On the theory of self-similar parabolic optical pulses

Self-similar optical pulses (or “similaritons”) of parabolic intensity profile can be found as asymptotic solutions of the nonlinear Schr¨odinger equation in a gain medium such as a fiber amplifier or laser resonator. These solutions represent a wide-ranging significance example of dissipative nonlinear structures in optics. Here, we address some issues related to the formation and evolution of parabolic pulses in a fiber gain medium by means of semi-analytic approaches. In particular, the effect of the third-order dispersion on the structure of the asymptotic solution is examined. Our analysis is based on the resolution of ordinary differential equations, which enable us to describe the main properties of the pulse propagation and structural characteristics observable through direct numerical simulations of the basic partial differential equation model with sufficient accuracy.

On the theory of self-similar parabolic optical pulses

Self-similar optical pulses (or “similaritons”) of parabolic intensity profile can be found as asymptotic solutions of the nonlinear Schr¨odinger equation in a gain medium such as a fiber amplifier or laser resonator. These solutions represent a wide-ranging significance example of dissipative nonlinear structures in optics. Here, we address some issues related to the formation and evolution of parabolic pulses in a fiber gain medium by means of semi-analytic approaches. In particular, the effect of the third-order dispersion on the structure of the asymptotic solution is examined. Our analysis is based on the resolution of ordinary differential equations, which enable us to describe the main properties of the pulse propagation and structural characteristics observable through direct numerical simulations of the basic partial differential equation model with sufficient accuracy.

On Parabolic Subgroups and Hecke Algebras of some Fractal Groups

* The authors thank the “Swiss National Science Foundation” for its support.

Performance of the Bayesian online algorithm for the perceptron

In this letter, we derive continuum equations for the generalization error of the Bayesian online algorithm (BOnA) for the one-layer perceptron with a spherical covariance matrix using the Rosenblatt potential and show, by numerical calculations, that the asymptotic performance of the algorithm is the same as the one for the optimal algorithm found by means of variational methods with the added advantage that the BOnA does not use any inaccessible information during learning. © 2007 IEEE.

Branching Processes with Immigration and Integer-valued Time Series

In this paper, we indicate how integer-valued autoregressive time series Ginar(d) of ordre d, d ≥ 1, are simple functionals of multitype branching processes with immigration. This allows the derivation of a simple criteria for the existence of a stationary distribution of the time series, thus proving and extending some results by Al-Osh and Alzaid [1], Du and Li [9] and Gauthier and Latour [11]. One can then transfer results on estimation in subcritical multitype branching processes to stationary Ginar(d) and get consistency and asymptotic normality for the corresponding estimators. The technique covers autoregressive moving average time series as well.

Time-spatial drift of decelerating electromagnetic pulses

A time dependent electromagnetic pulse generated by a current running laterally to the direction of the pulse propagation is considered in paraxial approximation. It is shown that the pulse envelope moves in the time-spatial coordinates on the surface of a parabolic cylinder for the Airy pulse and a hyperbolic cylinder for the Gaussian. These pulses propagate in time with deceleration along the dominant propagation direction and drift uniformly in the lateral direction. The Airy pulse stops at infinity while the asymptotic velocity of the Gaussian is nonzero. © 2013 Optical Society of America.

On Li’s Coefficients for Some Classes of L-Functions

AMS Subj. Classification: 11M41, 11M26, 11S40

Accuracy and optimal sampling in Monte Carlo solution of population balance equations

Implementation of a Monte Carlo simulation for the solution of population balance equations (PBEs) requires choice of initial sample number (N0), number of replicates (M), and number of bins for probability distribution reconstruction (n). It is found that Squared Hellinger Distance, H2, is a useful measurement of the accuracy of Monte Carlo (MC) simulation, and can be related directly to N0, M, and n. Asymptotic approximations of H2 are deduced and tested for both one-dimensional (1-D) and 2-D PBEs with coalescence. The central processing unit (CPU) cost, C, is found in a power-law relationship, C= aMNb0, with the CPU cost index, b, indicating the weighting of N0 in the total CPU cost. n must be chosen to balance accuracy and resolution. For fixed n, M × N0 determines the accuracy of MC prediction; if b > 1, then the optimal solution strategy uses multiple replications and small sample size. Conversely, if 0 < b < 1, one replicate and a large initial sample size is preferred. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2394–2402, 2015

Estimation of the Offspring and Immigration Mean Vectors for Bisexual Branching Processes with Immigration of Females and Males

2000 Mathematics Subject Classification: 60J80, 62M05.

Two-Type Age-Dependent Branching Processes with Inhomogeneous Immigration as Models of Renewing Cell Population

2000 Mathematics Subject Classification: primary 60J80; secondary 60J85, 92C37.

The Number of Parts of Given Multiplicity in a Random Integer Partition

2000 Mathematics Subject Classification: 05A16, 05A17.

Design of nonlinear regenerative transmission systems with high capacity

We present the design of nonlinear regenerative communication channels that have capacity above the classical Shannon capacity of the linear additive white Gaussian noise channel. The upper bound for regeneration efficiency is found and the asymptotic behavior of the capacity in the saturation regime is derived. © 2013 IEEE.

