We extend the notion of a Sasakian structure from the classical setting of a cooriented contact manifold, where it is given by a compatibility between a contact form $η$ and a Riemannian metric $g_M$ on $M$, to the case of an arbitrary contact structure understood as a contact distribution. In the cooriented case, this compatibility can be equivalently expressed by the fact that the symplectic form $ω=\mathrm{d}(s^2η)$ and the cone metric $g(x,s)=\mathrm{d} s\otimes\mathrm{d} s+s^2g_M(x)$ define a Kähler structure on the cone $\mathcal{M}=M\times\mathbb{R}_+$. Since general contact structures admit canonical realizations as homogeneous symplectic structures $ω$ on principal $\mathbb{R}^\times$-bundles $P\to M$, it is natural to interpret Sasakian geometry in full generality in terms of suitable homogeneous Kähler structures on $P$. We characterize homogeneous Kähler structures on symplectizations $(P,ω)$ associated with arbitrary contact structures on $M$, and show that they canonically determine a two-sheeted covering $\tilde M$ of $M$ equipped with a contact form. This reduces the problem to the cooriented case and leads to a notion of a generalized Sasakian structure on $M$ associated with a homogeneous Kähler structure on $(P,ω)$. Moreover, since products of Kähler manifolds are again Kähler, our framework naturally yields a concept of a product of Sasakian manifolds. The whole constructions are intrinsic and conceptual, avoiding any ad hoc choices.

We present TOI-2155\,b, a massive transiting companion, discovered using data from NASA's Transiting Exoplanet Survey Satellite (TESS) mission and confirmed with ground-based RV measurements from the Tillinghast Reflector Echelle Spectrograph (TRES). We also analyze ground-based follow-up photometric data from the Wendelstein Observatory (WST), Las Cumbres Observatory Global Telescope (LCOGT), and Wild Boar Remote Observatory (WBR). TOI-2155\,b is a short-period companion with {$P= 3.7246950 \pm{0.0000014}$}~days. The radius and mass of TOI-2155\,b are found to be $R_b = 0.972^{+0.009}_{-0.008} \,\mathrm{R_J}$ and $M_b = 80.6^{+1.0}_{-1.1} \,\mathrm{M_J}$, respectively, corresponding to a density of {$ρ_b= 109^{+3.1}_{-3.3}$ g cm$^{-3}$}. The F-type subgiant host star has an effective temperature of $T_{\rm eff} = 6085\pm 78$ K, a radius $R_{\thinstar} = 1.705^{+0.066}_{-0.064}$ $\mathrm{R_\odot}$ and a mass $M_\star = 1.33 \pm 0.008$~M$_\odot$. With a mass close to the hydrogen-burning minimum mass, TOI-2155\,b lies at the boundary between brown dwarfs and low-mass stars. Its measured mass, radius, and density place it in a transitional region, where distinguishing between a massive brown dwarf and a very low-mass star is not straightforward. TOI-2155\,b therefore provides a valuable benchmark for testing evolutionary models of stellar and substellar structure near the hydrogen-burning limit.

