A precise measurement of the two -photon exchange effect

The two-photon exchange phenomenon is believed to be responsible for the discrepancy observed between the ratio of proton electric and magnetic form factors, measured by the Rosenbluth and polarization transfer methods. This disagreement is about a factor of three at Q 2 of 5.6 GeV2. The precise knowledge of the proton form factors is of critical importance in understanding the structure of this nucleon. The theoretical models that estimate the size of the two-photon exchange (TPE) radiative correction are poorly constrained. This factor was found to be directly measurable by taking the ratio of the electron-proton and positron-proton elastic scattering cross sections, as the TPE effect changes sign with respect to the charge of the incident particle. A test run of a modified beamline has been conducted with the CEBAF Large Acceptance Spectrometer (CLAS) at Thomas Jefferson National Accelerator Facility. This test run demonstrated the feasibility of producing a mixed electron/positron beam of good quality. Extensive simulations performed prior to the run were used to reduce the background rate that limits the production luminosity. A 3.3 GeV primary electron beam was used that resulted in an average secondary lepton beam of 1 GeV. As a result, the elastic scattering data of both lepton types were obtained at scattering angles up to 40 degrees for Q2 up to 1.5 GeV2. The cross section ratio displayed an &epsis; dependence that was Q2 dependent at smaller Q2 limits. The magnitude of the average ratio as a function of &epsis; was consistent with the previous measurements, and the elastic (Blunden) model to within the experimental uncertainties. Ultimately, higher luminosity is needed to extend the data range to lower &epsis; where the TPE effect is predicted to be largest.

Proton Form Factor Puzzle and the CEBAF Large Acceptance Spectrometer (CLAS) two-photon exchange experiment

The electromagnetic form factors are the most fundamental observables that encode information about the internal structure of the nucleon. The electric (GE) and the magnetic ( GM) form factors contain information about the spatial distribution of the charge and magnetization inside the nucleon. A significant discrepancy exists between the Rosenbluth and the polarization transfer measurements of the electromagnetic form factors of the proton. One possible explanation for the discrepancy is the contributions of two-photon exchange (TPE) effects. Theoretical calculations estimating the magnitude of the TPE effect are highly model dependent, and limited experimental evidence for such effects exists. Experimentally, the TPE effect can be measured by comparing the ratio of positron-proton elastic scattering cross section to that of the electron-proton [R = σ(e +p)/σ(e+p)]. The ratio R was measured over a wide range of kinematics, utilizing a 5.6 GeV primary electron beam produced by the Continuous Electron Beam Accelerator Facility (CEBAF) at Jefferson Lab. This dissertation explored dependence of R on kinematic variables such as squared four-momentum transfer (Q2) and the virtual photon polarization parameter (&epsis;). A mixed electron-positron beam was produced from the primary electron beam in experimental Hall B. The mixed beam was scattered from a liquid hydrogen (LH2) target. Both the scattered lepton and the recoil proton were detected by the CEBAF Large Acceptance Spectrometer (CLAS). The elastic events were then identified by using elastic scattering kinematics. This work extracted the Q2 dependence of R at high &epsis;(&epsis; > 0.8) and the $&epsis; dependence of R at ⟨Q 2⟩ approx 0.85 GeV2. In these kinematics, our data confirm the validity of the hadronic calculations of the TPE effect by Blunden, Melnitchouk, and Tjon. This hadronic TPE effect, with additional corrections contributed by higher excitations of the intermediate state nucleon, largely reconciles the Rosenbluth and the polarization transfer measurements of the electromagnetic form factors.

