Does a new product spread faster among heterogeneous or homogeneous consumers? We analyze this qu... more Does a new product spread faster among heterogeneous or homogeneous consumers? We analyze this question using the stochastic discrete Bass model in which consumers may differ in their individual external influence rates [Formula: see text] and in their individual internal influence rates [Formula: see text]. When the network is complete and the heterogeneity is only manifested in [Formula: see text] or only in [Formula: see text], it always slows down the diffusion, compared with the corresponding homogeneous network. When, however, consumers are heterogeneous in both [Formula: see text] and [Formula: see text], heterogeneity slows down the diffusion in some cases but accelerates it in others. Moreover, the dominance between the heterogeneous and homogeneous adoption levels is global in time in some cases but changes with time in others. Perhaps surprisingly, global dominance between two networks is not always preserved under “additive transformations”, such as adding an identical n...
We study private-value auctions with n risk-averse bidders, where n is a large number. We first u... more We study private-value auctions with n risk-averse bidders, where n is a large number. We first use asymptotic techniques to calculate explicit approximations of the equilibrium bids and of the seller’s revenue in any k-price auction (k = 1, 2, . . . ), and use these explicit approximations to show that all large k-price auctions with risk-averse bidders are O(1/n) revenue equivalent. We then prove that there exist auction mechanisms for which the limiting revenue as n −→ ∞ in the case of risk-averse bidders is strictly below the risk-neutral limit. Therefore, these auction mechanisms are not revenue equivalent to large k-price auctions even in the limit as n −→ ∞. Finally, we formulate a general condition under which the limiting revenue with risk-averse bidders is equal to the risk-neutral limit. School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, fibich@tau.ac.il Department of Industrial Engineering and Management, Faculty of Engineering Sciences, Ben-Gu...
In mathematical models with uncertainties and noise, the calculation of a deterministic "qua... more In mathematical models with uncertainties and noise, the calculation of a deterministic "quantity of interest" (model output) is often replaced by the calculation of its moments (mean, standard deviation, etc.) and probability density function. Standard methods for these tasks are either statistical (Monte-Carlo, kernel density estimators, etc.) or spectral approximations (e.g., generalized polynomial chaos). In this paper we present a novel spline-based algorithm for these tasks. Our method offers significant advantages over the existing methods for density estimation, including a guaranteed convergence rate which is at least cubic in the number of samples. Furthermore, although spectral methods can approximate moments with exponential accuracy, the spline-based approximation is often more accurate when the sample size is small. We also show how to approximate the moments and density of non-smooth quantities of interest, which is often prohibitive in spectral methods. Fin...
ABSTRACT We show experimentally for ultrashort laser pulses propagating in air, that the filament... more ABSTRACT We show experimentally for ultrashort laser pulses propagating in air, that the filamentation distance of intense laser pulses in the atmosphere can be extended and controlled with a simple double‐lens setup. Using this method we were able to achieve a 20‐fold delay of the filamentation distance of non‐chirped 120 fs pulses propagating in air, from 16 m to 330 m. At 330 m, the collapsing pulse is sufficiently powerful to create plasma filaments. We also show that the scatter of the filaments at 330 m can be significantly reduced by tilting the second lens. We derive a simple formula for the filamentation distance, and confirm its agreement with the experimental results. We also observe that delaying the onset of filamentation increases the filament length. To the best of our knowledge, this is the longest distance reported in the literature at which plasma filaments were created and controlled. Finally, we show that the peak power at the onset of collapse is significantly higher with the double‐lens setup, compared with the standard negative chirping approach.
Two random traffic streams are competing for the service time of a single server (multiplexer). T... more Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling ...
