Recent discoveries in asymptotically good quantum codes have intensified research on their application in quantum computation and fault-tolerant operations. This study focuses on the addressability problem within CSS codes: we ask what circuits might implement logical gates on strict subsets of logical qubits. With some notion of fault-tolerance, we prove several impossibility results: for CSS codes with non-zero rate, one cannot address a logical $H$, $HS$, $SH$, or $\mathsf{CNOT}$ to any non-empty strict subset of logical qubits using a circuit made only from 1-local Clifford gates. Furthermore, we show that one cannot permute the logical qubits in a code purely by permuting the physical qubits, if the rate of the code is (asymptotically) greater than 1/3 and the distance is at least 3. We can show a similar no-go result for $\mathsf{CNOT}$s and $\mathsf{CZ}$s between two such high-rate codes, albeit under a more restrictive assumption on the circuit, which we call "global" (though recent addressable CCZ gates use global circuits). This work pioneers the study of distance-preserving addressability in quantum codes, mainly by considering automorphisms of the code. This perspective offers new insights and potential directions for future research. We argue that studying this trade off between addressability and efficiency of the codes is essential to understand better how to do efficient quantum computation.

Differential equations posed on quadratic matrix Lie groups arise in the context of classical mechanics and quantum dynamical systems. Lie group numerical integrators preserve the constants of motions defining the Lie group. Thus, they respect important physical laws of the dynamical system, such as unitarity and energy conservation in the context of quantum dynamical systems, for instance. In this article we develop a high-order commutator free Lie group integrator for non-autonomous differential equations evolving on quadratic Lie groups. Instead of matrix exponentials, which are expensive to evaluate and need to be approximated by appropriate rational functions in order to preserve the Lie group structure, the proposed method is obtained as a composition of Cayley transforms which naturally respect the structure of quadratic Lie groups while being computationally efficient to evaluate. Unlike Cayley--Magnus methods the method is also free from nested matrix commutators.

Bit commitment is a fundamental cryptographic primitive and a cornerstone for numerous two-party cryptographic protocols, including zero-knowledge proofs. However, it has been proven that unconditionally secure bit commitment, both classical and quantum, is impossible. In this work, we demonstrate that imposing a restriction on the committing party to perform only separable operations enables secure quantum bit commitment schemes. Specifically, we prove that in any perfectly hiding bit commitment protocol, an honestly-committing party limited to separable operations will be detected with high probability if they attempt to alter their commitment. To illustrate our findings, we present an example protocol.

What is the role of product managers in the responsible use of generative AI (genAI) in products and everyday work -- and what enables or constrains their ability to take action? Past literature has examined the ways in which organizational policies can become decoupled from practices when incentives for responsible action are misaligned or impeded by profit motives. While the role of engineers and professional ethicists in the context of AI has been examined in detail, the role of product managers -- who are frequently portrayed as "gatekeepers" or critical decision-makers in product teams -- remains unclear. In this paper, we examine what organizational conditions promote responsible use of genAI by product managers by drawing on twenty-five interviews and a global survey of over three hundred respondents in product management-related roles. We find that uncertainty around responsible AI and a sense of diffused responsibility constrain ethical action, while leadership commitment and organizational principles enable ethical action -- making some responsible practices up to fourteen times more likely. Further, we find two sets of actions product managers take to "recouple" ethical commitments and practices. The first includes low-resource, individual actions product managers can implement without explicit organizational incentives. The second includes high-resource, collective actions that require organizational incentives. Our research suggests recoupling ethical policies and practices at the level of product teams requires institutional buy-in and higher level leadership commitment. Nevertheless, we show that individual actors are able to exhibit agency through some meaningful, low resource actions, even in the absence of organizational incentives, though this alone is insufficient to operationalize responsible AI at scale.

This paper studies the design of cluster experiments to estimate the global treatment effect in the presence of network spillovers. We provide a framework to choose the clustering that minimizes the worst-case mean-squared error of the estimated global effect. We show that optimal clustering solves a novel penalized min-cut optimization problem computed via off-the-shelf semi-definite programming algorithms. Our analysis also characterizes simple conditions to choose between any two cluster designs, including choosing between a cluster or individual-level randomization. We illustrate the method's properties using unique network data from the universe of Facebook's users and existing data from a field experiment.

