We consider Schrödinger operators in $\ell^2(\mathbb{Z})$ with real-valued potentials in $\ell^\infty(\mathbb{Z})$ and show that the generic spectral type is pure point. More specifically, we show that for a generic bounded potential, the essential spectrum of the associated Schrödinger operator is a Cantor set and has zero weight with respect to all spectral measures.

We investigate the influence of asymmetric magnetic fields on the atmospheric circulation of hot Jupiters based on a Shallow-Water Magnetohydrodynamic (SWMHD) model. The Shallow-Water Hydrodynamic (SWHD) models predict eastward equatorial jets and hotspot offsets, while some observations have revealed westward hotspots, suggesting that magnetic fields may play an important role. We incorporate asymmetric magnetic fields between hemispheres, and analyze their effects through linear perturbation analysis and numerical calculations. Our results indicate that strong magnetic fields play a dominant role in momentum transport. Asymmetric magnetic field configurations lead to hemispheric temperature contrasts, with the dayside temperature maxima in the hemisphere of stronger magnetic field located closer to the equator. With the magnetic field fixed in one hemisphere, the equatorial hotspots shift westward then eastward as the other hemisphere's field strengthens, exhibiting a pronounced westward offset only at moderate field strengths and weak hemispheric asymmetry. These findings highlight the significance of magnetic field geometry in explaining observed atmospheric dynamics and hotspot variability in hot Jupiters.

Compliant element systems with ultra-large deformation display rich nonlinear dynamics and pose challenging control problems, which, when solved, could enable enhancements in several mechatronics applications, such as soft robotics, MEMS, and biomedical applications. This paper considers post-buckled dynamic analysis of an inverted ultra-flexible pendulum actuated by a rotary hub. We first derive a complete set of equations capturing the dynamics of the system, essential for control development, using the assumed modes method framework, considering ultra-large deformations. Constrained Lagrange formulation is used for the same. In the perfect inverted configuration with zero hub angle, the buckled beam would display two symmetric stable equilibria and one unstable. However, as the hub angle changes on either side, the equilibrium positions shift, and eventually two of them vanish, and we are left with only one stable equilibrium. We use the dynamic equations to characterize this interesting phenomenon, demonstrating the continuous state dependence of multiple equilibria. Furthermore, experimental counterparts of the equilibrium results are meticulously obtained and discussed. Moreover, simulation results capture the nonlinear dynamics of this system. Overall, the work establishes a solid mathematical foundation with a control-amenable model for futuristic ultra-compliant mechatronic systems.

Recent observations of 55 Cancri e suggest an atmosphere rich in CO or CO$_2$ (Hu et al. 2024); other observations indicate the planet's eclipse depth is highly variable (e.g. Patel et al. 2024). So far, these observations have only been interpreted using 1D models without self-consistent heat redistribution, as the planet's extreme temperatures make it inaccessible to most 3D models. Here we perform cloud-free GCM simulations of 55 Cancri e using custom correlated-$k$ coefficients developed from the ExoMol database. Our best-fit simulations match the JWST spectra from Hu et al. (2024) well, favoring an atmosphere that is both thick ($\ge$ 10 bar) and CO$_2$-rich ($>1\%$ CO$_2$ volume mixing ratio), while ruling out thin ($<$ 10 bar) and pure-CO/CO$_2$-poor atmosphere, which were previously proposed based on 1D models (Hu et al. 2024; Zilinskas et al. 2025). We also find large-scale atmospheric dynamics, i.e. weather, is insufficient to explain the observed variability. A thick, CO$_2$-rich atmosphere implies that 55 Cancri e likely formed with significantly more volatiles than Earth and Venus. In addition, a thick atmosphere makes it unlikely that the planet's variability is caused by transient outgassing (Heng 2023), favoring other variability mechanisms (e.g. clouds). Our work provides model constraints for upcoming JWST observations of 55 Cancri e, and highlights the importance of interpreting thermal emission observations with self-consistent 3D models.

