Logarithmic Terms Research Articles

Nonradial symmetry and blow-up of ground states for a Schrödinger–Poisson system with logarithmic term

Abstract We consider the following Schrödinger–Poisson system { − Δ u + V ( x ) u + λ ϕ ( x ) u = u log u 2 , x ∈ ℝ 3 , (0.1) − Δ ϕ = u 2 , lim | x | → + ∞ ϕ ( x ) = 0 , where λ ∈ R is a parameter, V ∈ C ( R 3 , R + ) is a coercive potential. We prove that, if V ( x ) ∼ ( log ⁡ | x | ) 1 2 at infinity, then the energy functional I λ associated with (0.1) fails to be C 1, and there is λ 0 > 0 such that (0.1) has a ground state u λ 0 for any λ ∈ ( 0 , λ 0 ) , which blows up as λ → 0 + , but if V ( x ) = V ( | x | ) ∼ ( log ⁡ | x | ) α with α ∈ ( 0 , 1 ) at infinity, then there exists a sequence λ n → 0 + such that each u λ n 0 must be non-radially symmetric. However, if V(x) grows like | x | γ (γ > 0) at infinity, we show that the functional I λ is of C 1 and (0.1) has a ground state u λ 1 for each λ < 0 which is radially symmetric. Also, the limit behavior of u λ 1 as λ → 0 − is discussed.

A strain rate-dependent distortional hardening model for nonlinear strain paths

In this paper, a strain rate-dependent distortional hardening model is firstly proposed to describe strain rate-dependent material behaviors under linear and nonlinear strain paths changes in 0≤θpathchange≤180∘. The proposed model is formulated based on the simplified strain rate-independent distortional hardening model (Choi and Yoon, 2023). Any yield function could be used for the strain rate-dependent isotropic and anisotropic yielding. For the linear strain path, the strain rate-dependent isotropic hardening behavior could be explained by two state variables representing rate-dependent yielding and convergence rate of flow stress under monotonically increasing loading condition, respectively. For the nonlinear strain paths, the strain rate-dependent material behaviors such as Bauschinger effect, yield surface contraction, permanent softening, and nonlinear transient behavior could be described by modifying the evolution equations of the simplified strain rate-independent distortional hardening model with a logarithmic term of strain rate. For the verification purpose, it was used the strain-rate dependent tension-compression experiments of TRIP980 and TWIP980 (Joo et al., 2019). In addition, a high speed U-draw bending test was conducted with original and pre-strained specimens. The springback prediction in high speed U-draw bending test was performed by using strain rate-independent isotropic, strain rate-dependent isotropic-kinematic and distortional hardening models. It is identified that the proposed model showed the most accurate prediction for the pre-strained specimen where the possible bilinear and trilinear path change in 0≤θpathchange≤180∘ is observed while it showed the same accuracy for the original specimen where main strain path change occur in forward-reverse manner (θpathchange=180∘).

Logarithmic soft theorems and soft spectra

Using universal predictions provided by classical soft theorems, we revisit the energy emission spectrum for gravitational scatterings of compact objects in the low-frequency expansion. We calculate this observable beyond the zero-frequency limit, retaining an exact dependence on the kinematics of the massive objects. This allows us to study independently the ultrarelativistic or massless limit, where we find agreement with the literature, and the small-deflection or post-Minkowskian (PM) limit, where we provide explicit results up to O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(G5). These confirm that the high-velocity limit of a given PM order is smoothly connected to the corresponding massless result whenever the latter is analytic in the Newton constant G. We also provide explicit expressions for the waveforms to order ω−1, log ω, ω(log ω)2 in the soft limit, ω → 0, expanded up to sub-subleading PM order, as well as a conjecture for the logarithmic soft terms of the type ωn−1(log ω)n with n ≥ 3.

Solutions of the minimal surface equation and of the Monge–Ampère equation near infinity

Abstract Classical results assert that, under appropriate assumptions, solutions near infinity are asymptotic to linear functions for the minimal surface equation and to quadratic polynomials for the Monge–Ampère equation for dimension n ≥ 3 n\geq 3 , with an extra logarithmic term for n = 2 n=2 . Via Kelvin transforms, we characterize remainders in the asymptotic expansions by a single function near the origin. Such a function is smooth in the entire neighborhood of the origin for the minimal surface equation in every dimension and for the Monge–Ampère equation in even dimension, but only C n − 1 , α C^{n-1,\alpha} for the Monge–Ampère equation in odd dimension 𝑛, for any α ∈ ( 0 , 1 ) \alpha\in(0,1) .