Proton Form Factor Puzzle and the CEBAF Large Acceptance Spectrometer(CLAS) Two-Photon Exchange Experiment

The electromagnetic form factors are the most fundamental observables that encode information about the internal structure of the nucleon. The electric ($G_{E}$) and the magnetic ($G_{M}$) form factors contain information about the spatial distribution of the charge and magnetization inside the nucleon. A significant discrepancy exists between the Rosenbluth and the polarization transfer measurements of the electromagnetic form factors of the proton. One possible explanation for the discrepancy is the contributions of two-photon exchange (TPE) effects. Theoretical calculations estimating the magnitude of the TPE effect are highly model dependent, and limited experimental evidence for such effects exists. Experimentally, the TPE effect can be measured by comparing the ratio of positron-proton elastic scattering cross section to that of the electron-proton $\large(R = \frac{\sigma (e^{+}p)}{\sigma (e^{-}p)}\large)$. The ratio $R$ was measured over a wide range of kinematics, utilizing a 5.6 GeV primary electron beam produced by the Continuous Electron Beam Accelerator Facility (CEBAF) at Jefferson Lab. This dissertation explored dependence of $R$ on kinematic variables such as squared four-momentum transfer ($Q^{2}$) and the virtual photon polarization parameter ($\varepsilon$). A mixed electron-positron beam was produced from the primary electron beam in experimental Hall B. The mixed beam was scattered from a liquid hydrogen (LH$_{2}$) target. Both the scattered lepton and the recoil proton were detected by the CEBAF Large Acceptance Spectrometer (CLAS). The elastic events were then identified by using elastic scattering kinematics. This work extracted the $Q^{2}$ dependence of $R$ at high $\varepsilon$ ($\varepsilon > $ 0.8) and the $\varepsilon$ dependence of $R$ at $\langle Q^{2} \rangle \approx 0.85$ GeV$^{2}$. In these kinematics, our data confirm the validity of the hadronic calculations of the TPE effect by Blunden, Melnitchouk, and Tjon. This hadronic TPE effect, with additional corrections contributed by higher excitations of the intermediate state nucleon, largely reconciles the Rosenbluth and the polarization transfer measurements of the electromagnetic form factors.

Nonlinear optical effects manifested by electromagnetic induced transparency in cold atoms

This dissertation reports experimental studies of nonlinear optical effects manifested by electromagnetically induced transparency (EIT) in cold Rb atoms. The cold Rb atoms are confined in a magneto-optic trap (MOT) obtained with the standard laser cooling and trapping technique. Because of the near zero Doppler shift and a high phase density, the cold Rb sample is well suited for studies of atomic coherence and interference and related applications, and the experiments can be compared quantitatively with theoretical calculations. It is shown that with EIT induced in the multi-level Rb system by laser fields, the linear absorption is suppressed and the nonlinear susceptibility is enhanced, which enables studies of nonlinear optics in the cold atoms with slow photons and at low light intensities. Three independent experiments are described and the experimental results are presented. First, an experimental method that can produce simultaneously co-propagating slow and fast light pulses is discussed and the experimental demonstration is reported. Second, it is shown that in a three-level Rb system coupled by multi-color laser fields, the multi-channel two-photon Raman transitions can be manipulated by the relative phase and frequency of a control laser field. Third, a scheme for all-optical switching near single photon levels is developed. The scheme is based on the phase-dependent multi-photon interference in a coherently coupled four-level system. The phase dependent multi-photon interference is observed and switching of a single light pulse by a control pulse containing ∼20 photons is demonstrated. These experimental studies reveal new phenomena manifested by quantum coherence and interference in cold atoms, contribute to the advancement of fundamental quantum optics and nonlinear optics at ultra-low light intensities, and may lead to the development of new techniques to control quantum states of atoms and photons, which will be useful for applications in quantum measurements and quantum photonic devices.

Effects of irradiance and spectral quality on leaf structure and function in seedlings of two Southeast Asian Hopea species