Let $X \subset ¶^5_{\C}$ be a smooth cubic fourfold.The Kuznetsov component $\sA_X$ is contained in the derived category $D^b(X)$ and the transcendental motive $t(X)$ is contained in the category of Chow motives $\sM_{rat}(\C))$. If $X$ and $Y$ are {\it Fourier -Mukai partners} and hence the categories $\sA_X$ and $\sA_Y$ are equivalent, then their transcendental motives $t(X)$ and $t(Y)$ are isomorphic. The aim of this note is to consider families of special cubic fourfolds $X$ with their FM-partners $Y$ and to give an explicit description of the isomorphism between the transcendental motives, in the case $X$ and $Y$ are rational and when they are conjecturally irrational. We also prove that ,for special cubic fourfolds $X $ in countably many Hassett divisors, with a symplectic automorphism of order 3, there exists another special cubic fourfold $Y$, an equivalence of categories $\sA^G_X \simeq \sA_{Y}$, where $\sA^G_X$ is the equivariant Kuznetsov component, and an isomorphism $t(X) \simeq t(Y)$.

SN 1006 is a historical Type Ia supernova remnant that exhibits non-thermal emission ranging from radio to multi-TeV $γ$-rays. Most of this emission (particularly X-rays and $γ$-rays) is concentrated in polar caps aligned with the ambient magnetic field, which makes it an ideal laboratory for studying cosmic ray (CR) acceleration at different shock obliquities and the hadronic/leptonic nature of the $γ$-ray emission. We model SN 1006's morphology, multi-wavelength spectrum, and radial profile using a self-consistent multi-zone kinetic model of particle acceleration that accounts for: CR-driven shock modification, magnetic field amplification, drift in magnetic fluctuations, and temporal dynamics including adiabatic and synchrotron losses. Our model can reproduce both the observed spectral and spatial properties, with the exception of the radio profile that we argue requires 3D hydrodynamic effects to replicate. We find that quasi-parallel regions (where the shock normal aligns with the ambient magnetic field) exhibit very prominent CR acceleration ($\sim$20% efficiency), while quasi-perpendicular regions exhibit efficiencies below 1%, consistent with the results of kinetic simulations. We also find that electrons are responsible for the majority of the $γ$-ray emission from SN 1006 (i.e., it is a leptonic source), with the exception of the northwest region due to an encounter with a dense cloud.

The development of simple and scalable fabrication strategies for cost-effective electrodes is crucial to advance water splitting in alkaline water electrolyzers (AWEs). Here, we present a coating-annealing method to conformally coat a MoSx catalyst layer onto a porous Ni foam (NF) substrate. By controlling the annealing process, the composition of the MoSx layer could be tuned from MoS2 to MoS3 and its catalytic performance for hydrogen evolution reaction (HER) in alkaline media was optimized. The MoS3@NF synthesized by this method achieved industrially relevant HER current densities of 200 mA/cm2 at a low overpotential of 246 mV, maintaining stable operation for over 240 h. The MoS3@NF cathode, combined with a stainless steel anode, enabled an alkaline water electrolyzer (AWE) cell to operate steadily at 1.96 V and 1 A/cm2 for 1000 h. This performance surpasses that of most of the previously reported water electrolyzers employing MoSx-based cathodes. Our work demonstrates the potential of MoS3 (with its abundant edge-sulfur atoms serving as active sites) as a high-performance cathode material for industrial AWEs.

Lyapunov feedback laws can be constructed for performing orbital transfer maneuvers based on Gauss Variational Equations, or GVEs, while satisfying specified state and control constraints. These state and control constraints are enforced using barrier functions and saturation, respectively, while the reference governor, employed as a convergence governor, is used to avoid getting stuck at spurious equilibria that may be created by barrier functions. In this article, these Lyapunov feedback laws are extended to rendezvous maneuvers where not only five orbital elements are matched to prescribed target values, but also the true anomaly matches its time dependent target value. The modification involves altering the commanded semi major axis with an outer loop feedback law, also designed using Lyapunov techniques. We illustrate the resulting safe closed loop rendezvous maneuvers in simulations for conventional thrust based propulsion and Lorentz force-based propulsion. In the latter case, only the current through the tether is controlled, subject to current limits, to accomplish prescribed orbital transfer and rendezvous maneuvers.

