This study investigates the thermodynamic and quantum properties of Einstein-Power-Yang-Mills (EPYM) black holes in an Anti-de Sitter background, focusing on the effects of the nonlinear Yang-Mills charge parameter $γ$. We derive the metric function, analyze Hawking radiation through boson tunneling, and calculate thermodynamic properties including temperature and phase transitions. The quantum tunneling of $W^+$ bosons is examined using the WKB approximation and Hamilton-Jacobi formalism, revealing how nonlinearity modifies the radiation spectrum. We compute the effective potential governing photon orbits and null geodesics, demonstrating significant alterations in light behavior in strong gravitational fields. Additionally, we explore the Aschenbach effect, showing that this phenomenon, which is typically associated with rotating black holes, can emerge in spherically symmetric EPYM spacetimes because of non-linear field interactions. Our results may yield observational markers that can be identified with instruments such as the Event Horizon Telescope and upcoming gravitational wave detectors.

We prove $L^p$-Hardy inequalities with distance to the boundary for domains in the Heisenberg group ${\mathbb{H}}^n$, $n\geq 1$. Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is then implemented for the distance defined by the gauge quasi-norm related to the fundamental solution of the horizontal Laplacian when the domain is a half-space or a convex polytope. Finally it is implemented for the Carnot-Carathéodory distance on half-spaces and arbitrary bounded convex domains of ${\mathbb{H}}^n$. In all cases the constant $((p-1)/p)^p$ is obtained. In the more general context of a stratified Lie group of step two we study the superharmonicity and the weak $H$-concavity of the Euclidean distance to the boundary, thus obtaining a proof of the $L^p$-Hardy inequality on convex domains.

We study controllability and constructive synthesis for control-affine systems. We introduce trajectory-dependent Gramian maps that extend the linear time-varying Gramian and yield explicit fixed-point synthesis maps. On feasible coercivity classes (uniform eigenvalue lower bounds), the Gramian map is Lipschitz, synthesis iterates exhibit factorial decay, and the Caccioppoli fixed point theorem gives a unique fixed point that steers the system and satisfies an energy identity. When, in addition, an orthogonality condition holds, this fixed point coincides with the unique global minimum-energy control on the feasible set; if the coercivity bound holds uniformly for all bounded controls, the same conclusion holds on the full bounded-control space. We provide structural conditions on the input matrix that ensure the nonemptiness of the feasible class (and, in fully actuated regimes, equality with the full space) and sufficient conditions for underactuated systems via bounded-amplitude reference controls. Case studies on Hopfield network dynamics illustrate refined estimates that enlarge reachable targets. A trajectory-freezing and compactness step extends the synthesis to general nonlinear control-affine systems. The results yield verifiable controllability criteria with explicit, numerically implementable controllers.

The question of whether interactions can break dynamical localization in quantum kicked rotor systems has been the subject of a long--standing debate. Here, we introduce an extended mapping from the kicked Lieb--Liniger model to a high--dimensional lattice model and reveal universal features: on--site pseudorandomness and hybrid exponential--algebraic decay couplings with increasing momenta. We find that the exponent and the amplitude of the algebraic decay undergo a crossover as the interaction strength increases. This mapping uncovers the origin of dynamical localization and the interaction effect on the integrability of the system. An analysis of the generalized fractal dimension and level--spacing ratio supports these findings, highlighting the presence of near integrability and multifractality in different regions of parameter space. Our results offer an explanation for the occurrence of many--body dynamical localization, particularly in strongly correlated quantum gases, and are anticipated to generalize to systems of many particles.

We prove that there are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, fixed dimension $n\neq5$, and finite fundamental group, up to foliated diffeomorphism. In addition, we construct various infinite families of isoparametric foliations that are mutually not foliated diffeomorphic, for instance on a fixed sphere.

We show that the coupling of a massive spin-2 field to undeformed N=1 supergravity in four dimensions is unique, leading to a specific non-minimal coupling to the Riemann tensor. The massive spin-2 coupling reproduces the one of an oscillator mode in open string theory, while the massive spin-1 coupling includes a higher-derivative term that is expected to violate causality in the background of a gravitational shock wave. We argue that the resolution of causality and the unitarity bound in the Regge limit require the introduction of infinitely many higher-spin fields similar to the Regge trajectories in string theory, therefore providing an argument in favour of the string lamppost principle with minimal supersymmetry in four dimensions. To obtain this result, we construct the general stress-energy tensor multiplet for the massive spin-2 multiplet with N=1 supersymmetry.

