Synthetic control methods are commonly used in panel data settings to evaluate the effect of an intervention. In many of these cases, the treated and control units correspond to spatial units such as regions or neighborhoods. Our approach addresses the challenge of understanding how an intervention applied at specific locations influences the surrounding area. Traditional synthetic control applications may struggle with defining the effective area of impact, the extent of treatment propagation across space, and the variation of effects with distance from the treatment sites. To address these challenges, we introduce Spatial Vertical Regression (SVR) within the Bayesian paradigm. This innovative approach allows us to accurately predict the outcomes in varying proximities to the treatment sites, while meticulously accounting for the spatial structure inherent in the data. Specifically, rooted on the vertical regression framework of the synthetic control method, SVR employs a Gaussian process to ensure that the imputation of missing potential outcomes for areas of different distance around the treatment sites is spatially coherent, reflecting the expectation that nearby areas experience similar outcomes and have similar relationships to control areas. This approach is particularly pertinent to our study on the Florentine tramway's first line construction. We study its influence on the local commercial landscape, focusing on how business prevalence varies at different distances from the tram stops.

This paper improves the session typing theory to support the modelling and verification of processes that implement federated learning protocols. To this end, we build upon the asynchronous ``bottom-up'' session typing approach by adding support for input/output operations directed towards multiple participants at the same time. We further enhance the flexibility of our typing discipline and allow for safe process replacements by introducing a session subtyping relation tailored for this setting. We formally prove safety, deadlock-freedom, liveness, and session fidelity properties for our session typing system. Moreover, we highlight the nuances of our session typing system, which (compared to previous work) reveals interesting interplays and trade-offs between safety, liveness, and the flexibility of the subtyping relation.

Let $G$ be a connected graph with $n$ vertices. The resistance distance $Ω_{G}(i,j)$ between any two vertices $i$ and $j$ of $G$ is defined as the effective resistance between them in the electrical network constructed from $G$ by replacing each edge with a unit resistor. The resistance matrix of $G$, denoted by $R_G$, is an $n \times n$ matrix whose $(i,j)$-entry is equal to $Ω_{G}(i,j)$. The resistance curvature $κ_i$ in the vertex $i$ is defined as the $i$-th component of the vector $(R_G)^{-1}\mathbf{1}$, where $\mathbf{1}$ denotes the all-one vector. If all the curvatures in the vertices of $G$ are equal, then we say that $G$ has constant resistance curvature. Recently, Devriendt, Ottolini and Steinerberger \cite{kde} conjectured that the cycle $C_n$ is extremal in the sense that its curvature is minimum among graphs with constant resistance curvature. In this paper, we confirm the conjecture. As a byproduct, we also solve an open problem proposed by Xu, Liu, Yang and Das \cite{kxu} in 2016. Our proof mainly relies on the characterization of maximum value of the sum of resistance distances from a given vertex to all the other vertices in 2-connected graphs.

A new model based on the relativistic Boltzmann equation in the isotropization time approximation is developed to investigate the collective behavior of the quark-gluon plasma produced in high-energy heavy-ion collisions. The equation is solved in (2+2)D (two spatial and two momentum-space dimensions). This framework couples pre-equilibrium dynamics with hydrodynamic evolution to simulate the dynamics of quasiparticle evolution. A numerical scheme based on the method of characteristics enables the evolution to begin from a specified initial Boltzmann distribution. In this work, the spatial structure of the initial distribution is modeled using the TrENTo framework. Our results show that a medium initialized at $τ_0$ on the order of 1 [fm/$c$] with a small shear viscosity to entropy density ratio ($η/s = 0.008$) evolves consistently with hydrodynamic simulations, such as those performed using the VISH2+1 code, while discrepancies arise for a medium with $η/s = 0.8$. Furthermore, when initialized with a highly anisotropic momentum distribution in the longitudinal direction at early times, the system exhibits spatially non-uniform thermalization in the transverse plane, leading to the emergence of a nontrivial hypersurface that marks the onset of hydrodynamic applicability. Finally, we compute the $p_T$-spectra for a non-fluctuating initial condition using the hybrid version of CoMBolt-ITA. In this hybrid setup, the description is switched from quasiparticles to hadrons, and UrQMD is used to model the hadron gas dynamics. We compare these results with those obtained from the hybrid VISH2+1 initialized within the same setup. For a small shear viscosity, $η/s = 0.08$, the two results show a good level of consistency, whereas for a larger value, $η/s = 0.8$, a noticeable discrepancy emerges.