We study proximity-induced unconventional superconductivity in a twisted graphene/NbSe$_2$ van der Waals heterostructure using the Bogoliubov-de Gennes formalism. The normal-state parameters of proximitized graphene are extracted from ab initio calculations at a twist angle of $23.4^\circ$, which reduces the common symmetry of the heterostructure to $\mathbf{C}_3$. We construct symmetry-allowed superconducting gap functions of the graphene layer according to the irreducible representations of the $\mathbf{C}_3$ group, containing singlet and triplet pairing channels and their mixtures. Computing the topological invariants as a function of the mixing parameters, we find a rich phase diagram of chiral topological superconducting phases, characterized by nonzero Chern numbers $C\in\{-4,-2,2,4\}$. While the nature of the superconducting order parameter of NbSe$_2$ remains debated, the formation of the van der Waals heterostructure and the related symmetry reduction can alter the relative stability of competing pairing channels, potentially stabilizing a chiral component that is proximity-induced into graphene and triggers the topological phases identified here, making the twisted graphene/NbSe$_2$ heterostructure a promising platform for chiral topological superconductivity detectable via quasiparticle interference imaging and transport measurements.

Smart factories are evolving into Cyber-Physical Systems (CPS), demanding increased autonomy. This necessitates real-time decision making, facilitated by insights derived from sensor data. Process mining offers a valuable approach to gain such insights and guide actions. The edge computing paradigm supports this real-time requirement by enabling network communication between sensors and leveraging nearby computing resources. This paper investigates the implications of performing real-time process mining algorithms on the edge. Within this paper, we first propose a formalism to describe relevant datasets and the computing topology. We then evaluate the edge computing approach through a case study involving an edge-based conformance checking algorithm. The results demonstrate the feasibility and benefits of edge-based real-time process mining for enhanced autonomous control in smart factories.

The integral representation of completely monotone functions of several real variables as Laplace or Stieltjes-Fantappié transforms of positive measures opens a Hilbert space path toward their finite-point interpolation by simpler functions. We combine, within a non-commutative Radon transform framework, the matrix pencil realization of the positive semi-definite Hankel kernel associated with the sampling of a completely monotone function with Weyl's operational calculus and Fantappiè's analytic calculus. The interpolation is achieved by finitely determined entire or rational functions, respectively, which are directionally completely monotone. In our relaxation scheme, the original positive measure is approximated by a sequence of specific Wigner distributions, which can also be regarded as analytic functionals. Throughout the interpolation process, tight bounds are enforced on the modulus or the real part of the holomorphic extension to the underlying tube domain.

We study Dirichlet Green's functions associated with second-order parabolic systems with rapidly oscillating periodic coefficients that are symmetric and independent of time. For bounded $C^{1,1}$ domains, we obtain a sharp zeroth-order convergence estimate from the oscillating Green's function to its homogenized counterpart, with the optimal rate $O(\varepsilon)$ and Gaussian off-diagonal decay. For bounded $C^{2,1}$ domains, we also prove a first-order expansion for the spatial gradient in terms of Dirichlet correctors, with an $O(\varepsilon)$ error up to a logarithmic factor. In this time-independent symmetric setting, these results improve the convergence rates established by Geng in [Calc. Var. Partial Differ. Equ., 62(6), 2023] for parabolic systems with time-dependent periodic coefficient matrices.

Sparse polynomial multiplication is a fundamental problem in computer algebra and the theory of computation, and the development of a quasi-linear time output-sensitive multiplication algorithm has been posed as an open challenge. In this paper, a counterexample is provided to a previously claimed solution to this open problem for integer coefficients. By employing the existing quasi-linear modular-black-box interpolation algorithm, we are able to provide an algorithm with quasi-linear bit complexity for the integer coefficients setting. Furthermore, in the case of coefficients over a finite field, we obtain an algorithm whose bit complexity is linear in the number of terms, the logarithm of the degree, and the logarithm of the size of the finite field.