Global existence and decay of a viscoelastic wave equation with logarithmic source under acoustic, fractional, and nonlinear delay conditions

This study is concerned with a nonlinear viscoelastic wave equation with logarithmic source term. By considering the nonlinear distributed delay feedback influenced by acoustic and fractional boundary conditions at the boundary coupling, we establish the global existence and general decay results under appropriate assumptions, even in the case of a general kernel. This result extends and complements some previous results.

Asymptotics of the partition function of the perturbed Gross–Witten–Wadia unitary matrix model

AbstractWe consider the asymptotics of the partition function of the extended Gross–Witten–Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a ‐function sequence of the Painlevé equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta‐function and the Barnes ‐function. A third‐order phase transition in the leading terms of the asymptotic expansions is also observed.

Local existence and blow up for the wave equation with nonlinear logarithmic source term and nonlinear dynamical boundary conditions combined with distributed delay

Local existence and blow up for the wave equation with nonlinear logarithmic source term and nonlinear dynamical boundary conditions combined with distributed delay

Logarithmic terms in discrete heat kernel expansions in the quadrant

In the context of lattice walk enumeration in cones, we consider the number of walks in the quarter plane with fixed starting and ending points, prescribed step-set, and given length. After renormalisation, this number may be interpreted as a discrete heat kernel in the quadrant. We propose a new method to compute complete asymptotic expansions of these numbers of walks as their length tends to infinity, based on two main ingredients: explicit expressions for the underlying generating functions in terms of elliptic Jacobi theta functions along with a duality known as Jacobi transformation. This duality allows us to pass from a classical Taylor expansion of the series to an expansion at the critical point of the model. We work through two examples. First, we present our approach on the well-known Kreweras model, which is algebraic, and show how to obtain a complete asymptotic expansion in this case. We then consider a more generic (so-called infinite group) model and find the associated complete asymptotic expansion. In this second case, we prove the existence of logarithmic terms in the asymptotic expansion, and we relate the coefficients appearing in the expansion to polyharmonic functions. To our knowledge, this is the first time that logarithmic terms have been observed in the asymptotics of a class of lattice walks confined to a quadrant.

On the speed of convergence of discrete Pickands constants to continuous ones

Abstract In this manuscript, we address open questions raised by Dieker and Yakir (2014), who proposed a novel method of estimating (discrete) Pickands constants $\mathcal{H}^\delta_\alpha$ using a family of estimators $\xi^\delta_\alpha(T)$ , $T>0$ , where $\alpha\in(0,2]$ is the Hurst parameter, and $\delta\geq0$ is the step size of the regular discretization grid. We derive an upper bound for the discretization error $\mathcal{H}_\alpha^0 - \mathcal{H}_\alpha^\delta$ , whose rate of convergence agrees with Conjecture 1 of Dieker and Yakir (2014) in the case $\alpha\in(0,1]$ and agrees up to logarithmic terms for $\alpha\in(1,2)$ . Moreover, we show that all moments of $\xi_\alpha^\delta(T)$ are uniformly bounded and the bias of the estimator decays no slower than $\exp\{-\mathcal CT^{\alpha}\}$ , as T becomes large.

Parameter estimation for Gibbs distributions

A central problem in computational statistics is to convert a procedure for sampling combinatorial objects into a procedure for counting those objects, and vice versa. We consider sampling problems coming from Gibbs distributions , which are families of probability distributions over a discrete space \(\Omega\) with probability mass function of the form \(\mu^{\Omega}_{\beta}(\omega)\propto e^{\beta H(\omega)}\) for \(\beta\) in an interval \([\beta_{\min},\beta_{\max}]\) and \(H(\omega)\in\{0\}\cup[1,n]\) . Two important parameters are the partition function , which is the normalization factor \(Z(\beta)=\sum_{\omega\in\Omega}e^{\beta H(\omega)}\) , and the vector of preimage counts \(c_{x}=|H^{-1}(x)|\) . We develop black-box sampling algorithms to estimate the counts using roughly \(\tilde{O}(\frac{n^{2}}{\varepsilon^{2}})\) samples for integer-valued distributions and \(\tilde{O}(\frac{q}{\varepsilon^{2}})\) samples for general distributions, where \(q=\log\frac{Z(\beta_{\max})}{Z(\beta_{\min})}\) (ignoring some second-order terms and parameters). We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we estimate all values of the partition function using \(\tilde{O}(\frac{n^{2}}{\varepsilon^{2}})\) samples for integer-valued distributions and \(\tilde{O}(\frac{q}{\varepsilon^{2}})\) samples for general distributions. This improves over a prior algorithm of Huber (2015) which computes a single point estimate \(Z(\beta_{\max})\) and which uses a slightly larger amount of samples. We show matching lower bounds, demonstrating this complexity is optimal as a function of \(n\) and \(q\) up to logarithmic terms.

