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This work demonstrates that repeated weak measurements together with post-selection can produce sharp dynamical discontinuities in meter observables, even in minimal quantum systems. The discontinuous behavior is governed by the polar angle…
Necessary conditions for high-order optimality in smooth nonlinear constrained optimization are explored and their inherent intricacy discussed. A two-phase minimization algorithm is proposed which can achieve approximate first-, second-…
Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree…
In this article, we construct semiparametrically efficient estimators of linear functionals of a probability measure in the presence of side information using an easy empirical likelihood approach. We use estimated constraint functions and…
The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of…
To predict allowable time-step size for the fully discretized nonlinear differential equations, a stability theory is developed using exact determination of an infinite perturbation series. Mathematical induction is used to determine the…
We develop a higher-order asymptotic analysis for the semi-hard triplet loss using the Edgeworth expansion. It is known that this loss function enforces that embeddings of similar samples are close while those of dissimilar samples are…
We consider Dirichlet Laplacians on straight strips in R^2 or layers in R^3 with a weak local deformation. First we generalize a result of Bulla et al. to the three-dimensional situation showing that weakly coupled bound states exist if the…
We introduce a simple model of deterministic particles in weakly disordered media which exhibits a transition from normal to anomalous diffusion. The model consists of a set of non-interacting overdamped particles moving on a disordered…
We propose a simple alternative proof of a famous result of Gallay regarding the nonlinear asymptotic stability of the critical front of the Fisher-KPP equation which shows that perturbations of the critical front decay algebraically with…
Discontinuity with respect to data perturbations is common in algebraic computation where solutions are often highly sensitive. Such problems can be modeled as solving systems of equations at given data parameters. By appending auxiliary…
Let $A=A^*$ be a linear operator in a Hilbert space $H$. Assume that equation $Au=f \quad (1)$ is solvable, not necessarily uniquely, and $y$ is its minimal-norm solution. Assume that problem (1) is ill-posed. Let $f_\d$, $||f-f_d||\leq…
Modern science increasingly relies on ever-growing observational datasets and automated inference pipelines, under the implicit belief that accumulating more data makes scientific conclusions more reliable. Here we show that this belief can…
Approximating nonlinear dynamics with a truncated perturbative expan- sion may be accurate for a while, but it in general breaks down at a long time scale that is one over the small expansion parameter. There are interesting occasions in…
In many applications, one needs to learn a dynamical system from its solutions sampled at a finite number of time points. The learning problem is often formulated as an optimization problem over a chosen function class. However, in the…
We consider compact composite linear operators in Hilbert space, where the composition is given by some compact operator followed by some non-compact one possessing a non-closed range. Focus is on the impact of the non-compact factor on the…
Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble, whereas canonical ones fail in the most interesting, mostly inhomogeneous, situations like phase separations or away from the thermodynamic…
Configuration interaction in many-electron atoms may cause anomalies in the fine structure which make the intervals small and very sensitive to variation of the fine structure constant. Repeated precision measurements of these intervals…
When a corrosive solution reaches the limits of a solid sample, a chemical fracture occurs. An analytical theory for the probability of this chemical fracture is proposed and confirmed by extensive numerical experiments on a two dimensional…
We study a variant of the Strang splitting for the time integration of the semilinear wave equation under the finite-energy condition on the torus $\mathbb{T}^3$. In the case of a cubic nonlinearity, we show almost second-order convergence…