High-fidelity quantum entanglement enables key quantum networking capabilities such as secure communication and distributed quantum computing, but long-distance entanglement distribution is limited by noise and loss. Entanglement distillation protocols address this problem by extracting high-fidelity Bell pairs from multiple noisy ones. The primary objective is minimizing the resource overhead: the number of noisy input pairs needed to distill each high-fidelity output pair. While protocols achieving optimal overhead are known in theory, they often require complex decoding operations that make practical implementation challenging. We circumvent this challenge by introducing protocols that use quantum scrambling -- the spreading of quantum information under chaotic dynamics -- through random Clifford operations. Based on this scrambling mechanism, our protocol maintains asymptotically \emph{constant} overhead, independent of the desired output error rate $\bar{\varepsilon}$, and can be implemented with shallow quantum circuits of depth $O(\mathrm{poly} \log \log \bar{\varepsilon}^{-1})$ and memory $O(\mathrm{poly} \log \bar{\varepsilon}^{-1})$. Our protocol remains effective even with noisy quantum gates. By incorporating error correction, our protocol achieves state-of-the-art performance: starting with pairs of 10% initial infidelity, we require only 7 noisy inputs per output pair to distill a single Bell pair with infidelity $\bar{\varepsilon}=10^{-12}$, substantially outperforming existing schemes. We demonstrate the utility of our protocols for quantum repeater networks.

In this paper, we quantify the resilience of nonlinear dynamical systems by studying the increased energy used by all inputs of a system that suffers a partial loss of control authority, either through actuator malfunctions or through adversarial attacks. To quantify the maximal increase in energy, we introduce the notion of an energetic resilience metric. Prior work in this particular setting does not consider general nonlinear dynamical systems. In developing this framework, we first consider the special case of linear driftless systems and recall the energies in the control signal in the nominal and malfunctioning systems. Using these energies, we derive a bound on the energetic resilience metric. For general nonlinear systems, we first obtain a condition on the mean value of the control signal in both the nominal and malfunctioning systems, which allows us to approximate the energy in the control. We then obtain a worst-case approximation of this energy for the malfunctioning system, over all malfunctioning inputs. Assuming this approximation is exact, we derive bounds on the energetic resilience metric when control authority is lost over one actuator. A set of simulation examples demonstrate that the metric is useful in quantifying the resilience of the system without significant conservatism, despite the approximations used in obtaining control energies for nonlinear systems.

The interplay of magnetic excitations and itinerant charge carriers is a ubiquitous phenomenon in strongly correlated electron systems. In the vicinity of magnetically ordered phases, strong interactions between itinerant quasiparticles and magnetic excitations can result in the dramatic renormalization of both. A key theoretical question is understanding the renormalization of the magnon quasiparticle, a collective spin excitation, upon doping a magnetic insulator. Here, we report the observation of a new type of quasiparticle arising from the dressing of a magnon with the doped holes of a cold atom Fermi-Hubbard system, i.e. a magnon-Fermi-polaron. Utilizing Raman excitation with controlled momentum in a doped, spin-polarized band insulator, we address the spectroscopic properties of the magnon-polaron. In an undoped system with strong interactions, photoexcitation produces magnons, whose properties are accurately described by spin wave theory. We measure the evolution of the photoexcitation spectra as we move away from this limit to produce magnon-polarons due to dressing of the magnons by charge excitations. We observe a shift in the polaron energy with doping that is strongly dependent on the injected momentum, accompanied by a reduction of spectral weight in the probed energy window. We anticipate that the technique introduced here, which is the analog of inelastic neutron scattering, will provide atomic quantum simulators access to the dynamics of a wide variety of excitations in strongly correlated phases.

