rely on real-time state measurements for feedback control, but this can be challenging in many real-world systems, such as the brain, due to their complex and dynamic nature. This paper utilizes vibrational control -- an open-loop strategy -- to regulate network stability. Unlike conventional methods targeting network nodes, our approach focuses on manipulating network edges through vibrational inputs. We establish sufficient graph-theoretic conditions for vibration-induced functional modifications of network edges and stabilization of network systems as a whole. Additionally, we provide methods for designing effective vibrational control inputs and validate our theoretical findings through numerical simulations., Comment: 12 pages, 9 figures

The brain is immensely complex, with diverse components and dynamic interactions building upon one another to orchestrate a wide range of functions and behaviors. Understanding patterns of these complex interactions and how they are coordinated to support collective neural activity and function is critical for parsing human and animal behavior, treating mental illness, and developing artificial intelligence. Rapid experimental advances in imaging, recording, and perturbing neural systems across various species now provide opportunities and challenges to distill underlying principles of brain organization and function. Here, we take stock of recent progresses and review methods used in the statistical analysis of brain networks, drawing from fields of statistical physics, network theory and information theory. Our discussion is organized by scale, starting with models of individual neurons and extending to large-scale networks mapped across brain regions. We then examine the organizing principles and constraints that shape the biological structure and function of neural circuits. Finally, we describe current opportunities aimed at improving models in light of recent developments and at bridging across scales to contribute to a better understanding of brain networks., Comment: Submitted to Annual Review of Biophysics. Comments welcome. When citing this paper, please use the following: Kulkarni S, Bassett DS. 2025. Towards Principles of Brain Network Organization and Function. Annu. Rev. Biophys. 54: Submitted. DOI: 10.1146/annurev-biophys-030722-110624

In this paper, we introduce a novel semidefinite programming framework for designing custom frugal resolvent splitting algorithms which find a zero in the sum of n monotone operators. This framework features a number of design choices which facilitate creating resolvent splitting algorithms with specific communication structure. We illustrate these design choices using a variety of constraint sets and objective functions, as well as the use of a mixed-integer SDP to minimize time per iteration or required number of communications between the resolvents which define the splitting. Using the Performance Estimation Problem (PEP) framework, we provide parameter selections, such as step size, which for high dimensional problems provide optimal contraction factors in the algorithms. Among the algorithm design choices we introduce, we provide a characterization of algorithm designs which provide minimal convergence time given structural properties of the monotone operators, resolvent computation times, and communication latencies.

We present a general strategy for the inverse design of metasurfaces composed of elementary shapes. We use it to design a structure that collects and collimates light from nitrogen-vacancy centers in diamond. Such metasurfaces constitute scalable optical interfaces for solid-state qubits, enabling efficient photon coupling into optical fibers and eliminating free-space collection optics. The many-body shape optimization strategy is a practical alternative to topology optimization that explicitly enforces material and fabrication constraints throughout the optimization, while still achieving high performance. The metasurface is easily adaptable to other solid-state qubits, and the optimization method is broadly applicable to fabrication-constrained photonic design problems., Comment: Accepted version. 16 pages, 11 figures (main text plus supplementary information)

Probabilistic representation spaces convey information about a dataset, and to understand the effects of factors such as training loss and network architecture, we seek to compare the information content of such spaces. However, most existing methods to compare representation spaces assume representations are points, and neglect the distributional nature of probabilistic representations. Here, instead of building upon point-based measures of comparison, we build upon classic methods from literature on hard clustering. We generalize two information-theoretic methods of comparing hard clustering assignments to be applicable to general probabilistic representation spaces. We then propose a practical method of estimation that is based on fingerprinting a representation space with a sample of the dataset and is applicable when the communicated information is only a handful of bits. With unsupervised disentanglement as a motivating problem, we find information fragments that are repeatedly contained in individual latent dimensions in VAE and InfoGAN ensembles. Then, by comparing the full latent spaces of models, we find highly consistent information content across datasets, methods, and hyperparameters, even though there is often a point during training with substantial variety across repeat runs. Finally, we leverage the differentiability of the proposed method and perform model fusion by synthesizing the information content of multiple weak learners, each incapable of representing the global structure of a dataset. Across the case studies, the direct comparison of information content provides a natural basis for understanding the processing of information., Comment: Code: https://github.com/murphyka/representation-space-info-comparison