We studied the development of leaf characters in two Southeast Asian dipterocarp forest trees under different photosynthetic photon flux densities (PFD) and spectral qualities (red to far-red, R:FR). The two species, Hopea helferi and H. odorata, are taxonomically closely related but differ in their ecological requirements; H. helferi is more drought tolerant and H. odorata more shade tolerant. Seedlings were grown in replicated shadehouse treatments of differing PFD and R:FR. We measured or calculated (1) leaf and tissue thicknesses; (2) mesophyll parenchyma, air space, and lignified tissue volumes; (3) mesophyll air volumes (Vmes/Asurf) and surfaces (Ames/Asurf); (4) palisade cell length and width; (5) chlorophyll/cm2 and a/ b; (6) leaf absorption; and (7) attenuance/absorbance at 652 and 550 nm. These characters varied in response to light conditions in both taxa. Characters were predominantly affected by PFD, and R:FR slightly influenced many characters. Leaf characters of H. odorata were more plastic in response to treatment conditions. Characters were correlated with each other in a complex fashion. Variation in leaf anatomy is most likely a consequence of increasing leaf thickness in both taxa, which may increase mechanical strength and defense against herbivory in more exposed environments. Variation in leaf optical properties was most likely affected by pigment photo-bleaching in treatments of more intense PFD and was not correlated with Amax. The greater plasticity of leaf responses in H. odorata helps explain the acclimation over the range of light conditions encountered by this shade-tolerant taxon. The dense layer of scales on the leaf undersurface and other anatomical characters in H. helferi reduced gas exchange and growth in this drought-tolerant tree.

Leaf Gas Exchange and Nutrient Use Efficiency Help Explain the Distribution of Two Neotropical Mangroves under Contrasting Flooding and Salinity

Rhizophora mangle and Laguncularia racemosa cooccur along many intertidal floodplains in the Neotropics. Their patterns of dominance shift along various gradients, coincident with salinity, soil fertility, and tidal flooding. We used leaf gas exchange metrics to investigate the strategies of these two species in mixed culture to simulate competition under different salinity concentrations and hydroperiods. Semidiurnal tidal and permanent flooding hydroperiods at two constant salinity regimes (10 g L−1 and 40 g L−1) were simulated over 10 months. Assimilation ( ), stomatal conductance ( ), intercellular CO2 concentration ( ), instantaneous photosynthetic water use efficiency (PWUE), and photosynthetic nitrogen use efficiency (PNUE) were determined at the leaf level for both species over two time periods. Rhizophora mangle had significantly higher PWUE than did L. racemosa seedlings at low salinities; however, L. racemosa had higher PNUE and and, accordingly, had greater intercellular CO2 (calculated) during measurements. Both species maintained similar capacities for A at 10 and 40 g L−1 salinity and during both permanent and tidal hydroperiod treatments. Hydroperiod alone had no detectable effect on leaf gas exchange. However, PWUE increased and PNUE decreased for both species at 40 g L−1 salinity compared to 10 g L−1. At 40 g L−1 salinity, PNUE was higher for L. racemosa than R. mangle with tidal flooding. These treatments indicated that salinity influences gas exchange efficiency, might affect how gases are apportioned intercellularly, and accentuates different strategies for distributing leaf nitrogen to photosynthesis for these two species while growing competitively.

The Spectral Distribution of Radiation in Two Neotropical Rainforests

The spectral quality of radiation in the understory of two neotropical rainforests, Barro Colorado Island in Panama and La Selva in Costa Rica, is profoundly affected by the density of the canopy. Understory light conditions in both forests bear similar spectral characteristics. In both the greatest changes in spectral quality occur at low flux densities, as in the transition from extreme shade to small light flecks. Change in spectral quality, as assessed by the red: far-red (R:FR) ratio, the ratio of radiant energy 400-700: 300-1100 nm, and the ratio of quantum flux density 400-700:300-1100 nm, is strongly correlated with a drop in percentage of solar radiation as measurable by a quantum radiometer. Thus, by knowing the percentage of photosynthetic photon flux density (PPFD) in relation to full sunlight, it is possible to estimate the spectral quality in the forest at a particular time and microsite.