The shadow of an abstract simplicial complex $K$ with vertices in $\mathbb{R}^N$ is a subset of $\mathbb{R}^N$ defined as the union of the convex hulls of simplices of $K$. The Vietoris--Rips complex of a metric space $(S,d)$ at scale $β$ is an abstract simplicial complex whose each $k$-simplex corresponds to $(k+1)$ points of $S$ within diameter $β$. In case $S\subset\mathbb R^2$ and $d(a,b)=\|a-b\|$ the standard Euclidean metric, the natural shadow projection of the Vietoris--Rips complex is already proved by Chambers et al. to induce isomorphisms on $π_0$ and $π_1$. We extend the result beyond the standard Euclidean distance on $S\subset\mathbb R^N$ to a family of path-based metrics, $d^\varepsilon_{S}$. From the pairwise Euclidean distances of points in $S$, we introduce a family (parametrized by $\varepsilon$) of path-based Vietoris--Rips complexes $R^\varepsilon_β(S)$ for a scale $β>0$. If $S\subset\mathbb{R}^2$ is Hausdorff-close to a planar Euclidean graph $G$, we provide quantitative bounds on scales $β,\varepsilon$ for the shadow projection map of the Vietoris--Rips complex of $(S,d^\varepsilon_S)$ at scale $β$ to induce $π_1$-isomorphism. This paper first studies the homotopy-type recovery of $G\subset\mathbb R^N$ using the abstract Vietoris--Rips complex of a Hausdorff-close sample $S$ under the $d^\varepsilon_S$ metric. Then, our result on the $π_1$-isomorphism induced by the shadow projection lends itself to providing also a geometrically close embedding for the reconstruction. Based on the length of the shortest loop and large-scale distortion of the embedding of $G$, we quantify the choice of a suitable sample density $\varepsilon$ and a scale $β$ at which the shadow of $R^\varepsilon_β(S)$ is homotopy-equivalent and Hausdorff-close to $G$.

Dynamical quantum phase transitions (DQPTs) are non-equilibrium transitions characterized by the orthogonality between an initial quantum state and its time-evolved counterpart following a sudden quench. Recently, studies of this phenomenon have been extended beyond closed quantum systems to include environmental interactions, often modeled through non-Hermitian effects. However, because non-Hermitian descriptions neglect both quantum jump processes and interaction effects, the ultimate fate of interacting quantum systems under full open-system quantum dynamics remains an open question. In this paper, by incorporating both interactions and full Liouvillian dynamics, we prove that DQPTs in open quantum systems remain robust when subject to either particle loss or particle gain alone, but are generically smeared out when both processes coexist, as a result of many-body particle backflow. Furthermore, we uncover a non-perturbative dynamical effect: even a weak admixture of gain (loss) into a system with loss (gain) can dramatically reshape the long-time behavior of DQPT dynamics, leading to substantial deviations over time. These phenomena--including the universal smearing of DQPTs and the emergence of large dynamical deviations in the long-time limit--arise intrinsically from non-equilibrium many-body effects in open quantum systems. Our findings are general and substantiated by both analytical arguments and numerical simulations.

In this paper, we introduce the notions of quasi-triangular and factorizable Poisson bialgebras. A factorizable Poisson bialgebra induces a factorization of the underlying Poisson algebra. We prove that the Drinfeld classical double of a Poisson bialgebra naturally admits a factorizable Poisson bialgebra structure. Furthermore, we introduce the notion of quadratic Rota-Baxter Poisson algebras and show that a quadratic Rota-Baxter Poisson algebra of zero weight induces a triangular Poisson bialgebra. Moreover, we establish a one-to-one correspondence between factorizable Poisson bialgebras and quadratic Rota-Baxter Poisson algebras of nonzero weights. Finally, we establish the quasi-triangular and factorizable theories for differential antisymmetric infinitesimal (ASI) bialgebras, and construct quasi-triangular and factorizable Poisson bialgebras from quasi-triangular and factorizable (commutative and cocommutative) differential ASI bialgebras respectively.

A great amount of endeavor has recently been devoted to activity detection for massive machine-type communications in cell-free multiple-input multiple-output (MIMO) systems. However, as the number of antennas at the access points (APs) increases, the Rayleigh distance that separates the near-field and far-field regions also expands, rendering the conventional assumption of far-field propagation alone impractical. To address this challenge, this paper establishes a covariance-based formulation that can effectively capture the statistical property of hybrid near-far field channels. Based on this formulation, we theoretically reveal that increasing the proportion of near-field channels enhances the detection performance. Furthermore, we propose a distributed algorithm, where each AP performs local activity detection and only exchanges the detection results to the central processing unit, thus significantly reducing the computational complexity and the communication overhead. Not only with convergence guarantee, the proposed algorithm is unified in the sense that it can handle single-cell or cell-free systems with either near-field or far-field devices as special cases. Simulation results validate the theoretical analyses and demonstrate the superior performance of the proposed approach compared with existing methods.