Neural Machine Translation models are extremely data and compute-hungry. However, not all data points contribute equally to model training and generalization. Data pruning to remove the low-value data points has the benefit of drastically reducing the compute budget without significant drop in model performance. In this paper, we propose a new data pruning technique: Checkpoints Across Time (CAT), that leverages early model training dynamics to identify the most relevant data points for model performance. We benchmark CAT against several data pruning techniques including COMET-QE, LASER and LaBSE. We find that CAT outperforms the benchmarks on Indo-European languages on multiple test sets. When applied to English-German, English-French and English-Swahili translation tasks, CAT achieves comparable performance to using the full dataset, while pruning up to 50% of training data. We inspect the data points that CAT selects and find that it tends to favour longer sentences and sentences with unique or rare words., Comment: Accepted to ACL 2024 Findings

Nuclear quadrupolar resonance (NQR) spectroscopy reveals chemical bonding patterns in materials and molecules through the unique coupling between nuclear spins and local fields. However, traditional NQR techniques require macroscopic ensembles of nuclei to yield a detectable signal, which precludes the study of individual molecules and obscures molecule-to-molecule variations due to local perturbations or deformations. Optically active electronic spin qubits, such as the nitrogen-vacancy (NV) center in diamond, facilitate the detection and control of individual nuclei through their local magnetic couplings. Here, we use NV centers to perform NQR spectroscopy on their associated nitrogen-14 ($^{14}$N) nuclei at room temperature. In mapping the nuclear quadrupolar Hamiltonian, we resolve minute variations between individual nuclei. The measurements further reveal correlations between the parameters in the NV center's electronic spin Hamiltonian and the $^{14}$N quadropolar Hamiltonian, as well as a previously unreported Hamiltonian term that results from symmetry breaking. We further design pulse sequences to initialize, readout, and control the quantum evolution of the $^{14}$N nuclear state using the nuclear quadrupolar Hamiltonian., Comment: 15 pages, 10 figures

Prions are misfolded proteins that transmit their structural arrangement to neighboring proteins. In biological systems, prion dynamics can produce a variety of complex functional outcomes. Yet, an understanding of prionic causes has been hampered by the fact that few computational models exist that allow for experimental design, hypothesis testing, and control. Here, we identify essential prionic properties and present a biologically inspired model of prions using simple mechanical structures capable of undergoing complex conformational change. We demonstrate the utility of our approach by designing a prototypical mechanical prion and validating its properties experimentally. Our work provides a design framework for harnessing and manipulating prionic properties in natural and artificial systems., Comment: Added supplements, 25 pages, 11 figures

Since the turn of the century, astronomers have been exploiting the rich information afforded by combining stellar kinematic maps and imaging in an attempt to recover the intrinsic, three-dimensional (3D) shape of a galaxy. A common intrinsic shape recovery method relies on an expected monotonic relationship between the intrinsic misalignment of the kinematic and morphological axes and the triaxiality parameter. Recent studies have, however, cast doubt about underlying assumptions relating shape and intrinsic kinematic misalignment. In this work, we aim to recover the 3D shape of individual galaxies using their projected stellar kinematic and flux distributions using a supervised machine learning approach with mixture density network (MDN). Using a mock dataset of the EAGLE hydrodynamical cosmological simulation, we train the MDN model for a carefully selected set of common kinematic and photometric parameters. Compared to previous methods, we demonstrate potential improvements achieved with the MDN model to retrieve the 3D galaxy shape along with the uncertainties, especially for prolate and triaxial systems. We make specific recommendations for recovering galaxy intrinsic shapes relevant for current and future integral field spectroscopic galaxy surveys., Comment: Accepted for publication in PASA. 18 pages, 12 figures, 2 tables

