This paper investigates the impact of carbon pricing under the EU Emissions Trading System (EU ETS) on the Italian electricity market, focusing on the carbon cost pass-through rate (CPTR) across market zones during Phases 3 and 4 (2016-2024). Using daily data, the study applies an econometric framework based on a linear regression model with autoregressive dynamics to estimate the extent to which carbon costs are reflected in wholesale electricity prices. It further incorporates robustness checks and quantile regression to assess how the CPTR varies across different fuel spread levels. The results show that carbon costs are positively and significantly transmitted to electricity prices, confirming the relevance of carbon pricing as a key market driver. However, pass-through is incomplete, with CPTR values consistently below 100%. At the national level, the pass-through estimate is around 32%, with no statistically significant change between Phase 3 and Phase 4. Substantial heterogeneity emerges across market zones: pass-through increases in the North, Centre-North, and Sardinia during Phase 4, while it declines in the Centre-South and Sicily, reflecting differences in generation mix, carbon intensity, and market conditions. Overall, the findings highlight the importance of market zones factors in shaping the effectiveness of carbon pricing in electricity markets.

The article establishes a rigorous shift operator calculus for stationary time series modeling, addressing a certain gap in the literature. It provides proofs of existence and isometry for the transfer function operators $f(B)$ and $f(T)$ where $B$ is the bilateral shift operator and $T$ is the unilateral shift operator for different families of functions $f$. The article establishes convergence of the power series of $f(B)$ and $f(T)$ under the operator norm for the Wiener algebra $\mathbb{W}_+$, and convergence under strong operator topology for $f$ in $H^{\infty}$, based on the use of Abel sums. Based on this calculus, it unifies the notion of stationary process invertibility with the operator invertibility of the transfer function $f(T)$.

We propose a minimal extension of the Newtonian action by introducing a time-dependent fractional kernel characterized by a single deformation parameter $α$. This kernel admits a natural interpretation as a nontrivial integration measure defined by a time-dependent kernel, placing the formulation within measure-based approaches to anomalous or fractal dynamics. Despite the appearance of a friction-like term in the equations of motion, a conserved quantity is still obtained, containing a memory-like fractional kinetic energy contribution. Moreover, by generalizing the standard Newtonian potential to an effective $α$-dependent potential induced by the underlying measure, the resulting cosmological equations exhibit an effective correspondence with relativistic FLRW cosmology at the level of background dynamics. In the limit $α=1$, the framework reduces to standard Newtonian cosmology. We show that, with a single unified potential, the matter-dominated, radiation-dominated, and present accelerated phases are obtained self-consistently, while the latter two epochs cannot be described within standard Newtonian cosmology. The structural presence of $α$ in all physical observables allows theoretical and observational constraints to be imposed, indicating compatibility with observational data in the regime where $α$ is close to unity. Within this framework, an effective cosmological constant naturally arises, controlled by the small deviation of $α$ from the Newtonian limit. These results show that the proposed fractional framework can effectively reproduce the main background dynamical features of $Λ$CDM cosmology through a simple measure-induced deformation of the Newtonian action.

This paper introduces Triosecuris, a formally verified defense against Spectre BTB, RSB, and PHT that combines CET-style hardware-assisted control-flow integrity with compiler-inserted speculative load hardening (SLH). Triosecuris is based on the novel observation that in the presence of CET-style protection, we can precisely detect BTB misspeculation for indirect calls and RSB misspeculation for returns and set the SLH misspeculation flag. We formalize Triosecuris as a transformation in Rocq and provide a machine-checked proof that it achieves relative security: any transformed program running with speculation leaks no more than what the source program leaks without speculation. This strong security guarantee applies to arbitrary programs, even those not following the cryptographic constant-time programming discipline.