We study the impact of the carbon steel rebars and the steel TT bridge within the JUNO structure on the shielding effect of the coils. Our simulations demonstrate that despite the presence of carbon steel structures of the rebars of the water pool and the TT bridge within the central detector vicinity, the residual magnetic field experienced by the PMTs remains within the acceptable limit established by the JUNO experiment of 10% for CD-PMTs and 20% for Veto-PMTs, compared to the geomagnetic field. The maximum magnetic fields experienced by the CD-PMTs and Veto-PMTs are 9% and 18% of the geomagnetic field strength, respectively. These findings indicate that the residual magnetic field has minimal impacts on the PMTs detection efficiency.

We develop new methods for approximating conformal blocks as positive functions times polynomials, with applications to the numerical bootstrap. We argue that to obtain accurate bootstrap bounds, conformal block approximations should minimize a certain error norm related to the asymptotics of dispersive functionals. This error norm can be made small using interpolation nodes with an appropriate optimal density. The optimal density turns out to satisfy a kind of force-balance equation for charges in one dimension, which can be solved using standard techniques from large-N matrix models. We also describe how to use optimal density interpolation nodes to improve condition numbers inside the semidefinite program solver SDPB. Altogether, our new approximation scheme and improvements to condition numbers lead to more accurate bootstrap bounds with fewer computational resources. They were crucial in the recent bootstrap study of stress tensors in the 3d Ising CFT.

Here we first present an alternative formulation of the Lewis & Riesenfeld theorem for solving the Schrödinger equation with nonautonomous Hermitian and pseudo-Hermitian Hamiltonians. We then employ this framework to characterize the spontaneous breaking of time-dependent antilinear symmetries of these Hamiltonians. We demonstrate that, under unbroken antilinear symmetries, the Lewis & Riesenfeld phases are real and odd functions of time, which allows us to recover the well-known real spectra of time-independent pseudo-Hermitian Hamiltonians. However, in the spontaneously broken regime, imaginary components of the Lewis & Riesenfeld phases arise, leading to coalescence effects analogous to those in the time-independent scenario. Finally, we present an illustrative example of unbroken and broken PT-symmetry for a time-dependent Hamiltonian modeling the non-Hermitian dynamical Casimir effect.

Let $F$ be a locally compact non-Archimedean field of characteristic $0$, and let $G$ be either the split special orthogonal group $\mathrm{SO}_{2n+1}(F)$ or the symplectic group $\mathrm{Sp}_{2n}(F)$. The goal of this paper is to give an explicit description of the Aubert-Zelevinsky duality for $G$ in terms of Langlands parameters. We present a new algorithm, inspired by the Moeglin-Waldspurger algorithm for $\mathrm{GL}_n(F)$, which computes the dual Langlands data in a recursive and combinatorial way. Our method is simple enough to be carried out by hand and provides a practical tool for explicit computations. Interestingly, the algorithm was discovered with the help of machine learning tools, guiding us toward patterns that led to its formulation.

In this paper, we introduce an immersed $C^0$ interior penalty method for solving two-dimensional biharmonic interface problems on unfitted meshes. To accommodate the biharmonic interface conditions, high-order immersed finite element (IFE) spaces are constructed in the least-squares sense. We establish key properties of these spaces including unisolvency and partition of unity are, and verify their optimal approximation capability. These spaces are further incorporated into a modified $C^0$ interior penalty scheme with additional penalty terms on interface segments. The well-posedness of the discrete solution is proved. Numerical experiments with various interface geometries confirm optimal convergence of the proposed method in $L^2$, $H^1$ and $H^2$ norms.