In this chapter we review some results describing the high-frequency eigenmodes of the Laplacian on compact manifolds, or Euclidean domains, for which the geodesic flow is chaotic. We focus on the macroscopic distribution of these eigenmodes, which is described by the concept of semiclassical measure. The main result on the question is the Quantum Ergodicity theorem, originally due to Schnirelman. We provide the detailed proof of this theorem, including the adjustments necessary to treat the case of manifolds with boundary. We also discuss the Quantum Unique Ergodicity conjecture, and some progress towards this conjecture for strongly chaotic (Anosov) systems. In particular, we describe the constraints on admissible semiclassical measures, in terms of their Kolmogorov-Sinai entropy, as well as more recent delocalization results.

We present a one-month-ahead conditional probabilistic framework for wind-power forecasting at ten-minute resolution. Monthly Weibull shape and scale parameters are estimated from serially dependent SCADA wind-speed data, corrected through a Godambe covariance, and forecast by a heteroskedastic Kalman filter on a bivariate VAR(1) state-space model. Conditional on the MMSE forecasted Weibull invariant law, we construct and compare three positive wind-speed SDE models: an Ornstein-Uhlenbeck-Weibull transform, a Fokker-Planck drift-first diffusion, and a Fokker-Planck diffusion-first model. The simulated wind-speed ensembles are mapped to power through a calibrated XGBoost power curve. Applied to January 2021 data from a Senvion MM92 turbine at Kelmarsh Wind Farm, the three SDE formulations are statistically indistinguishable in probabilistic accuracy, with mean CRPS values between 1.569 and 1.575 m/s. The diffusion-first model is therefore preferred on computational grounds, reducing runtime by about a factor of seven relative to the OU-Weibull model. In the power domain, the Wasserstein distance between simulated and observed distributions is 26.1-27.6 kW, below $1.4\%$ of rated capacity, while the monthly energy-yield bias is about $-7.3\%$ for the examined month. Exceedance-probability errors remain below 1.6 percentage points over the 0-1500 kW range and about 2.2 percentage points near rated power. These quantities provide decision-relevant probabilistic inputs for downstream operational problems, rather than completed reserve, storage, market, or fatigue-optimization decisions. Full marginalisation over the Kalman predictive law of the Weibull parameters is left as a natural extension.

Any quantum computation consists of a sequence of unitary evolutions described by a finite set of Hamiltonians. For the case where this set consists of only products of Pauli operators, known as Pauli strings, we provide a necessary and sufficient condition for it to generate $\mathfrak{su}(2^n)$, i.e., to be universal for quantum computation on $n$ qubits. When combining Pauli strings with a general Hamiltonian, we show a sufficient (and in certain circumstances even necessary) condition for universality based on the Pauli-basis expansion of the Hamiltonian. As an application of these results, we prove two corollaries: (i) a necessary and sufficient condition for the universality of a general Hamiltonian given arbitrary single-qubit control on all qubits, and (ii) the universality of an XYZ Heisenberg Hamiltonian with local control of just two adjacent qubits.

In this paper, we propose two fully decoupled, linear and structure-preserving block-centered finite difference schemes for the classical Keller-Segel chemotaxis system on staggered non-uniform spatial grids. Both novel schemes are second-order accurate in space; one is first-order accurate in time, while the other achieves second-order temporal accuracy. Moreover, we show that the schemes preserve several inherent physical laws at the discrete level: (i) the positivity of both the cell density and the chemoattractant concentration; (ii) the conservation of total cell mass; and (iii) a discrete energy dissipation property for the first-order scheme. In particular, the temporally first-order scheme unconditionally preserves positivity, mass conservation, and energy dissipation, whereas the second-order scheme ensures positivity under a sufficient (but not necessary) time-step condition. The proposed methods yield more accurate and efficient simulations of chemotactic dynamics, especially in the presence of rapid blow-up phenomena, on specified non-uniform spatial grids. Numerical experiments are conducted to validate the theoretical findings and to illustrate the accuracy and reliability of the proposed schemes.