We consider a parabolic-elliptic type of Keller-Segel equations with generalized diffusion and logistic source under homogeneous Neumann-Neumann boundary conditions. We construct bounded weak solutions globally in time in an unbalanced optimal transport framework, provided that the magnitude of the chemotactic sensitivity can be restricted depending on parameters. In the case of subquadratic degradation of the logistic source, we quantify the chemotactic sensitivity, in particular, in terms of the power of degradation and the pointwise bound of the initial density.

The Coherent Ising Machine (CIM) is a quantum network of optical parametric oscillators (OPOs) intended to find ground states of the Ising model. This is an NP-hard problem, related to several important minimization problems, including the max-cut graph problem. In order to enhance its potential performance, we analyze the coherent coupling strategy for the CIM in a highly quantum regime. To explore this limit, without assuming gaussianity, we employ accurate numerical simulations. Due to the inherent complexity of the system, the maximum network size is limited. While master equation methods can be used, their scalability diminishes rapidly for larger systems. Instead, we use Monte Carlo wave-function methods, which scale as the wave-function dimension, and use large numbers of samples. These simulations involve Hilbert spaces exceeding $10^{7}$ dimensions. To evaluate success probabilities, we use quadrature probabilities. We demonstrate the potential for quantum computational advantage by reducing the time required to reach maximum success probability in a low-dissipation regime enabled by initial quantum superpositions and entanglement. Furthermore, we demonstrate that tailored time-dependent couplings can amplify these quantum effects. Comparisons with classical CIM models give evidence that quantum tunneling effects in this strong coupling limit can overcome trapping in false minima. This can greatly increase success rates, indicating a potential for quantum advantage. Finally, we perform a coherence analysis based on the state purity to examine the role of quantum coherence in CIM performance and to determine how state purity correlates with improved optimization outcomes.

Entanglement is one of the fundamental properties of a quantum state and is a crucial differentiator between classical and quantum computation. There are many ways to define entanglement and its measure, depending on the problem or application under consideration. Each of these measures may be computed or approximated by multiple methods. However, hardly any of these methods can be run on near-term quantum hardware. This work presents a quantum adaptation of the iterative high-order power method for estimating the geometric measure of entanglement of multi-qubit pure states using rank-1 tensor approximation. This method is executable on early fault-tolerant (hybrid) quantum hardware and does not depend on quantum memory. We simulate this algorithm and mitigate the effects of noise on the results of the computation using a theoretical model based on a known mitigation approach, which assumes a global depolarising noise channel.

This paper studies a type of periodic utility maximization problem for portfolio management in incomplete stochastic factor models with convex trading constraints. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealth levels over an infinite horizon, featuring the dynamic adjustments in portfolio decision according to past achievements. Under power utility, we transform the original infinite horizon optimal control problem into an auxiliary terminal wealth optimization problem under a modified utility function. To cope with the convex trading constraints, we further introduce an auxiliary unconstrained optimization problem in a modified market model and develop the martingale duality approach to establish the existence of the dual minimizer such that the optimal unconstrained wealth process can be obtained using the dual representation. With the help of the duality results in the auxiliary problems, the relationship between the constrained and unconstrained models as well as some fixed point arguments, we derive and verify the optimal constrained portfolio process for the original problem over an infinite horizon.

We present a unified theoretical framework for kernel-based measures of dependence on product spaces. Building on the ideas underlying distance covariance, distance multivariance, and the Hilbert-Schmidt Independence Criterion (HSIC), we define a new family of kernels on an $n$-fold Cartesian product, termed positive definite independent of order $k$ (PDI$_{k}$ kernels). These kernels extend the concepts of positive definite and conditionally negative definite kernels to higher orders and provide the foundation for generalized independence and interaction tests, such as the generalized Lancaster interaction of order $k$ ($Λ_{k}^{n}$), and the Streitberg interaction ($Σ$). Our analysis focuses on the continuous setting, where we prove a Kernel Mean Embedding Theorem for PDI$_{k}$ kernels and establish the corresponding integrability restrictions. Based on these results, we characterize how the Kronecker products of PDI kernels behave.

