We implemented ab initio Hubbard parameter calculation schemes in the k-point sampling real-time TDDFT (RT-TDDFT) program in CP2K. We propose a new linear-response-based calculation scheme for energy-dependent Hubbard parameters. Our scheme extends the minimum-tracking linear-response method proposed in [Moynihan et al., arXiv preprint arXiv:1704.08076(2017); E. B. Linscott et al., Phys. Rev. B 98, 235157 (2018)] to realize the calculation of energy-dependent Hubbard parameters that reflect the exchange-correlation (xc) effects included in the xc-functional. We discuss the properties of the minimum-tracking linear-response method in comparison to another promising scheme, ACBN0 [Agapito et al., Phys. Rev. X, 5, 011006 (2015)]. We show that, while neither clearly outperforms the other in the accuracy of static property calculations, each has a distinct dynamical application depending on its theoretical formulation.

Data centers (DCs) are emerging as large, geographically distributed, controllable loads whose participation in electricity markets can significantly affect grid operation, especially when cloud platforms shift workloads across sites to exploit energy-arbitrage opportunities. This paper analyzes and seeks to mitigate the grid impacts of geographically distributed multi-site DCs under exogenous electricity prices. It develops a detailed job-level scheduling framework for market-driven DCs, formulated as a mixed-integer model that preserves execution logic and captures a unified set of implementable control actions. It also incorporates service-side quality-of-service (QoS) constraints and penalty terms to improve fidelity. Case studies on a modified IEEE 14-bus system, complemented by a more realistic network based on Travis County, Texas, show that purely price-driven scheduling improves economic performance, but also increases voltage-security risk and congestion exposure by inducing localized demand concentration and sharp site-level load variation. To mitigate these effects, this work introduces load-redistribution policies that curb extreme load shifting and support grid operators in managing such conditions.

Optically levitated micro- and nanoparticles are an ideal optomechanical platform for precision measurements, particularly enabling the detection of ultraweak forces. Nevertheless, quantum backaction and inherent instabilities induced by the trapping laser fundamentally restrict further improvements in force sensitivity and resolution. To circumvent these bottlenecks, we actively drive the levitated nanoparticle's mechanical motion in a phase-locked phonon laser mode and integrate a carrier-modulation measurement architecture to enhance force sensing capabilities. The stable and high-amplitude oscillation of the phonon laser allows for the robust trapping under 1 mW-level laser power, which in turn reduces the force noise to 4.0(3)*10^-22 N/Hz^1/2. Furthermore, by using phase-locked phonon laser, the measurement system achieves active stabilization and extended coherence time with the measured signal to 12,500 seconds, realizing a measurement resolution of 8(4)*10^-24 N with a sensitivity of 9.3(7)*10^-22 N/Hz^1/2 under a loaded force. These results establish the phonon laser as a low-noise, long-coherence-time, self-stabilizing platform for precision measurements, as well as in quantum and fundamental physics tests.

Blazars, the jet dominated class of AGN comprising flat spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs) are now increasingly identified as potential sources of high energy neutrinos. Such neutrino blazars are ideal targets to investigate the high energy emission processes and to understand their role as neutrino sources. We report results on four candidate neutrino blazars, PKS 0446+112, TXS 0506+056, PKS 1424$-$418 and PKS 1502+106. We carried out $γ$-ray spectral and timing analysis on three time periods that comprise a quiescent epoch, an epoch that corresponds to neutrino detection and a flaring epoch. We also carried out modeling of the broadband pectral energy distribution (SED) on those three epochs. We found that the $γ$-ray spectra of the BL Lac TXS 0506+056 can be adequately described by a power-law, while the spectra of the other three FSRQs require a log-parabola model. On shorter timescales, we observed flux variability with doubling/halving timescales of 4.70 hrs, 9.24 hrs, 30.76 hrs and 15.42 hrs for PKS 0446+112, TXS 0506+056, PKS 1424$-$418 and PKS 1502+106, respectively. The SEDs of most of the epochs for the sources are well explained by a leptonic scenario. However, the quiescent epoch of PKS 1502+106 and the neutrino-emission epoch of PKS 0446+112 required an additional hadronic component to reproduce the observed SEDs. Our analysis reveals a complex interplay of leptonic and hadronic processes. While certain neutrino-associated epochs align with a leptonic model, others necessitate a hadronic component to explain the emission features.