Micro-objects levitated in a vacuum are an exciting platform for precision sensing due to their low dissipation motion and the potential for control at the quantum level. Arrays of such sensors would offer increased sensitivity, directionality, and in the quantum regime the potential to exploit correlation and entanglement. We use neuromorphic detection via a single event based camera to record the motion of an array of levitated microspheres. We present a scalable method for arbitrary multiparticle tracking and control by implementing real-time feedback to {simultaneously cool the motion of three uncoupled microscale objects

Multigraded linear series generalize the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We investigate the collection of the natural cornering morphisms into elementary bigraded linear series obtained from direct summands of the original globally generated vector bundle. Our main result is a condition on the injectivity of the product morphism. We apply our result in two examples: modules over the reconstruction algebra and equivariant Hilbert and Quot schemes of quotient stacks.

We show that every closed connected non-orientable PL $4$-manifold $X$ is a simple branched covering of $\RP^4$. We also show that $X$ is a simple branched covering of the twisted $S^3$-bundle $S^1 \simtimes S^3$ if and only if the first Stiefel--Whitney class $w_1(X)$ admits an integral lift. In both cases, the degree of the covering can be any number $d \geq 4$, provided that $d$ has the same parity of the Stiefel--Whitney number $w_1^4[X]$ in the case of $\RP^4$. Moreover, the branch set can be assumed to be non-singular if $d \geq 5$ and to have just nodal singularities if $d=4$.

The "classical" (weak) greedy algorithm is widely used within model order reduction in order to compute a reduced basis in the offline training phase: An a posteriori error estimator is maximized and the snapshot corresponding to the maximizer is added to the basis. Since these snapshots are determined by a sufficiently detailed discretization, the offline phase is often computationally extremely costly. We suggest to replace the serial determination of one snapshot after the other by a parallel approach. In order to do so, we introduce a batch size $b$ and add $b$ snapshots to the current basis in every greedy iteration. These snapshots are computed in parallel. We prove convergence rates for this new batch greedy algorithm and compare them to those of the classical (weak) greedy algorithm in the Hilbert and Banach space case. Then, we present numerical results where we apply a (parallel) implementation of the proposed algorithm to the linear elliptic thermal block problem. We analyze the convergence rate as well as the offline and online wall-clock times for different batch sizes. We show that the proposed variant can significantly speed-up the offline phase while the size of the reduced problem is only moderately increased. The benefit of the parallel batch greedy increases for more complicated problems.

Implementation security, higher generation rate, and lower cost are primary missions in the domain of quantum key distributions in recent years. However, simultaneously achieving robust security, high speed, and low cost often resembles an ``impossible triangle''. This is largely because the modulation system imposes a strict bandwidth limitation. Pushing a low-cost modulator to a high repetition frequency inevitably introduces correlations and misalignment, which can create security loopholes. Conversely, operating at a conservative rate fails to exploit the system's potential, while adopting ultra-high-bandwidth components is often expensive for practical implementation, forcing a perpetual trade-off among implementation security, key rate, and cost. In this work, we propose a comprehensive countermeasure to overcome this modulation bandwidth bottleneck. We present a protocol specifically designed to address the security loopholes arising from modulation imperfections, ensuring security even in overclocked modulation systems. Furthermore, we develop two practical techniques to characterize and mitigate the detrimental correlations. Our experimental setup demonstrates that the proposed method achieves the lowest correlated deviation reported in similar studies, while maintaining a high secret key rate using a bandwidth-limited modulation system. By simultaneously enhancing security, performance, and practicality, this work releases QKD systems from the traditional performance-cost trade-off in the near term, paving the way for widespread deployment. In the long run, this work can be readily integrated with high-bandwidth components to further push the boundaries of system performance.

