Let $(X_1,\ldots,X_n)$ be independent nonnegative random variables with $\mathbb{E} X_i\le1$, and write $S=\sum_iX_i$. For $δ>0$, we prove that \[ \mathbb{P}\left(S<\mathbb{E} S+δ\right)\ge b_{n,δ}, \] where $b_{n,δ}=δ(n/(n+δ))^n$ for $0<δ<1$ and $b_{n,δ}=(1-1/(n+δ))^n$ for $δ\ge1$. The bound is sharp for every $n$ and $δ\ge 1$. In particular, since $b_{n,δ} \ge e^{-1}$ for $δ\ge 1$, our result proves Feige's conjecture [Feige, 2004] in the affirmative for $δ\ge 1$. The proof is found by ChatGPT 5.6 Pro. It combines the exact Dirichlet calibration theorem of Vlassis and Thomas [Vlassis and Thomas, 2026], which resolves Gaffke's conjecture in statistics, with results in convex geometry including Grünbaum's centroid theorem [Grünbaum, 1960] and its generalization by Letwin and Yaskin [Letwin and Yaskin, 2024].

Modern machine learning classifiers often exhibit vanishing classification error on the training set. They achieve this by learning nonlinear representations of the inputs that maps the data into linearly separable classes. Motivated by these phenomena, we revisit high-dimensional maximum margin classification for linearly separable data. We consider a stylized setting in which data $(y_i,{\boldsymbol x}_i)$, $i\le n$ are i.i.d. with ${\boldsymbol x}_i\sim\mathsf{N}({\boldsymbol 0},{\boldsymbol \Sigma})$ a $p$-dimensional Gaussian feature vector, and $y_i \in\{+1,-1\}$ a label whose distribution depends on a linear combination of the covariates $\langle {\boldsymbol \theta}_*,{\boldsymbol x}_i \rangle$. While the Gaussian model might appear extremely simplistic, universality arguments can be used to show that the results derived in this setting also apply to the output of certain nonlinear featurization maps. We consider the proportional asymptotics $n,p\to\infty$ with $p/n\to \psi$, and derive exact expressions for the limiting generalization error. We use this theory to derive two results of independent interest: $(i)$ Sufficient conditions on $({\boldsymbol \Sigma},{\boldsymbol \theta}_*)$ for `benign overfitting' that parallel previously derived conditions in the case of linear regression; $(ii)$ An asymptotically exact expression for the generalization error when max-margin classification is used in conjunction with feature vectors produced by random one-layer neural networks.

This article extends the solution proposed by Ait-Sahalia, Fan, and Li for the leverage effect puzzle, which refers to a fact that empirical correlation between daily asset returns and the changes of daily volatility estimated from high frequency data is nearly zero. Complementing the analysis in Ait-Sahalia, Fan, and Li via the Heston model, we work with a generic semi-nonparametric stochastic volatility model via an operator-based expansion method. Under such a general setup, we identify a new source of bias due to the flexibility of variance dynamics, distinguishing the leverage effect parameter from the instantaneous correlation parameter. For estimating the leverage effect parameter, we show that the main results on analyzing the various sources of biases as well as the resulting statistical procedures for biases correction in Ait-Sahalia, Fan, and Li hold true and are thus indeed theoretically robust. For estimating the instantaneous correlation parameter, we developed a new nonparametric estimation method.

Distributing points on a (possibly high-dimensional) sphere with minimal energy is a long-standing problem in and outside the field of mathematics. This paper considers a novel energy function that arises naturally from statistics and combinatorial optimization, and studies its theoretical properties. Our result solves both the exact optimal spherical point configurations in certain cases and the minimal energy asymptotics under general assumptions. Connections between our results and the L1-Principal Component analysis and Quasi-Monte Carlo methods are also discussed.