In this paper we introduce a new fix point iteration scheme for solving nonlinear electromagnetic scattering problems. The method is based on a spectral formulation of Maxwell's equations called the Bidirectional Pulse Propagation Equations. The scheme can be applied to a wide array of slab-like geometries, and for arbitrary material responses. We derive the scheme and investigated how it performs with respect to convergence and accuracy by applying it to the case of light scattering from a simple slab whose nonlinear material response is a sum a very fast electronic vibrational response, and a much slower molecular vibrational response.

We provide a new proof of the analogue of the Artin-Springer theorem for groups of type $\mathsf{D}$ that can be represented by similitudes over an algebra of Schur index $2$: an anisotropic generalized quadratic form over a quaternion algebra $Q$ remains anisotropic after generic splitting of $Q$, hence also under odd degree field extensions of the base field. Our proof is characteristic free and does not use the excellence property.

We study the spin-flavor precession of neutrinos in a magnetic field within the quantum field theory approach in which neutrinos are virtual particles. Neutrinos are taken to be Majorana particles having a nonzero transition magnetic moment. We derive the dressed propagators of the neutrino mass eigenstates exactly accounting for the magnetic field contribution. The matrix element and the transition probability for the spin-flavor precession are obtained in the approximation of the forwardly scattered charged leptons. The leading term in the transition probability is shown to coincide with the result of the standard quantum mechanical description of neutrino oscillations. We also discuss the quantum field theory contributions to the neutrino dressed propagators and demonstrate that these contributions result in a small correction to the transition probability. The case of charged leptons with arbitrary energies is considered.

Dicke superradiance describes an ensemble of $N$ permutationally invariant two-level systems collectively emitting radiation with a peak radiated intensity scaling as $N^2$. Although individual Dicke states are typically entangled, the density matrix during superradiant decay is a mixture of such states, raising the subtle question of whether the total state is entangled or separable. We resolve this by showing analytically that for all $N$, starting from the fully excited state, the collective decay preserves separability for all times. This answers a longstanding question on the role of entanglement in Dicke superradiance and underscores that, despite collective dissipation, separable states remain separable under these dynamics.

Finite-sample upper bounds on the estimation error of a winsorized mean estimator of the population mean in the presence of heavy tails and adversarial contamination are established. In comparison to existing results, the winsorized mean estimator we study avoids a sample splitting device and winsorizes substantially fewer observations, which improves its applicability and practical performance.

Recent years have witnessed much progress on Gaussian and bootstrap approximations to the distribution of sums of independent random vectors with dimension $d$ large relative to the sample size $n$. However, for any number of moments $m>2$ that the summands may possess, there exist distributions such that these approximations break down if $d$ grows faster than the polynomial barrier $n^{\frac{m}{2}-1}$. In this paper, we establish Gaussian and bootstrap approximations to the distributions of winsorized and trimmed means that allow $d$ to grow at an exponential rate in $n$ as long as $m>2$ moments exist. The approximations remain valid under some amount of adversarial contamination. Our implementations of the winsorized and trimmed means do not require knowledge of $m$. As a consequence, the performance of the approximation guarantees ``adapts'' to $m$.

We consider a tick-by-tick model of price formation, in which buy and sell orders are modeled as self-exciting point processes (Hawkes process), similar to the one in [Bacry, Delattre, Hoffmann, Muzy, Modelling microstructure noise with mutually exciting point processes, Quantitative Finance, 2013] and [El Euch, Fukasawa, Rosenbaum, The microstructural foundations of leverage effect and rough volatility, Finance and Stochastics, 2018]. We adopt an agent based approach by studying the aggregation of a large number of these point processes, mutually interacting in a mean-field sense. The financial interpretation of the model is that of an asset on which several labeled agents place buy and sell orders following these point processes, influencing the price. The mean-field interaction introduces positive correlations between order volumes coming from different agents that reflect features of real markets such as herd behavior and contagion. When the large scale limit of the aggregated asset price is computed, if parameters are set to a critical value, a singular phenomenon occurs: the aggregated model converges to a stochastic volatility model with leverage effect and faster-than-linear mean reversion of the volatility process. The faster-than-linear mean reversion of the volatility process is supported by econometric evidence, and we have linked it in [Dai Pra, Pigato, Multi-scaling of moments in stochastic volatility models, Stochastic Processes and their Applications, 2015] to the observed multifractal behavior of assets prices and market indices. This seems connected to the Statistical Physics perspective that expects anomalous scaling properties to arise in the critical regime.