Type Ia supernovae (SNe) occur when a white dwarf (WD) explodes via runaway thermonuclear burning. Till date, major uncertainties remain regarding the nature of the explosion mechanism and its observable signatures. In this work, we study how the double detonation explosion mechanism, or a helium shell detonation in a sub-Chandrasekhar WD followed by a core detonation, shapes supernova remnants (SNRs) and encodes information about the WD progenitor. We evolve a suite of double-detonation SN models to the remnant phase, up to several centuries after explosion, and measure the characteristic sizes of substructures formed in the SNR due to turbulent mixing. By comparing our models to high-resolution optical observations of the young Type Ia SNR 0509-67.5, we find that the size distribution of its small-scale substructures is consistent with a double detonation explosion mechanism and further places constraints on the carbon-oxygen core mass and helium shell mass of the WD progenitor. The observed sizes of iron-dominated and sulfur-dominated substructures in SNR 0509-67.5 indicate a progenitor core mass and a shell mass of 1 solar mass and greater than 0.05 solar mass, respectively.

We demonstrate a high-efficiency, free space optical delay line utilizing a nested multipass cell architecture. This design supports extended optical paths with low loss, aided by custom broadband dielectric coating that provides high reflectivity across a wide spectral bandwidth. The cell is characterized using polarization-entangled photon pairs, with signal photons routed through the delay line and idler photons used as timing reference. Quantum state tomography performed on the entangled pair reveals entanglement preservation with a fidelity of $99.6(9)\%$ following a single-transit delay of up to $687$~ns, accompanied by a photon retrieval efficiency of $95.390(5)%$. The delay is controllable and can be set between $1.8$~ns to $687$~ns in $\sim12.6$~ns increments. The longest delay and wide spectral bandwidth result in a time-bandwidth product of $3.87\times 10^7$. These results position this delay line as a strong candidate for all-optical quantum memories and synchronization modules for scalable quantum networks.

We prove that there is a Borel quasi-kernel in any locally countable Borel directed graph with finite Borel chromatic number. We prove that the Borel chromatic number of a Borel directed graph with bounded out-degree $n$ is either infinite or less than or equal to $\frac{(n+1)(n+2)}{2}$. This is an alternative proof of Palamourdas' theorem.

We use classical tools from calculus of variations to formally derive necessary conditions for a Markov control to be optimal in a standard finite time horizon stochastic control problem. As an example, we solve the well-known Merton portfolio optimization problem.

Public discourse around climate change remains polarized despite scientific consensus on anthropogenic climate change (ACC). This study examines how "believers" and "skeptics" of ACC differ in their YouTube comment discourse. We analyzed 44,989 comments from 30 videos using a large language model (LLM) as a qualitative annotator, identifying ten distinct topics. These annotations were combined with social network analysis to examine engagement patterns. A linear mixed-effects model showed that comments about government policy and natural cycles generated significantly lower interaction compared to misinformation, suggesting these topics are ideologically settled points within communities. These patterns reflect motivated reasoning, where users selectively engage with content that aligns with their identity and beliefs. Our findings highlight the utility of LLMs for large-scale qualitative analysis and highlight how climate discourse is shaped not only by content, but by underlying cognitive and ideological motivations.

We propose a discrete model to determine the metabolic scaling exponent based on Fibonacci growth patterns and discrete biological development phases. In contrast to continuous fractal models such as the West-Brown-Enquist (WBE) theory, the present approach describes metabolic scaling as the cumulative result of successive discrete stages, each incrementally contributing to metabolic activity. The scaling exponent b(n) emerges naturally from the logarithmic relationship between consecutive Fibonacci numbers, varying systematically with the organism's developmental stage. A refined logarithmic formulation significantly enhances quantitative agreement with empirical metabolic data across various mammalian species. This discrete framework effectively captures deviations from classical scaling laws, directly connecting recursive hierarchical structures with metabolic processes. Our model provides an alternative to traditional fractal transport approaches and can be naturally extended to hierarchical physical systems, opening new avenues to explore stage-dependent scaling phenomena in complex adaptive systems.