The rapid adoption of generative artificial intelligence (GenAI) in schools raises concerns about students' uncritical reliance on its outputs. Effective use of large language models (LLMs) requires not only technical knowledge but also the ability to monitor, evaluate, and regulate one's interaction with the system, processes closely tied to metacognitive regulation. These skills are still developing in middle school, making students particularly vulnerable to over-trust and premature acceptance of AI outputs. Because classroom time and teacher training resources are constrained, there is a pressing need to develop and evaluate AI literacy interventions that can be implemented under realistic school conditions. We report a controlled classroom study examining whether a two-hour AI literacy workshop improves students' interaction strategies and quality of final answers in LLM-supported science problem solving. A total of 116 students (grades 8-9; ages 13-15) completed six science investigation tasks using a generative AI system. Two days prior, the intervention group attended the workshop, which combined information about how LLMs work and fail with practical guidance on prompting and response evaluation; the control group received no training. Trained students showed less uncritical reliance on the system: they more often reformulated queries, asked follow-up questions, and more accurately judged response correctness, leading to better performance. In contrast, GenAI and metacognitive self-report scores did not predict performance, suggesting that effective use of generative AI depends less on self-reported measures and more on explicit training in interaction regulation. Overall, the results show that brief, scalable AI literacy instruction can meaningfully improve how middle-school students use generative AI in school-like learning activities.

Fast radio bursts (FRBs) are millisecond-duration radio transients. As they propagate through the interstellar medium, they interact with free electrons, resulting in dispersion. The corresponding dispersion measure (DM) is referred to as the real DM (DM$_{\rm real}$). In practice, however, the dispersion measure derived from modeling (DM$_{\rm model}$) is often contaminated by intrinsic burst morphology, giving rise to a pseudo DM component (DM$_{\rm pseudo} = {\rm DM}_{\rm model} - {\rm DM}_{\rm real}$). In this work, we focus on the highly active repeating FRB~20220912A and utilize its microshots -- extremely short-duration (typically tens of microseconds), broadband emissions -- to investigate its DM$_{\rm real}$ and DM$_{\rm pseudo}$. We adopt two assumptions: first, that FRB~20220912A resides in a non-magneto-ionic environment and that its DM$_{\rm real}$ variation is smaller than $10^{-2}$\,pc\,cm$^{-3}$ over a few years; and second, that microshots have a negligible intrinsic morphological time delay. By identifying two new microshots and combining them with previously reported ones, we find that all four microshots exhibit remarkably consistent DM values over a one-month timescale, with an average of $219.380 \pm 0.004\,\mathrm{pc\,cm^{-3}}$. We define this value as the DM$_{\rm real}$ of FRB~20220912A. We further show that bright, narrow bursts with a width of less than 2\,ms also yield DM estimates consistent with the microshot-based DM$_{\rm real}$. A survey of five repeating FRBs reveals that DM$_{\rm pseudo}$ is a common phenomenon, with variations typically spanning a range of approximately $10\,\mathrm{pc\,cm^{-3}}$ at 1.2\,GHz. These findings highlight the importance of accounting for morphological contributions in DM interpretation and demonstrate that microshots and narrow bursts are powerful tools for probing DM$_{\rm real}$.

Exclusive semileptonic and nonleptonic $J/ψ$ decays are investigated in the framework of the relativistic quark model based on the quasipotential approach and quantum chromodynamics. The form factors parameterizing the hadronic matrix element of the weak current are calculated with the complete account of the relativistic effects. These form factors are expressed as the overlap integrals of the meson wave functions and are determined in the whole accessible kinematic range. On this basis the semileptonic decay branching fractions are evaluated for decays involving both electrons and muons. The nonleptonic decays are considered in the factorization approximation in the limit for the number of colors $N_c\to \infty$. The obtain branching fractions are found to be of the order $10^{-9}\sim 10^{-12}$. They are compared with the previous theoretical predictions and available experimental upper bounds.

For a non-zero parameter $q$, we define Hermite trace polynomials, which are multivariate polynomials indexed by permutations. We prove several combinatorial properties for them, such as expansions and product formulas. The linear functional determined by these trace polynomials is a state for $q = \frac{1}{N}$ for $N$ a non-zero integer. For such $q$, Hermite trace polynomials of different degrees are orthogonal. The product formulas extend to the closure with respect to the state. The state can be identified with the expectation induced by the $N \times N$ Hermitian Brownian motion. Hermite trace polynomials are martingales for this Brownian motion, while the elements in the closure can be interpreted as stochastic integrals with respect to it. Using the grading on the algebra, we prove several chaos decompositions for such integrals, as well as analyze corresponding creation and annihilation operators. In the univariate, pure trace polynomial case, trace Hermite polynomials can be identified with the Hermite polynomials of matrix argument.

We give an explicit description of the BPS Lie algebra of any quiver with zero potential, by relating the perverse filtration on the cohomological Hall algebra with certain limit conditions on polynomials. Our results also give a partial description of the perverse filtration for arbitrary potential, which we conjecture is complete in the case of tripled quivers with canonical cubic potential.