In quantum mechanics, the time evolution of particles is given by the Schrödinger equation. It is valid in a nonrelativistic regime where the interactions with the particle can be modelled by a potential and quantised fields are not required. This has been verified in countless experiments when the interaction is of electromagnetic origin, but also corrections due to the quantised field are readily observed. When the interaction is due to gravity, then one cannot expect to see effects of the quantised field in current-technology Earth-bound experiments. However, this does not yet guarantee that in the accessible regime, the time evolution is accurately given by the Schrödinger equation. Here we propose to measure the effects of an asymmetric mass configuration on a quantum particle in an interferometer. For this setup we show that with parameters within experimental reach, one can be sensitive to possible deviations from the Schrödinger equation, beyond the already verified lowest-order regime. Performing this experiment will hence directly test the nonclassical behaviour of a quantum particle in the gravitational field.

We study a population of $N$ individuals evolving according to a biparental Moran model with two types, one being advantaged compared to the other. The advantage is conferred by a Mendelian mutation, which reduces the death probability of individuals carrying it. We assume that a proportion $a$ of individuals initially carry this mutation, which therefore eventually gets fixed with high probability. After a long time, we sample a gene uniformly from the population, at a new locus, independent of the locus under selection, and calculate the probability that this gene originated from one of the initially advantaged individuals, when the population size is large. Our theorem provides quantitative insights, such as the observation that under strong viability selection, if only $1\%$ of the individuals are initially advantaged, up to $19\%$ of the population's genome will originate from them after a long time.

We show that Banyaga's Hofer-like norm, a generalization of the Hofer norm coincides with the classical Hofer norm when restricted to Hamiltonian diffeomorphisms on compact symplectic manifolds. This result proves a conjecture of Banyaga and fills the gap between Hofer and Hofer-like geometries: the refined Hofer-like structure degenerates to standard Hofer geometry within the Hamiltonian subgroup. This equality allows the straightforward extension of essential results from Hofer geometry to the Hofer-like setting.

Quantifying causal effects in the presence of complex and multivariate outcomes remains a key challenge in treatment evaluation. For hierarchical multivariate outcomes, the FDA recommends the Win Ratio and Generalized Pairwise Comparisons approaches \citep{Pocock2011winratio,Buyse2010}. However, commonly used estimators can yield treatment recommendations that target a population-level estimand (the probability that a randomly sampled patient under treatment fares better than another randomly sampled patient under control), which can contradict conclusions drawn from an ideal estimand (the probability that an individual would fare better with treatment than without), especially in heterogeneous populations. This discrepancy arises from the non-identifiability of the latter estimand and underscores both the influence of the chosen causal measure on the resulting conclusions and the necessity of articulating the underlying causal framework with clarity. We propose a novel, individual-level yet identifiable causal effect measure that more closely approximates the ideal individual-level estimand. We show that computing the Win Ratio or Net Benefit via nearest-neighbor pairing between treated and control patients, which can be seen as an extreme form of stratification, yields an estimator of our new causal measure in both randomized controlled trials and observational settings. We then develop a distributional regression framework, alongside semiparametric efficient estimators. Our methods are simple to implement and readily applicable in practice. We evaluate the proposed approach through simulations and apply it to the CRASH-3 trial \citep{crash3}, a major study assessing the effects of tranexamic acid in patients with traumatic brain injury.

Given an extended real-valued submeasure $ν$ defined on a field of subsets $Σ$ of a given set, we provide necessary and sufficient conditions for which the pseudometric $d_ν$ defined by $d_ν(A,B):=\min\{1,ν(A\bigtriangleup B)\}$ for all $A,B \in Σ$ is complete. As an application, we show that if $φ: \mathcal{P}(ω)\to [0,\infty]$ is a lower semicontinuous submeasure and $ν(A):=\lim_n φ(A\setminus \{0, 1, \ldots, n-1\})$ for all $A\subseteq ω$, then $d_ν$ is complete. This includes the case of all weighted upper densities, fixing a gap in a proof by Just and Krawczyk in [Trans.~Amer.~Math.~Soc.~\textbf{285} (1984), 803--816]. In contrast, we prove that if $ν$ is the upper Banach density (or an upper density greater than or equal to the latter) then $d_ν$ is not complete. We conclude with several characterizations of completeness in terms of the Stone space of the Boolean algebra $Σ/ν$.

We study a relationship between the Heegaard Floer homology correction terms of integral homology spheres and the word metric on the Torelli group. For example, we give an elementary proof that the Cayley graph of the Torelli group has infinite diameter in the word metric induced by the generating set of all separating twists and bounding pair maps. On the other hand, we show that many subsets of the Torelli group are bounded with respect to this metric. Finally, we address the case of rational homology spheres by ruling out a certain Morita-type formula for congruence subgroups of mapping class groups.