The nonlinear Schrödinger equation (NLS) is the standard model for propaga- tion of intense laser... more The nonlinear Schrödinger equation (NLS) is the standard model for propaga- tion of intense laser beams in Kerr media. The NLS is derived from the non- linear Helmholtz equation (NLH) by employing the paraxial approximation and neglecting the backscattered waves. In this study we use a fourth-order finite- difference method supplemented by special two-way artificial boundary condi- tions (ABCs) to
Does a new product spread faster among heterogeneous or homogeneous consumers? We analyze this qu... more Does a new product spread faster among heterogeneous or homogeneous consumers? We analyze this question using the stochastic discrete Bass model in which consumers may differ in their individual external influence rates [Formula: see text] and in their individual internal influence rates [Formula: see text]. When the network is complete and the heterogeneity is only manifested in [Formula: see text] or only in [Formula: see text], it always slows down the diffusion, compared with the corresponding homogeneous network. When, however, consumers are heterogeneous in both [Formula: see text] and [Formula: see text], heterogeneity slows down the diffusion in some cases but accelerates it in others. Moreover, the dominance between the heterogeneous and homogeneous adoption levels is global in time in some cases but changes with time in others. Perhaps surprisingly, global dominance between two networks is not always preserved under “additive transformations”, such as adding an identical n...
We study private-value auctions with n risk-averse bidders, where n is a large number. We first u... more We study private-value auctions with n risk-averse bidders, where n is a large number. We first use asymptotic techniques to calculate explicit approximations of the equilibrium bids and of the seller’s revenue in any k-price auction (k = 1, 2, . . . ), and use these explicit approximations to show that all large k-price auctions with risk-averse bidders are O(1/n) revenue equivalent. We then prove that there exist auction mechanisms for which the limiting revenue as n −→ ∞ in the case of risk-averse bidders is strictly below the risk-neutral limit. Therefore, these auction mechanisms are not revenue equivalent to large k-price auctions even in the limit as n −→ ∞. Finally, we formulate a general condition under which the limiting revenue with risk-averse bidders is equal to the risk-neutral limit. School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, fibich@tau.ac.il Department of Industrial Engineering and Management, Faculty of Engineering Sciences, Ben-Gu...
In mathematical models with uncertainties and noise, the calculation of a deterministic "qua... more In mathematical models with uncertainties and noise, the calculation of a deterministic "quantity of interest" (model output) is often replaced by the calculation of its moments (mean, standard deviation, etc.) and probability density function. Standard methods for these tasks are either statistical (Monte-Carlo, kernel density estimators, etc.) or spectral approximations (e.g., generalized polynomial chaos). In this paper we present a novel spline-based algorithm for these tasks. Our method offers significant advantages over the existing methods for density estimation, including a guaranteed convergence rate which is at least cubic in the number of samples. Furthermore, although spectral methods can approximate moments with exponential accuracy, the spline-based approximation is often more accurate when the sample size is small. We also show how to approximate the moments and density of non-smooth quantities of interest, which is often prohibitive in spectral methods. Fin...
ABSTRACT We show experimentally for ultrashort laser pulses propagating in air, that the filament... more ABSTRACT We show experimentally for ultrashort laser pulses propagating in air, that the filamentation distance of intense laser pulses in the atmosphere can be extended and controlled with a simple double‐lens setup. Using this method we were able to achieve a 20‐fold delay of the filamentation distance of non‐chirped 120 fs pulses propagating in air, from 16 m to 330 m. At 330 m, the collapsing pulse is sufficiently powerful to create plasma filaments. We also show that the scatter of the filaments at 330 m can be significantly reduced by tilting the second lens. We derive a simple formula for the filamentation distance, and confirm its agreement with the experimental results. We also observe that delaying the onset of filamentation increases the filament length. To the best of our knowledge, this is the longest distance reported in the literature at which plasma filaments were created and controlled. Finally, we show that the peak power at the onset of collapse is significantly higher with the double‐lens setup, compared with the standard negative chirping approach.
Two random traffic streams are competing for the service time of a single server (multiplexer). T... more Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling ...
The nonlinear Schrödinger equation (NLS) is the standard model for propaga- tion of intense laser... more The nonlinear Schrödinger equation (NLS) is the standard model for propaga- tion of intense laser beams in Kerr media. The NLS is derived from the non- linear Helmholtz equation (NLH) by employing the paraxial approximation and neglecting the backscattered waves. In this study we use a fourth-order finite- difference method supplemented by special two-way artificial boundary condi- tions (ABCs) to
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