The sparse analogue of Szemerédi's regularity method has played a central role in the development of extremal results for random graphs. While the sparse embedding lemma (the KLR conjecture) has been resolved, the corresponding sparse counting lemma remains widely open. The conjecture, formulated by Gerke, Marciniszyn, and Steger, states that for every fixed graph $H$ and any $β>0$, there exists $\varepsilon>0$ such that the following holds. Consider a balanced blow-up of $H$ with vertex classes of size $n$, where each pair corresponding to an edge of $H$ forms an $(\varepsilon)$-regular bipartite graph with exactly $m$ edges. Assume that $m$ is above the natural threshold $m \gg n^{2-1/m_2(H)}$, then all but a $β^m$ proportion of such graphs contain at least $(1-δ)$ times the expected number of copies of $H$. In this paper, we establish the $H=K_4$ case of the conjecture.

We investigate Diophantine pairs and triples of triangular numbers with the property $D(a)$ for a non-zero integer $a$. We prove that if a triangular number belongs to a $D(a)$-pair, it can be extended to infinitely many $D(a)$-triples of triangular numbers. Additionally, we determine infinite families of integers $a$ that admit such pairs, as well as families for which no $D(a)$-pairs can exist.

We develop the theory of transfer and norm maps for finite group schemes, extending classical results from finite group theory to a context where induction and restriction are not necessarily bi-adjoint. In the additive setting, we construct a transfer map for both modules and $\rm Ext $ groups and prove that its surjectivity characterizes relative projectivity, establishing a generalization of Higman's criterion. In the multiplicative setting, we define a relative norm map for algebras with a group scheme action. We compare this norm with other versions in the literature, proving that it coincides with Mumford's norm for finite morphisms and on fields is a power of the classical field norm.

Optical thermodynamics has recently emerged as a theoretical framework describing a Rayleigh-Jeans (RJ) modal power distribution of multimoded nonlinear photonic circuits. However, its applicability is constrained to systems exhibiting weak nonlinear mode-mode interactions. Here, by employing a Transfer Integral Operator, we circumvent this limitation and establish a steady-state interacting RJ modal distribution -- referred to as non-ideal RJ (NIRJ) -- with renormalized temperature and optical chemical potential. This also builds a natural bridge with earlier work on grand-canonical statistical-mechanical formulations of discrete nonlinear systems. The theory derives the optical analogue of the compressibility factor, which controls the transition from an ideal, non-interacting equation of state (EoS) to a van der Waals-like interacting EoS.

Azimuthal anisotropy scaling functions are constructed from species-resolved anisotropy measurements in Pb+Pb ($\sqrt{s_{NN}}$=2.76, 5.02~TeV) and Au+Au ($\sqrt{s_{NN}}$=7.7--200~GeV) collisions to probe baryon transport and medium response at finite baryon chemical potential ($μ_B$). Within this data-driven framework, meson and baryon anisotropies spanning the collective-flow and quenching regimes collapse onto common scaling curves, enabling quantitative separation of viscous attenuation, radial flow, and hadronic re-scattering. The attenuation scale $k_β$ exhibits a non-monotonic beam-energy dependence, coincident with the low-energy rise of hadronic re-scattering, consistent with a temperature-dependent specific shear viscosity featuring a near-minimum near the QCD critical region. A charge-odd baryon--antibaryon separation in the effective radial-flow response is negligible at LHC energies but grows toward lower $\sqrt{s_{NN}}$. This species-uniform, baryon-number-scaling separation across $p,Λ,Ξ,Ω$, and $d$ disfavors a purely hadronic origin and supports junction-driven net-baryon transport at finite $μ_B$, enhancing the experimental visibility of critical dynamics in finite, rapidly evolving systems. Together, these results establish species-resolved scaling functions as a compact and robust tool for constraining baryon stopping, medium opacity, and QGP transport properties.

Cut-diagrams are diagrammatic objects, defined in dimensions 1 and 2, that generalize links in 3-space and surface-links in 4-space; in dimension 1, this coincides with the theory of welded links. Using cut-diagrams, we introduce an equivalence relation called cut-concordance, which encompasses the topological notion of concordance for classical links. Our main result is that the nilpotent peripheral system of 1-dimensional cut-diagrams is an invariant of cut-concordance, giving along the way a combinatorial version of a theorem of Stallings. We also investigate the relationship with several other equivalence relations in diagrammatic knot theory, in particular in connection with link-homotopy.