Freedman and Krushkal introduced a triple torsion linking form for rational homology $3$-spheres and used it to obstruct locally flat embeddings in $S^4$. For every odd prime $p$, we identify their triple torsion form, computed with parameter $t=p$ on rational homology $3$-spheres whose first homology has exponent $p$, with the mod-$p$ triple cup product under torsion-linking duality. For algebraically split $\pm p$-framed surgery links, this gives a signed formula in terms of Milnor's integral length-three invariants $\barμ_{ijk}$, with the framing-sign factor dictated by torsion-linking duality. We then use Borromean band-sums to realize arbitrary mod-$p$ triple cup tensors on rational homology $3$-spheres with $H_1\cong(\mathbb Z/p)^6$ and fixed hyperbolic ordinary torsion linking form. Finally, using the classical spinor/Klein model for the split six-dimensional quadratic space, we classify the tensors with no dual null Hantzsche pair. This produces, for every odd prime $p$, a rational homology $3$-sphere with hyperbolic ordinary torsion linking form but with no locally flat embedding in $S^4$, and indeed no locally flat embedding in any integer homology $4$-sphere.

For integers $k\ge 1$ and $n\ge k+1$, let $\operatorname{ex}^{\mathrm{HIST}}_k(n)$ denote the maximum number of edges in a $k$-connected graph of order $n$ which contains no homeomorphically irreducible spanning tree (or briefly HIST). We determine these extremal numbers for $k=1$ and $k=2$. More precisely, we prove that $\operatorname{ex}^{\mathrm{HIST}}_1(n)=\binom{n-2}{2}+2$ for $n\ge 9$, with $L_n$ as the unique extremal graph, and that $\operatorname{ex}^{\mathrm{HIST}}_2(n)=\binom{n-3}{2}+4$ for $n\ge 13$, with $B_n$ as the unique extremal graph. This provides a Turán-type extremal result for spanning trees with no vertices of degree two.

In this work, we systematically study the gravitational lensing properties and observational signatures of compact boson stars. Unlike black holes, the photon effective potential of a compact boson star develops a nearly flat region, whose width increases with the compactness of the star. This flat structure significantly broadens the range of impact parameters that can produce large-angle deflections, leading to noticeably wider lensing rings of all orders. Photons constituting these rings traverse more complex paths, rendering the resulting images more sensitive to the spatial distribution of the accretion flow. Ray tracing results show that, compared to black hole models, the image topology and visibility amplitudes of compact boson stars exhibit a stronger dependence on the accretion flow structure. These results highlight qualitative differences in the observational properties of compact boson stars and black holes.

A specific heat treatment at 300 C, named medium-temperature baking (Mid-T baking) is applied to superconducting radio-frequency (SRF) accelerating cavities to improve the quality factor (Qo) at medium accelerating fields (10 to 20 MV/m). This treatment is very successful when done properly as it can reduce by almost a factor of two the power dissipations in this field range. However, surface contamination can lead to the degradation of Qo instead. Plasma-based surface treatment provides an effective approach to eliminate contaminants from the Niobium surface. In this study an insitu plasma cleaning process with argon containing 10 % O2 was performed to remove hydrocarbons from Niobium surface. The treatment was applied before and after a heat treatment at 500 C under ultra-high vacuum conditions (Mid-T baking). Changes in chemical speciation and oxide layer alteration induced by plasma processing were analyzed using insitu X-ray photoelectron spectroscopy (XPS) and exsitu scanning electron microscopy (SEM). The results show that plasma treatment modifies the composition of Niobium oxides, converting a Nb2O5 layer into NbO2. Furthermore, a plasma treatment before Mid-T baking helps reduce unstable oxides such as NbxO and significantly increases the proportion of metallic Niobium at the surface. The Niobium sample treated by plasma prior to Mid-T baking showed a 53 % reduction in carbide formation. Moreover, the C1s component attributed to NbC bonds shifts toward lower binding energy, indicating the formation of a more metallic NbC phase. Whereas without plasma treatment, the higher binding energy component observed after Mid T baking is consistent with Nb2C.