Epilepsy is one of the most common neurological disorders globally, affecting millions of individuals. Despite significant advancements, the precise mechanisms underlying this condition remain largely unknown, making accurately predicting and preventing epileptic seizures challenging. In this paper, we employ a bistable model, where a stable equilibrium and a stable limit cycle coexist, to describe epileptic dynamics. The equilibrium captures normal steady-state neural activity, while the stable limit cycle signifies seizure-like oscillations. The noise-driven switch from the equilibrium to the limit cycle characterizes the onset of seizures. The differences in the regions of attraction of these two stable states distinguish epileptic brain dynamics from healthy ones. We analytically construct the regions of attraction for both states. Further, using the notion of input-to-state stability, we theoretically show how the regions of attraction influence the stability of the system subject to external perturbations. Generalizing the bistable system into coupled networks, we also find the role of network parameters in shaping the regions of attraction. Our findings shed light on the intricate interplay between brain networks and epileptic activity, offering mechanistic insights into potential avenues for more predictable treatments., Comment: 6 pages, 4 figures, submitted to IEEE CDC 2024

We investigate the vulnerability of computer-vision-based signal classifiers to adversarial perturbations of their inputs, where the signals and perturbations are subject to physical constraints. We consider a scenario in which a source and interferer emit signals that propagate as waves to a detector, which attempts to classify the source by analyzing the spectrogram of the signal it receives using a pre-trained neural network. By solving PDE-constrained optimization problems, we construct interfering signals that cause the detector to misclassify the source even though the perturbations to the spectrogram of the received signal are nearly imperceptible. Though such problems can have millions of decision variables, we introduce methods to solve them efficiently. Our experiments demonstrate that one can compute effective and physically realizable adversarial perturbations for a variety of machine learning models under various physical conditions.

We present the full Hubble diagram of photometrically-classified Type Ia supernovae (SNe Ia) from the Dark Energy Survey supernova program (DES-SN). DES-SN discovered more than 20,000 SN candidates and obtained spectroscopic redshifts of 7,000 host galaxies. Based on the light-curve quality, we select 1635 photometrically-identified SNe Ia with spectroscopic redshift 0.10$< z <$1.13, which is the largest sample of supernovae from any single survey and increases the number of known $z>0.5$ supernovae by a factor of five. In a companion paper, we present cosmological results of the DES-SN sample combined with 194 spectroscopically-classified SNe Ia at low redshift as an anchor for cosmological fits. Here we present extensive modeling of this combined sample and validate the entire analysis pipeline used to derive distances. We show that the statistical and systematic uncertainties on cosmological parameters are $\sigma_{\Omega_M,{\rm stat+sys}}^{\Lambda{\rm CDM}}=$0.017 in a flat $\Lambda$CDM model, and $(\sigma_{\Omega_M},\sigma_w)_{\rm stat+sys}^{w{\rm CDM}}=$(0.082, 0.152) in a flat $w$CDM model. Combining the DES SN data with the highly complementary CMB measurements by Planck Collaboration (2020) reduces uncertainties on cosmological parameters by a factor of 4. In all cases, statistical uncertainties dominate over systematics. We show that uncertainties due to photometric classification make up less than 10% of the total systematic uncertainty budget. This result sets the stage for the next generation of SN cosmology surveys such as the Vera C. Rubin Observatory's Legacy Survey of Space and Time., Comment: 39 pages, 19 figures; Submitted to ApJ; companion paper Dark Energy Collaboration et al. on consecutive arxiv number 2401.02929