The quadratic theory of gravity is the unique renormalizable theory of quantum gravity in 4 dimensions, as proved by K. S. Stelle in 1977. Over the decades, the theory has been understood to contain a massive tensor ghost, and several attempts have been made to evade its adverse effects by proposing new quantization prescriptions and interpretations. In this paper, we show that the additional spin--2 of quadratic gravity can be turned into a healthy inverted harmonic oscillator (IHO)-like instability, which can be quantized consistently with direct-sum quantum field theory (DQFT), which incorporates geometric superselection sectors. Such modes possess a well-defined quantum description yet do not admit a particle interpretation and are not part of the asymptotic spectrum, being characterized by hyperbolic evolution and spacelike momentum support. We argue that, as a consequence, the extra spin--2 degree of freedom remains off-shell and effectively decoupled from ordinary matter fields, avoiding unitarity violations in observable processes. We argue that this IHO instability is a prevalent feature of fundamental physics, whether it concerns quantum fields on curved spacetimes or the Higgs $\mathbb{Z}_2$ symmetry breaking in the Standard Model of particle physics. Thus, we demonstrate that our new understanding of quadratic gravity offers a fundamental pathway to quantum gravity and a safe beginning for the Universe. Furthermore, we derive key observational predictions of this construction in the view of primordial gravitational waves with new bounds on the tensor-to-scalar ratio and the parity asymmetric features on the large angular scales.

Build systems become an indispensable part of the software implementation and deployment process. New programming languages are released with the build system integrated into the language tools, for example, Go, Rust, or Zig. However, in the hardware description domain, no official build systems have been released with the predominant Hardware Description Languages (HDL) such as VHDL or SystemVerilog. Moreover, hardware design projects are often multilingual. The paper characterizes and compares two common approaches for hardware build system implementations. The first one, the direct-Tcl approach, in which the build system code is executed directly by the EDA tool during the design build flow. The second one, the indirect-abstract approach, in which the build system produces a Tcl script, which is later run by a proper EDA tool. As none of the existing direct-Tcl build systems was close to the indirect-abstract build systems in terms of supported functionalities, the paper also presents a new direct-Tcl hardware build system called HBS. The implemented build system was used as a representative of direct-Tcl build systems in comparison with indirect-abstract build systems.

For a traveling salesperson problem (TSP) of n cities, we present a compact quantum encoding based on a time-register representation of tours. A candidate route is represented as a sequence of n-1 city labels over discrete time steps, with one fixed start city and the remaining cities encoded in binary registers. We describe three ingredients of the construction: uniform route generation over the route register, a reversible validity oracle, and a phase oracle that encodes the total tour cost. The validity oracle checks both that the non-start city labels form a permutation and, for incomplete graphs, that every directed edge used by the route exists. The cost oracle then accumulates the start-edge, intermediate-transition, and return-edge costs into a tour-dependent phase for valid routes. This yields a coherent superposition of candidate routes with feasibility and tour-length information embedded directly in the quantum state. The complete construction uses O(n log n) qubits, while a naive implementation has worst-case elementary-gate complexity O(n^3 log n). The encoding is compatible with amplitude amplification or spectral filtering techniques such as the quantum singular value transform (QSVT) or Grover's algorithm. However, due to the exponentially small fraction of valid tours, the overall complexity remains exponential even when combined with amplitude amplification.

Oscillation modes of neutron stars, a key target for third-generation gravitational wave detectors, encode key information about their constituent nuclear matter. In this work, we study the effect of viscosity on oscillations of cold, polytropic, spherically symmetric neutron stars. We focus on purely radial oscillations and work perturbatively to linear order within two hydrodynamic frameworks: the acausal covariant generalization of the Navier-Stokes equations proposed by Eckart, and the causal generalization formulated by Bemfica, Disconzi, Noronha, and Kovtun (BDNK). We find that viscosity damps the radial modes on millisecond timescales and induces fractional shifts in the oscillation frequency which increase both with the compactness and viscosity of the star, reaching up to the percent level for the fundamental mode with bulk viscosities $ζ\sim10^{30}\mathrm{g}/\mathrm{cm}/\mathrm{s}$. For more viscous stars, the oscillation frequency decreases, becoming zero (i.e., an overdamped mode) for $ζ\gtrsim10^{31}\mathrm{g}/\mathrm{cm}/\mathrm{s}$. We also study the linear threshold of gravitational collapse. Consistent with recent analytic results in the zero heat conductivity limit, we find that viscosity in Eckart theory cannot stabilize an unstable inviscid star. We provide numerical evidence that viscosity in BDNK theory is similarly unable to prevent gravitational collapse, but it slightly modifies the threshold of collapse. Overall, our results advance our understanding of the impact of viscosity on the oscillation modes of neutron stars, a key component of viscous asteroseismology with next-generation gravitational wave detectors.