One approach to risk-limiting audits (RLAs) compares randomly selected cast vote records (CVRs) to votes read by human auditors from the corresponding ballot cards. Historically, such methods reduce audit sample sizes by considering how each sampled CVR differs from the corresponding true vote, not merely whether they differ. Here we investigate the latter approach, auditing by testing whether the total number of mismatches in the full set of CVRs exceeds the minimum number of CVR errors required for the reported outcome to be wrong (the "CVR margin"). This strategy makes it possible to audit more social choice functions and simplifies RLAs conceptually, which makes it easier to explain than some other RLA approaches. The cost is larger sample sizes. "Mismatch-based RLAs" only require a lower bound on the CVR margin, which for some social choice functions is easier to calculate than the effect of particular errors. When the population rate of mismatches is low and the lower bound on the CVR margin is close to the true CVR margin, the increase in sample size is small. However, the increase may be very large when errors include errors that, if corrected, would widen the CVR margin rather than narrow it; errors affect the margin between candidates other than the reported winner with the fewest votes and the reported loser with the most votes; or errors that affect different margins.

We introduce a framework for understanding the impact of generative AI on human work, which we call the human-AI task tensor. A tensor is a structured framework that organizes tasks along multiple interdependent dimensions. Our human-AI task tensor introduces a systematic approach to studying how humans and AI interact to perform tasks, and has eight dimensions: task definition, AI integration, interaction modality, audit requirement, output definition, decision-making authority, AI structure, and human persona. After describing the eight dimensions of the tensor, we provide illustrative frameworks (derived from projections of the tensor) and a human-AI task canvas that provide analytical tractability and practical insight for organizational decision-making. We demonstrate how the human-AI task tensor can be used to organize emerging and future research on generative AI. We propose that the human-AI task tensor offers a starting point for understanding how work will be performed with the emergence of generative AI.

Constructing efficient risk-limiting audits (RLAs) for multiwinner single transferable vote (STV) elections is a challenging problem. An STV RLA is designed to statistically verify that the reported winners of an election did indeed win according to the voters' expressed preferences and not due to mistabulation or interference, while limiting the risk of accepting an incorrect outcome to a desired threshold (the risk limit). Existing methods have shown that it is possible to form RLAs for two-seat STV elections in the context where the first seat has been awarded to a candidate in the first round of tabulation. This is called the first winner criterion. We present an assertion-based approach to conducting full or partial RLAs for STV elections with three or more seats, in which the first winner criterion is satisfied. Although the chance of forming a full audit that verifies all winners drops substantially as the number of seats increases, we show that we can quite often form partial audits that verify most, and sometimes all, of the reported winners. We evaluate our method on a dataset of over 500 three- and four-seat STV elections from the 2017 and 2022 local council elections in Scotland.

The Vulnerability Exploitability eXchange (VEX) format has been introduced to complement Software Bill of Materials (SBOM) with security advisories of known vulnerabilities. VEX gives an accurate understanding of vulnerabilities found in the dependencies of third-party software, which is critical for secure software development and risk analysis. In this paper, we present a study that analyzes state-of-the-art VEX-generation tools (Trivy, Grype, DepScan, Scout, Snyk, OSV, Vexy) applied to containers. Our study examines how consistently different VEX-generation tools perform. By evaluating their performance across multiple datasets, we aim to gain insight into the overall maturity of the VEX-generation tool ecosystem, beyond any single implementation. We use the Jaccard and Tversky indices to produce similarity scores of tool results for three different datasets created from container images. Overall, our results show a low level of consistency among the tools, thus indicating a low level of maturity in the VEX tool space. We perform a number of experiments to explore the impact of different factors on the consistency of the results, with the difference in vulnerability databases queried showing the largest impact.

Nuclear magnetic resonance (NMR) spectroscopy is widely used in fields ranging from chemistry, material science to neuroscience. Nanoscale NMR spectroscopy using Nitrogen-vacancy (NV) centers in diamond has emerged as a promising platform due to an unprecedented sensitivity down to the single spin level. At the nanoscale, high nuclear spin polarization through spin fluctuations (statistical polarization) far outweighs thermal polarization. However, until now efficient NMR detection using coherent averaging techniques could not be applied to the detection of statistical polarization, leading to long measurement times. Here we present two protocols to enable coherent averaging of statistical oscillating signals through rectification. We demonstrate these protocols on an artificial radiofrequency signal detected with a single NV center at 2.7 T. Through this, the signal-to-noise scaling with number of measurements $N$ is improved from $N^{0.5}$ to $N^1$, improving the measurement time significantly. The relevance of rectification for the detection of statistical polarization using NV ensembles is outlined, paving the way for efficient nanoscale NMR spectroscopy.