We present cosmological constraints from the sample of Type Ia supernovae (SN Ia) discovered during the full five years of the Dark Energy Survey (DES) Supernova Program. In contrast to most previous cosmological samples, in which SN are classified based on their spectra, we classify the DES SNe using a machine learning algorithm applied to their light curves in four photometric bands. Spectroscopic redshifts are acquired from a dedicated follow-up survey of the host galaxies. After accounting for the likelihood of each SN being a SN Ia, we find 1635 DES SNe in the redshift range $0.100.5$ SNe compared to the previous leading compilation of Pantheon+, and results in the tightest cosmological constraints achieved by any SN data set to date. To derive cosmological constraints we combine the DES supernova data with a high-quality external low-redshift sample consisting of 194 SNe Ia spanning $0.025

The COVID-19 pandemic has highlighted the need to upgrade systems for infectious disease surveillance and forecasting and modeling of the spread of infection, both of which inform evidence-based public health guidance and policies. Here, we discuss requirements for an effective surveillance system to support decision making during a pandemic, drawing on the lessons of COVID-19 in the U.S., while looking to jurisdictions in the U.S. and beyond to learn lessons about the value of specific data types. In this report, we define the range of decisions for which surveillance data are required, the data elements needed to inform these decisions and to calibrate inputs and outputs of transmission-dynamic models, and the types of data needed to inform decisions by state, territorial, local, and tribal health authorities. We define actions needed to ensure that such data will be available and consider the contribution of such efforts to improving health equity.

Deterministic chaos permits a precise notion of a "perfect measurement" as one that, when obtained repeatedly, captures all of the information created by the system's evolution with minimal redundancy. Finding an optimal measurement is challenging, and has generally required intimate knowledge of the dynamics in the few cases where it has been done. We establish an equivalence between a perfect measurement and a variant of the information bottleneck. As a consequence, we can employ machine learning to optimize measurement processes that efficiently extract information from trajectory data. We obtain approximately optimal measurements for multiple chaotic maps and lay the necessary groundwork for efficient information extraction from general time series., Comment: Project page: https://distributed-information-bottleneck.github.io

While machine learning (ML) has made significant contributions to the biopharmaceutical field, its applications are still in the early stages in terms of providing direct support for quality-by-design based development and manufacturing of biopharmaceuticals, hindering the enormous potential for bioprocesses automation from their development to manufacturing. However, the adoption of ML-based models instead of conventional multivariate data analysis methods is significantly increasing due to the accumulation of large-scale production data. This trend is primarily driven by the real-time monitoring of process variables and quality attributes of biopharmaceutical products through the implementation of advanced process analytical technologies. Given the complexity and multidimensionality of a bioproduct design, bioprocess development, and product manufacturing data, ML-based approaches are increasingly being employed to achieve accurate, flexible, and high-performing predictive models to address the problems of analytics, monitoring, and control within the biopharma field. This paper aims to provide a comprehensive review of the current applications of ML solutions in a bioproduct design, monitoring, control, and optimisation of upstream, downstream, and product formulation processes. Finally, this paper thoroughly discusses the main challenges related to the bioprocesses themselves, process data, and the use of machine learning models in biopharmaceutical process development and manufacturing. Moreover, it offers further insights into the adoption of innovative machine learning methods and novel trends in the development of new digital biopharma solutions., Comment: 155 pages

Although neural-network-based emulators enable efficient parameter estimation in 21-cm cosmology, the accuracy of such constraints is poorly understood. We employ nested sampling to fit mock data of the global 21-cm signal and high-$z$ galaxy ultraviolet luminosity function (UVLF) and compare for the first time the emulated posteriors obtained using the global signal emulator ${\tt globalemu}$ to the `true' posteriors obtained using the full model on which the emulator is trained using ${\tt ARES}$. Of the eight model parameters we employ, four control the star formation efficiency (SFE), and thus can be constrained by UVLF data, while the remaining four control UV and X-ray photon production, and the minimum virial temperature of star-forming halos ($T_{\rm min}$), and thus are uniquely probed by reionization and 21-cm measurements. For noise levels of 50 and 250 mK in the 21-cm data being jointly-fit, the emulated and `true' posteriors are consistent to within $1\sigma$. However, at lower noise levels of 10 and 25 mK, ${\tt globalemu}$ overpredicts $T_{\rm min}$ and underpredicts $\gamma_{\rm lo}$, an SFE parameter, by $\approx3-4\sigma$, while the `true' ${\tt ARES}$ posteriors capture their fiducial values within $1\sigma$. We find that jointly-fitting the mock UVLF and 21-cm data significantly improves constraints on the SFE parameters by breaking degeneracies in the ${\tt ARES}$ parameter space. Our results demonstrate the astrophysical constraints that can be expected for global 21-cm experiments for a range of noise levels from pessimistic to optimistic, and also the potential for probing redshift evolution of SFE parameters by including UVLF data., Comment: 19 pages, 8 figures, 2 tables. Accepted by ApJ