We introduce the toroh -- formally designated an Extreme Orographic Convective Event (EOCE) -- as a previously uncharacterised class of atmospheric hazard distinct from downbursts and conventional hailstorms. The toroh is a coherent hydraulic ice-piston formed when a convective system with anomalously narrow drop size distribution (mu ~ 20) undergoes explosive secondary ice production via the Hallett-Mossop mechanism, triggered by marine iodine ice-nucleating particles injected by a vortex or orographic resonance with an inselberg (Planalto Mirador, SC, Brazil). The piston collapses coherently into canyon terrain, producing a two-phase acoustic signature, seismic tremor M_L ~ 2-3, and a diagnostic erosion scar with zero fine-grained matrix. Piston cohesion is justified by ice sintering kinetics: tau_p ~ 0.04 s << tau_sint = 1-10 s, and sintered tensile strength ~10^4 Pa exceeds aerodynamic fragmentation pressure ~10^3 Pa. We propose EOCE as a candidate mechanism for four persistent geological anomalies: (1) erosional amphitheatres in resistant bedrock; (2) heavy mineral concentration in canyon lag deposits; (3) spatiotemporal heterogeneity of the Great Unconformity, incompatible with uniform Snowball Earth glaciation; (4) the nutrient pulse preceding the Cambrian Explosion. In pristine pre-industrial conditions, EOCE frequency is estimated at 1-10 per century per canyon -- sufficient for independent geomythological encoding in ~18 culturally isolated traditions. The Adams Event (Laschamp, ~42 ka) amplified this baseline 10-100x; 20th-century tetraethyl-lead aerosols suppressed it below observational threshold. A 3D anelastic bin-microphysics model and testable predictions are presented. Code: https://github.com/reinaldohaas/toro-model

A quasiconformal tree is a doubling (compact) metric tree in which the diameter of each arc is comparable to the distance of its endpoints. We show that for each integer $n\geq 2$, the class of all quasiconformal trees with uniform branch separation and valence at most $n$, contains a quasisymmetrically ''universal'' element, that is, an element of this class into which every other element can be embedded quasisymmetrically. We also show that every quasiconformal tree with uniform branch separation quasisymmetrically embeds into $\mathbb{R}^2$. Our results answer two questions of Bonk and Meyer from 2022, in higher generality, and partially answer one question of Bonk and Meyer from 2020.

Mixed-state phases of matter under local decoherence have recently garnered significant attention due to the ubiquitous presence of noise in current quantum processors. One of the key issues is understanding how topological quantum memory is affected by realistic coherent noise, such as random rotation noise and amplitude-damping noise. In this work, we investigate the intrinsic error threshold of the two-dimensional toric code (TC), a paradigmatic topological quantum memory, under these types of coherent noise by employing both analytical and numerical methods based on the doubled-Hilbert-space formalism. A connection between the mixed-state phase of the decohered TC and a non-Hermitian Ashkin-Teller-type statistical-mechanics model is established, and the mixed-state phase diagrams under the coherent noise are obtained. We find remarkable stability of mixed-state topological order under random rotation noise with axes near the $Y$-axis of qubits. We also identify intriguing extended critical regions at the phase boundaries, highlighting a connection with non-Hermitian physics. We argue that these phase boundaries provide upper bounds for the intrinsic error threshold, beyond which quantum error correction becomes impossible. We complement these findings by estimating the error thresholds for random rotation noise under standard quantum error correction, thereby providing lower bounds on the intrinsic error threshold.

Recently, there has been growing interest in simulating time-dependent Hamiltonians using quantum algorithms, driven by diverse applications, such as quantum adiabatic computing. While techniques for simulating time-independent Hamiltonian dynamics are well-established, time-dependent Hamiltonian dynamics is less explored and it is unclear how to systematically organize existing methods and to find new methods. Sambe-Howland's continuous clock elegantly transforms time-dependent Hamiltonian dynamics into time-independent Hamiltonian dynamics, which means that by taking different discretizations, existing methods for time-independent Hamiltonian dynamics can be exploited for time-dependent dynamics. In this work, we systemically investigate how Sambe-Howland's clock can serve as a unifying framework for simulating time-dependent Hamiltonian dynamics. Firstly, we demonstrate the versatility of this approach by showcasing its compatibility with analog quantum computing and digital quantum computing. Secondly, for digital quantum computers, we illustrate how this framework, combined with time-independent methods (e.g., product formulas, multi-product formulas, qDrift, and LCU-Taylor), can facilitate the development of efficient algorithms for simulating time-dependent dynamics. This framework allows us to (a) resolve the problem of finding minimum-gate time-dependent product formulas; (b) establish a unified picture of both Suzuki's and Huyghebaert and De Raedt's approaches; (c) generalize Huyghebaert and De Raedt's first and second-order formula to arbitrary orders; (d) answer an unsolved question in establishing time-dependent multi-product formulas; (e) and recover continuous qDrift on the same footing as time-independent qDrift. Thirdly, we demonstrate the efficacy of our newly developed higher-order Huyghebaert and De Raedt's algorithm through digital adiabatic simulation.

