Biomolecular condensates are cellular phase-separated droplets that usually exhibit a viscoelastic mechanical response. A behavior rationalized by modeling the complex molecules that make up a condensate as stickers and spacers, which assemble into a network-like structure. Condensates usually exhibit a solidification over a long period of time (days), a phenomenon described as aging.The emergence of such a long timescale of evolution from microscopic processes, as well as the associated microscopic reorganization leading to aging, remains mostly an open question. In this article, we explore the connection between the mechanical properties of the condensates and their microscopic structure. We propose a minimal model for the dynamic of stickers and spacers, and show that entropy maximization of spacers leads to an attractive force between stickers. Our system displays a surprisingly slow relaxation toward equilibrium, reminiscent of glassy systems and consistent with the liquid-to-solid transition observed. To explain this behavior, we study the clustering dynamic of stickers and successfully explain the origin of glassy relaxation.

This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. Numerous renowned algorithms for tackling the compressed sensing problem employ an alternating strategy, which typically involves data matching in one module and denoising in another. We present a novel approach, the Alternating Subspace Method (ASM), which integrates the principles of the greedy methods (e.g., the orthogonal matching pursuit type methods) and the splitting methods (e.g., the approximate message passing type methods). Crucially, ASM enhances the splitting method by achieving fidelity in a subspace-restricted fashion. \textcolor{black}{We reveal that such a restriction strategy guarantees global convergence via proximal residual control and establish its local geometric convergence on the LASSO problem.} Numerical experiments on the LASSO, channel estimation, and dynamic compressed sensing problems demonstrate its high convergence rate and its capacity to incorporate different prior distributions. Overall, the proposed method is promising in terms of efficiency, accuracy, and flexibility, and has the potential to be competitive in different sparse recovery applications.

``Einstein from noise" (EfN) is a prominent example of the model bias phenomenon: systematic errors in the statistical model that lead to spurious but consistent estimates. In the EfN experiment, one falsely believes that a set of observations contains noisy, shifted copies of a template signal (e.g., an Einstein image), whereas in reality, it contains only pure noise observations. To estimate the signal, the observations are first aligned with the template using cross-correlation, and then averaged. Although the observations contain nothing but noise, it was recognized early on that this process produces a signal that resembles the template signal! This pitfall was at the heart of a central scientific controversy about validation techniques in structural biology. This paper provides a comprehensive statistical analysis of the EfN phenomenon above. We show that the Fourier phases of the EfN estimator (namely, the average of the aligned noise observations) converge to the Fourier phases of the template signal, explaining the observed structural similarity. Additionally, we prove that the convergence rate is inversely proportional to the number of noise observations and, in the high-dimensional regime, to the Fourier magnitudes of the template signal. Moreover, in the high-dimensional regime, the Fourier magnitudes converge to a scaled version of the template signal's Fourier magnitudes. This work not only deepens the theoretical understanding of the EfN phenomenon but also highlights potential pitfalls in template matching techniques and emphasizes the need for careful interpretation of noisy observations across disciplines in engineering, statistics, physics, and biology.

Heavy vector boson dark matter at the TeV scale or higher may be produced non-thermally in a first-order phase transition taking place at a lower energy scale. While the production of vector dark matter has previously been studied for bubble wall collisions, here we calculate production by bubble wall expansion in a plasma, which can be the dominant production mechanism. We compute the results numerically and provide an analytical fit for the vector dark matter density. The numerical fit is also validated for scalar dark matter production, obtaining results in agreement with past literature. We find that vector pair production leads to bubble wall friction with a novel boost factor scaling behaviour compared to transition radiation emission of a single vector. We conclude that TeV-scale WIMP vector dark matter can be efficiently produced non-thermally by first-order phase transitions in a wide region of parameter space where thermal freeze-out is inefficient. In this scenario, the phase transition scale is predicted to be in the sub-GeV to $\mathcal{O}(10)$ TeV range and could therefore be accessible to future gravitational wave detectors.