The multiconfiguration Dirac–Hartree–Fock (MCDHF) and relativistic configuration interaction methods are used to provide excitation energies, lifetimes, and radiative transition data for the 604 (699, 702, 704, 704, 704, and 699) lowest levels of the 3s 23p 2, 3s3p 3, 3s 23p3d, 3p 4, 3s3p 23d, 3s 23d 2, 3p 33d, 3s3p3d 2, 3s3d 3, 3p3d 3, 3p 23d 2, 3s 23p4s, 3s 23p4p, 3s 23p4d, 3s 23p4f, 3s3p 24s, 3s3p 24p, 3s3p 24d, 3s3p 24f, 3s 23d4s, 3s 23d4p, 3p 34s, 3p 34p, 3s3p3d4s, 3s 23p5s, and 3s 23p5p configurations in Cr xi, (Mn xii, Fe xiii, Co xiv, Ni xv, Cu xvi, and Zn xvii). Previous line identifications of Fe xiii and Ni xv in the EUV and X-ray wavelength ranges are reviewed by comprehensively comparing the MCDHF theoretical results with available experimental data. Many recent identifications of Fe xiii and Ni xv lines are confirmed, and several new identifications for these two ions are proposed. A consistent atomic data set with spectroscopic accuracy is provided for the lowest hundreds of levels for Si-like ions of iron-group elements of astrophysical interest, for which experimental values are scarce. The uncertainty estimation method suggested by Kramida, applied to the comparison of the length and velocity line strength values, is used for ranking the transition data. The correlation of the latter with the gauge dependency patterns of the line strengths is investigated.

Using the multiconfiguration Dirac–Hartree–Fock and the relativistic configuration interaction methods, a consistent set of transition energies and radiative transition data for the main states of the 2s22p4, 2s2p5, 2p6, 2s22p33s, 2s22p33p, 2s22p33d, 2s2p43s, 2s2p43p, and 2s2p43d configurations in O-like Ions between Ar XI (Z=18) and Cr XVII (Z=24) is provided. Our data set is compared with the NIST compiled values and previous calculations. The data are accurate enough for identification and deblending of new emission lines from hot astrophysical and laboratory plasmas. The amount of data of high accuracy is significantly increased for the n=3 states of several O-like ions, where experimental data are very scarce.

We present a framework to calculate large deviations for nonlinear functions of independent random variables supported on compact sets in Banach spaces, by extending the result in Chatterjee and Dembo [6]. Previous research on nonlinear large deviations has only focused on random variables supported on $\{-1,+1\}^{n}$, a small subset of random objects people usually study, thus it is of natural interest and need to research the corresponding theory for random variables with general distributions. Since our results put fewer constraints on the random variables, it has considerable flexibility in application. To show this, we provide examples with continuous and high dimensional random variables. Our framework could also be used to verify the mathematical rigor of the mean field approximation method; to demonstrate, we verify the mean field approximation for a class of spin vector models.

Using the multiconfiguration Dirac–Hartree–Fock and the relativistic configuration interaction methods, a consistent set of transition energies and radiative transition data for the lowest 546 (623, 701, and 745) states of the , , , , , , , , , , , , , , , and configurations in Mn xi (Fe xii, Co xiii, and Ni xiv) is provided. The comparison between calculated excitation energies for the n = 4 states and available experimental values for Fe xii indicate that the calculations are highly accurate, with uncertainties of only a few hundred cm−1. Lines from these states are prominent in the soft X-rays. With the present calculations, several recent new identifications are confirmed. Other identifications involving levels in Fe xii that were found to be questionable are discussed and a few new assignments are recommended. As some n = 4 states of the other ions also show large discrepancies between experimental and calculated energies, we reassess their identification. The present study provides highly accurate atomic data for the n = 4 states of P-like ions of astrophysical interest, for which experimental data are scarce.

The stability of an Auto-Regressive (AR) time sequence of finite order L, is determined by the maximal modulus r^⋆ among all zeros of its generating polynomial. If r^⋆<1 then the effect of input and initial conditions decays rapidly in time, whereas for r^⋆>1 it is exponentially magnified (with constant or polynomially growing oscillations when r^⋆=1). Persistence of such AR sequence (namely staying non-negative throughout [0,N]) with decent probability, requires the largest positive zero of the generating polynomial to have the largest multiplicity among all zeros of modulus r^⋆. These objects are behind the rich spectrum of persistence probability decay for AR_L with zero initial conditions and i.i.d. Gaussian input, all the way from bounded below to exponential decay in N, with intermediate regimes of polynomial and stretched exponential decay. In particular, for AR_3 the persistence decay power is expressed via the tail probability for Brownian motion to stay in a cone, exhibiting the discontinuity of such power decay between the AR_3 whose generating polynomial has complex zeros of rational versus irrational angles.

