Discretization strategies for computing Conley indices and Morse decompositions of flows

Conley indices and Morse decompositions of flows can be found by using algorithms which rigorously analyze discrete dynamical systems. This usually involves integrating a time discretization of the flow using interval arithmetic. We compare the old idea of fixing a time step as a parameters to a time step continuously varying in phase space. We present an example where this second strategy necessarily yields better numerical outputs and prove that our outputs yield a valid Morse decomposition of the given flow

Equivalence of Sobolev norms involving generalized Hardy operators

We consider the fractional Schr\"odinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schr\"odinger operators and have an important application in the treatment of large relativistic atoms.Comment: 16 pages; v2 contains improved results for positive coupling constant

Nonlinear optical signals and spectroscopy with quantum light

Conventional nonlinear spectroscopy uses classical light to detect matter properties through the variation of its response with frequencies or time delays. Quantum light opens up new avenues for spectroscopy by utilizing parameters of the quantum state of light as novel control knobs and through the variation of photon statistics by coupling to matter. We present an intuitive diagrammatic approach for calculating ultrafast spectroscopy signals induced by quantum light, focusing on applications involving entangled photons with nonclassical bandwidth properties - known as "time-energy entanglement". Nonlinear optical signals induced by quantized light fields are expressed using time ordered multipoint correlation functions of superoperators. These are different from Glauber's g- functions for photon counting which use normally ordered products of ordinary operators. Entangled photon pairs are not subjected to the classical Fourier limitations on the joint temporal and spectral resolution. After a brief survey of properties of entangled photon pairs relevant to their spectroscopic applications, different optical signals, and photon counting setups are discussed and illustrated for simple multi-level model systems

CUP: Comprehensive User-Space Protection for C/C++

Memory corruption vulnerabilities in C/C++ applications enable attackers to execute code, change data, and leak information. Current memory sanitizers do no provide comprehensive coverage of a program's data. In particular, existing tools focus primarily on heap allocations with limited support for stack allocations and globals. Additionally, existing tools focus on the main executable with limited support for system libraries. Further, they suffer from both false positives and false negatives. We present Comprehensive User-Space Protection for C/C++, CUP, an LLVM sanitizer that provides complete spatial and probabilistic temporal memory safety for C/C++ program on 64-bit architectures (with a prototype implementation for x86_64). CUP uses a hybrid metadata scheme that supports all program data including globals, heap, or stack and maintains the ABI. Compared to existing approaches with the NIST Juliet test suite, CUP reduces false negatives by 10x (0.1%) compared to the state of the art LLVM sanitizers, and produces no false positives. CUP instruments all user-space code, including libc and other system libraries, removing them from the trusted code base

Relativistic Strong Scott Conjecture: A Short Proof

We consider heavy neutral atoms of atomic number $Z$ modeled with kinetic energy $(c^2p^2+c^4)^{1/2}-c^2$ used already by Chandrasekhar. We study the behavior of the one-particle ground state density on the length scale $Z^{-1}$ in the limit $Z,c\to\infty$ keeping $Z/c$ fixed. We give a short proof of a recent result by the authors and Barry Simon showing the convergence of the density to the relativistic hydrogenic density on this scale

The Scott conjecture for large Coulomb systems: a review

We review some older and more recent results concerning the energy and particle distribution in ground states of heavy Coulomb systems. The reviewed results are asymptotic in nature: they describe properties of many-particle systems in the limit of a large number of particles. Particular emphasis is put on models that take relativistic kinematics into account. While non-relativistic models are typically rather well understood, this is generally not the case for relativistic ones and leads to a variety of open questions.Comment: 62 page

Reproducibility of Novel Soft-Tissue Landmarks on Three-Dimensional Human Facial Scan Images in Caucasian and Asian

BACKGROUND Three-dimensional surface imaging is established in many disciplines for objective facial acquisition regarding anthropometry. Former studies addressed the validation of landmark-based measurements for single race. In order to distinguish racial difference, the reproducibility of the landmark measurements must first be validated. OBJECTIVES Our purpose is to validate the reproducibility of 46 facial soft-tissue landmarks on x, y, z axes to prove their reliability as 3D reference points. METHODS The study included 80 European Caucasian and 80 Chinese volunteers. Standardized 3D surface imaging was performed using Vectra 3D system. Two raters identified and defined 46 landmarks (138 coordinates), then repeatedly 3D-imaged volunteers' facial region in separate sessions. Coordinates' reproducibility of landmarks is divided into three categories (1 mm) for intra- and inter-rater reproducibility assessments. RESULTS Coordinates' reproducibility of 160 samples was distributed as follows: Intra-rater: 1 mm (13%); inter-rater: 1 mm (26.8%). The reproducibility of landmarks in nasal tip region differs slightly between Caucasians and Asians. Compared to females, males typically have higher landmark reproducibility in lip and chin region. However, there were no differences in the reproducibility ranking of landmarks by gender. CONCLUSION The majority of the 46 landmarks in the 3D plane are reproducible to 1 mm, which is clinically acceptable. All selected landmarks showed strong consistency across race and gender, suggesting their potential use as reference points in prospective clinical practice. LEVEL OF EVIDENCE IV This journal requires that authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors www.springer.com/00266