In this paper, we prove new existence and multiplicity results for critical points of lower semicontinuous functionals in Banach spaces, complementing the nonsmooth critical point theory set forth by Szulkin and avoiding the need of the Palais-Smale condition. We apply our abstract results to get entire solutions with finite energy to Born-Infeld type autonomous equations. More precisely, under almost optimal conditions on the nonlinearity, we construct a positive solution and infinitely many solutions both in the classes of radially symmetric functions and nonradiallly symmetric ones.

The contact process is a simple model for the spread of an infection in a structured population. We investigate the case when the underlying structure evolves dynamically as a degree-dependent dynamical percolation model. Starting with a connected locally finite base graph we initially declare edges independently open with a connection probability that is allowed to depend on the degree of the adjacent vertices and closed otherwise. Edges are independently updated with a rate depending on the degrees and then are again declared open and closed with the same probabilities. We are interested in the contact process, where infections are only allowed to spread via open edges. Our aim is to analyse the impact of the update speed and the connection probability on the existence of a phase transition. For a general connected locally finite graph, our first result gives sufficient conditions for the critical value for survival to be strictly positive. Furthermore, in the setting of Bienaymé-Galton-Watson trees, we show that the process survives strongly with positive probability for any infection rate if the offspring distribution has a stretched exponential tail with an exponent depending on the connection probability and the update speed. In particular, if the offspring distribution follows a power law and the connection probability is given by a product kernel and the update speed exhibits polynomial behaviour, we provide a complete characterisation of the phase transition.

Several monads of probability measures have been shown to have presentations as codensity monads over small categories of stochastic maps. This paper studies how three key properties of these probability monads, relevant to categorical approaches to probability, can arise from their codensity presentations. We first derive the existence of a Kleisli law into the Giry monad, which provides a formal connection to measurable probability. In particular, from their codensity presentations, we prove a novel universal property of several probability monads as terminal liftings of the Giry monad. This generalises a result by Van Breugel on the Kantorovich monad, and proves the existence of such Kleisli laws. We additionally provide sufficient conditions for a codensity monad to be lax monoidal and affine, which provides a connection to the theory of Markov categories. In particular, we introduce the condition for a codensity monad to be exactly pointwise monoidal, which is then lax monoidal, and prove a characterisation of this condition in terms of Day convolution. We show that the Radon monad is exactly pointwise monoidal, and use our characterisation to give a description of the tensor product of free algebras of the Radon monad in terms of Day convolution. Finally, we show that the Giry monad is only exactly pointwise monoidal when restricted to standard Borel spaces, due to the existence of probability bimeasures that do not extend to measures.

We produce the gravitational waveforms for the extreme mass ratio inspiral systems (EMRIs) of binary stars moving around central supermassive black hole (SBH), or called B-EMRIs. We calculate the external orbits of the binary stars via the commonly used Hamilton-Jacobi (HJ) approach, and calculate the internal orbits of the binary stars via Lagrangian approach. To improve accuracy we adopt the quadrupole-octupole expression of gravitational wave (GW) and study the contribution of radiation reaction. Compared to the waveforms of EMRIs, there are higher frequency oscillations superposed on the waveforms of B-EMRIs. We perform frequency spectrum analysis of the GW waveforms, and find that higher frequency signals give their prominency in the waveforms of B-EMRIs. To obtain high precise result for future observation of GWs from space-based detector, we take into account gravito-electromagnetic (GEM) force, and compare the waveforms of B-EMRIs with GEM effects against those of B-EMRIs without GEM effects and against those of EMRIs. The result of mismatch shows that the waveforms of B-EMRIs are credibly distinguishable by the space-based GW detectors when GEM force is considered.