This paper proposes a novel non-parametric multidimensional convex regression estimator which is designed to be robust to adversarial perturbations in the empirical measure. We minimize over convex functions the maximum (over Wasserstein perturbations of the empirical measure) of the absolute regression errors. The inner maximization is solved in closed form resulting in a regularization penalty involves the norm of the gradient. We show consistency of our estimator and a rate of convergence of order $ \widetilde{O}\left( n^{-1/d}\right) $, matching the bounds of alternative estimators based on square-loss minimization. Contrary to all of the existing results, our convergence rates hold without imposing compactness on the underlying domain and with no a priori bounds on the underlying convex function or its gradient norm.

We present a consistent set of calculated energies and El, Ml, E2, M2 radiative transition data for the main n = 3 levels from the 3s(2)3p(4), 3p(6), 3s3p(4)3d, 3s(2)3p(2)3d(2), 3s3p(5), 3s(2)3p(3)3d, and 3s3p(3)3d(2) configurations for S-like ions from Cr TX to Cu XIV. The fully relativistic multiconfiguration Dirac Hartree Fock method implemented in the GRASP2K code is used to perform the present calculations. The excitation energies of the lowest 47 levels from the 3s(2)3p(4), 3s3p(5), and 3s(2)3p3 3d configurations, producing the strongest lines, are found to be in good agreement, reaching spectroscopic accuracy, with the latest experimental values for Fe XI evaluated by Del Zanna. Our energies can reliably be used to identify in astrophysical and laboratory spectra the 3s(2)3p(3) 3d levels in other S-like ions, which are mostly unknown. On the contrary, significant discrepancies with the 3s3p(4)3d levels were found, emphasizing the need for more detailed experimental studies. A few new tentative identifications are suggested. The benchmarks we present indicate that our consistent set of radiative data is accurate and can be used for spectral line modeling.

Employing two state-of-the-art methods, multiconfiguration Dirac–Hartree–Fock and second-order many-body perturbation theory, highly accurate calculations are performed for the lowest 272 fine-structure levels arising from the 2s22p3, 2s2p4, 2p5, 2s22p23l (l=s,p,d), 2s2p33l (l=s,p,d), and 2p43l (l=s,p,d) configurations in nitrogen-like Ge XXVI. Complete and consistent atomic data, including excitation energies, lifetimes, wavelengths, hyperfine structures, Landé gJ-factors, and E1, E2, M1, M2 line strengths, oscillator strengths, and transition rates among these 272 levels are provided. Comparisons are made between the present two data sets, as well as with other available experimental and theoretical values. The present data are accurate enough for identification and deblending of emission lines involving the n=3 levels, and are also useful for modeling and diagnosing fusion plasmas.

We introduce an algebra model to study higher-order sum rules for orthogonal polynomials on the unit circle. We build the relation between the algebra model and sum rules, and prove an equivalent expression on the algebra side for the sum rules involving a Hall–Littlewood type polynomial. By this expression, we recover an earlier result by Golinskii and Zlatoš and prove a new case - half of the Lukic conjecture in the case of a single critical point with arbitrary order.

Excitation energies and lifetimes for the 213 lowest states of the n ≤ 5 configurations in B-like S xii are calculated using highly correlated wave functions, optimized with the fully relativistic multiconfiguration Dirac–Hartree–Fock method. Multipole transition rates and associated radiative data (line strengths and oscillator strengths) for transitions connecting these levels are also reported. The theoretical excitation energies are systematically compared with the NIST Atomic Spectra Database in which misidentifications are pointed out. After eliminating the latter, a mean energy difference with the standard deviation between computed and observed energies of 12 ± 341 cm−1 is obtained for the n ≥ 3 high-lying states. This level of accuracy confirms that elaborate ab initio calculations can assist in the identification of new emission lines in the solar and other astrophysical spectra. The present work provides atomic data of high accuracy for an ion of astrophysical interest, B-like S xii, for which experimental data are scarce.