Cosmic rays may scatter off dark matter particles in active galactic nuclei, where both the densities of cosmic rays and dark matter are expected to be very large. These scatterings could yield a flux of boosted dark matter particles directly detectable on Earth, which enhances the sensitivity of dark matter direct detection and neutrino experiments to light and inelastic dark matter models. Here we calculate the cosmic-ray boosted dark matter flux from the neutrino-emitting active galactic nuclei, NGC 1068 and TXS 0506+056, by considering realistic cosmic-ray distributions, deep inelastic scatterings, and mass splittings in the dark sector. From this we derive novel bounds from these sources on light and/or inelastic dark matter models with Super-K and XENONnT. We find that cosmic-ray boosted dark matter from neutrino-emitting active galactic nuclei can test regions of parameter space favored to reproduce the observed relic abundance of dark matter in the Universe, and that are otherwise experimentally inaccessible.

Quantum entanglement is essential for modern quantum information processing. Entanglement gates convert initially non-entangled states into entangled ones by applying time-dependent parametric pulses. While Bell state preparation has been experimentally validated in various platforms, its stability and fidelity are constrained by environmental decoherence and parametric fluctuations.Here, we propose a dynamical framework for preparing robust Bell states by leveraging time-boundary engineering and momentum-space projective measurements within Su-Schrieffer-Heeger (SSH) systems. Employing Lindblad master equation, we theoretically demonstrate that the prepared Bell states exhibit remarkable robustness against both environmental decoherence and parametric time fluctuations, achieving a nearly perfect quantum fidelity, with momentum conservation law governing this robust behavior. To enrich Bell states in momentum space, multi-band SSH models are designed to induce multifold time scattering processes. This time-boundary engineering framework is applicable to both fermionic and bosonic excitations, offering a robust paradigm for generating Bell states in quantum communication and quantum computation.

We address the stabilization of linear, time-varying parabolic PDEs using finite-dimensional receding horizon controls (RHCs) derived from reduced-order models (ROMs). We first prove exponential stability and suboptimality of the continuous-time full-order model (FOM) RHC scheme in Hilbert spaces. A Galerkin model reduction is then introduced, along with a rigorous a posteriori error analysis for the associated finite-horizon optimal control problems. This results in a ROM-based RHC algorithm that adaptively constructs reduced-order controls, ensuring exponential stability of the FOM closed-loop state and providing computable performance bounds with respect to the infinite-horizon FOM control problem. Numerical experiments with a non-smooth cost functional involving the squared l1-norm confirm the methods effectiveness, even for exponentially unstable systems.

We define the family of Maya-Tupi graphs as those graphs that admit a partition $(A,B)$ of their vertex sets such that $A$ induces a complete multipartite graph where each part has size at most two, and $B$ induces a graph where every connected component is $K_1$ or $K_2$. The family of Maya-Tupi graphs is self complementary, generalizes split graphs, falls into the sparse-dense partitioning schema and is characterized by finitely many forbidden induced subgraphs. Unfortunately, our computational experiments show that the number of minimal forbidden induced subgraphs to characterize Maya-Tupi graphs is greater than 2000. In this work, we find characterizations in terms of minimal forbidden induced subgraphs for disconnected graphs, which imply the same for cographs; our results imply linear-time certifying recognition algorithms for Maya-Tupi graphs within these classes. We also show that Maya-Tupi graphs can be recognized in $\mathcal{O}(n^3)$-time in $C_4$-free graphs and in graphs with bounded neighborhood diversity; in $\mathcal{O}(n^4)$-time for triangle-free graphs; and in $\mathcal{O}(n^2)$-time for graphs with bounded clique-width. We provide efficient algorithms to calculate the clique, the independence, the chromatic, and the treewidth numbers, as well as a minimum fill-in for Maya-Tupi graphs.