Quantum computing is a highly abstract scientific discipline, which, however, is expected to have great practical relevance in future information technology. This forces educators to seek new methods to teach quantum computing for students with diverse backgrounds and with no prior knowledge of quantum physics. We have developed an online course built around an interactive quantum circuit simulator designed to enable easy creation and maintenance of course material with ranging difficulty. The immediate feedback and automatically evaluated tasks lowers the entry barrier to quantum computing for all students, regardless of their background.

We prove a priori and a posteriori Hölder bounds and Schauder $C^{1,α}$ estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form $$ \mathrm{div}\left(|u|^a A\nabla w\right)=0\qquad\mathrm{in \ }Ω\subset\mathbb R^2,\quad a\in\mathbb R, $$ where the weight $u$ is itself a solution to an elliptic equation of the type $\mathrm{div}(A \nabla u) = 0$, with $A$ a Lipschitz-continuous, uniformly elliptic matrix. The function $u$ is allowed to have a nontrivial, possibly singular nodal set. The estimates are uniform with respect to $u$ within a class of normalized solutions having bounded Almgren frequency. In the special case $a = 2$, our results apply to the ratio of two solutions to the same elliptic equation sharing a common zero set. Precisely, we prove higher-order boundary Harnack principles on nodal domains, via the derived Schauder estimates for the associated degenerate equations. The results are based upon a fine blow-up argument, a Liouville theorem, and quasiconformal maps.

In functionally complex systems, higher-order connectivity is often revealed in the underlying geometry of networked units. Furthermore, such systems often show signatures of self-organized criticality, a specific type of non-equilibrium collective behaviour associated with an attractor of internal dynamics with long-range correlations and scale invariance, which ensures the robust functioning of complex systems, such as the brain. Here, we highlight the intertwining of features of higher-order geometry and self-organized critical dynamics as a plausible mechanism for the emergence of new properties on a larger scale, representing the central paradigm of the physical notion of complexity. Considering the time scale of the structural evolution with the known separation of the time scale in self-organized criticality, i.e., internal dynamics and external driving, we distinguish three classes of geometries that can shape the self-organized dynamics on them differently. We provide an overview of current trends in the study of collective dynamics phenomena, such as the synchronization of phase oscillators and discrete spin dynamics with higher-order couplings embedded in the faces of simplicial complexes. For a representative example of self-organized critical behaviour induced by higher-order structures, we present a more detailed analysis of the dynamics of field-driven spin reversal on the hysteresis loops in simplicial complexes composed of triangles. These numerical results suggest that two fundamental interactions representing the edge-embedded and triangle-embedded couplings must be taken into account in theoretical models to describe the influence of higher-order geometry on critical dynamics.

We show that every Hardy field extends to an $ω$-free Hardy field. This result relates to classical oscillation criteria for second-order homogeneous linear differential equations. It is essential in [10], and here we apply it to answer questions of Boshernitzan, and to generalize a theorem of his.

Given a smooth projective variety $X$ and a smooth nef divisor $D$, we identify genus zero relative Gromov--Witten invariants of $(X,D)$ with $(n+1)$ relative markings with genus zero orbifold Gromov--Witten invariants of multi-root stacks over the $\mathbb P^1$-bundle $P:=\mathbb P(\mathcal O_X(-D)\oplus \mathcal O_X)$ with $n$ orbifold markings. This is a generalization of the local-relative correspondence beyond maximal contacts. Repeating this process, we identify genus zero relative Gromov--Witten invariants of ambient insertions with genus zero absolute Gromov--Witten invariants of toric bundles. We also present how this correspondence can be used to compute genus zero two-point relative Gromov--Witten invariants.

