Asymptotic Assumptions Research Articles

Continuous-variable quantum key distribution with noisy squeezed states

Abstract We address the crucial role of noisy squeezing in security and performance of
continuous-variable (CV) quantum key distribution (QKD) protocols. Squeezing has
long been recognized for its numerous advantages in CV QKD, such as enhanced
robustness against channel noise and loss, and improved secret key rates. However, the
excess noise of the squeezed states, that unavoidably originates already from optical
loss as well as other imperfections in the source, raises concerns about its potential
exploitation by an eavesdropper. For the widely adopted trust assumption on the excess
noise in the signal states, we confirm the stability of the protocol against the noisy
squeezing in both purely attenuating as well as noisy channels in the asymptotic limit,
which implies perfect parameter estimation. In the finite-size regime we show that
this stability largely holds at up to 107 data points using optimal biased homodyne
detection for key distribution and parameter estimation. Untrusted assumption on
the noisy squeezing, on the other hand, introduces additional security bounds on the
squeezing excess noise already in the asymptotic regime, which is further enforced
by the finite-size effects. Additionally, we show the critical negative role of noisy
squeezing in the case of atmospheric free-space channels, already in the trusted-noise
and asymptotic assumptions, which emphasizes the importance of squeezing purity
in the free-space quantum channels. Our results pave the way towards practical
realization of squeezed-state CV QKD protocols in both fibre and free-space channels.

On the turnpike to design of deep neural networks: Explicit depth bounds

It is well-known that the training of Deep Neural Networks (DNN) can be formalized in the language of optimal control. In this context, this paper leverages classical turnpike properties of optimal control problems to attempt a quantifiable answer to the question of how many layers should be considered in a DNN. The underlying assumption is that the number of neurons per layer—i.e., the width of the DNN—is kept constant. Pursuing a different route than the classical analysis of approximation properties of sigmoidal functions, we prove explicit bounds on the required depths of DNNs based on asymptotic reachability assumptions and a dissipativity-inducing choice of the regularization terms in the training problem. Numerical results obtained for the two spiral task data set for classification indicate that the proposed constructive estimates can provide non-conservative depth bounds.

Metric compatibility and determination in complete metric spaces

The local slope operator was introduced in De Giorgi et al. (Atti Accad Naz Lincei Rend Cl Sci Fis Mat Nat 68:180–187, 1980) to study gradient flow dynamics in metric spaces. This tool has now become a cornerstone in metric evolution equations (see, e.g., Ambrosio et al. (Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics. Birkhauser, 2008). Very recently, in Daniilidis and Salas (Proc Am Math Soc 150:4325–4333, 2022), it was established that Lipschitz inf-compact functions are uniquely determined by their local slope and critical values. Compactness played a paramount role in this result, ensuring in particular the existence of critical points. We hereby emancipate from this restriction and establish a determination result for merely bounded from below functions, by adding an assumption controlling the asymptotic behavior. This assumption is trivially fulfilled if f is inf-compact. In addition, our result is not only valid for the (De Giorgi) local slope, but also for the main paradigms of average descent operators as well as for the global slope, case in which the asymptotic assumption becomes superfluous. Therefore, the present work extends simultaneously the metric determination results of Daniilidis and Salas (Proc Am Math Soc 150:4325–4333, 2022) and Thibault and Zagrodny (Commun Contemp Math 25(7):2250014, 2023).

Singularity analysis for V‐notches in a piezoelectric multimaterial and functionally graded piezoelectric materials

AbstractThis study develops a novel singularity analysis method that addresses the limitations posed by piezoelectric material quantity and the variation patterns of property functions in functionally graded piezoelectric materials (FGPMs). Given that piezoelectric composite materials consist of multiple materials, the material properties of FGPMs vary with respect to angle. By introducing the asymptotic assumption that governs the physical fields close to the notch vertex into the static equilibrium equations, a comprehensive set of characteristic equations is formulated. All singularity orders and corresponding characteristic angle functions of the notch are determined by numerically solving the established characteristic equations. The effects of the notch opening angle, piezoelectric material polarization direction, and boundary conditions on the electromechanical field singularity at the notch are assessed. The judicious selection of material variation patterns can alleviate notch singularity in FGPMs.

