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We consider application of the self-similarity principle in approximation theory under the conditions of asymptotic scale-invariance. For the effective summation of the asymptotic series methods, an iterative Borel summation with self-similar iterated roots is applied. The approximants follow from the self-similarity considerations and behave asymptotically as a power-law satisfying the asymptotic scale invariance. Optimal conditions on convergence of the sequence of approximants are imposed through the critical indices defined from the approximants. The indices are understood as control parameters for the optimal convergence of the asymptotic series. Such interpretation of the indices leads to an overall improvement of accuracy in calculations of the indices. The statement is supported by fifteen examples from condensed matter physics, quantum mechanics and field theory.

The method of Fractional Borel Summation is suggested in conjunction with self-similar factor approximants. The method used for extrapolating asymptotic expansions at small variables to large variables, including the variables tending to infinity, is described. The method is based on the combination of optimized perturbation theory, self-similar approximation theory, and Borel-type transformations. General Borel Fractional transformation of the original series is employed. The transformed series is resummed in order to adhere to the asymptotic power laws. The starting point is the formulation of dynamics in the approximations space by employing the notion of self-similarity. The flow in the approximation space is controlled, and “deep” control is incorporated into the definitions of the self-similar approximants. The class of self-similar approximations, satisfying, by design, the power law behavior, such as the use of self-similar factor approximants, is chosen for the reasons of transparency, explicitness, and convenience. A detailed comparison of different methods is performed on a rather large set of examples, employing self-similar factor approximants, self-similar iterated root approximants, as well as the approximation technique of self-similarly modified Padé–Borel approximations.

A new analytical procedure is developed for the deduction of the asymptotic series of the singular solutions in displacements and stresses near the vertex of the linear elastic isotropic corners with the Dirichlet–Robin (fixed-spring) and Neumann–Robin (free-spring) boundary conditions. Under the assumption of antiplane shear loading, the corresponding elastic problem reduces to the Laplace equation for the out-of-plane displacement. In the deduction of such singular solution, the complex variable is used to propose a harmonic function in the form of an asymptotic series including both power and logarithmic terms. This original procedure is suitable for its implementation in a computer algebra software which makes all the necessary symbolic computing, simplifications and rearrangements. This is a key issue due to the fact that the complexity of terms in these series may increase with increasing order of terms. These series are composed by the main terms (also called main singularities), solutions of the corresponding Dirichlet–Neumann or Neumann–Neumann problems, and the associated finite or infinite series of the so-called shadow terms (also called shadow singularities). These terms are determined by solving systems of recursive inhomogeneous Dirichlet–Neumann or Neumann–Neumann problems, respectively. A general classification of the behaviours of the asymptotic series covering all the considered corner problems is introduced. A few examples of the asymptotic series for corners with Dirichlet–Robin and Neumann–Robin boundary conditions are presented to illustrate the capabilities of this procedure.

We develop a perturbative Liouville approach to celestial amplitudes and present its first concrete result: a systematic small-b expansion of the DOZZ threepoint structure constant in the light-operator regime αi = bσi (i = 1; 2; 3). In this regime the DOZZ constant factorizes into a universal prefactor times a power series in b2 whose constant term equals one; the higher-order coefficients Ωn(σi) are symmetric polynomials in the σi. Using Thorn’s asymptotic expansion of ϒb we obtain closed-form expressions for Ω1, Ω2, Ω3. These coefficients provide explicit quantum corrections to the light semiclassical Liouville three-point function and hence supply practical input for reconstructing loop-level three-gluon amplitudes via the inverse Mellin transform. We conclude by outlining future work, notably the explicit computation of the one-loop contribution to the three-point amplitude and an extension of this framework to four-point correlators.

A bstract We initiate the systematic study of flux scalar potentials and their vacua by using asymptotic Hodge theory. To begin with, we consider F-theory compactifications on Calabi-Yau fourfolds with four-form flux. We argue that a classification of all scalar potentials can be performed when focusing on regions in the field space in which one or several fields are large and close to a boundary. To exemplify the constraints on such asymptotic flux compactifications, we explicitly determine this classification for situations in which two complex structure moduli are taken to be large. Our classification captures, for example, the weak string coupling limit and the large complex structure limit. We then show that none of these scalar potentials admits de Sitter critical points at parametric control, formulating a new no-go theorem valid beyond weak string coupling. We also check that the recently proposed asymptotic de Sitter conjecture is satisfied near any infinite distance boundary. Extending this strategy further, we generally identify the type of fluxes that induce an infinite series of Anti-de Sitter critical points, thereby generalizing the well-known Type IIA settings. Finally, we argue that also the large field dynamics of any axion in complex structure moduli space is universally constrained. Displacing such an axion by large field values will generally lead to severe backreaction effects destabilizing other directions.