A Posteriori Error Analysis of Defect Correction Method for Singular Perturbation Problems With Discontinuous Coefficient and Point Source

Abstract The paper presents a defect correction method to solve singularly perturbed problems with discontinuous coefficient and point source. The method combines an inexpensive, lower-order stable, upwind difference scheme and a higher-order, less stable central difference scheme over a layer-adapted mesh. The mesh is designed so that most mesh points remain in the regions with rapid transitions. A posteriori error analysis is presented. The proposed numerical method is analyzed for consistency, stability, and convergence. The error estimates of the proposed numerical method satisfy parameter-uniform second-order convergence on the layer-adapted grid. The convergence obtained is optimal because it is free from any logarithmic term. The numerical analysis confirms the theoretical error analysis.

A critical Kirchhoff equation with a logarithmic type perturbation in dimension 4

In this paper, a critical Kirchhoff type elliptic equation involving a logarithmic type perturbation is considered in a bounded domain in . Three main features of the problem bring essential difficulties when proving the existence of weak solutions. The first one is the nonlocal term which makes the structure of the corresponding energy functional more complicated, the second one is that the problem is critical in the sense that the energy functional lacks compactness, and the third one is the appearance of the logarithmic term which satisfies neither the standard monotonicity condition nor the Ambrosetti–Rabinowitz condition. Moreover, the boundedness of the sequence is hard to obtain for Kirchhoff problems in dimension 4. By combining a result by Jeanjean (Proc. Roy. Soc. Edinburgh Sect. A, 1999, 129: 787–809) and a recent estimate by Deng et al. (Adv. Nonlinear Stud., 2023, 23: No. 20220049) with the mountain pass lemma and Brézis‐Lieb's lemma, it is proved that either the norm of the sequence of approximation solution goes to infinity or the problem admits a nontrivial weak solution, under appropriate assumptions on the parameters.

Trade-Offs Between Energy and Depth of Neural Networks.

We present an investigation on threshold circuits and other discretized neural networks in terms of the following four computational resources-size (the number of gates), depth (the number of layers), weight (weight resolution), and energy-where the energy is a complexity measure inspired by sparse coding and is defined as the maximum number of gates outputting nonzero values, taken over all the input assignments. As our main result, we prove that if a threshold circuit C of size s, depth d, energy e, and weight w computes a Boolean function f (i.e., a classification task) of n variables, it holds that log( rk (f))≤ed(logs+logw+logn) regardless of the algorithm employed by C to compute f, where rk (f) is a parameter solely determined by a scale of f and defined as the maximum rank of a communication matrix with regard to f taken over all the possible partitions of the n input variables. For example, given a Boolean function CD n(ξ) =⋁i=1n/2ξi∧ξn/2+i, we can prove that n/2≤ed( log s+logw+logn) holds for any circuit C computing CD n. While its left-hand side is linear in n, its right-hand side is bounded by the product of the logarithmic factors of s,w,n and the linear factors of d,e. If we view the logarithmic terms as having a negligible impact on the bound, our result implies a trade-off between depth and energy: n/2 needs to be smaller than the product of e and d. For other neural network models, such as discretized ReLU circuits and discretized sigmoid circuits, we also prove that a similar trade-off holds. Thus, our results indicate that increasing depth linearly enhances the capability of neural networks to acquire sparse representations when there are hardware constraints on the number of neurons and weight resolution.

Asymptotic behavior of Riemann solutions for the one-dimensional mean-field games in conservative form with the logarithmic coupling term

Construction of Riemann solutions for a hyperbolic system in conservative form arising from the one-dimensional mean-field games with the quadratic Hamiltonian and the logarithmic coupling term are provided in detail. Moreover, the delta shock formation is concretely analyzed from the limit of double-shock-solution as well as the emergence of vacuum state is also specifically discussed from the limit of double-rarefaction-solution when the coupling coefficient drops to zero. Accordingly, the remarkable cavitation and concentration phenomena can be closely observed and explored. Additionally, the numerical experiments are also presented in correspondence to authenticate the theoretical analysis results.