Covariate-adjusted generalized pairwise comparisons in small samples.

Semiparametric probabilistic index models allow for the comparison of two groups of observations, whilst adjusting for covariates, thereby fitting nicely within the framework of generalized pairwise comparisons (GPC). As with most regression approaches in this setting, the limited amount of data results in invalid inference as the asymptotic normality assumption is not met. In addition, separation issues might arise when considering small samples. In this article, we show that the parameters of the probabilistic index model can be estimated using generalized estimating equations, for which adjustments exist that lead to estimators of the sandwich variance-covariance matrix with improved finite sample properties and that can deal with bias due to separation. In this way, appropriate inference can be performed as is shown through extensive simulation studies. The known relationships between the probabilistic index and other GPC statistics allow to also provide valid inference for example, the net treatment benefit or the success odds.

Statistical considerations in model-based dose finding for binary responses under model uncertainty.

The statistical methodology for model-based dose finding under model uncertainty has attracted increasing attention in recent years. While the underlying principles are simple and easy to understand, developing and implementing an efficient approach for binary responses can be a formidable task in practice. Motivated by the statistical challenges encountered in a phase II dose finding study, we explore several key design and analysis issues related to the hybrid testing-modeling approaches for binary responses. The issues include candidate model selection and specifications, optimal design and efficient sample size allocations, and, notably, the methods for dose-response testing and estimation. Specifically, we consider a class of generalized linear models suited for the candidate set and establish D-optimal designs for these models. Additionally, we propose using permutation-based tests for dose-response testing to avoid asymptotic normality assumptions typically required for contrast-based tests. We perform trial simulations to enhance our understanding of these issues.

Individualized empirical null estimation for exact tests of healthcare quality.

United States federal agencies evaluate healthcare providers to identify, flag, and potentially penalize those that deliver low-quality care compared to national expectations. In practice, evaluation metrics are inevitably impacted by unobserved confounding factors, which reduce flagging accuracy and cause the statistics to be overdispersed relative to the theoretical null distributions. In response to this issue, several authors have proposed individualized empirical null (IEN) methods to estimate an appropriate null distribution for each provider's evaluation statistic while taking into account the provider's effective size. However, existing IEN methods require that the statistics asymptotically follow normal distributions, which often does not hold in applications with small providers or misspecified models. In this article, we develop an IEN framework for exact hypothesis tests that accounts for the impact of unobserved confounding without making any asymptotic assumptions. Simulations show that the proposed IEN method has greater flagging accuracy compared to conventional approaches. We apply these methods to evaluate dialysis facilities and transplant centers that are monitored by the Centers for Medicare and Medicaid Services.

Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture

AbstractIn this paper, we consider the general dual fractional parabolic problem∂tαu(x,t)+Lu(x,t)=f(t,u(x,t))inRn×R.${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$We show that the bounded entire solutionusatisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition onf. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operatorL$\mathcal{L}$and the Marchaud time derivative∂tα${\partial }_{t}^{\alpha }$, we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operatorL$\mathcal{L}$, which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case ofL$\mathcal{L}$, the fractional Laplacian (−Δ)s, and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.

Correction to: Necessary Conditions in Infinite-Horizon Control Problems That Need No Asymptotic Assumptions

Correction to: Necessary Conditions in Infinite-Horizon Control Problems That Need No Asymptotic Assumptions

Modified DFA for a robust discrimination between short-term and long-range correlations in short time series

Since the standard detrended fluctuation analysis (s-DFA) method depends on asymptotic power-law scaling assumption, it has been questioned on its ability in characterizing the correlation structure at the small scales, let alone discriminating the correlation structures of short time series. A modified DFA (m-DFA) with a novel defined fluctuation function is employed in this study to discriminate and characterize the distinct (short-termed or long-ranged) correlation structures in short time series. Detailed results show that the m-DFA is able not only to distinguish the short-termed, or long-ranged correlation, but also to well quantify the correlation strengths in both output from classical models and real-world series (for example, DTR variability over China) of data length as short as 2000. Moreover, m-DFA can work effectively in time series with strong correlation of even shorter length as 500. The correlation structures inferred by m-DFA in short intervals contribute greatly to better understanding of local and evolutionary correlation features of an underlying process, not limited to only global one from s-DFA.