Elastically accommodated grain-boundary sliding (EAGBS) is a plausible source of upper-mantle seismic attenuation and dispersion, yet classical theory predicts a localized Debye-like peak that is absent or only weakly expressed in dry olivine experiments. Here we test whether microstructural heterogeneity can explain this discrepancy using 2-D finite-element simulations on periodic Voronoi tessellations. We find that irregular grain geometry changes the baseline EAGBS response relative to the regular hexagonal benchmark, but increasing grain-size variance alone produces only modest changes in modulus and peak height, with little spectral broadening. In contrast, a broad distribution of grain-boundary viscosities progressively suppresses and broadens the Debye-like loss peak into a weak background spanning a wide frequency interval. This broadening arises from the superposition of many localized relaxation processes with distinct characteristic timescales and motivates a reduced-order 0-D description of the aggregate response. These results suggest that the absence of a pronounced EAGBS peak in dry olivine does not necessarily imply the absence of EAGBS mechanism itself. If grain boundaries sample a sufficiently broad viscosity distribution, the macroscopic EAGBS contribution may appear experimentally only as part of a broad attenuation background, while still remaining relevant for upper-mantle seismic attenuation and velocity dispersion.

Non-Hermitian systems can evade Anderson localization and exhibit delocalized states even in one dimension. Here, we show that such non-Hermitian delocalized states under periodic boundary conditions (PBC) are intrinsically critical, realizing the universality class of one-dimensional random Dirac fermions. By linking spectral winding to topological Anderson transitions via Hermitization, we demonstrate that the delocalized PBC states exhibit a Dirac-type criticality with universal algebraic correlations. In contrast to Hermitian systems, where this criticality occurs only at fine-tuned transition points, it emerges generically in non-Hermitian systems as a consequence of spectral topology. These results identify a universal mechanism by which non-Hermiticity promotes criticality, providing a unified description of non-Hermitian delocalization in one dimension.

Peer-to-peer (P2P) energy trading requires network-aware coordination because transactions are physically realized through distribution networks. However, sensitivity-based coordination causes a confidentiality-verifiability tradeoff, as network sensitivities may reveal vulnerable components while undisclosed sensitivities prevent participants from verifying utility-provided transaction guides. This paper proposes a zero-knowledge-proof-based method for verifying the computational integrity of network-constrained transaction guides with respect to committed private network data, without exposing network-sensitivity information. The guide defines admissible injection and withdrawal volumes derived from sign-decomposed sensitivity matrices while satisfying balance, voltage, line-flow, and optimality conditions. These conditions are encoded in an arithmetic circuit, represented as R1CS constraints and a quadratic arithmetic program, and verified using a bilinear pairing. Blockchain commitments bind the approved circuit, public inputs, statement identifiers, proof, and verification result for tamper-evident auditability. The proposed proof certifies correct guide computation from committed network data; the authenticity of the committed network data is handled through an explicit registration and attestation assumption. Case studies on a modified IEEE 33-bus system show satisfaction of network constraints after clearing, rejection of public-input and witness-inconsistency attacks, and practical on-chain overhead, with an 806-byte proof.

The Schiff moment of a nucleus is a symmetry-violating nuclear moment that indicates new physics beyond the Standard Model. We place the limit, $|\mathscr{S}({}^{153}$Eu)$| < 1.7 \times 10^{-8}$ $e\,$fm$^3$ (95\% confidence), on the Schiff moment of the $^{153}$Eu nucleus, using nuclear spin resonances in two ensembles of oppositely-polarized $^{153}$Eu$^{3+}$ ions in a Y${}_2$SiO${}_5$ crystal. This measurement using octupolar nuclei in a mm-scale crystal constrains new physics at the TeV energy-scale.