Many bacteria share the fascinating ability to sense Earth's magnetic field -- a process known as magnetotaxis. These bacteria synthesize magnetic nanoparticles, called magnetosomes, within their own cell body and arrange them to form a linear magnetic chain. The chain, which behaves like a compass needle, aligns the microorganisms with the geomagnetic field. Here, we measure the magnetic hysteresis of an individual bacterium of the species Magnetospirillum gryphiswaldense via ultrasensitive torque magnetometry. These measurements, in combination with transmission electron microscopy and micromagnetic simulations, reveal the magnetic configurations of the magnetosomes, their progression as a function of applied field, as well as the total remanent magnetic moment and effective magnetic anisotropy of a chain within a single bacterium. Knowledge of magnetic properties is crucial both for understanding the mechanisms behind magnetotaxis and for the design of systems exploiting magnetotactic bacteria in biomedical applications.

Multimodal datasets, where measurements are obtained from multiple sensors, have become central to many scientific domains. In unsupervised settings, most representation learning methods focus on identifying shared latent structures, such as clusters or continuous processes that appear across modalities. However, some aspects of the data may be observed only through a single modality. For example, in computational biology, certain cell-subtypes may appear in genetic profiles but not in epigenetic markers. In this paper, we present DELVE, a spectral method for extracting modality-specific (differential) latent variables. Our approach constructs a graph for each modality and leverages differences in their connectivity patterns to design a graph filter that attenuates shared signals while preserving modality-specific components. We provide an asymptotic convergence analysis for our method under a product manifold model. To evaluate the performance of our method, we test its ability to recover differential latent structures in several synthetic and real datasets.

In this study, we explore the geometric construction of the Klein bottle and the real projective plane ($\mathrm{RP}^2$) within the framework of tensor networks, focusing on the implementation of crosscap and rainbow boundaries. Previous investigations have applied boundary matrix product state techniques to study these boundaries. We introduce an approach that incorporates such boundaries into the tensor renormalization group methodology, facilitated by an efficient representation of a spatial reflection operator. This advancement enables us to compute the crosscap and rainbow free energy terms and the one-point function on $\mathrm{RP}^2$ with enhanced efficiency and for larger system sizes. Additionally, our method is capable of calculating the partition function under isotropic conditions of space and imaginary time. The versatility of this approach is further underscored by its applicability to constructing other (non)orientable surfaces of higher genus.

The Learning with Errors (\LWE) problem has been widely utilized as a foundation for numerous cryptographic tools over the years. In this study, we focus on an algebraic variant of the \LWE problem called \emph{Group ring} \LWE ($\GRLWE$). We select group rings (or their direct summands) that underlie specific families of finite groups constructed by taking the semi-direct product of two cyclic groups. Unlike the Ring-\LWE problem described in \cite{lyubashevsky2010ideal}, the multiplication operation in the group rings considered here is non-commutative. As an extension of Ring-$\LWE$, it maintains computational hardness and can be potentially applied in many cryptographic scenarios. In this paper, we present two polynomial-time quantum reductions. Firstly, we provide a quantum reduction from the worst-case shortest independent vectors problem (\SIVP) in ideal lattices with polynomial approximate factor to the search version of $\GRLWE$. This reduction requires that the underlying group ring possesses certain mild properties; Secondly, we present another quantum reduction for two types of group rings, where the worst-case \SIVP problem is directly reduced to the (average-case) decision $\GRLWE$ problem. The pseudorandomness of $\GRLWE$ samples guaranteed by this reduction can be consequently leveraged to construct semantically secure public-key cryptosystems.

We propose a simple approach that provides accurate uncertainty quantification for Bayesian inference in misspecified or approximate models, and for generalized (Gibbs) posteriors. While existing solutions in this context are based on explicit Gaussian approximations or post-processing procedures, we demonstrate that correct uncertainty quantification can be achieved by substituting the usual posterior with an intuitively appealing alternative that conveys the same information. This solution applies to both likelihood-based and loss-based posteriors, and is formally demonstrated to reliably quantify uncertainty. This new approach is demonstrated through a range of examples, including generalized linear models, and doubly intractable models.