The geometry of [formula omitted]-harmonic maps

In this paper, we motivate and extend the study of harmonic maps or Φ(1)-harmonic maps (cf. Eells and Sampson, 1964, Remark 1.3 (iii)), Φ-harmonic maps or Φ(2)-harmonic maps (cf. Han and Wei, 2022, Remark 1.3 (v)), and explore geometric properties of Φ(3)-harmonic maps by unified geometric analytic methods. We define the notion of Φ(3)-harmonic maps and obtain the first variation formula and the second variation formula of the Φ(3)-energy functional EΦ(3). By using a stress–energy tensor and the asymptotic assumption of maps at infinity, we prove Liouville type results for Φ(3)-harmonic maps. We introduce the notion of Φ(3)-Superstrongly Unstable (Φ(3)-SSU) manifold and provide many interesting examples. By using an extrinsic average variational method in the calculus of variations (cf. Wei, 1998; Wei, 1984), we find Φ(3)-SSU manifold and prove that any stable Φ(3)-harmonic maps from a compact Φ(3)-SSU manifold (into any compact Riemannian manifold) or (from any compact Riemannian manifold) into a compact Φ(3)-SSU manifold must be constant. We also prove that the homotopy class of any map from a compact Φ(3)-SSU manifold (into any compact Riemannian manifold) or (from any compact Riemannian manifold) into a compact Φ(3)-SSU manifold contains elements of arbitrarily small Φ(3)-energy. We call a compact Riemannian manifold M to be Φ(3)-strongly unstable (Φ(3)-SU) if it is not the target or domain of a nonconstant stable Φ(3)-harmonic map (from or into any compact Riemannian manifold) and also the homotopy class of any map to or from M (from or into any compact Riemannian manifold) contains elements of arbitrarily small Φ(3)-energy. We prove that every compact Φ(3)-SSU manifold is Φ(3)-SU. As consequences, we obtain topological vanishing theorems and sphere theorems by employing a Φ(3)-harmonic map as a catalyst. This is in contrast to the approaches of utilizing a geodesic (Synge, 1936), minimal surface, stable rectifiable current (Lawson Jr. and Simons, 1973; Howard and Wei, 2015; Wei, 1984), p-harmonic map (cf. Wei, 1998), etc., as catalysts. These mysterious phenomena are analogs of harmonic maps or Φ(1)-harmonic maps, p-harmonic maps, ΦS-harmonic maps, ΦS,p-harmonic maps, Φ(2)-harmonic maps, etc., (cf. Howard and Wei, 1986; Wei, 1992; Wei and Yau, 1994; Wei, 1998; Feng-Han-Li-Wei, 2021; Feng-Han-Wei, 2021).

A hierarchical Bayesian non‐asymptotic extreme value model for spatial data

AbstractSpatial maps of extreme precipitation are crucial in flood prevention. With the aim of producing maps of precipitation return levels, we propose a novel approach to model a collection of spatially distributed time series where the asymptotic assumption, typical of the traditional extreme value theory, is relaxed. We introduce a Bayesian hierarchical model that accounts for the possible underlying variability in the distribution of event magnitudes and occurrences, which are described through latent temporal and spatial processes. Spatial dependence is characterized by geographical covariates and effects not fully described by the covariates are captured by spatial structure in the hierarchies. The performance of the approach is illustrated through simulation studies and an application to daily rainfall extremes across North Carolina (USA). The results show that we significantly reduce the estimation uncertainty with respect to state of the art techniques.

On the asymptotic assumptions for Milne-like spacetimes

Milne-like spacetimes are a class of hyperboloidal FLRW spacetimes which admit continuous spacetime extensions through the big bang, $$\tau = 0$$ . In a previous paper [27], it was advocated that the existence of this big bang extension could have applications to fundamental problems in cosmology, which illustrates the physical importance of such extensions. By definition, the scale factor for a Milne-like spacetime satisfies the past asymptotic assumption $$a(\tau ) = \tau + o(\tau ^{1+\varepsilon })$$ as $$\tau \rightarrow 0$$ for some $$\varepsilon > 0$$ . The existence of the big bang extension follows from writing the metric in conformal Minkowskian coordinates and using the past asymptotic assumption of the scale factor. This asymptotic assumption implies $$a(\tau ) = \tau + o(\tau )$$ as $$\tau \rightarrow 0$$ . In this paper, we show that $$a(\tau ) = \tau + o(\tau )$$ is not sufficient to achieve a big bang extension, but it is necessary (provided its derivative converges as $$\tau \rightarrow 0$$ ). We also show that the $$\varepsilon $$ in $$a(\tau ) = \tau + o(\tau ^{1+\varepsilon })$$ is not necessary to achieve a big bang extension.