In this paper, we define a new class of distributed tasks, called SOS tasks (for Set of Output Sets tasks), defined by the set $O$ of distinct output sets of values that can be produced. We then demonstrate that this class of tasks is decidable: there exists an effective procedure that determines whether any SOS task is solvable asynchronously under $t$ crashes. The decision rule is as follows. Every SOS task is solvable when $t=0$. For $t > 0$, an SOS task is solvable if and only if its SOS graph $G=(O,\subset)$ is connected. In this graph, each vertex is an output set in $O$, and two vertices are linked by an edge whenever one output set includes the other. One of the surprising implications of our results is that, without a validity property, $k$-set agreement is solvable under any number of crashes $t \geq 0$ for $k>1$, and unsolvable under $t >0$ crashes only for $k=1$ (consensus). Finally, we study a novel family of tasks called $d$-disagreement, which requires the system to always produce $d$ different output values, and we show that its implementability condition is related to the harmonic series.

We introduce a variational formulation of the homogeneous Boltzmann equation, with hard-sphere cross section, which selects the unique energy conserving solution. We prove that this solution arises from the microscopic dynamics, namely Kac's walk, and we establish the propagation in time of entropic chaoticity, under the minimal assumption that the initial distribution is entropically chaotic.

We develop a novel multi-layer predictor-feedback to achieve exact compensation of state-dependent input delay of general nonlinear integro-differential equations. The system of interest is an unconventional mixed Partial Differential Equation (PDE)-Ordinary Differential Equation (ODE) system, in which a nonlinear ODE is actuated through an inhomogeneous advection PDE. Moreover, the propagation speed of the PDE depends on a moving window integral of the ODE state. The two above features are not addressed yet in standard PDE backstepping-based predictor-feedback designs. Unlike the conventional Lyapunov-based approaches used in the field, our stability and well-posedness analysis rely on the characteristic method and a fixed-point argument. Both of our designs achieve global asymptotic stability (GAS) in the supremum norm of the PDE and ODE states under the mild assumption that the nonlinearity in the PDE governing equation is uniformly Lipschitz continuous. The transport speed, governed by the integral of the ODE state, models systems such as production or queuing processes in which the state of a finite buffer-namely, the inventory level-determines the production or service rate. Numerical simulations demonstrate the effectiveness of the proposed control design for buffer-regulated production lines and queuing systems, ensuring asymptotic stability under a locally safe softened bang-bang feedback law that preserves the positivity of both the system state and the actuation signal.

In qualitative statistics, permutation tests are very popular, mainly because of their finite-sample exactness under exchangeability. However, in non-exchangeable settings, the covariance structure of permuted statistics typically differs from that of the original statistic. A common solution is studentization, which restores asymptotic correctness for general hypotheses while preserving exactness under exchangeability. In multiple testing settings, however, standard studentization fails to provide the correct joint limiting distribution. Existing solutions such as prepivoting address this issue but are computationally expensive and therefore rarely used in practice. We propose a general, computationally more efficient methodology that overcomes this fundamental limitation. By appropriately correcting the covariance matrix of multiple permutation statistics, our approach restores the correct joint asymptotic dependence structure, enabling asymptotically valid permutation tests in broad multiple testing frameworks. The proposed method is highly flexible: it accommodates singular covariance structures and is not tied to specific parameters, test statistics, or permutation schemes. This generality makes it applicable across a wide range of problems. Extensive simulation studies demonstrate that our approach results in reliable inference and outperforms existing methods across diverse settings.

We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations, i.e. equations that change their nature from elliptic to hyperbolic through the computational domain. Well-posedness of the discrete problem is established via coercivity in an energy norm, achieved through the Morawetz multiplier technique. We derive $hp$-a priori error estimates in the energy norm, which we use to prove convergence rates for standard and quasi-Trefftz polynomial spaces. Numerical experiments validate the theoretical results.

The Krasnosel'ski\uı-Mann algorithm is a well-known method for finding fixed points of a nonexpansive mapping with strong theoretical guarantees. However, there are practical large-scale problems to which this algorithm cannot be applied. Here, to resolve the issue caused by the computational difficulty of the mapping, we define a computable mini-batch stochastic mapping, which is a unbiased estimator of the nonexpansive mapping, and implement it in the Krasnosel'ski\uı-Mann algorithm. We show that the algorithm with increasing batch sizes converges almost surely to a fixed point of the nonexpansive mapping. We also perform a convergence rate analysis on the algorithm.

Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.

Graph theory provides powerful tools for modeling concepts in number theory, leading to the introduction of graphs derived from arithmetic properties. One such structure is the divisor prime graph, $G_{Dp(n)}$. For any positive integer $n$, let $D(n)$ be the set of its positive divisors. The vertex set of $G_{Dp(n)}$ consists of the elements of $D(n)$, with the adjacency condition that two vertices $x$ and $y$ share an edge if and only if their greatest common divisor is $1$. The primary focus of this study is to evaluate the topological characteristics of $G_{Dp(n)}$. To achieve this, we analyze and compute various distance and degree-based indices, specifically focusing on the Wiener, Harary, hyper-Wiener, First and Second Zagreb, Schultz, Gutman, and Eccentric connectivity indices.

In this paper, we study the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data. Our contributions are twofold: first we introduce a Dirichlet-to-Neumann map for this operator and analyze an associated inverse problem; and second we establish an additional density result for the spectral fractional Laplacian.

Large language models (LLMs) are increasingly applied in computer science education for tasks such as tutoring, content generation, and code assessment. However, systematic evaluations aligned with formal curricula and certification standards remain limited. This study benchmarked four recent models, including GPT-5, DeepSeek-R1, Qwen-Plus, and Llama-3.3-70B-Instruct, using a dataset of 1,068 questions derived from six certification exams covering networking, office applications, and Java programming. We evaluated performance across language (Chinese vs. English), cognitive levels based on Bloom's Taxonomy, domain knowledge, confidence-accuracy alignment, and robustness to input masking. Results showed that GPT-5 performed best on English-language certifications, while Qwen-Plus performed better in Chinese contexts. DeepSeek-R1 achieved the most balanced cross-lingual performance, whereas Llama-3.3 showed clear limitations in higher-order reasoning and robustness. All models performed worse on more complex tasks. These findings provide empirical support for the integration of LLMs into computer science education and offer practical implications for curriculum design and assessment.

Exciton dissociation via the excitonic Mott transition (EMT) governs the high-density optical response of semiconductors and sets fundamental limits for optoelectronic devices. The EMT is conventionally linked to the onset of population inversion and the emergence of optical gain. Here, we demonstrate that this paradigm can break down under ultrafast non-equilibrium excitation. Using femtosecond pump-probe optical spectroscopy, we drive a monolayer transition metal dichalcogenide into a dense photoexcited state in which the excitonic resonance is completely quenched within ~100 fs, while the optical gain is entirely absent across the explored fluence range. State-of-the-art real-time ab initio simulations reveal that the EMT is governed by an interplay of strongly nonthermal carrier populations and nonequilibrium dynamical screening of the Coulomb interaction. The quantitative agreement between theory and experiment identifies a distinct, ultrafast pathway to exciton ionization beyond quasi-equilibrium descriptions and demonstrates that population inversion is not a universal prerequisite for the EMT.

Many complex systems exhibit interactions that depend not only on pairwise connections, but also group structures and memory effects. To capture such effects, we develop a unified tensor framework for modeling higher-order Markov chains with memory. Our formulation introduces an even-order paired tensor that links folded and unfolded dynamics and characterizes their steady states and convergence. We further show that a Markov chain with memory can be approximated by a low-dimensional nonlinear tensor-based system and then provide a full system analysis. As an application, we define random walks on hypergraphs where memory naturally arises from the hyperedge structure, providing new tools for analyzing higher-order networks with time-dependent effects.

Understanding regional Consumer Price Index (CPI) dynamics is essential for timely and effective economic policymaking. However, traditional modeling procedures typically rely only on parametric panel modeling with low-frequency and high-cost macroeconomic indicators, which often fail to capture rapid market fluctuations and lead to inaccurate predictions. To this end, we propose a residual-joint-modeling framework that integrates large language model (LLM) analyses and social media narratives via a new deep neural network based panel modeling. Specifically, we construct a large narrative corpus from a newly collected {\it Sina Weibo} dataset, and develop a prompt-based GPT model and a series of fine-tuned BERT models to generate high-frequency LLM-induced surrogates for regional CPI. A novel joint modeling strategy is then advocated to transfer the information from these surrogates to the target regional CPI data and hence empower CPI prediction. To solve the joint objectives, we further introduce a new deep panel learning procedure with region-wise homogeneity pursuit, which has its own significance in panel data analysis literature. In addition, conformal-based panel prediction intervals are provided to quantify the uncertainty of the LLM-powered prediction. The proposed approach significantly reduces short-term forecasting errors and more effectively captures abrupt inflationary shifts compared to traditional econometric models. While demonstrated for regional CPI forecasting, the proposed framework is broadly applicable for incorporating insights from LLMs to enhance traditional statistical modeling.