Hexagonal boron nitride (h-BN) hosts pure single-photon emitters that have shown evidence of optically detected electronic spin dynamics. However, the electrical and chemical structure of these optically addressable spins is unknown, and the nature of their spin-optical interactions remains mysterious. Here, we use time-domain optical and microwave experiments to characterize a single emitter in h-BN exhibiting room temperature optically detected magnetic resonance. Using dynamical simulations, we constrain and quantify transition rates in the model, and we design optical control protocols that optimize the signal-to-noise ratio for spin readout. This constitutes a necessary step towards quantum control of spin states in h-BN., Comment: 15 pages, 15 figures. Updates in version 2 include modifications to figure 4, an updated estimate for T2*, and minor syntax edits to the main text and supplement for clarity

Humans are constantly exposed to sequences of events in the environment. Those sequences frequently evince statistical regularities, such as the probabilities with which one event transitions to another. Collectively, inter-event transition probabilities can be modeled as a graph or network. Many real-world networks are organized hierarchically and understanding how humans learn these networks is an ongoing aim of current investigations. While much is known about how humans learn basic transition graph topology, whether and to what degree humans can learn hierarchical structures in such graphs remains unknown. We investigate how humans learn hierarchical graphs of the Sierpi\'nski family using computer simulations and behavioral laboratory experiments. We probe the mental estimates of transition probabilities via the surprisal effect: a phenomenon in which humans react more slowly to less expected transitions, such as those between communities or modules in the network. Using mean-field predictions and numerical simulations, we show that surprisal effects are stronger for finer-level than coarser-level hierarchical transitions. Surprisal effects at coarser levels of the hierarchy are difficult to detect for limited learning times or in small samples. Using a serial response experiment with human participants (n=$100$), we replicate our predictions by detecting a surprisal effect at the finer-level of the hierarchy but not at the coarser-level of the hierarchy. To further explain our findings, we evaluate the presence of a trade-off in learning, whereby humans who learned the finer-level of the hierarchy better tended to learn the coarser-level worse, and vice versa. Our study elucidates the processes by which humans learn hierarchical sequential events. Our work charts a road map for future investigation of the neural underpinnings and behavioral manifestations of graph learning., Comment: 22 pages, 10 figures, 1 table

Active Inference is a recent framework for modeling planning under uncertainty. Empirical and theoretical work have now begun to evaluate the strengths and weaknesses of this approach and how it might be improved. A recent extension - the sophisticated inference (SI) algorithm - improves performance on multi-step planning problems through recursive decision tree search. However, little work to date has been done to compare SI to other established planning algorithms. SI was also developed with a focus on inference as opposed to learning. The present paper has two aims. First, we compare performance of SI to Bayesian reinforcement learning (RL) schemes designed to solve similar problems. Second, we present an extension of SI - sophisticated learning (SL) - that more fully incorporates active learning during planning. SL maintains beliefs about how model parameters would change under the future observations expected under each policy. This allows a form of counterfactual retrospective inference in which the agent considers what could be learned from current or past observations given different future observations. To accomplish these aims, we make use of a novel, biologically inspired environment designed to highlight the problem structure for which SL offers a unique solution. Here, an agent must continually search for available (but changing) resources in the presence of competing affordances for information gain. Our simulations show that SL outperforms all other algorithms in this context - most notably, Bayes-adaptive RL and upper confidence bound algorithms, which aim to solve multi-step planning problems using similar principles (i.e., directed exploration and counterfactual reasoning). These results provide added support for the utility of Active Inference in solving this class of biologically-relevant problems and offer added tools for testing hypotheses about human cognition., Comment: 31 pages, 5 figures