We present a comprehensive analysis of the host galaxies of 11 luminous fast blue optical transients (LFBOTs). We model new and archival host photometry and spectroscopy with Prospector. We determine that all LFBOT hosts are actively star-forming with recent bursts of star formation and have a median stellar mass of $\log(M_*/M_\odot)=9.61^{+0.74}_{-1.61}$, present-day star formation rate SFR=$0.99^{+14.85}_{-0.95}$~$M_\odot$~yr$^{-1}$, and gas-phase oxygen abundance metallicity 12+log(O/H)=$8.59^{+0.18}_{-0.22}$. To contextualize these results, we compare them to the host properties of Hydrogen-poor superluminous supernovae (SLSNe-I), several core-collapse supernova subtypes (CCSN; SNe Ibc, II, and Ibn) and long gamma-ray bursts (LGRBs). We find that LFBOT hosts are more star-forming than CCSN hosts, but less star-forming than SLSN-I hosts. We further show that LFBOT hosts are more metal-poor than SN Ibc and II hosts, but more metal-rich than SLSN-I and LGRB hosts. Finally, we find that, similar to SLSNe-I and unlike CCSNe and LGRBs, a large fraction of LFBOTs occur in their hosts' faintest pixel or outside their host galaxy's light. Our results indicate that LFBOTs have a massive stellar origin that do not trace active star-forming regions within their hosts and have a weaker metallicity-dependence than other extreme transients. For these reasons, we favor a compact-object and Wolf-Rayet star merger progenitor scenario over other previously proposed models, such as tidal disruption events and failed or successful CCSN. Future discoveries of LFBOTs with the Rubin observatory will help to increase their sample size and place firmer constraints on their environments and progenitors.

There has been recent interest in the relaxational modes of small-scale fully connected systems of aligning self-propelled particles (Spera et al., Phys. Rev. Lett. {\bf 132}: 078301 (2024)). We revisit the classical connection between Fokker-Planck and Schrödinger equations to address this by means of exact diagonalization, allowing for rigorous analytical insight into the full spectrum. This allows us to extract exact results which we compare to the existing result from linearized statistical field theory. We derive asymptotically correct analytical results that improve upon the prior approximations. We show that this methodology can fruitfully be extended to the case of non-reciprocal interactions which gives rise to a non-Hermitian Schrödinger problem akin to those in open quantum mechanics. While the non-reciprocity can be chosen such as not to alter the stationary distribution, it fundamentally changes the nature of the steady state which we quantify via the entropy production. We discuss the case of low particle numbers as well as the emergence of mean-field dynamics at large numbers.

Let $\g$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected complex Lie group with $\mathrm{Lie}(G)=\g$ and compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\OAztG$ of the quantized function algebra $\OtG$, establishing that $\OAztG=A_0\text{-alg}<\RAzp \cup \RAzm>.$ This extends the triangular decomposition recently obtained for types $A_n, B_n, C_n, D_n, E_6$, and $E_7$ in~\cite{DDPa} to all simple complex Lie algebras. As a consequence, we obtain: (i) the inclusion $\OAztG\subseteq\OAztK$ conjectured by Matassa-Yuncken and (ii) the crystal limit $\CpKo$ is a compact quantum semigroup with a unique bi-invariant (Haar) state.