Lattice QCD (LQCD) calculations play a key role in the establishment of flavor anomalies. One of the most recent advancements in LQCD to this end has been the publication of several calculations of the $B\to D^\ast\ellν$ form factors, but despite all the anticipation, the LQCD results have been unable to give a final answer to the questions it was destined to answer. In this work I briefly review what is the current status of heavy-to-heavy and heavy-to-light semileptonic decays calculations in LQCD, and what we can expect for the near and not-so-near future.

The quantum nature of gravity remains experimentally unverified, despite recent proposals to probe it using tabletop experiments such as gravity-mediated entanglement schemes. In parallel, consistent formulations of classical--quantum dynamics have been developed as alternative descriptions of gravity, in which quantum matter interacts with a classical mediator assumed to be fundamentally classical. In this work, we show that classical--quantum dynamics arise generically as an effective description of fully quantum systems under decoherence, providing a bridge between fully quantum and classical--quantum dynamics. We derive the reduced dynamics, which are generically non-Markovian, using an explicit hidden model in which the mediator is coupled to unobserved environmental degrees of freedom. We identify a concrete criterion for when a classical--quantum interpretation is valid: the semi-Wigner operator associated with the mediator sector must remain positive semidefinite, which can be expressed as a positivity condition on nonlocal kernels governing the evolution. In the short-memory limit, the reduced evolution reproduces Markovian classical--quantum dynamics of Oppenheim and collaborators. Our results imply that a classical mediator can arise effectively from decohered quantum dynamics, so that experimental agreement with classical-quantum models does not uniquely determine whether the mediator is fundamentally classical.

We investigate the nanoscale friction behaviour of MX2 monolayers (M = Mo, W; X = S, Se) on Au(111) and Ag(111) substrates with a silicon tip using classical molecular dynamics simulations with machine-learning-based force fields. This approach enables an accurate description of tip-surface interactions and friction mechanisms at the atomic scale. We observe a pronounced non-monotonic dependence of the friction force on the applied normal load, indicating a breakdown of Amontons's law at the nanoscale. Analysis of lateral force' signals and their spatial Fourier transforms reveals the coexistence of multiple sliding modes, including longitudinal sliding, lateral slip, and zig-zag motions. We show that the overall friction response is governed by the relative contributions of these motions. While the qualitative features of friction are largely substrate-independent, both the magnitude of friction and the balance between sliding modes depend sensitively on the substrate-monolayer combination. In particular, Au/MoSe2/Si exhibits significantly reduced friction due to suppression of lateral slip motion. Our results indicate that the method is broadly applicable for probing nanoscale friction in related heterostructures.

In the short block length regime, ensemble decoding schemes with their inherently parallel structure can improve error correction performance and reduce latency compared to stand-alone suboptimal decoders such as belief propagation (BP). In this work, we introduce affine subcode ensemble decoding (aSCED), which uses an ensemble of decoders operating on linear block codes and both linear and strictly affine subcodes. This generalizes the recently proposed subcode ensemble decoding (SCED), which is restricted to linear subcodes. We derive BP update rules for affine subcodes and show that aSCED simplifies ensemble design compared to SCED, multiple bases BP, and automorphism ensemble decoding. Monte-Carlo simulations of two low-density parity-check codes and two Bose-Chaudhuri-Hocquenghem (BCH) codes demonstrate improved error correction performance of aSCED over competing existing ensemble schemes. Notably, for one BCH code, when combining ensemble design with algorithms for constructing high-performance parity-check matrices, aSCED achieves near-maximum likelihood performance using only 64 BP decoding paths.