Collective migration is an important component of many biological processes, including wound healing, tumorigenesis, and embryo development. Spatial agent-based models (ABMs) are often used to model collective migration, but it is challenging to thoroughly predict these models' behavior throughout parameter space due to their random and computationally intensive nature. Modelers often coarse-grain ABM rules into mean-field differential equation (DE) models. While these DE models are fast to simulate, they suffer from poor (or even ill-posed) ABM predictions in some regions of parameter space. In this work, we describe how biologically-informed neural networks (BINNs) can be trained to learn interpretable BINN-guided DE models capable of accurately predicting ABM behavior. In particular, we show that BINN-guided partial DE (PDE) simulations can 1.) forecast future spatial ABM data not seen during model training, and 2.) predict ABM data at previously-unexplored parameter values. This latter task is achieved by combining BINN-guided PDE simulations with multivariate interpolation. We demonstrate our approach using three case study ABMs of collective migration that imitate cell biology experiments and find that BINN-guided PDEs accurately forecast and predict ABM data with a one-compartment PDE when the mean-field PDE is ill-posed or requires two compartments. This work suggests that BINN-guided PDEs allow modelers to efficiently explore parameter space, which may enable data-driven tasks for ABMs, such as estimating parameters from experimental data. All code and data from our study is available at https://github.com/johnnardini/Forecasting_predicting_ABMs.

Quadratization refers to a transformation of an arbitrary system of polynomial ordinary differential equations to a system with at most quadratic right-hand side. Such a transformation unveils new variables and model structures that facilitate model analysis, simulation, and control and offers a convenient parameterization for data-driven approaches. Quadratization techniques have found applications in diverse fields, including systems theory, fluid mechanics, chemical reaction modeling, and mathematical analysis. In this study, we focus on quadratizations that preserve the stability properties of the original model, specifically dissipativity at given equilibria. This preservation is desirable in many applications of quadratization including reachability analysis and synthetic biology. We establish the existence of dissipativity-preserving quadratizations, develop an algorithm for their computation, and demonstrate it in several case studies.

This paper establishes a complete time-harmonic scattering theory for hyperbolic space, formulating it within the classical Sommerfeld-Rellich paradigm centered on far-field patterns--a foundational framework that has been absent despite the well-developed spectral and time-dependent theories for this geometry. We explicitly construct the ingoing and outgoing fundamental solutions for the Helmholtz operator and perform a precise asymptotic analysis at the conformal boundary to derive a {hyperbolic Sommerfeld radiation condition}. This condition, which is a local criterion at infinity, uniquely selects physically admissible outgoing solutions. We prove a {hyperbolic Rellich theorem} guaranteeing the uniqueness of the scattered field and its far-field pattern from asymptotic data. Within this rigorous framework, we solve the direct scattering problem for compact sources, penetrable media, and impenetrable obstacles, providing explicit representations for the corresponding {far-field patterns}. As a principal application and demonstration of the framework's utility, we initiate the study of inverse scattering on hyperbolic space, formulating both the inverse obstacle and inverse medium problems where the objective is to reconstruct the scatterer from measurements of its far-field pattern. Our work lays the groundwork for a far-field-based approach to scattering and inversion in hyperbolic geometry. Due to the local feature of the {hyperbolic Sommerfeld radiation condition}, our study can be readily extended to the broader asymptotically hyperbolic manifolds.

It was proved by Maksimova in 1977 that exactly eight varieties of Heyting algebras have the amalgamation property, and hence exactly eight axiomatic extensions of intuitionistic propositional logic have the deductive interpolation property. The prevalence of the deductive interpolation property for axiomatic extensions of substructural logics and the amalgamation property for varieties of pointed residuated lattices, their equivalent algebraic semantics, is far less well understood, however. Taking as our starting point a formulation of intuitionistic propositional logic as the full Lambek calculus with exchange, weakening, and contraction, we investigate the role of the exchange rule--algebraically, the commutativity law--in determining the scope of these properties. First, we show that there are continuum-many varieties of idempotent semilinear residuated lattices that have the amalgamation property and contain non-commutative members, and hence continuum-many axiomatic extensions of the corresponding logic that have the deductive interpolation property in which exchange is not derivable. We then show that, in contrast, exactly sixty varieties of commutative idempotent semilinear residuated lattices have the amalgamation property, and hence exactly sixty axiomatic extensions of the corresponding logic with exchange have the deductive interpolation property. From this latter result, it follows also that there are exactly sixty varieties of commutative idempotent semilinear residuated lattices whose first-order theories have a model completion.