Exchange rate overvaluation under hyperinflation: The case of Peru

This dissertation examines the behavior of the exchange rate under two different scenarios. The first one is characterized by, relatively, low inflation or a situation where prices adjust sluggishly. The second is a high inflation economy where prices respond very rapidly even to unanticipated shocks. In the first one, following a monetary expansion, the exchange rate overshoots, i.e. the nominal exchange rate depreciates at a faster pace than the price level. Under high levels of inflation, prices change faster than the exchange rate so the exchange rate undershoots its long run equilibrium value.^ The standard work in this area, Dornbusch (1976), explains the overshooting process in the context of perfect capital mobility and sluggish adjustment in the goods market. A monetary expansion will make the exchange rate increase beyond its long run equilibrium value. This dissertation expands on Dornbusch's model and provides an analysis of the exchange rate under conditions of currency substitution and price flexibility, characteristics of the Peruvian economy during the hyper inflation process that took place at the end of the 1980's. The results of the modified Dornbusch model reveal that, given a monetary expansion, the change in the price level will be larger than the change in the exchange rate if prices react more than proportionally to the monetary shock.^ We will expect this over-reaction in circumstances of high inflation when the velocity of money is increasing very rapidly. Increasing velocity of money, gives rise to a higher relative price variability which in turn contributes to the appearance of new financial (and also non-financial) instruments that report a higher return than the exchange rate, causing people to switch their demand for foreign exchange to this new assets. In the context of currency substitution, economic agents hoard and use foreign exchange as a store of value. The big decline in output originated by hyper inflation induces people to sell this hoarded money to finance current expenses, increasing the supply of foreign exchange in the market. Both, the decrease in demand and the increase in supply reduce the price of foreign exchange i.e. the real exchange rate. The findings mentioned above are tested using Peruvian data for the period January 1985-July 1990, the results of the econometric estimation confirm our findings in the theoretical model. ^

Empirical study of exchange rates in Brazil, Canada and the Caribbean

This dissertation examines the monetary models of exchange rate determination for Brazil, Canada, and two countries in the Caribbean, namely, the Dominican Republic and Jamaica. With the exception of Canada, the others adopted the floating regime during the past ten years.^ The empirical validity of four seminal models in exchange rate economics were determined. Three of these models were entirely classical (Bilson and Frenkel) or Keynesian (Dornbusch) in nature. The fourth model (Real Interest Differential Model) was a mixture of the two schools of economic theory.^ There is no clear empirical evidence of the validity of the monetary models. However, the signs of the coefficients of the nominal interest differential variable were as predicted by the Keynesian hypothesis in the case of Canada and as predicted by the Chicago theorists in the remaining countries. Moreover, in case of Brazil, due to hyperinflation, the exchange rate is heavily influenced by domestic money supply.^ I also tested the purchasing power parity (PPP) for this same set of countries. For both the monetary as well as the PPP hypothesis, I tested for co-integration and applied ordinary least squares estimation procedure. The error correction model was also used for the PPP model, to determine convergence to equilibrium.^ The validity of PPP is also questionable for my set of countries. Endogeinity among the regressors as well as the lack of proper price indices are the contributing factors. More importantly, Central Bank intervention negate rapid adjustment of price and exchange rates to their equilibrium value. However, its forecasting capability for the period 1993-1994 is superior compared to the monetary models in two of the four cases.^ I conclude that in spite of the questionable validity of these models, the monetary models give better results in the case of the "smaller" economies like the Dominican Republic and Jamaica where monetary influences swamp the other determinants of exchange rate. ^

Computational modeling of intermediate temperature proton exchange membrane (PEM) fuel cells

A two-phase three-dimensional computational model of an intermediate temperature (120--190°C) proton exchange membrane (PEM) fuel cell is presented. This represents the first attempt to model PEM fuel cells employing intermediate temperature membranes, in this case, phosphoric acid doped polybenzimidazole (PBI). To date, mathematical modeling of PEM fuel cells has been restricted to low temperature operation, especially to those employing Nafion ® membranes; while research on PBI as an intermediate temperature membrane has been solely at the experimental level. This work is an advancement in the state of the art of both these fields of research. With a growing trend toward higher temperature operation of PEM fuel cells, mathematical modeling of such systems is necessary to help hasten the development of the technology and highlight areas where research should be focused.^ This mathematical model accounted for all the major transport and polarization processes occurring inside the fuel cell, including the two phase phenomenon of gas dissolution in the polymer electrolyte. Results were presented for polarization performance, flux distributions, concentration variations in both the gaseous and aqueous phases, and temperature variations for various heat management strategies. The model predictions matched well with published experimental data, and were self-consistent.^ The major finding of this research was that, due to the transport limitations imposed by the use of phosphoric acid as a doping agent, namely low solubility and diffusivity of dissolved gases and anion adsorption onto catalyst sites, the catalyst utilization is very low (∼1--2%). Significant cost savings were predicted with the use of advanced catalyst deposition techniques that would greatly reduce the eventual thickness of the catalyst layer, and subsequently improve catalyst utilization. The model also predicted that an increase in power output in the order of 50% is expected if alternative doping agents to phosphoric acid can be found, which afford better transport properties of dissolved gases, reduced anion adsorption onto catalyst sites, and which maintain stability and conductive properties at elevated temperatures.^