We present a low-redshift-agnostic compression of anisotropic baryon acoustic oscillation (BAO) distances to constrain the normalized dark-energy density evolution, $X(z)\equiv ρ_{\rm DE}(z)/ρ_{\rm DE}(0)$, above the lowest BAO redshift node $z_1$. Standard BAO summaries include the transverse comoving distance $D_{\rm M}/r_{\rm d}$, which depends on the integral of $H^{-1}(z)$ from $z=0$ to $z$ and therefore mixes the expansion history at $z<z_1$ with the higher-redshift signal. We instead replace the set $D_{\rm M}(z_i)/r_{\rm d}$ by adjacent increments $ΔD_{\rm M}(z_i,z_{i+1})/r_{\rm d}$ while retaining the radial distances $D_{\rm H}(z_i)/r_{\rm d}$. The mapping is linear, so the covariance propagates exactly. This compression intentionally removes one absolute transverse-distance mode, namely the additive contribution to $D_{\rm M}/r_{\rm d}$ below the first BAO node, and preserves the remaining information relevant to reconstructing the expansion history above $z_1$. Applied to DESI DR1 and DR2 anisotropic BAO measurements, the method yields almost uncorrelated constraints on piecewise-constant interval parameters $X_j$. In this sense, the compressed likelihood provides a conservative band-power-like estimate of dark-energy evolution: each interval is constrained mainly by BAO information from its own redshift range, while one nonlocal transverse mode and stronger global assumptions are deliberately projected out or marginalized over. Because our baseline analysis also marginalizes over bin-local matter-density and distance-scale parameters with broad external priors, the resulting $X_j$ constraints should be interpreted as a low-redshift-agnostic BAO baseline rather than as a fully prior-free reconstruction. All bins are consistent with $X=1$ within current uncertainties.

The unique combination of outstanding optical quality and attractive spin properties opens new avenues for optical spin control in hybrid organic-inorganic perovskite semiconductors. Using the optically detected magnetic resonance technique, we study the spins of electrons and holes in mixed-cation MA$_x$FA$_{1-x}$PbI$_3$ single crystals with $x = 0.4$ and 0.8. Multiple distinct spin subensembles with $g$-factors spanning from 2.9 to 3.6 for electrons and from 0.5 to 1.2 for holes are resolved, revealing diverse localization environments. We measure the longitudinal spin relaxation times, $T_1$, reaching 2 ms and remaining in the $μ$s range even for weakly localized carriers at the cryogenic temperature of 1.6 K. The magnetic-field dependence of $T_1$ is dominated by the random nuclear (Overhauser) fields with strengths of $\sim 0.4-0.8$ mT for electrons and $\sim 4-12$ mT for holes, corresponding to $μ$s-long correlation times of the hyperfine field determined by carrier hopping between shallow localization sites. The temperature dependence of $T_1$ reveals a weak localization potential of the charge carriers and shows a correlation between $T_1$ and the inhomogeneity of the spin ensemble. These results establish mixed-A-site perovskite single crystals as a promising solid-state platform with long-lived spin states for quantum information applications.

The present manuscript consists of inverse problems for a coupled system of wave equations with potential in $\mathbb{R}^3$. By establishing the fundamental solution to the aforementioned operator, we study the uniqueness aspects of the inverse problem of recovering the matrix-valued potential coefficient from time-dependent measurements. We consider these inverse problems in two different cases: (i) the {\it coincident} setup, where the source and receiver are located at a single point, and (ii) the {\it non-coincidence or separated} setup, in which case source and receiver are situated at distinct locations. The problems considered here are under-determined; hence, some additional assumptions for the potential are expected to guarantee the uniqueness of the inverse problems considered in this article. We proved the desired uniqueness results under some extra assumptions on the coefficients.

We present the status of calculations of the form factors of the most relevant heavy-to-heavy and heavy-to-light decay channels. Using seven $N_f = 2+1+1$ HISQ ensembles, with lattice spacings ranging from 0.15 fm down to 0.06 fm, we calculate the form factors of the decays, including correlations among them. More than half of our ensembles feature physical pion masses, and the heavy quarks are simulated at their physical masses using the Wilson-clover action with the Fermilab interpretation. Even though we have recently seen huge qualitative and quantitative leaps in the characterization of heavy-to-heavy decays, these advances have failed to translate into improvements for the inclusive vs exclusive question, or the matter of the Lepton Flavor Universality ratios. In particular, in the $B\to D^{\ast}\ellν$ channel, the current situation of the lattice-QCD form factors is far from clear. Further, the latest lattice-QCD results on the heavy-to-light form factors display unexplained tensions that must urgently be resolved. The work presented here is an attempt to address these issues.