We show that the physical consistency of magic state distillation imposes new constraints on the weight enumerators of classical error-correcting codes. We establish that for $|T\rangle$-state distillation protocols based on linear self-orthogonal $GF(4)$ codes, the distillation threshold and noise-suppression exponent are directly determined by the code's simple weight enumerator. By enforcing the physical consistency of the distillation process -- specifically, that the probability of successfully projecting onto the target state must be non-negative -- we derive a new set of constraints on classical weight enumerators. These ``quantum consistency'' constraints prove to be strictly stronger than those derived from classical invariant theory, yielding new upper bounds on the minimum distance of certain classical and quantum codes. Most notably, we show that these new constraints resolve a long-standing open problem in classical coding theory by proving the non-existence of extremal Hermitian self-dual codes over $GF(4)$ with parameters $[12m, 6m, 4m+2]$. Additionally, we use our formalism to perform an exhaustive search of distillation protocols based on linear $GF(4)$ codes with $n < 20$, finding no protocols with thresholds exceeding the 5-qubit code, and we derive analytic upper bounds on the noise-suppression exponents of such distillation routines as a function of $n$.

The electromagnetic (EM) scattering by non-symmetric wavelength-scale particles on a planar surface has numerous applications in the remote sensing of planetary bodies, both in planetary and geo-sciences. We conduct numerical simulations of EM scattering by rough polyhedral particles (with 12 or 20 faces) using the discrete-dipole approximation and contrast the results to that of spheres. The particles have permittivities corresponding to common minerals in the microwave regime ($ε_r=4.7 + 0.016$i and $7.8 + 0.09$i), and a size-frequency distribution (SFD) consistent with the observed scattering properties (power-law distribution of size parameters between 0.5 and 8 with an index from $-2.5$ to $-3.5$). The assumed substrate permittivity $2.4 + 0.012$i corresponds to a powdered regolith. We present what roles the particle roundness, permittivity, and SFD for a realistic range of parameters play in the EM scattering properties as a function of incidence angle with a focus on backscattering in microwave-remote-sensing applications. The particle roundness and SFD have a clearly observable effect on the polarimetric properties, while the role of permittivity is relatively minor (in the studied range). Among various backscattering observables, the circular polarization ratio is the least sensitive to the decrease of the upper boundary (down to a size parameter of 3) and the index of the SFD.

For a system of partial differential equations (PDEs) $F = 0$ admitting a local (point, contact, or higher) symmetry $X$ with the characteristic $φ$, invariant solutions satisfy the reduced system $F = φ= 0$. We propose a framework that allows, for every $X$-invariant conservation law, presymplectic structure, variational principle, or another geometric structure of the given PDE system $F = 0$, to systematically calculate its corresponding reduced form that describes the corresponding structure for the reduced system $F = φ= 0$. In particular, we show in what way Noether's theorem holding for the given PDE system is inherited by the reduced PDE system. We consider several detailed examples, including cases of point and higher symmetry invariance. The proposed framework is directly applicable to a wide range of PDE models, including complex PDE systems of contemporary interest arising across disciplines, where symmetry reduction is essential for analysis and simulation, as well as to integrable, Lagrangian, and gauge systems.

In this paper we are concerned with the homogenization property of stochastic non-homogeneous incompressible Navier-Stokes equations with rapid oscillation in a smooth bounded domain of $\mathbb{R}^d$, $d=2,3$, and driven by multiplicative infinite-dimensional Wiener noise. Using two-scale convergence, stochastic compactness and the martingale representation theory, we first show the solutions of original equations converge to the solution of a stochastic non-homogeneous incompressible homogenized system. Also, the energy equation of the homogenized system is established. Furthermore, a corrector result is proved which strengthens the two-scale convergence from weak to strong in the regularity space $H^1(\mathcal{O})$. Since the continuity equation which is of transport type cannot confer any regularization effect, there are some issues for proving the two results, including the difficulties for establishing the stochastic compactness and passing to the limit. We develop new regularity estimates, a stochastic version of lower semicontinuity as well as energy equation to overcome these difficulties.