Human-centered AI (HCAI) puts the user in the driver's seat of so-called human-centered AI-infused tools (HCAI tools): interactive software tools that amplify, augment, empower, and enhance human performance using AI models. We discuss how interactive visualization can be a key enabling technology for creating such human-centered AI tools. To validate our approach, we surveyed the existing literature on HCI, AI, and visualization and interviewed researchers in relevant fields to define the characteristics of HCAI tools. We then present several examples of HCAI tools using visualization and use the examples to extract guidelines on how interactive visualization can support future HCAI tool research and development.

While multimodal recommendation models have effectively integrated visual and textual information, their reliance on unique ID embeddings constitutes a fundamental performance bottleneck. Specifically, ID-based paradigms suffer from three limitations: (1) \textbf{Information Isolation}, where unique IDs prevent semantic information exchange among related items; (2) \textbf{Cold-Start Vulnerability}, as ID embeddings are difficult to optimize with sparse interactions; and (3) \textbf{Storage Inefficiency}, where parameter costs scale linearly with item quantity. To overcome these challenges, we propose \textbf{MOTOR}, a novel \textbf{ID-free MultimOdal TOken Representation} scheme. MOTOR replaces explicit item IDs with learnable, shared multimodal tokens, fundamentally transforming the recommender into an ID-free framework. Methodologically, we first employ product quantization to discretize raw multimodal features into compact token IDs. These tokens serve as implicit item features, which are then synthesized via a novel \textbf{Token Cross Network (TCN)} to capture high-order interaction patterns. This "discretize-and-interact" mechanism enables semantic sharing across items and significantly compresses the model size without introducing complex auxiliary losses. Extensive experiments across nine mainstream models demonstrate the significant performance improvement achieved by MOTOR. Further, MOTOR improves the capability of these models to recommend items in cold-start scenarios.

In this paper, we introduce the notions of tight closure of ideals on Witt rings and quasi-tightly closedness of system of parameters. By using the notions, we obtain a characterization of quasi-$F$-rationality. Furthermore, we study the relationship between the closure operator and integrally closure.

We define the adelic hypergeometric function of special Gaussian type by means of a tower of hypergeometric curves. This function takes values in an adelic completed group ring and interpolates all the hypergeometric functions of the same type over all finite fields. It specializes at the unit argument to the adelic beta function of Ihara and Anderson. We prove some transformation formulas and a summation formula for the adelic hypergeometric function, which are known classically for complex hypergeometric functions.

We study the existence of bounds on the expected $p$-Wasserstein distance between a random measure and its mean under the assumption that the $p$-th centered moments of the counting statistics are controlled uniformly in space. The average Wasserstein transport cost is shown to be bounded from above and from below by some multiples of the number of points. $D$-dimensional versions of those results are also obtained. As a corollary, we prove that for any value of $p\geq 1$ the Ginibre point process can be seen as a perturbed lattice with identically distributed perturbations with a finite $p$-th moment.

In this paper, we characterize the vanishing of twisted central $L$-values attached to newforms of square-free level in terms of certain polynomials of quadratic forms introduced by Zagier and the action of finitely many Hecke operators thereon. To be more precise, we establish that a twisted central $L$-value attached to a newform vanishes if and only if a certain explicitly computable polynomial is constant. We describe these constants explicitly in two different ways. One of the descriptions involves the generalized Hurwitz class numbers, which were introduced by Pei and Wang in $2003$. We provide some numerical examples and conclude by offering some questions for future work.

