Solutions for certain classes of Riccati differential equation

We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are C^{\infty}-functions. We show that if \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1} and s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1}, n=1,2,..., then The Riccati equation has a solution given by y(x)=\mp s_{n-1}(x)/\lambda_{n-1}(x). Extension to the generalized Riccati equation y'(x)+P(x)y(x)+Q(x)y^2(x)=R(x) is also investigated.Comment: 10 page

Physical applications of second-order linear differential equations that admit polynomial solutions

Conditions are given for the second-order linear differential equation P3 y" + P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of degree n. Several application of these results to Schroedinger's equation are discussed. Conditions under which the confluent, biconfluent, and the general Heun equation yield polynomial solutions are explicitly given. Some new classes of exactly solvable differential equation are also discussed. The results of this work are expressed in such way as to allow direct use, without preliminary analysis.Comment: 13 pages, no figure

Construction of exact solutions to eigenvalue problems by the asymptotic iteration method

We apply the asymptotic iteration method (AIM) [J. Phys. A: Math. Gen. 36, 11807 (2003)] to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue problems which includes Schroedinger problems with Coulomb, harmonic oscillator, or Poeschl-Teller potentials, as well as the special eigenproblems studied recently by Bender et al [J. Phys. A: Math. Gen. 34 9835 (2001)] and generalized in the present paper to higher dimensions.Comment: 10 page

Benford's law: what does it say on adversarial images?

Convolutional neural networks (CNNs) are fragile to small perturbations in the input images. These networks are thus prone to malicious attacks that perturb the inputs to force a misclassification. Such slightly manipulated images aimed at deceiving the classifier are known as adversarial images. In this work, we investigate statistical differences between natural images and adversarial ones. More precisely, we show that employing a proper image transformation and for a class of adversarial attacks, the distribution of the leading digit of the pixels in adversarial images deviates from Benford's law. The stronger the attack, the more distant the resulting distribution is from Benford's law. Our analysis provides a detailed investigation of this new approach that can serve as a basis for alternative adversarial example detection methods that do not need to modify the original CNN classifier neither work on the raw high-dimensional pixels as features to defend against attacks

Globally optimal parameters for on-line learning in multilayer neural networks

We present a framework for calculating globally optimal parameters, within a given time frame, for on-line learning in multilayer neural networks. We demonstrate the capability of this method by computing optimal learning rates in typical learning scenarios. A similar treatment allows one to determine the relevance of related training algorithms based on modifications to the basic gradient descent rule as well as to compare different training methods

On the Prospects for Laser Cooling of TlF

We measure the upper state lifetime and two ratios of vibrational branching fractions f_{v'v} on the B^{3}\Pi_{1}(v') - X^{1}\Sigma^{+}(v) transition of TlF. We find the B state lifetime to be 99(9) ns. We also determine that the off-diagonal vibrational decays are highly suppressed: f_{01}/f_{00} < 2x10^{-4} and f_{02}/f_{00} = 1.10(6)%, in excellent agreement with their predicted values of f_{01}/f_{00} < 8x10^{-4} and f_{02}/f_{00} = 1.0(2)% based on Franck-Condon factors calculated using Morse and RKR potentials. The implications of these results for the possible laser cooling of TlF and fundamental symmetries experiments are discussed.Comment: 5 pages, 2 figure

Solitary coherent structures in viscoelastic shear flow: computation and mechanism

Starting from stationary bifurcations in Couette-Dean flow, we compute nontrivial stationary solutions in inertialess viscoelastic circular Couette flow. These solutions are strongly localized vortex pairs, exist at arbitrarily large wavelengths, and show hysteresis in the Weissenberg number, similar to experimentally observed ``diwhirl'' patterns. Based on the computed velocity and stress fields, we elucidate a heuristic, fully nonlinear mechanism for these flows. We propose that these localized, fully nonlinear structures comprise fundamental building blocks for complex spatiotemporal dynamics in the flow of elastic liquids.Comment: 5 pages text and 4 figures. Submitted to Physical Review Letter

The role of biases in on-line learning of two-layer networks

The influence of biases on the learning dynamics of a two-layer neural network, a normalized soft-committee machine, is studied for on-line gradient descent learning. Within a statistical mechanics framework, numerical studies show that the inclusion of adjustable biases dramatically alters the learning dynamics found previously. The symmetric phase which has often been predominant in the original model all but disappears for a non-degenerate bias task. The extended model furthermore exhibits a much richer dynamical behavior, e.g. attractive suboptimal symmetric phases even for realizable cases and noiseless data

Radiation arteritis: A contraindication to carotid stenting?

BackgroundCarotid artery stenting (CAS) for high-risk anatomic lesions is accepted practice. Neck irradiation and radiotherapy-induced arteritis are common indications. The clinical outcomes of CAS for radiation arteritis have been poorly defined.MethodsA prospective database of patients undergoing CAS at a tertiary referral academic medical center was maintained from 1999 to 2006. Patients undergoing primary carotid artery stenting for significant atherosclerotic (ASOD) and radiotherapy (XRT)-induced occlusive disease were analyzed. Life-table analyses were performed to assess time-dependent outcomes. Cox proportional hazard analysis or Fisher’s exact test was performed to identify factors associated with outcomes. Data are presented as the mean ± SEM unless otherwise indicated.ResultsDuring the study period, 150 patients underwent primary CAS, 75% with embolic protection. Fifty-eight percent were symptomatic. One hundred twenty-seven (85%) were treated for ASOD, and 23 (15%) had XRT. The 30-day all-cause mortality rate was 1% for ASOD and 0% for XRT (P = NS); overall survival at 3 years was equivalent. There was no significant difference in major adverse event rates as defined by the Stenting and Angioplasty with Protection in Patients at High Risk for Endarterectomy (SAPPHIRE) trial between the groups. The 3-year neurologic event-free rate was 85% for ASOD and 87% for XRT (P = NS). Late asymptomatic occlusions were seen only in XRT patients. The 3-year freedom from restenosis rate was significantly worse for the XRT group, at 20%, vs 74% for the ASOD group (P < .05). Likewise, the 3-year patency rate was also worse for the XRT group, at 91%, vs 100% for ASOD by Kaplan-Meier analysis (P < .05). No factor was predictive of occlusion or stenosis by Cox proportional hazards analysis.ConclusionCAS for radiation arteritis has poor long-term anatomic outcome and can present with late asymptomatic occlusions. These findings suggest that these patients require closer postoperative surveillance and raise the question of whether CAS is appropriate for carotid occlusive lesions caused by radiation arteritis

Statistical Mechanics Analysis of LDPC Coding in MIMO Gaussian Channels

Using analytical methods of statistical mechanics, we analyse the typical behaviour of a multiple-input multiple-output (MIMO) Gaussian channel with binary inputs under LDPC network coding and joint decoding. The saddle point equations for the replica symmetric solution are found in particular realizations of this channel, including a small and large number of transmitters and receivers. In particular, we examine the cases of a single transmitter, a single receiver and the symmetric and asymmetric interference channels. Both dynamical and thermodynamical transitions from the ferromagnetic solution of perfect decoding to a non-ferromagnetic solution are identified for the cases considered, marking the practical and theoretical limits of the system under the current coding scheme. Numerical results are provided, showing the typical level of improvement/deterioration achieved with respect to the single transmitter/receiver result, for the various cases.Comment: 25 pages, 7 figure