The in-plane anomalous Hall effect (IPAHE) with planar Hall current and magnetization/magnetic fields in various quantum materials has received increasing attention. Most of the current efforts are devoted to the intrinsic part due to the Berry curvature of electronic bands, however, how disorder scattering affects the extrinsic part (the skew scattering and side jump) remains largely elusive. Here we theoretically investigate the three universal classes of disorder scattering (scalar, spin-conserving, and spin-flipping) for the IPAHE, based on the prototypical two-dimensional massive Dirac fermion model with warping term under generic Zeeman fields. We find that the different disorder scattering results in a distinct dependence of the anomalous Hall conductivity on disorder strength, and we recover previously known results within some limits. Remarkably, the spin-flipping scattering could give rise to nontrivial contributions featuring sinusoidal oscillations with periods of \textgreek{π} and 2\textgreek{π} to the extrinsic part, in contrast to the standard two-dimensional massive Dirac fermions. Our work unveils the rich features of anomalous transport in planar Hall geometry in the presence of disorder scattering and provides some useful insights into the magnetotransport phenomena.

It is shown that longitudinally polarized protons can be used to induce chirality in the final states of the $(\vec{p},pN)$ reaction at intermediate energies, when there exist three final-state particles with non-coplanar momentum vectors. The analyzing power $A_z$ is proposed as a measure of this effect. Theoretical descriptions to obtain $A_z$ based on an intuitive picture as well as a distorted wave impulse approximation are presented, showing that the helicity of incident protons is coupled to the chirality of the orbital motion of a single-particle wave function, resulting in the chirality of the final states and a large $A_z$ value.

Photocatalytic water splitting has attracted considerable attention for renewable energy production. Since the first reported photocatalytic water splitting by titanium dioxide, this material remains one of the most promising photocatalysts, due to its suitable band gap and band-edge positions. However, predicting both of these properties is a challenging task for existing computational methods. Here we show how Koopmans spectral functionals can accurately predict the band structure and level alignment of rutile, anatase, and brookite TiO$_2$ using a computationally efficient workflow that only requires (a) a DFT calculation of the photocatalyst/vacuum interface and (b) a Koopmans spectral functional calculation of the bulk photocatalyst. The success of this approach for TiO$_2$ suggests that this strategy could be deployed for assessing the suitability of novel photocatalyst candidates.

We prove that if the zero set of the Fourier transform of $A\subseteq\mathbb Z_n\times\mathbb Z_n$ contains an element of prime power order, then there is an equi-distribution relation in subsets of $A$ with respect to certain hyperplanes. With this we further show that if $A$ is a tiling complement of the subgroup generated by $(p,0)$ and $(0,p)$ in $\mathbb Z_{p^m}\times\mathbb Z_{p^m}$, then the zero set of its Fourier transform is disjoint with the orthogonal rotation of $A$. These results are motivated by a casual observation in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$.

Using only a binary intervention and outcome and the design of the randomization within an experiment, we construct a design-based likelihood of the joint distribution of potential outcomes in the sample -- the numbers of always takers, compliers, defiers, and never takers. We develop a visualization to show that samples with defiers can sometimes generate the data in more ways than samples without, yielding a higher likelihood. This likelihood can vary within the Frechet bounds, even though the traditional likelihood does not. Evidence is weak, but it exists, as we illustrate with health applications and our dbmle package.

We study the symmetry of the coupling between surface acoustic waves and ferromagnetic resonance in a thin magnetic film of CoFeB deposited on top of a piezoelectric Z-cut LiNbO3 substrate. We vary the orientation of the applied magnetic field with respect to the wavevector of the surface acoustic wave. Experiments indicate an unexpected 2-fold symmetry of the absorption of the SAW energy by the magnetic film. We discuss whether this symmetry can arise from the magnetoelastic torque of the longitudinal strain and the magnetic susceptibility of ferromagnetic resonance. We find that one origin of the 2-fold symmetry can be the weak in-plane uniaxial anisotropy present within the magnetic film. This phenomena adds to the previously identified other source of 2-fold symmetry but shall persist for ultrathin films when the dipolar interactions cease to contribute to the anisotropy of the slope of the spin wave dispersion relation.