International stock portfolio selection and performance measures recognizing higher moments of return distributions

Since the seminal works of Markowitz (1952), Sharpe (1964), and Lintner (1965), numerous studies on portfolio selection and performance measure have been based upon the mean-variance framework. However, several researchers (e.g., Arditti (1967, and 1971), Samuelson (1970), and Rubinstein (1973)) argue that the higher moments cannot be neglected unless there is reason to believe that: (i) the asset returns are normally distributed and the investor's utility function is quadratic, or (ii) the empirical evidence demonstrates that higher moments are irrelevant to the investor's decision. Based on the same argument, this dissertation investigates the impact of higher moments of return distributions on three issues concerning the 14 international stock markets.^ First, the portfolio selection with skewness is determined using: the Polynomial Goal Programming in which investor preferences for skewness can be incorporated. The empirical findings suggest that the return distributions of international stock markets are not normally distributed, and that the incorporation of skewness into an investor's portfolio decision causes a major change in the construction of his optimal portfolio. The evidence also indicates that an investor will trade expected return of the portfolio for skewness. Moreover, when short sales are allowed, investors are better off as they attain higher expected return and skewness simultaneously.^ Second, the performance of international stock markets are evaluated using two types of performance measures: (i) the two-moment performance measures of Sharpe (1966), and Treynor (1965), and (ii) the higher-moment performance measures of Prakash and Bear (1986), and Stephens and Proffitt (1991). The empirical evidence indicates that higher moments of return distributions are significant and relevant to the investor's decision. Thus, the higher moment performance measures should be more appropriate to evaluate the performances of international stock markets. The evidence also indicates that various measures provide a vastly different performance ranking of the markets, albeit in the same direction.^ Finally, the inter-temporal stability of the international stock markets is investigated using the Parhizgari and Prakash (1989) algorithm for the Sen and Puri (1968) test which accounts for non-normality of return distributions. The empirical finding indicates that there is strong evidence to support the stability in international stock market movements. However, when the Anderson test which assumes normality of return distributions is employed, the stability in the correlation structure is rejected. This suggests that the non-normality of the return distribution is an important factor that cannot be ignored in the investigation of inter-temporal stability of international stock markets. ^

Examining moderators of the hindsight bias in the context of civil legal decision -making: Counterfactuals, causal proximity, and self -referencing

The current research sought to clarify the diverging relationships between counterfactual thinking and hindsight bias observed in the literature thus far. In a non-legal context, Roese and Olson (1996) found a positive relationship between counterfactuals and hindsight bias, such that counterfactual mutations that undid the outcome also increased participants’ ratings of the outcome’s a priori likelihood. Further, they determined that this relationship is mediated by causal attributions about the counterfactually mutated antecedent event. Conversely, in the context of a civil lawsuit, Robbennolt and Sobus (1997) found that the relationship between counterfactual thinking and hindsight bias is negative. The current research sought to resolve the conflicting findings in the literature within a legal context. ^ In Experiment One, the manipulation of the normality of the defendant’s target behavior, designed to manipulate participants’ counterfactual thoughts about said behavior, did moderate the hindsight effect of outcome knowledge on mock jurors’ judgments of the foreseeability of that outcome as well as their negligence verdicts. Although I predicted that counterfactual thinking would increase, or exacerbate, the hindsight bias, as found by Roese and Olson (1996), my results provided some support for Robbenolt and Sobus’s (1997) finding that counterfactual thinking decreases the hindsight bias. Behavior normality did not moderate the hindsight effect of outcome knowledge in Experiment Two, nor did causal proximity in Experiment Three. ^ Additionally, my hypothesis that self-referencing may be an effective hindsight debiasing technique received little support across the three experiments. Although both the self-referencing instructions and self-report measure consistently decreased mock jurors’ likelihood of finding the defendant negligent, and self-referencing instructions decreased their foreseeability ratings in studies two and three, the self-referencing manipulation did not interact with outcome knowledge to moderate a hindsight bias effect on either foreseeability or negligence judgments. The consistent pattern of results across the three experiments, however, suggests that self-referencing may be an effective technique in reducing the likelihood of negligence verdicts.^

Examining Moderators of the Hindsight Bias in the Context of Civil Legal Decision-Making: Counterfactuals, Causal Proximity, and Self-Referencing

The current research sought to clarify the diverging relationships between counterfactual thinking and hindsight bias observed in the literature thus far. In a non-legal context, Roese and Olson (1996) found a positive relationship between counterfactuals and hindsight bias, such that counterfactual mutations that undid the outcome also increased participants’ ratings of the outcome’s a priori likelihood. Further, they determined that this relationship is mediated by causal attributions about the counterfactually mutated antecedent event. Conversely, in the context of a civil lawsuit, Robbennolt and Sobus (1997) found that the relationship between counterfactual thinking and hindsight bias is negative. The current research sought to resolve the conflicting findings in the literature within a legal context. In Experiment One, the manipulation of the normality of the defendant’s target behavior, designed to manipulate participants’ counterfactual thoughts about said behavior, did moderate the hindsight effect of outcome knowledge on mock jurors’ judgments of the foreseeability of that outcome as well as their negligence verdicts. Although I predicted that counterfactual thinking would increase, or exacerbate, the hindsight bias, as found by Roese and Olson (1996), my results provided some support for Robbenolt and Sobus’s (1997) finding that counterfactual thinking decreases the hindsight bias. Behavior normality did not moderate the hindsight effect of outcome knowledge in Experiment Two, nor did causal proximity in Experiment Three. Additionally, my hypothesis that self-referencing may be an effective hindsight debiasing technique received little support across the three experiments. Although both the self-referencing instructions and self-report measure consistently decreased mock jurors’ likelihood of finding the defendant negligent, and self-referencing instructions decreased their foreseeability ratings in studies two and three, the self-referencing manipulation did not interact with outcome knowledge to moderate a hindsight bias effect on either foreseeability or negligence judgments. The consistent pattern of results across the three experiments, however, suggests that self-referencing may be an effective technique in reducing the likelihood of negligence verdicts.