This study focuses on Pristine$\_180956.78$$-$$294759.8$ (hereafter P180956, $[Fe/H] =-1.95\pm0.02$), a star selected from the Pristine Inner Galaxy Survey (PIGS), and followed-up with the recently commissioned Gemini High-resolution Optical SpecTrograph (GHOST) at the Gemini South telescope. The GHOST spectrograph's high efficiency in the blue spectral region ($3700-4800$~\AA) enables the detection of elemental tracers of early supernovae (\eg Al, Mn, Sr, Eu). The star exhibits chemical signatures resembling those found in ultra-faint dwarf systems, characterised by very low abundances of neutron-capture elements (Sr, Ba, Eu), which are uncommon among stars in the Milky Way halo. Our analysis suggests that P180956 bears the chemical imprints of a small number (2 or 4) of low-mass hypernovae ($\sim10-15 M_{\odot}$), which are needed to mostly reproduce the abundance pattern of the light-elements (\eg [Si, Ti/Mg, Ca] $\sim0.6$), and one fast-rotating intermediate-mass supernova ($\sim300\kms$, $\sim80-120 M_{\odot}$), which is the main channel contributing to the high [Sr/Ba] ($\sim +1.2$). The small pericentric ($\sim0.7$ kpc) and apocentric ($\sim13$ kpc) distances and its orbit confined to the plane ($\lesssim 2$ kpc), indicate that this star was likely accreted during the early Galactic assembly phase. Its chemo-dynamical properties suggest that P180956 formed in a system similar to an ultra-faint dwarf galaxy accreted either alone, as one of the low-mass building blocks of the proto-Galaxy, or as a satellite of Gaia-Sausage-Enceladus. The combination of Gemini's large aperture with GHOST's high efficiency and broad spectral coverage makes this new spectrograph one of the leading instruments for near-field cosmology investigations., Comment: Accepted version, minor editing. New figure showing Sr and Ba lines. Section 4.7 revised

Cluster synchronization is of paramount importance for the normal functioning of numerous technological and natural systems. Deviations from normal cluster synchronization patterns are closely associated with various malfunctions, such as neurological disorders in the brain. Therefore, it is crucial to restore normal system functions by stabilizing the appropriate cluster synchronization patterns. Most existing studies focus on designing controllers based on state measurements to achieve system stabilization. However, in many real-world scenarios, measuring system states, such as neuronal activity in the brain, poses significant challenges, rendering the stabilization of such systems difficult. To overcome this challenge, in this paper, we employ an open-loop control strategy, vibrational control, which does not requires any state measurements. We establish some sufficient conditions under which vibrational inputs stabilize cluster synchronization. Further, we provide a tractable approach to design vibrational control. Finally, numerical experiments are conducted to demonstrate our theoretical findings., Comment: Submitted to Open Journal of Control Systems

Many natural and man-made network systems need to maintain certain patterns, such as working at equilibria or limit cycles, to function properly. Thus, the ability to stabilize such patterns is crucial. Most of the existing studies on stabilization assume that network systems states can be measured online so that feedback control strategies can be used. However, in many real-world scenarios, systems states, e.g., neuronal activity in the brain, are often difficult to measure. In this paper, we take this situation into account and study the stabilization problem of linear network systems with an open-loop control strategy (vibrational control). We derive a graph-theoretic sufficient condition for structural vibrational stabilizability, under which network systems can always be stabilized. We further provide an approach to select the locations in the network for control placement and design corresponding vibrational inputs to stabilize systems that satisfy this condition. Finally, we provide some numerical results that demonstrate the validity of our theoretical findings.