A microscopic model of the Brout-Englert-Higgs (BEH) mechanism is proposed. Massless fermions and antifermions do not belong to the Fock space with definite particle-number distribution, but belong to a non-Fock space with indefinite one. From this vast space, their ground-state is selected by a kinematical condition. Due to the interaction between them, the Fock state in which fermions and antifermions are massive is restored, but this state has the vestiges of the non-Fock state in the massive gauge boson and the Higgs boson. In the non-Fock state, massless fermions and antifermions exist as pairs, and behave as quasi bosons within a small space-time region. Due to Bose statistics, the direction of their center-of-mass motion is parallel to each other, and their transverse excitations are suppressed by an energy gap, making the gauge boson coupled to them massive. The Higgs boson is not an elementary particle, but a quasiparticle appearing after the Fock vacuum is restored.

Let $G$ be a graph and $T$ be a spanning tree of $G$. We use $Q(G)=D(G)+A(G)$ to denote the signless Laplacian matrix of $G$, where $D(G)$ is the diagonal degree matrix of $G$ and $A(G)$ is the adjacency matrix of $G$. The signless Laplacian spectral radius of $G$ is denoted by $q(G)$. A necessary and sufficient condition for a connected bipartite graph $G$ with bipartition $(A,B)$ to have a spanning tree $T$ with $d_T(v)\geq k$ for any $v\in A$ was independently obtained by Frank and Gyárfás (A. Frank, E. Gyárfás, How to orient the edges of a graph?, Colloq. Math. Soc. Janos Bolyai 18 (1976) 353--364), Kaneko and Yoshimoto (A. Kaneko, K. Yoshimoto, On spanning trees with restricted degrees, Inform. Process. Lett. 73 (2000) 163--165). Based on the above result, we establish a lower bound on the signless Laplacian spectral radius $q(G)$ of a connected bipartite graph $G$ with bipartition $(A,B)$, in which the bound guarantees that $G$ has a spanning tree $T$ with $d_T(v)\geq k$ for any $v\in A$.

We present a generalized linear response theory for mixed jump-diffusion models -- combining Gaussian and Lévy noise interacting with nonlinear dynamics -- by deriving comprehensive response formulas accounting for perturbations to both the drift term and the jumps law. This class of models is particularly relevant for parameterizing the effects of unresolved scales in complex systems. Our formulas thus quantify uncertainties in parameterized components (e.g., jump laws) or measure dynamical changes due to drift term perturbations (e.g., parameter variations). By generalizing the concepts of Kolmogorov operators and Green's functions, we obtain new forms of fluctuation-dissipation relations. The resulting response is decomposed into contributions from the eigenmodes of the Kolmogorov operator, revealing the intimate relationship between a system's natural and forced variability. We demonstrate the theory's predictive power with two distinct climate-centric applications. First, we apply our framework to a paradigmatic ENSO model subject to state-dependent jumps and additive white noise, showing how the theory accurately predicts the system's response to perturbations and how Kolmogorov modes can be used to diagnose its complex time variability. In a second, more challenging application, we use our linear response theory to perform accurate climate change projections in the Ghil-Sellers energy balance climate model, a spatially-extended model forced by a spatio-temporal $α$-stable process. This work provides a comprehensive approach to climate modeling and prediction that enriches Hasselmann's program, with implications for understanding climate sensitivity, detection and attribution of climate change, and assessing climate tipping points. Our results may find applications beyond climate, and are relevant for epidemiology, biology, finance, and quantitative social sciences.

We study linearly independent complex line fields on almost-complex manifolds, which is a topic of long-standing interest in differential topology and complex geometry. A necessary condition for the existence of such fields is the vanishing of appropriate virtual Chern classes. We prove that this condition is also sufficient for the existence of one, two, or three linearly independent complex line fields over certain manifolds. More generally, our results hold for a wider class of complex bundles over CW complexes.

Dirac operators and Dirac cohomology for Lie superalgebras of Riemannian type, introduced by Huang and Pandžić, provide an effective tool for the study of unitarizable supermodules. In this article, we study these objects for Lie superalgebras of type $A$ and relate them systematically to unitarity. In the first part, we establish the basic structure of the theory in this setting. We relate unitarity to the Dirac operator, derive the corresponding Dirac inequality, and show that Dirac cohomology determines unitarizable supermodules. We also determine explicitly the Dirac cohomology of unitarizable simple supermodules. In the second part, we turn to applications. We obtain a new characterization of unitarity, establish a relation with Kostant's cohomology, derive a formula for formal characters, and introduce a Dirac index.