Double-copy supertranslations

In the framework of the convolutional double copy, we investigate the asymptotic symmetries of the gravitational multiplet stemming from the residual symmetries of its single-copy constituents at null infinity. We show that the asymptotic symmetries of Maxwell fields in D=4 imply “double-copy supertranslations”, i.e., BMS supertranslations and two-form asymptotic symmetries, together with the existence of infinitely many conserved charges involving the double-copy scalar. With the vector fields in Lorenz gauge, the double-copy parameters display a radial expansion involving logarithmic subleading terms, essential for the corresponding charges to be nonvanishing. Published by the American Physical Society 2024

The asymptotic behavior of Riemann solutions for the Aw-Rascle-Zhang traffic flow model with the polytropic and logarithmic combined pressure term

The exact Riemann solutions for a specially designated Aw-Rascle-Zhang traffic flow model are explicitly obtained by using the combination of 1-rarefaction or 1-shock wave with 2-contact discontinuity. The vacuum and delta shock formation is carefully identified and analyzed in the Riemann solutions when both the two perturbed parameters in the combined pressure term drop to zero simultaneously, where the intrinsic cavitation and concentration phenomena are concretely surveyed and explored. Additionally, our theoretical analysis results are also validated through several representative numerical simulations.

Convergence and stability analysis of energy stable and bound‐preserving numerical schemes for binary fluid‐surfactant phase‐field equations

AbstractIn this article, we develop stable and efficient numerical schemes for a binary fluid‐surfactant phase‐field model which consists of two Cahn–Hilliard type equations with respect to the free energy containing a Ginzburg–Landau double‐well potential, a logarithmic Flory–Huggins potential and a nonlinear coupling entropy. The numerical schemes, which are decoupled and linear, are established by the central difference spatial approximation in combination with the first‐ and second‐order exponential time differencing methods based on the convex splitting of the free energy. For the sake of the linearity of the schemes, the nonlinear terms, especially the logarithmic term, are approximated explicitly, which requires the bound preservation of the numerical solution to make the algorithm robust. We conduct the convergence analysis and prove the bound‐preserving property in details for both first‐ and second‐order schemes, where the high‐order consistency analysis is applied to the first‐order case. In addition, the energy stability is also obtained by the nature of the convex splitting. Numerical experiments are performed to verify the accuracy and stability of the schemes and simulate the dynamics of phase separation and surfactant adsorption.

Symmetry resolution in non-Lorentzian field theories

Starting from the computation of Symmetry Resolved Entanglement Entropy (SREE) for boosted intervals in a two dimensional Conformal Field Theory, we compute the same in various non-Lorentzian limits, viz, Galilean and Carrollian Conformal Field Theory in same number of dimensions. We approach the problem both from a limiting perspective and by using intrinsic symmetries of respective non-Lorentzian conformal algebras. In particular, we calculate the leading order terms, logarithmic terms, and the O1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}(1) $$\\end{document} terms and explicitly show exact compliance with equipartition of entanglement, even in the non-Lorentzian system. Keeping in mind the holographic origin of SREE for the Carrollian limit, we further compute SREE for BMS3-Kac-Moody, which couples a U(1) × U(1) theory with bulk gravity.

Entanglement entropy and the boundary action of edge modes

We consider an antisymmetric gauge field in the Minkowski space of d-dimension and decompose it in terms of the antisymmetric tensor harmonics and fix the gauge. The Gauss law implies that the normal component of the field strength on the spherical entangling surface will label the superselection sectors. From the two-point function of the field strength on the sphere, we evaluate the logarithmic divergent term of the entanglement entropy of edge modes of p-form field. We observe that the logarithmic divergent term in entanglement entropy of edge modes coincides with the edge partition function of co-exact p-form on the sphere when expressed in terms of the Harish-Chandra characters. We also develop a boundary path integral of the antisymmetric p-form gauge field. From the boundary path integral, we show that the edge mode partition function corresponds to the co-exact (p − 1)-forms on the boundary. This boundary path integral agrees with the direct evaluation of the entanglement entropy of edge modes extracted from the two-point function of the normal component of the field strength on the entangling surface.

Deflection of light by a Reissner–Nordström black hole and Painlevé VI equation

We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner–Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard–Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.