We investigate the experimental bounds on the Flavor-Violating (FV) couplings of the $Z$ boson to the charged leptons. In addition to the direct LHC searches for FV $Z$ decays to leptons, we investigate indirect bounds from flavor-conserving $Z$ decays to leptons at 1-loop, bounds from LEP searches, Electroweak Precision Observables (EWPO), $\ell_{i}\to\ell_{j}γ$ decays, $\ell_{i}\to3\ell_{j}$ decays, $\ell_{i}\to\ell_{j}+\text{inv.}$ decays, FV meson decays to leptons, FV $τ$ decays to $μ(e)$ + mesons, muon conversion in nuclei, and from muonium-antimuonium oscillations. For FV $Z$ couplings to $τμ$, we find that $τ\toμγ$ yields the strongest bounds, with a level reaching $\mathcal{O}(10^{-5})$, followed by bounds from $τ\to3μ(μee)$. For FV $Z$ couplings to $τe$, we find that the strongest bounds arise from the decay $τ\toμμe$, reaching $\mathcal{O}(10^{-7})$ as well, with bounds from $τ\to3e$ also yielding strong bounds. For FV $Z$ couplings to $μe$, we find that the strongest bounds are obtained from the decay $μ\to3e$, reaching $\mathcal{O}(10^{-11})$, with bounds from $μ\to eγ$, muon conversion, $K_{L}^{0}\rightarrowμe$ and $μ\to e+\text{inv.}$ also providing strong bounds. We also study projections from future experiments, such as the FCC-ee, Belle II and the Mu2e experiment. For the $Z$ couplings to $τμ$, we find that future experiments could improve the bound to $\mathcal{O}(10^{-6})$, whereas for the $Z$ couplings to $τe$, we find that future experiments could improve the bound to $\mathcal{O}(10^{-8})$, and for the $Z$ couplings to $μe$, they could improve the bound to $\mathcal{O}(10^{-13})$

We consider the solution of the Sylvester equation $AX+XB=C$ in mixed precision. We derive a new iterative refinement scheme to solve perturbed quasi-triangular Sylvester equations; our rounding error analysis provides sufficient conditions for convergence and a bound on the attainable relative residual. We leverage this iterative scheme to solve the general Sylvester equation. The new algorithms compute the Schur decomposition of the coefficient matrices $A$ and $B$ in lower than working precision, use the low-precision Schur factors to obtain an approximate solution to the perturbed quasi-triangular equation, and iteratively refine it to obtain a working-precision solution. In order to solve the original equation to working precision, the unitary Schur factors of the coefficient matrices must be unitary to working precision, but this is not the case if the Schur decomposition is computed in low precision. We propose two effective approaches to address this: one is based on re-orthonormalization in working precision, and the other on explicit inversion of the almost-unitary factors. The two mixed-precision algorithms thus obtained are tested on various Sylvester and Lyapunov equations from the literature. Our numerical experiments show that, for both types of equations, the new algorithms are at least as accurate as existing ones. Our cost analysis, on the other hand, suggests that they would typically be faster than mono-precision alternatives if implemented on hardware that natively supports low precision.

Automata over infinite alphabets have emerged as a convenient computational model for processing structures involving data, such as nonces in cryptographic protocols or data values in XML documents. We introduce active learning methods for bar automata, a species of automata that process finite data words represented as bar strings, which are words with explicit name binding letters. Bar automata have pleasant algorithmic properties. We develop a framework in which every learning algorithm for standard deterministic or nondeterministic finite automata over finite alphabets can be used to learn bar automata, with a query complexity determined by that of the chosen learner. The technical key to our approach is the algorithmic handling of $α$-equivalence of bar strings, which allows bridging the gap between finite and infinite alphabets. The principles underlying our framework are generic and also apply to bar Büchi automata and bar tree automata, leading to the first active learning methods for data languages of infinite words and finite trees.