Necessary Conditions in Infinite-Horizon Control Problems that Need no Asymptotic Assumptions

Necessary Conditions in Infinite-Horizon Control Problems that Need no Asymptotic Assumptions

Spatial heterogeneity localizes Turing patterns in reaction-cross-diffusion systems

Motivated by bacterial chemotaxis and multi-species ecological interactions in heterogeneous environments, we study a general one-dimensional reaction-cross-diffusion system in the presence of spatial heterogeneity in both transport and reaction terms. Under a suitable asymptotic assumption that the transport is slow over the domain, while gradients in the reaction heterogeneity are not too sharp, we study the stability of a heterogeneous steady state approximated by the system in the absence of transport. Using a WKB ansatz, we find that this steady state can undergo a Turing-type instability in subsets of the domain, leading to the formation of localized patterns. The boundaries of the pattern-forming regions are given asymptotically by 'local' Turing conditions corresponding to a spatially homogeneous analysis parameterized by the spatial variable. We developed a general open-source code which is freely available, and show numerical examples of this localized pattern formation in a Schnakenberg cross-diffusion system, a Keller-Segel chemotaxis model, and the Shigesada-Kawasaki-Teramoto model with heterogeneous parameters. We numerically show that the patterns may undergo secondary instabilities leading to spatiotemporal movement of spikes, though these remain approximately within the asymptotically predicted localized regions. This theory can elegantly differentiate between spatial structure due to background heterogeneity, from spatial patterns emergent from Turing-type instabilities.

Triangular code: Near-optimal linear time fountain code

In this paper, we propose Triangular Code (TC), a new class of fountain code with near-zero redundancy and linear encoding and decoding computational complexities of O(Lklogk), where k is the packet batch size and L is the packet data length. Different from previous works where the optimal performance of codes has been shown under asymptotic assumption, TC enjoys near-zero redundancy even under non-asymptotic settings for small-moderate number of packets. These features make TC suitable for practical implementation in battery-constrained devices in IoT, D2D and M2M network paradigms to achieve scalable reliability, and minimize latency due to its low decoding delay. TC is a non-linear code, which is encoded using the simple shift and XOR addition operations, and decoded using the simple back-substitution algorithm. Although it is nonlinear code at the packet level, it remains linear code when atomized at the bit level. We use this property to show that the back-substitution decoder of TC is equivalent to the Belief Propagation (BP) decoder of LT code. Therefore, TC can benefit from rich prolific literature published on LT code, to design efficient code for various applications. Despite the equivalency between the decoders of TC and LT code, we show that compared to state-of-the-art optimized LT code, TC reduces the redundancy of LT code by 68%–99% for k reaching 1024.© 2022 Published by Elsevier Ltd.

Higher order asymptotic refinements in Bell regressions

The discrete Bell distribution and its associated regression model were introduced recently in the statistical literature. The Bell distribution has proved to be a useful alternative to the traditional Poisson distribution, mainly to deal with overdispersion. Likelihood‐based inference on the Bell regression parameters relies on asymptotic assumptions like the sample size going to infinity. In this paper, we focus on the small‐sample case and consider higher order asymptotic refinements in this class of regression models. In particular, we derive the second‐order biases of the maximum likelihood estimators, which are used to define bias‐corrected estimators. The preventive method to bias reducing is also considered. We provide a simple matrix formula for the skewness of the distributions of the maximum likelihood estimators, and also derive a simple matrix formula for the second‐order variance–covariance matrix of the maximum likelihood estimators in this class of regression models. We use Monte Carlo simulations to verify the performance of the proposed methods. Our simulation results suggest that the analytical expressions we derive deliver more reliable results in small‐sized samples. An empirical application to a real data set is considered for illustrative purposes.