Inspired by [Pan22], we give a new proof that for an overconvergent modular eigenform $f$ of weight $1+k$ with $k\in\mathbb{Z}_{\ge1}$, assuming that its associated global Galois representation $ρ_{f}$ is irreducible, then $f$ is classical if and only if $ρ_{f}$ is de Rham at $p$. For the proof, we prove that theta operator $θ^{k}$ coincides with Fontaine operator in a suitable sense.

We give a complete description of the Poisson boundary of wreath products $A\wr B= \bigoplus_{B} A\rtimes B$ of countable groups $A$ and $B$, for probability measures $μ$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $μ$ to $B$ is Liouville, we prove that the Poisson boundary of $(A\wr B,μ)$ is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich, and Lyons-Peres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $μ$ with a finite first moment.

We use kinematic data of proper motions from Gaia of forty-two globular and open clusters from Large Magellanic Cloud (LMC) to explore the possibility of them having extragalactic origins. We find the difference between the proper motions of cluster stars and a surrounding patch of young LMC stars in each case. We find five globular clusters towards the north-east showing a high difference (> 0.11 mas/yr, or > 25 km/s). We also examine the statistical significance of this difference taking into account both measurement errors of cluster and surrounding stars as well as inherent dispersion of stellar motions in the local galactic environment. The five globular clusters (NGC 2005, NGC 2210, NGC 1978, Hodge 3 and Hodge 11) have mean proper motions that lie outside the 85% confidence interval of the mean of surrounding young stars, with a clear outlier (NGC 1978 outside 99.96% confidence) whose difference cannot be accounted for by statistical noise. A young cluster (NGC 2100) also fitting the criteria is ruled out owing to contrary evidence from literature. This indicates a possible interaction with a dwarf galaxy resulting in the accretion/disruption in path of the five globular clusters, or possibly one or more past merger(s) of smaller galaxy/galaxies with LMC from its north-eastern region. This direction also coincides with the location of Tarantula Nebula, suggesting the possibility of the interaction event or merger having triggered its star formation activity.

Large Language Models (LLMs) undergo continuous updates to improve user experience. However, prior research on the security and safety implications of LLMs has primarily focused on their specific versions, overlooking the impact of successive LLM updates. This prompts the need for a holistic understanding of the risks in these different versions of LLMs. To fill this gap, in this paper, we conduct a longitudinal study to examine the adversarial robustness -- specifically misclassification, jailbreak, and hallucination -- of three prominent LLM families: GPT, Llama, and Qwen. Our study reveals that LLM updates do not consistently improve adversarial robustness as expected. For instance, a later version of GPT-3.5 degrades regarding misclassification and hallucination despite its improved resilience against jailbreaks. GPT-4 and GPT-4o demonstrate (incrementally) higher robustness overall. Larger Llama and Qwen models do not uniformly exhibit improved robustness across all three aspects studied. In addition, larger model sizes do not necessarily yield improved robustness. Minor updates lacking substantial robustness improvements can exacerbate existing issues rather than resolve them. We hope our study can offer valuable insights into navigating model updates and informed decisions in model development and usage.