Quantum statistics dictates the behavior of identical particles in the quantum world: Bosons like to congregate, whereas fermions avoid one another. However, strong interactions can cause a string of bosons to behave like fermions. This so-called fermionization phenomenon has been studied in equilibrium. Wilson et al. instead focused on dynamical fermionization in a nonequilibrium system consisting of tubes of strongly interacting bosonic rubidium atoms. After letting the tubes expand in the axial direction, the researchers monitored the momentum distribution of the atoms and found that it evolved from bosonic-like to fermionic-like. Science , this issue p. 1461 The momentum distribution of an expanding one-dimensional Bose gas evolves from bosonic to fermionic. The wave function of a Tonks-Girardeau (T-G) gas of strongly interacting bosons in one dimension maps onto the absolute value of the wave function of a noninteracting Fermi gas. Although this fermionization makes many aspects of the two gases identical, their equilibrium momentum distributions are quite different. We observed dynamical fermionization, where the momentum distribution of a T-G gas evolves from bosonic to fermionic after its axial confinement is removed. The asymptotic momentum distribution after expansion in one dimension is the distribution of rapidities, which are the conserved quantities associated with many-body integrable systems. Our measurements agree well with T-G gas theory. We also studied momentum evolution after the trap depth is suddenly changed to a new nonzero value, and we observed the theoretically predicted bosonic-fermionic oscillations.

We consider the minimization of a strongly convex objective function given access to unbiased estimates of its gradient through stochastic gradient descent (SGD) with constant step size. While the detailed analysis was only performed for quadratic functions, we provide an explicit asymptotic expansion of the moments of the averaged SGD iterates that outlines the dependence on initial conditions, the effect of noise and the step size, as well as the lack of convergence in the general (nonquadratic) case. For this analysis we bring tools from Markov chain theory into the analysis of stochastic gradient. We then show that Richardson–Romberg extrapolation may be used to get closer to the global optimum, and we show empirical improvements of the new extrapolation scheme.

A bstract We present a novel expression for an integrated correlation function of four superconformal primaries in SU( N ) N = 4 supersymmetric Yang-Mills ( N = 4 SYM) theory. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. In this paper the correlator is re-expressed as a sum over a two dimensional lattice that is valid for all N and all values of the complex Yang-Mills coupling τ = θ / 2 π + 4 πi / g YM 2 . In this form it is manifestly invariant under SL(2 , ℤ) Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the SU( N ) correlator to the SU( N + 1) and SU( N − 1) correlators. For any fixed value of N the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series, E s τ τ ¯ with s ∈ ℤ, and rational coefficients that depend on the values of N and s . The perturbative expansion of the integrated correlator is an asymptotic but Borel summable series, in which the n -loop coefficient of order ( g YM / π ) 2 n is a rational multiple of ζ (2 n + 1). The n = 1 and n = 2 terms agree precisely with results determined directly by integrating the expressions in one-loop and two-loop perturbative N = 4 SYM field theory. Likewise, the charge- k instanton contributions (| k | = 1 , 2 , . . . ) have an asymptotic, but Borel summable, series of perturbative corrections. The large- N expansion of the correlator with fixed τ is a series in powers of N 1 2 − ℓ ( ℓ ∈ ℤ) with coefficients that are rational sums of E s τ τ ¯ with s ∈ ℤ + 1/2. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider the ’t Hooft topological expansion of large- N Yang-Mills theory in which λ = g YM 2 N is fixed. The coefficient of each order in the 1 /N expansion can be expanded as a series of powers of λ that converges for |λ| < π 2 . For large λ this becomes an asymptotic series when expanded in powers of 1 / λ with coefficients that are again rational multiples of odd zeta values, in agreement with earlier results and providing new ones. We demonstrate that the large- λ series is not Borel summable, and determine its resurgent non-perturbative completion, which is O exp − 2 λ .

The aim of this paper is to establish some asymptotic expansions and inequalities related to BBP- and Ramanujan-type approximation series of the constant ...

Characterizing how neural network depth, width, and dataset size jointly impact model quality is a central problem in deep learning theory. We give here a complete solution in the special case of linear networks with output dimension one trained using zero noise Bayesian inference with Gaussian weight priors and mean squared error as a negative log-likelihood. For any training dataset, network depth, and hidden layer widths, we find nonasymptotic expressions for the predictive posterior and Bayesian model evidence in terms of Meijer-G functions, a class of meromorphic special functions of a single complex variable. Through asymptotic expansions of these Meijer-G functions, a rich new picture of the joint role of depth, width, and dataset size emerges. We show that linear networks make provably optimal predictions at infinite depth: the posterior of infinitely deep linear networks with data-agnostic priors is the same as that of shallow networks with evidence-maximizing data-dependent priors. This yields a principled reason to prefer deeper networks when priors are forced to be data-agnostic. Moreover, we show that with data-agnostic priors, Bayesian model evidence in wide linear networks is maximized at infinite depth, elucidating the salutary role of increased depth for model selection. Underpinning our results is an emergent notion of effective depth, given by the number of hidden layers times the number of data points divided by the network width; this determines the structure of the posterior in the large-data limit.