Safety-critical multi-agent systems, from cooperative guidance to collision avoidance, must often reach a coordinated decision by a hard deadline rather than merely converge to one eventually. This paper proposes the first fully distributed algorithm that solves the generalized Nash equilibrium (GNE) problem, a non-cooperative game with shared coupling constraints and general cost coupling, at a user-prescribed time $T$ independent of initial conditions. The foundation is a centralized, prescribed-time result built on the optimization Lyapunov function framework and implemented via unnormalized Hessian-gradient feedback, chosen because, unlike the Newton and normalized Hessian-gradient realizations, it naturally splits into per-agent computations. Distributing this feedback requires each agent to run three coupled processes simultaneously: a prescribed-time observer of the global state, a local optimization law, and a dual-consensus mechanism that enforces the shared multipliers of the variational GNE. Their simultaneous operation is the core difficulty, as the optimization continually displaces the states the observers track, while estimation errors corrupt the gradients that drive the optimization. We resolve this coupling with a multi-rate gain schedule whose observer and dual-consensus layers contract strictly faster than the optimization layer, so that every error component vanishes exactly at $T$. A Fischer-Burmeister reformulation keeps the design projection-free while enforcing the constraints at the deadline. Numerical results for a Cournot game and a time-critical sensor-coverage problem validate the approach and demonstrate its use as a solver-in-the-loop for time-critical autonomy.

Folkman's theorem asserts the existence of graphs $G$ which are $K_4$-free, but which have the property that every two-coloring of $E(G)$ contains a monochromatic triangle. The quantitative aspects of $f(2,3,4)$, the least $n$ such that there exists an $n$-vertex graph with both properties above, are notoriously difficult; a series of improvements over the span of two decades witnessed the solution to two \$100 Erdős problems, and the current record due to Lange, Radziszowski, and Xu now stands at $f(2,3,4) \leq 786$,with another \$100 problem of Graham asking for a proof that $f(2,3,4) < 100$. In this paper, we study Folkman-like properties of a sequence $H_q$ of finite geometric graphs constructed using Hermitian unitals in projective planes and present some evidence that the graph $H_3$, which has 63 vertices, might contain a Folkman graph as a proper subgraph. More precisely, we first prove that for all prime powers $q \geq 3$, there exists a system $\mathscr{T}_q$ of triangles in $H_q$ such that no four span a $K_4$ in $H_q$, but every two-coloring of $E(H_q)$ induces a monochromatic triangle in $\mathscr{T}_q$. We then show that a certain random alteration of $H_q$ which destroys all of its $K_4$'s will, for large $q$, maintain the Ramsey property with high probability.

There is a growing literature adopting a stochastic optimal control (SOC) perspective to fine-tune diffusion models and related generative policies. A prominent class of methods, known as iterative diffusion optimization, solves the SOC problem by simulating the diffusion process, evaluating a loss function, and applying stochastic optimization algorithms, with adjoint matching emerging as a state-of-the-art approach. However, the adjoint process used in these methods is not adapted to the forward diffusion filtration, which can lead to unstable or high-variance gradient estimates. In this paper, we revisit gradient estimation in diffusion models through the lens of backward stochastic differential equations (BSDEs). We propose an alternative estimator based on a time-reversed BSDE formulation introduced in our prior work, which produces an adjoint process adapted to the underlying filtration. This adapted structure leads to more stable gradient estimates with potentially lower variance. We analyze the accuracy of the proposed estimator and compare it with adjoint matching. Numerical experiments on fine-tuning toy diffusion models demonstrate improved gradient stability and competitive performance.

We study universality of Lipschitz quotients. First, we modify a construction of Johnson, Lindenstrauss, Preiss and Schechtman to obtain a complete separable metric space that has every complete separable metric space as a Lipschitz quotient. Our main result is in the compact setting, where we prove that no such universal metric space can exist. We deduce this impossibility result by studying the curve-flat index, an ordinal index which provides a measure of the complexity of the curve-fragment structure in a metric space. We show that Lipschitz quotients cannot increase this index in compact domains; while there exist compact spaces with arbitrarily high countable curve-flat index. The main technical part of the paper is dedicated to proving a strong version of the latter fact: for every ordinal $α$ and every compact metric space $M$, there exists a compact metric space $N$ such that the curve-flat quotient of $N$ of order $α$ is almost-isometric to $M$.

Designing clinical trials requires evaluating multiple operating characteristics (OCs), such as the likelihood of an early stopping decision, the probability of detecting a treatment effect, and the Type I error rate. In most cases, these evaluations are based on computationally intensive Monte Carlo simulations. As the complexity of clinical trials and the use of adaptive designs increase, the computational burden can quickly become prohibitive. We introduce a strategy for rapidly approximating OCs, called the Q-approximation. Our approach is based on quadratic approximations of the log-likelihood and asymptotic arguments. The main idea is to replace simulation of full trial datasets with simulation of the approximate likelihood functions that determine the trial's interim and final decisions. The Q-approximation approach can be applied to any trial design that uses data analysis methods coherent with the likelihood principle, including multistage designs with early stopping, adaptively randomized designs, and designs that leverage external data. We illustrate the approach with several examples and show that it provides an accurate approximation of important OCs while reducing the computation time compared to Monte Carlo simulations. In particular, in our experiments, the standard Monte Carlo approximation of OCs requires 150 to 1,900 times greater computing budget than Q-approximations to achieve comparable levels of accuracy. By enabling fast OC evaluations, Q-approximations can support the broader use of innovative trial designs in both applied trial planning and methodological development.