The goal of influence maximization (IM) is to select a small set of seed nodes which maximizes the spread of influence on a network. In this work, we propose BOPIM, a Bayesian Optimization (BO) algorithm for IM on temporal networks. The IM task is well-suited for a BO solution due to its expensive and complicated objective function. There are at least two key challenges, however, that must be overcome, primarily due to the inputs coming from a cardinality-constrained, non-Euclidean, combinatorial space. The first is constructing the kernel function for the Gaussian Process regression. We propose two kernels, one based on the Hamming distance between seed sets and the other leveraging the Jaccard coefficient between node's neighbors. The second challenge is the acquisition function. For this, we use the Expected Improvement function, suitably adjusting for noise in the observations, and optimize it using a greedy algorithm to account for the cardinality constraint. In numerical experiments on real-world networks, we prove that BOPIM outperforms competing methods and yields comparable influence spreads to a gold-standard greedy algorithm while being as much as ten times faster. In addition, we find that the Hamming kernel performs favorably compared to the Jaccard kernel in nearly all settings, a somewhat surprising result as the former does not explicitly account for the graph structure. Finally, we demonstrate two ways that the proposed method can quantify uncertainty in optimal seed sets. To our knowledge, this is the first attempt to look at uncertainty in the seed sets for IM.

We study free open fermionic strings on a non-commutative phase space. Modified super-Virasoro algebras in both Ramond and Neveu-Schwarz sectors acquire non-commutativity anomalies, and this noncommutativity also breaks Lorentz symmetry and give a non-diagonal mass operator. Redefining the Fock space diagonalizes the mass operator. Extra constraints on non-commutativity parameters cancel the anomalies, restore the standard spectrum and make the GSO projection possible.

The hidden geometry of simplicial complexes can influence the collective dynamics of nodes in different ways depending on the simplex-based interactions of various orders and competition between local and global structural features. We study a system of phase oscillators attached to nodes of 4-dimensional simplicial complexes and interacting via positive/negative edges-based pairwise $K_1$ and triangle-based triple $K_2\geq 0$ couplings. Three prototypal simplicial complexes are grown by aggregation of 5-cliques, controlled by the chemical affinity parameter $ν$, resulting in sparse, mixed, and compact architecture, all of which have 1-hyperbolic graphs but different spectral dimensions. By changing the interaction strength $K_1\in[-4,2]$ along the forward and backward sweeps, we numerically determine individual phases of each oscillator and a global order parameter to measure the level of synchronisation. Our results reveal how different architectures of simplicial complexes, in conjunction with the interactions and internal-frequency distributions, impact the shape of the hysteresis loop and lead to patterns of locally synchronised groups that hinder global network synchronisation.

Interdisciplinary research has emerged as a hotbed for innovation and a key approach to addressing complex societal challenges. The increasing dominance of grant-supported research in shaping scientific advances, coupled with growing interest in funding interdisciplinary work, raises fundamental questions about the effectiveness of interdisciplinary grants in fostering high-impact interdisciplinary research outcomes. Here, we quantify the interdisciplinarity of both research grants and publications, capturing 350,000 grants from 164 funding agencies across 26 countries and 1.3 million papers that acknowledged their support from 1985 to 2009. Our analysis uncovers two seemingly contradictory patterns: Interdisciplinary grants tend to produce interdisciplinary papers, which are generally associated with high impact. However, compared to disciplinary grants, interdisciplinary grants on average yield fewer papers and interdisciplinary papers they support tend to have substantially reduced impact. We demonstrate that the key to explaining this paradox lies in the power of disciplinary grants in propelling high-impact interdisciplinary research. Specifically, our results show that highly interdisciplinary papers supported by deeply disciplinary grants garner disproportionately more citations, both within their core disciplines and from broader fields. Moreover, disciplinary grants, particularly when combined with other similar grants, are more effective in producing high-impact interdisciplinary research. Amidst the rapid rise of support for interdisciplinary work across the sciences, these results highlight the hitherto unknown role of disciplinary grants in driving crucial interdisciplinary advances, suggesting that interdisciplinary research requires deep disciplinary expertise and investments.