The combinatorial design space of multilayer perovskite solar cells is vast, yet exhaustive experimental or computational screening of all possible material combinations remains impractical. Here, we integrate SCAPS-1D device simulations with machine learning to systematically explore 125 device architectures constructed from five electron transport layers (ETL), five absorbers (including lead-free double perovskites), and five hole transport layers (HTL). A representative subset of configurations is used to train a machine learning (ML) model, which predicts the power conversion efficiency (PCE) of the remaining unexplored structures. Leave-One-Group-Out cross-validation yields a Spearman rank correlation, demonstrating reliable ranking capability. SHAP (SHapley Additive exPlanations) analysis reveals that the HTL band gap, absorber band gap, and ETL electron affinity are the most influential descriptors, providing physical insights into interfacial recombination and charge extraction. The machine learning model identifies several high-performance configurations that are subsequently verified by full SCAPS-1D simulations. Among them, the device FTO/TiO$_2$/Cs$_2$AgBiBr$_6$/NiO/Ag achieves a PCE of 28.85%, and the ML-suggested structure FTO/SnO$_2$/Cs$_2$AgInBr$_6$/NiO/Ag exhibits 28.62%, outperforming a closely related literature architecture by approximately 4% absolute. Notably, eight of the top-11 structures employ the lead-free double perovskite Cs$_2$AgInBr$_6$. This work demonstrates that a physics-based, data-driven workflow combining SCAPS-1D, ML, and SHAP can accelerate the discovery of high-efficiency, environmentally friendly perovskite solar cells while providing transparent design rules. The approach is generalizable to other multilayer optoelectronic systems.

We solve exactly the relativistic finite-difference equation for the quantum three-dimensional ring-shaped Quesne oscillator potential. Our investigation is based on a finite-difference version of relativistic quantum mechanics. So-called relativistic configurational r-space is a key concept here. We show that the radial wavefunctions and angular wavefunctions are expressed through the continuous dual Hahn polynomials and Jacobi polynomials, respectively. A discrete energy spectrum has been found. The radial wave functions and energy spectrum have the correct nonrelativistic limit. We also build a dynamical symmetry group SU (1, 1) for the radial part of the equation of motion, which allows us to find the energy spectrum purely algebraically.

We describe the anlogue of the Einsidler, Kaydev, Polo and Sarig on Intrinsic Ergodicity in the simple setting of expanding maps and, in particular, $β$-transformations.

We prove a Bishop--Phelps--Bollobás theorem for operators into spaces of continuous scalar-valued functions on extremally disconnected compact Hausdorff spaces over both the real and complex scalar fields. The main result applies whenever the density character of the domain is strictly smaller than the Baire number of the underlying compact space. The proof also yields an explicit quadratic Bishop--Phelps--Bollobás modulus. In particular, every separable Banach space paired with such a function space has the Bishop--Phelps--Bollobás property for operators.

In quantum mechanics, a measurement applied to a state in general produces some amount of intrinsic randomness. This is not only a fundamental feature of the theory, but is also at the basis of any quantum process to generate random numbers. The simplest of such processes consists of a single, fully charaterized, measurement acting on a single, fully characterized, state. Unfortunately, no general method to estimate the intrinsic randomness produced in such setups is known. In this work, we address this issue by presenting a semidefinite programming formulation of the maximum probability with which an adversary, Eve, can guess the outcomes of characterized but untrusted prepare-and-measure setups. We then present several applications of this construction. First, we apply our method to a variety of specific setups, allowing us both to benchmark the approach and, more importantly, to determine the exact amount of certifiable randomness in scenarios where only upper bounds were previously available. Then, we show that the presence of entanglement between the device preparing the state and the measurement strictly increases Eve's predictive power, already in the most elementary setup of a binary measurement acting on a qubit state.