We investigate Ruelle-Pollicott resonances of smooth Anosov diffeomorphisms, acting on manifolds of every dimension, with respect to the measure of maximal entropy. We highlight a profound connection between resonances and eigenvalues of the action induced by the dynamics on de Rham cohomology. In particular, resonances appear as eigenvalues of a quasi-compact transfer operator acting on suitable anisotropic spaces of currents. After defining the anisotropic Banach spaces, we introduce the anisotropic de Rham cohomology and we show that it is isomorphic to the standard de Rham cohomology. The relation between resonances and cohomological eigenvalues is deduced from a comparison of spectra. We finally exploit these results to get information about the Ruelle-Pollicott asymptotics of the correlation function and to establish a cohomological bound for the speed of mixing of Anosov diffeomorphisms.

Random-matrix statistics are usually imposed at the level of matrix entries or spectral correlations. Here we formulate a complementary inverse problem: can a matrix with prescribed random-matrix moments be generated from a structured set of latent vectors? We introduce a pair-resolved vector ansatz consisting of two vector families, P and Q, construct a complex-symmetric non-Hermitian matrix M = a1P P T + a2QQT . The transpose is intentionally not a conjugate transpose; hence the reconstructed bilinear overlap matrices are not Hermitian Gram matrices once the algebraic parameters become complex. The free parameters of the vectors are fixed by complex algebraic constraints matching diagonal and off-diagonal random-matrix moments, together with a mixed-overlap condition suppressing systematic correlations between the two bilinear sectors. A fast machine-precision solve for N = 8 returns six complex branches. We therefore supplement moment matching with reproducible branch diagnostics: residual error, approximate vector orthogonality, non-Hermiticity, imaginary spectral weight, inverse participation ratio, maximum component weight, and eigenvector conditioning. Optional entanglement and low-weight Pauli-moment diagnostics can be added when N = 2n . This protocol constitutes a finite-dimensional inverse reconstruction of hidden vectorspace representations behind apparent random-matrix behavior. It is static and algebraic: it probes moment-induced delocalization, non-Hermitian branch structure, and complex spectral statistics, but it does not by itself establish dynamical chaos in the sense of sensitive dependence on nearby initial conditions.

Topological data analysis (TDA) uses persistent homology to quantify loops and higher-dimensional holes in data, making it particularly relevant for examining the characteristics of images of cells in the field of cell biology. In the context of a cell injury, as time progresses, a wound in the form of a ring emerges in the cell image and then gradually vanishes. Performing statistical inference on this ring-like pattern in a single image is challenging due to the absence of repeated samples. In this paper, we develop a novel framework leveraging TDA to estimate underlying structures within individual images and quantify associated uncertainties through confidence regions. Our proposed method partitions the image into the background and the damaged cell regions. Then pixels within the affected cell region are used to establish confidence regions in the space of persistence diagrams (topological summary statistics). The method establishes estimates on the persistence diagrams which correct the bias of traditional TDA approaches. A simulation study is conducted to evaluate the coverage probabilities of the proposed confidence regions in comparison to an alternative approach is proposed in this paper. We also illustrate our methodology by a real-world example provided by cell repair.

We developed an open-source scalar wave transport model to estimate the generalized scattering matrix (S matrix) of a disordered medium in the diffusion regime. The term generalized refers to the incorporation of evanescent wave field modes alongside propagating modes in the estimation of the S matrix. To achieve this, we employed the scalar Kirchhoff-Helmholtz boundary integral formulation together with the Green's function perturbation method, thereby extending the conventional Fisher-Lee relations to include evanescent modes. The estimated S matrix, which satisfies the generalized unitarity and reciprocity relations, is modeled for a 2D disordered waveguide. The generalized transmission matrix contained within the S matrix is utilized to estimate the optimal phase-conjugate wavefront for focusing onto an evanescent mode. The phenomenon of a universal transmission value of 2/3 for such an optimal phase conjugate wavefront is demonstrated in the context of evanescent wave mode focusing through a diffusive disorder. The presented code framework may be of interest to wavefront shaping researchers for visualizing and estimating wave transport properties in general.

We prove that the spaces of smooth and ultradifferentiable vectors associated with a representation of a real Lie group on a Fréchet space $E$ are quasinormable if $E$ is so. A similar result is shown to hold for the linear topological invariant $(Ω)$. In the ultradifferentiable case, our results particularly apply to spaces of Gevrey vectors of Beurling type. As an application, we study the quasinormability and the property $(Ω)$ for a broad class of Fréchet spaces of smooth and ultradifferentiable functions on Lie groups globally defined via families of weight functions.