We consider the problem of estimating an unknown matrix $\boldsymbol{X}\in {\mathbb R}^{m\times n}$, from observations $\boldsymbol{Y} = \boldsymbol{X}+\boldsymbol{W}$ where $\boldsymbol{W}$ is a noise matrix with independent and identically distributed entries, as to minimize estimation error measured in operator norm. Assuming that the underlying signal $\boldsymbol{X}$ is low-rank and incoherent with respect to the canonical basis, we prove that minimax risk is equivalent to $(\sqrt{m}\vee\sqrt{n})/\sqrt{I_W}$ in the high-dimensional limit $m,n\to\infty$, where $I_W$ is the Fisher information of the noise. Crucially, we develop an efficient procedure that achieves this risk, adaptively over the noise distribution (under certain regularity assumptions). Letting $\boldsymbol{X} = \boldsymbol{U}{\boldsymbol{\Sigma}}\boldsymbol{V}^{\sf T}$ --where $\boldsymbol{U}\in {\mathbb R}^{m\times r}$, $\boldsymbol{V}\in{\mathbb R}^{n\times r}$ are orthogonal, and $r$ is kept fixed as $m,n\to\infty$-- we use our method to estimate $\boldsymbol{U}$, $\boldsymbol{V}$. Standard spectral methods provide non-trivial estimates of the factors $\boldsymbol{U},\boldsymbol{V}$ (weak recovery) only if the singular values of $\boldsymbol{X}$ are larger than $(mn)^{1/4}{\rm Var}(W_{11})^{1/2}$. We prove that the new approach achieves weak recovery down to the the information-theoretically optimal threshold $(mn)^{1/4}I_W^{1/2}$.

Using the multiconfiguration Dirac–Fock method, calculations for the lowest 62 levels of the ([Ar] 3d10)4s24p5, ([Ar] 3d10)4s24p34d2, ([Ar] 3d10)4s24p44d, ([Ar] 3d10)4s4p6, and ([Ar] 3d10)4s4p54d configurations are performed for the bromine isoelectronic sequence La XXIII-Dy XXXII, W XL. Results of energy levels, lifetimes, wavelengths, and electric dipole, magnetic dipole, electric quadrupole, and magnetic quadrupole radiative rates are presented. In order to assess the accuracy of results, independent calculations for W XL have been carried out using the many-body perturbation theory (MBPT) method. Comparisons are made with available theoretical results from other calculations and the observed values of the Atomic Spectra Database of the National Institute of Standards and Technology. Energy levels are estimated to be accurate to better than 1%, and radiative rates (and lifetimes) are accurate to better than 20% for a majority of strong transitions. These results should be useful in many applications of lanthanide ions related to broad area of research such as applied physics, laser physics and fusion science.

Employing two state-of-the-art methods, multiconfiguration Dirac-Hartree-Fock and second-order many-body perturbation theory, the excitation energies and lifetimes for the lowest 200 states of the 2 s(2)2p(4), 2s2p(5), 2p(6), 2s(2)2p(3)3s, 2s(2)2p(3)3p, 2s(2)2p(3)3d, 2s(2)p(4)3s, 2s2p(4)3p, and 2s2p(4)3d configurations, and multipole (electric dipole (E1), magnetic dipole (M1), and electric quadrupole (E2)) transition rates, line strengths, and oscillator strengths among these states are calculated for each O-like ion, from Cr XVII to Zn XXIII. Our two data sets are compared with the National Institute of Standards and Technology and CHIANTI compiled values, and previous calculations. The data are accurate enough for identification and deblending of new emission lines from the Sun and other astrophysical sources. The amount of high-accuracy data is significantly increased for the n. =. 3 states of several O-like ions of astrophysical interest, where experimental data are very scarce.

Energy levels and transition rates for electric-dipole (E1), electric-quadrupole (E2), magnetic-dipole (M1), and magnetic-quadrupole (M2) transitions of the lowest 393 levels arising from the 3l3 (0≤l⩽2), 3s24l (0≤l⩽3), 3s3p4l (0≤l⩽3), 3p24l (0≤l⩽2), 3s3d4l (0≤l⩽1), and 3s25l (0≤l⩽4) configurations in Al-like Zn are calculated through the multi-configuration Dirac-Hartree-Fock (MCDHF) method and second-order many-body perturbation theory (MBPT). In the MCDHF calculation, valence-valence and core-valence correlations with the 2p and 2s electrons are taken into account. The effect of Breit interaction and quantum electrodynamics corrections on excitation level energies and level lifetimes are assessed though the MCDHF and MBPT calculations. The two sets of level energies are in excellent agreement of better than 0.1%, while the level lifetimes mostly agree to within 2%. Comparisons are also made with experimental measurements and other theoretical results to assess the accuracy of our calculations.