Characterization and calibration of quantum devices are necessary steps to achieve fault-tolerant quantum computing. As quantum devices become more sophisticated, it is increasingly essential to rely not only on physics-based models, but also on predictive models with open-loop optimization. Therefore, we choose the Graybox approach, which is composed of an explicit (whitebox) model describing the known dynamics and an implicit (blackbox) model describing the noisy dynamics in the form of a deep neural network, to characterize and calibrate superconducting-qubit devices. By sending a set of selected pulses to the devices and measuring Pauli expectation values, the Graybox approach can train the implicit model and optimize gates based on specified loss functions. We also benchmark our optimized gates on the devices and cross-testing predictive models with two types of loss functions, i.e., the mean squared errors (MSE) of expectation values and the absolute errors (AE) of average gate fidelities (AGF). While the Graybox method allows for flexibility of the implicit noise model, its construction relies on a finite measurement shots dataset. We thus apply the decomposition of expected MSE loss to show that the finite-shot estimation of expectation values is the main contribution to the minimum value achievable of the expected MSE loss. We also show that the expected loss is an upper bound of the expected absolute error of AGF between the exact value and model prediction. Our results provide insights for quantum device characterization and gate optimization in experiments where only finite shots of data are available.

In this paper, we investigate the problem of decentralized online resource allocation in the presence of Byzantine attacks. In this problem setting, some agents may be compromised due to external manipulations or internal failures, causing them to behave maliciously and disrupt the resource allocation process by sending incorrect messages to their neighbors. Given the non-consensual nature of the resource allocation problem, we formulate it under a primal-dual optimization framework, where the dual variables are aggregated among the agents, enabling the incorporation of robust aggregation mechanisms to mitigate Byzantine attacks. By leveraging the classical Byzantine attack model, we propose a class of Byzantine-resilient decentralized online resource allocation algorithms that judiciously integrate the adaptive robust clipping technique with the existing robust aggregation rules to filter out adversarial messages. We establish theoretical guarantees, showing that the proposed algorithms achieve tight linear dynamic regret and accumulative constraint violation bounds, where the constants depend on the properties of robust aggregation rules. Numerical experiments on decentralized online economic dispatch validate the effectiveness of our approach and support our theoretical results.

Non-signaling correlations, which (strictly) include quantum correlations, provide a tractable path to explore the potential impact of quantum nonlocality on the capacity of classical communication networks. Motivated by a recent discovery that certain wireless network settings benefit significantly from non-signaling (NS) correlations, various generalizations are considered. First, it is shown that for a point to point discrete memoryless channel with a non-causal channel state information at the transmitter (CSIT), the NS-assisted Shannon capacity matches the classical (without NS assistance) capacity of the channel for the setting where the state is also made available to the receiver. The key insight is summarized as 'virtual signaling of CSIT via NS-assistance' and is supported by further results as follows. For a discrete memoryless 2-user broadcast channel (BC), the Shannon capacity region with NS-assistance available only between the transmitter and User 1, is found next. Consistent with the aforementioned key insight, the result matches the classical capacity region for the setting where the desired message of User 2 is made available in advance as side-information to User 1. The latter capacity region is known from a result of Kramer and Shamai. Next, for a semi-deterministic BC, the Shannon capacity region with full (tripartite) NS-assistance is shown to be the same as if only bipartite NS-assistance was available between the transmitter and the non-deterministic user. Bipartite NS-assistance between the transmitter and only the deterministic user, does not improve the capacity region relative to the corresponding classical setting. Finally, the analysis is extended to a K-user BC with full NS-assistance among all parties.

The goal of this survey is to present intimate interactions between four branches of conformal dynamics: iterations of anti-rational maps, actions of Kleinian reflection groups, dynamics generated by Schwarz reflections in quadrature domains, and algebraic correspondences. We start with several examples of Schwarz reflections as well as algebraic correspondences obtained by matings between anti-rational maps and reflection groups, and examples of Julia set realizations for limit sets of reflection groups (including classical Apollonian-like gaskets). We follow up these examples with dynamical relations between explicit Schwarz reflection parameter spaces and parameter spaces of anti-rational maps and of reflection groups. These are complemented by a number of general results and illustrations of important technical tools, such as David surgery and straightening techniques. We also collect several analytic applications of the above theory.