In our previous paper, we gave a complete list of the finite non-abelian simple groups whose holomorph contains a solvable regular subgroup. In this paper, we refine our previous work by considering all finite almost simple groups. In particular, our result yields a complete characterization of the finite almost simple groups which occur as the type of a Hopf-Galois structure on a solvable extension, or equivalently, the additive group of a skew brace having a solvable multiplicative group.

We build models using an indiscernible model sub-structures of $κ \ge λ$ and related more complicated structures. We use this to build various Boolean algebras.

We investigate the behavior of the empirical neighborhood distribution of marked graphs in the framework of local weak convergence. Here we extend known results by considering uniform random graphs with given degree sequences and i.i.d. marks on half-edges and vertices. We establish a large deviation principle for such families of empirical measures. The proof builds on Bordenave and Caputo's seminal 2015 paper, and Delgosha and Anantharam's 2019 introduction of BC entropy, relying on combinatorial lemmas that allow one to construct suitable approximations of measures supported on marked trees. Possible applications of these results are in the study of interacting diffusions on top of random graphs.

We exhibit a relationship between projective duality and the sheaf of logarithmic vector fields along a reduced divisor $D$ of projective space, in that the push-forward of the ideal sheaf of the conormal variety in the point-hyperplane incidence, twisted by the tautological ample line bundle is isomorphic to logarithmic differentials along $D$. Then we focus on the adjoint discriminant $D$ of a simple Lie group with Lie algebra $\mathfrak{g}$ over an algebraically closed field $\mathbf{k}$ of characteristic zero and study the logarithmic module $\mathrm{Der}_{\mathbf{U}}(-\log(D))$ over $\mathbf{U} = \mathbf{k}[\mathfrak{g}]$. When $\mathfrak{g}$ is simply laced, we show that this module has two direct summands: the $G$-invariant part, which is free with generators in degrees equal to the exponents of $G$, and the $G$-variant part, which is of projective dimension one, presented by the Jacobian matrix of the basic invariants of $G$ and isomorphic to the image of the map $\mathbf{ad}\,: \mathfrak{g} \otimes \mathbf{U}(-1) \rightarrow \mathfrak{g} \otimes \mathbf{U}$ given by the Lie bracket. When $\mathfrak{g}$ is not simply laced, we give a length-one equivariant graded free resolution of $\mathrm{Der}_{\mathbf{U}}(-\log(D))$ in terms of the exponents of $G$ and of the quasi-minuscule representation of $G$.

In 1991, it was proposed that fourfold-degenerate Landau levels formed by a single species of electrons could host a non-Abelian fractional quantum Hall (FQH) state with Fibonacci anyons at filling fraction $ν= \frac{2}{3}$. In this work, we investigate how such degenerate Landau levels can be realized in rhombohedral-stacked tetralayer graphene. We identify the following key conditions which may stabilize the Fibonacci state: (1) A magnetic field of around 20 Tesla is required if surface and interior carbons have the same energy level. If substrate hybridization raises the surface carbon energy level by $Δ_2 = 30$\,meV relative to interior carbon, the required field will have a larger range: 15 -- 20 Tesla. For $Δ_2 = 45$\,meV, the range reaches a maximum: 7 -- 20 Tesla. (2) The displacement field must be tuned to achieve Landau level degeneracy. $ν= \frac{3}{5}$ Fibonacci FQH states may also be realized in pentalayer rhombohedral graphene with a magnetic field of 12 Tesla, and $ν= \frac12$ states with Ising anyons may occur in trilayer graphene for magnetic fields of 12 -- 20 Tesla at $Δ_2 = 0$ or 5 -- 20 Tesla at $Δ_2 = 45$\,meV. We also study a simple interaction model to explore spin/valley polarization effects, and we see that the Fibonacci statemay occur at $ν= 2/3 + $ integer filling fractions, where the integer is 0 and 4 for sufficiently weak interaction, or can shift to 2 and 5 under a stronger interaction. The case $Δ_2 = 45$\,meV also produces states at negative filling fraction, e.g. $-\frac23$, $-4\frac23$. Here $ν$ is defined with respect to the Hall conductance, $σ_{xy} = ν\frac{e^2}{h}$.