In this paper, we propose an invariant quantile regression (IQR) framework specifically designed for multi-environment datasets, which captures the invariance across different environments. This framework is closely related to transfer learning, causal inference, and fair machine learning, and is motivated by scenarios in which the conditional probability of the response given covariates varies, while certain key variables remain invariant. This perspective differs notably from previous works that restrict attention to the conditional mean, which is often insufficient to capture the full causal relationships between covariates and the response in heterogeneous environments. In contrast, quantile-based invariance naturally accommodates heterogeneity, and aligns more closely with structural causal models, in which variables invariant across environments at one or multiple quantile levels directly indicate potential and stable causal variables. Moreover, we show that IQR may yield a larger set of endogenous variables compared to the conditional mean framework, which in turn promotes more effective exclusion of spurious (non-causal) variables. To achieve this, we introduce a Kernel-Smoothed Invariant Quantile Regression (KS-IQR) estimator, which leverages the underlying invariance structure and heterogeneity among environments, ensuring stable estimation across multiple environments. We establish the causal discovery properties of our method, demonstrate its ability to overcome the ``curse of endogeneity'', and derive an $\ell_2$ error bound for our estimator, all in a non-asymptotic framework. We apply our method to real data for causal discovery and obtain biologically meaningful relationships, recovering known signaling pathways and revealing additional quantile-specific effects.

Pressure-relief valves, often the critical last line of defence in process engineering, are known to be susceptible to valve chatter. Such behaviour has been shown to arise from a flutter instability, or Hopf bifurcation, associated with the fundamental, quarter-wave acoustic mode of their inlet piping. Here, a novel design concept is proposed and analyzed for eliminating this instability. The concept involves using an oversized valve with reduced lift and adopting a discharge characteristic that enhances the blow-down effect, so that the valve opens immediately to its upper lift limit upon reaching set pressure. The concept is demonstrated numerically using an updated version of a 1D fluid pipe dynamics mathematical model solved using a Lax-Wendroff method. Stability properties are analysed using dynamical systems theory, applied to an improved reduced-order modal model. It is shown how the valve settles to a stable so-called pseudo equilibrium, in contact with the upper stop, provided the coefficient of restitution of is not too large. Such stable operation is reached despite the equivalent regular valve being unstable to the quarter-wave Hopf bifurcation. Parameter studies using the reduced-order model demonstrate the extent of the enhanced stability effect, which is confirmed using the full model for the case of gas service valves.

Identifying materials hosting an excitonic insulator ground state has been one of the major pursuits in condensed matter physics in recent years. Promising candidates in transition metal chalcogenide compounds (TMC), including $1T-\mathrm{TiSe_2}$, $\mathrm{Ta_2Pd_3Te_5}$, and $\mathrm{Ta_2NiSe_5}$, share a crucial common characteristic: their low-energy physics is governed by electrons in $d-$ orbitals subject to strong on-site Coulomb interactions. In this work, we investigate spatially indirect exciton condensation in two-dimensional semimetals on triangular lattice. Using a combination of dynamical mean-field theory and the determinant quantum Monte Carlo method, we study two- and three-orbital Hubbard models incorporating strong on-site ($U$) and inter-orbital interactions ($V$). Our results demonstrate that on-site Hubbard $U$ can strongly suppress the condensation temperature $T_c$, an effect that is particularly pronounced at higher electron-hole pair densities. This behavior contrasts sharply with the case without on-site $U$, where $T_c$ grows with pair density at fixed $V$. Moreover, we uncover competition among multiple electron-hole pairing channels in the three-orbital model, which also acts to suppress $T_c$ of exciton condensation. An orbital-selective electron-hole pairing state is identified. These findings may help explain the large discrepancy between strong binding-energy and relative low transition temperature for indirect excitons in TMCs materials, offering important insights for understanding and engineering exciton condensation in materials with strongly correlated $d-$ shell electrons.