Asymptotic Freeness of Layerwise Jacobians Caused by Invariance of Multilayer Perceptron: The Haar Orthogonal Case

Free Probability Theory (FPT) provides rich knowledge for handling mathematical difficulties caused by random matrices appearing in research related to deep neural networks (DNNs), such as the dynamical isometry, Fisher information matrix, and training dynamics. FPT suits these researches because the DNN’s parameter-Jacobian and input-Jacobian are polynomials of layerwise Jacobians. However, the critical assumption of asymptotic freeness of the layerwise Jacobian has not been proven mathematically so far. The asymptotic freeness assumption plays a fundamental role when propagating spectral distributions through the layers. Haar distributed orthogonal matrices are essential for achieving dynamical isometry. In this work, we prove asymptotic freeness of layerwise Jacobians of multilayer perceptron (MLP) in this case. A key to the proof is an invariance of the MLP. Considering the orthogonal matrices that fix the hidden units in each layer, we replace each layer’s parameter matrix with itself multiplied by the orthogonal matrix, and then the MLP does not change. Furthermore, if the original weights are Haar orthogonal, the Jacobian is also unchanged by this replacement. Lastly, we can replace each weight with a Haar orthogonal random matrix independent of the Jacobian of the activation function using this key fact.

Exact boundary controllability of 1D semilinear wave equations through a constructive approach

The exact controllability of the semilinear wave equation $$y_{tt}-y_{xx}+ f(y)=0$$ , $$x\in (0,1)$$ assuming that f is locally Lipschitz continuous and satisfies the growth condition $$\limsup _{\vert r\vert \rightarrow \infty } \vert f(r)\vert /(\vert r\vert \ln ^{p}\vert r\vert )\leqslant \beta $$ for some $$\beta $$ small enough and $$p=2$$ has been obtained by Zuazua (Ann Inst H Poincaré Anal Non Linéaire 10(1):109–129, 1993). The proof based on a non-constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized wave equation. Under the above asymptotic assumption with $$p=3/2$$ , by introducing a different fixed point application, we present a simpler proof of the exact boundary controllability which is not based on the cost of observability of the wave equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition $$\limsup _{\vert r\vert \rightarrow \infty } \vert f^\prime (r)\vert /\ln ^{3/2}\vert r\vert \leqslant \beta $$ for some $$\beta $$ small enough, we show that the above fixed point application is contracting yielding a constructive method to approximate the controls for the semilinear equation. Numerical experiments illustrate the results. The results can be extended to the multi-dimensional case and for nonlinearities involving the gradient of the solution.

Asymptotic properties of generalized D-solutions to the stationary axially symmetric Navier–Stokes equations

In this paper, we derive asymptotic properties of both the velocity and the vorticity fields to the 3-dimensional axially symmetric Navier–Stokes equations at infinity under the generalized D-solution assumption [Formula: see text] for [Formula: see text]. We do not impose any zero or non-zero constant vector asymptotic assumption to the solution at infinity. Our results generalize those in [H. Choe and B. Jin, Asymptotic properties of axis-symmetric D-solutions of the Navier–Stokes equations, J. Math. Fluid Mech. 11(2) (2009) 208–232; S. Weng, Decay properties of axially symmetric D-solutions to the steady Navier–Stokes equations, J. Math. Fluid Mech. 20(1) (2018) 7–25; B. Carrillo, X. Pan and Q. S. Zhang, Decay and vanishing of some axially symmetric D-solutions of the Navier–Stokes equations, J. Funct. Anal. 279(1) (2020) 108504], where the authors focused on the case [Formula: see text] and the velocity field approaches zero at infinity. Meanwhile, when [Formula: see text] and the velocity field approaches zero at infinity, our results coincide with the results in [H. Choe and B. Jin, Asymptotic properties of axis-symmetric D-solutions of the Navier–Stokes equations, J. Math. Fluid Mech. 11(2) (2009) 208–232; S. Weng, Decay properties of axially symmetric D-solutions to the steady Navier–Stokes equations, J. Math. Fluid Mech. 20(1) (2018) 7–25; B. Carrillo, X. Pan and Q. S. Zhang, Decay and vanishing of some axially symmetric D-solutions of the Navier–Stokes equations, J. Funct. Anal. 279(1) (2020) 108504].