Successful operation of large particle detectors like the Compact Muon Solenoid (CMS) at the CERN Large Hadron Collider requires rapid, in-depth assessment of data quality. We introduce the ``AutoDQM'' system for Automated Data Quality Monitoring using advanced statistical techniques and unsupervised machine learning. Anomaly detection algorithms based on the beta-binomial probability function, principal component analysis, and neural network autoencoder image evaluation are tested on the full set of proton-proton collision data collected by CMS in 2022. AutoDQM identifies anomalous ``bad'' data affected by significant detector malfunction at a rate 4 -- 6 times higher than ``good'' data, demonstrating its effectiveness as a general data quality monitoring tool.

Building on recent progress in the study of Anderson and many-body localization via the renormalization group (RG), we examine the scaling theory of localization in the quantum Random Energy Model (QREM). The QREM is known to undergo a localization-delocalization transition at finite energy density, while remaining fully ergodic at the center of the spectrum. At zero energy density, we show that RG trajectories consistently flow toward the ergodic phase, and are characterized by an unconventional scaling of the fractal dimension near the ergodic fixed point. When the disorder amplitude is rescaled, as suggested by the forward scattering approximation approach, a localization transition emerges also at the center of the spectrum, with properties analogous to the Anderson transition on expander graphs. At finite energy density, a localization transition takes place without disorder rescaling, and yet it exhibits a scaling behavior analogous to the one observed on expander graphs. The universality class of the model remains unchanged under the rescaling of the disorder, reflecting the independence of the RG from microscopic details. Our findings demonstrate the robustness of the scaling behavior of random graphs and offer new insights into the many-body localization transition.

When are two algorithms the same? How can we be sure a recently proposed algorithm is novel, and not a minor variation on an existing method? In this paper, we present a framework for reasoning about equivalence between a broad class of iterative algorithms, with a focus on algorithms designed for convex optimization. We propose several notions of what it means for two algorithms to be equivalent, and provide computationally tractable means to detect equivalence. Our main definition, oracle equivalence, states that two algorithms are equivalent if they result in the same sequence of calls to the function oracles (for suitable initialization). Borrowing from control theory, we use state-space realizations to represent algorithms and characterize algorithm equivalence via transfer functions. Our framework can also identify and characterize equivalence between algorithms that use different oracles that are related via a linear fractional transformation. Prominent examples include linear transformations and function conjugation. To support the paper, we have developed a software package named Linnaeus that implements the framework to identify other iterative algorithms that are equivalent to an input algorithm.

For databases consisting of many text documents, one of the most fundamental data analysis tasks is counting (i) how often a pattern appears as a substring in the database (substring counting) and (ii) how many documents in the collection contain the pattern as a substring (document counting). If such a database contains sensitive data, it is crucial to protect the privacy of individuals in the database. Differential privacy is the gold standard for privacy in data analysis. It gives rigorous privacy guarantees, but comes at the cost of yielding less accurate results. In this paper, we carry out a theoretical study of substring and document counting under differential privacy. We propose a data structure storing $ε$-differentially private counts for all possible query patterns with a maximum additive error of $O(\ell\cdot\mathrm{polylog}(n\ell|Σ|))$, where $\ell$ is the maximum length of a document in the database, $n$ is the number of documents, and $|Σ|$ is the size of the alphabet. We also improve the error bound for document counting with $(ε, δ)$-differential privacy to $O(\sqrt{\ell}\cdot\mathrm{polylog}(n\ell|Σ|))$. We show that our additive errors for substring counting and document counting are optimal up to an $O(\mathrm{polylog}(n\ell))$ factor both for $ε$-differential privacy and $(ε, δ)$-differential privacy. Our data structures immediately lead to improved algorithms for related problems, such as privately mining frequent substrings and q-grams. Additionally, we develop a new technique of independent interest for differentially privately computing a general class of counting functions on trees.

Induced seismicity caused by fluid extraction or injection in underground reservoirs is a major challenge for safe energy production and storage. This paper presents a robust output-feedback controller for induced seismicity mitigation in geological reservoirs described by a coupled 3D PDE-ODE model. The controller is nonlinear and robust (MIMO Super-Twisting design), producing a continuous control signal and requiring minimal model information, while accommodating parameter uncertainties and spatial heterogeneity. Two operational outputs are regulated simultaneously: regional pressures and seismicity rates computed over reservoir sub-regions. Closed-loop properties are established via explicit bounds on the solution and its time derivative for both the infinite-dimensional dynamics and the nonlinear ODE system, yielding finite-time or exponential convergence of the tracking errors. The method is evaluated on the Groningen gas-field case study in two scenarios: gas production while not exceeding the intrinsic seismicity of the region, and combined production with CO$_2$ injection toward net-zero carbon operation. Simulations demonstrate accurate tracking of pressure and seismicity targets across regions under significant parameter uncertainty, supporting safer reservoir operation while preserving production objectives.