In Milner's seminal book on communication and concurrency introducing CCS, a process algebra inherently non-deterministic, chapter 11 was completely devoted to introduce the notion of determinacy and confluence in order to identify a subcalculus of CCS in which all definable agents are confluent. At the same time, or shortly later, determinate semantics were given for programming languages that reconcile concurrency and determinacy, such as Esterel by Berry and Gonthier, or SL by Boussinot and de Simone. These dedicated semantics do not easily map to Milner's confluence theory for CCS, which is unable to express causality and shared memory multi-threading with reaction to absence in a compositional way. We present an extension of CCS with priority-guarded actions and clocks, and we exploit the added expressiveness to enrich Milner's original notion of confluence by the new concept of coherence which permits us to encode, in a compositional fashion, synchronous programming languages such as Esterel.

Modern web applications combine persistent state updates, concurrent interactions, and unreliable communication with external services. Failures such as timeouts can occur after partial state changes, producing temporary inconsistencies whose resolution depends on liveness properties that are often not verified in practice. Although formal methods offer rigorous guarantees for reasoning about complex software, they remain rarely adopted in enterprise settings due to their perceived complexity and lack of practical automation. Multiparty Session Types (MPST) offer strong guarantees for communication safety, yet they do not account for the interplay between state evolution, dynamic workflow structure, and failure behaviour that are essential for reasoning about the correctness of real web applications. This paper introduces a global-type framework that equips MPST with explicit failure semantics and dynamic participation. We define the syntax and operational semantics of these enriched global types and establish core properties, including coherence preservation. This foundation enables formal reasoning about communications in web applications where failures may occur, and lays the groundwork for future stateful extensions and automated verification of liveness properties.

Predicate subtypes provide an attractive mechanism to specify range constraints on variable declarations. This paper discusses how we add support for predicate subtypes to the VerCors program verifier. Our approach automatically generates appropriate specifications from predicate subtype declarations. It provides support to easily combine multiple subtypes for a single variable declaration. Moreover, in order to use predicate subtypes for overflow checking, a special strict mode is introduced, where every subexpression also has to stay within the declared subtype. A prototype implementation is integrated into the VerCors verifier.

Effpi is a framework for writing strongly-typed message-passing programs in Scala, where the compiler enforces the conformance of process implementations to specified protocol types. A compiler plugin is provided to verify properties of protocols, such as deadlock-freedom and liveness, by encoding the behavioural types into a variant of CCS. To address limitations in the expressiveness of the existing toolkit, we extend Effpi with external choice by introducing a branching operation. Upon accepting a message via a branch, protocols enforce a continuation which depends on the label (type) of the received message. We equip the branching operation with the ability to accept messages over more than one channel. Additionally, we introduce a "catch timeout" operation to allow processes to gracefully handle a lack of incoming messages. The enhanced expressiveness of Effpi is demonstrated through a number of examples, including an implementation of the Raft consensus algorithm.

Distributed systems have become increasingly prevalent in the software industry. Due to their intrinsic complexity, much research has focused on the verification of their behaviour. An active research line is around behaviour models that capture these protocols - e.g., session types, or typestates - allowing their static verification. Correctly designing distributed protocols is not trivial. Their communication behaviour is typically implicitly defined via asynchronous message handlers, making errors harder to detect until execution. While typestates can ease the design process by explicitly defining correct sequences of operations, they struggle in two ways: they lack the expressiveness to define quantitative constraints that govern distributed protocols (i.e., number of acknowledgements for a quorum); and they assume strict sequencing of operations, failing to capture concurrent input/output actions in a state, typical of the distributed setting. Furthermore, runtime network failures cannot be statically verified. We present a probabilistic runtime solution extending typestates with: (i) an internal mutable state for the expression of quantitative constraints; (ii) mixed sessions to represent concurrent input and output actions; (iii) expected ratios for the number of actions in a state, with monitoring semantics to detect deviations from an expected behaviour at runtime. We demonstrate the suitability of our solution with two examples that motivated our approach: an acknowledgement protocol with a participant that sends several messages while waiting for a response, effectively modelling input and output operations in a state; and a voting protocol whose participants try to achieve consensus on a single bit using a quorum, thus, requiring an internal mutable state, while respecting a pre-defined distribution for the volume of exchanged messages.