We introduce a dynamic model where the state space is the set of contractible cubical sets in the Euclidian space. The permissible state transitions, that is addition and removal of a cube to/from the set, are closest to Eden model with topological constraints, and, we show, are locally decidable. We prove that in the planar special case the state space is connected. We then define a continuous time Markov chain with a fugacity (tendency to grow) parameter. Using the correspondence between our model on the plane and the self-avoiding polygons, we prove that the Markov chain is irreducible (due to state connectivity), and is also ergodic if the fugacity is smaller than a threshold.

In commuting parametric quantum circuits, the Fourier series of the pairwise fidelity can be expressed as the characteristic function of random variables. Furthermore, expressiveness can be cast as the recurrence probability of a random walk on a lattice. This construction has been successfully applied to the group composed only of Pauli-Z rotations, and we generalize this probabilistic strategy to any commuting set of Pauli operators. We utilize an efficient algorithm by van den Berg and Temme (2020) using the tableau representation of Pauli strings to yield a unitary from the Clifford group that, under conjugation, simultaneously diagonalizes our commuting set of Pauli rotations. Furthermore, we fully characterize the underlying distribution of the random walk using stabilizer states and their basis state representations. This would allow us to tractably compute the lattice volume and variance matrix used to express the frame potential. Together, this demonstrates a scalable strategy to calculate the expressiveness of parametric quantum models.

We analyze the spectral properties of the Heisenberg spin-1/2 chain with random fields in light of recent works of the renormalization-group flow of the Anderson model in infinite dimension. We reconstruct the beta function of the order parameter from the numerical data, and show that it does not admit a one-parameter scaling form and a simple Wilson-Fisher fixed point. Rather, it is compatible with a two-parameter, Berezinskii-Kosterlitz-Thouless-like flow with a line of fixed points (the many-body localized phase), which terminates into the localization transition critical point. Therefore, we argue that previous studies, which assumed the existence of an isolated Wilson- Fisher fixed point and performed one-parameter finite-size scaling analysis, could not explain the numerical data in a coherent way.

This paper introduces a test for fractional integration in a model that possibly contains smooth deterministic trends. We model the trend component using a Chebyshev polynomial and specify the short-run dynamics semi-parametrically, accommodating a broad class of possibly nonlinear processes, including those with conditional heteroskedasticity. We use a local Whittle approach for constructing a Lagrange multiplier test statistic and for constructing a frequency-domain information criterion for the selection of the order of the Chebyshev polynomial. We show that widely used time-domain information criteria are generally inconsistent for the true order, whereas our frequency-domain criterion remains robust under both short- and long-memory behaviour. Monte Carlo simulations and an empirical application to the UK Great Ratios support our theoretical findings.

A quadratic lattice $M$ over a Dedekind domain $R$ with fraction field $F$ is defined to be a finitely generated torsion-free $R$-module equipped with a non-degenerate quadratic form on the $F$-vector space $F\otimes_{R}M$. Assuming that $F\otimes_{R}M$ is isotropic of dimension $\geq 3$ and that $2$ is invertible in $R$, we prove that a quadratic lattice $N$ can be embedded into a quadratic lattice $M$ over $R$ if and only if $S\otimes_{R}N$ can be embedded into $S\otimes_{R}M$ over $S$, where $S$ is the integral closure of $R$ in a finite extension of odd degree of $F$. As a key step in the proof, we establish several versions of the norm principle for integral spinor norms, which may be of independent interest.

The aim of this second part of the article is to study the absolute definition of the seawater entropy described in Part I with several concrete cases. Observed vertical profiles and polar transects, as well as analysed surface data, show that very different temperatures and salinity values can organise to create new isentropic regions. This can only be revealed by the absolute formulation of the entropy of seawater (Arctic Ocean; Bay of Bengal; Mediterranean, Black, and Caspian Seas). Existing hypotheses to explain these results include the possible impact of turbulent processes that must be applied to the entropies of the atmosphere and oceans.