Essays on exchange rate economics

Exchange rate economics has achieved substantial development in the past few decades. Despite extensive research, a large number of unresolved problems remain in the exchange rate debate. This dissertation studied three puzzling issues aiming to improve our understanding of exchange rate behavior. Chapter Two used advanced econometric techniques to model and forecast exchange rate dynamics. Chapter Three and Chapter Four studied issues related to exchange rates using the theory of New Open Economy Macroeconomics. ^ Chapter Two empirically examined the short-run forecastability of nominal exchange rates. It analyzed important empirical regularities in daily exchange rates. Through a series of hypothesis tests, a best-fitting fractionally integrated GARCH model with skewed student-t error distribution was identified. The forecasting performance of the model was compared with that of a random walk model. Results supported the contention that nominal exchange rates seem to be unpredictable over the short run in the sense that the best-fitting model cannot beat the random walk model in forecasting exchange rate movements. ^ Chapter Three assessed the ability of dynamic general-equilibrium sticky-price monetary models to generate volatile foreign exchange risk premia. It developed a tractable two-country model where agents face a cash-in-advance constraint and set prices to the local market; the exogenous money supply process exhibits time-varying volatility. The model yielded approximate closed form solutions for risk premia and real exchange rates. Numerical results provided quantitative evidence that volatile risk premia can endogenously arise in a new open economy macroeconomic model. Thus, the model had potential to rationalize the Uncovered Interest Parity Puzzle. ^ Chapter Four sought to resolve the consumption-real exchange rate anomaly, which refers to the inability of most international macro models to generate negative cross-correlations between real exchange rates and relative consumption across two countries as observed in the data. While maintaining the assumption of complete asset markets, this chapter introduced endogenously segmented asset markets into a dynamic sticky-price monetary model. Simulation results showed that such a model could replicate the stylized fact that real exchange rates tend to move in an opposite direction with respect to relative consumption. ^

QCD structure of nuclear interactions

The research presented in this dissertation investigated selected processes involving baryons and nuclei in hard scattering reactions. These processes are characterized by the production of particles with large energies and transverse momenta. Through these processes, this work explored both, the constituent (quark) structure of baryons (specifically nucleons and Δ-Isobars), and the mechanisms through which the interactions between these constituents ultimately control the selected reactions. The first of such reactions is the hard nucleon-nucleon elastic scattering, which was studied here considering the quark exchange between the nucleons to be the dominant mechanism of interaction in the constituent picture. In particular, it was found that an angular asymmetry exhibited by proton-neutron elastic scattering data is explained within this framework if a quark-diquark picture dominates the nucleon’s structure instead of a more traditional SU(6) three quarks picture. The latter yields an asymmetry around 90o center of mass scattering with a sign opposite to what is experimentally observed. The second process is the hard breakup by a photon of a nucleon-nucleon system in light nuclei. Proton-proton (pp) and proton-neutron (pn) breakup in 3He, and ΔΔ-isobars production in deuteron breakup were analyzed in the hard rescattering model (HRM), which in conjunction with the quark interchange mechanism provides a Quantum Chromodynamics (QCD) description of the reaction. Through the HRM, cross sections for both channels in 3He photodisintegration were computed without the need of a fitting parameter. The results presented here for pp breakup show excellent agreement with recent experimental data. In ΔΔ-isobars production in deuteron breakup, HRM angular distributions for the two ΔΔ channels were compared to the pn channel and to each other. An important prediction fromthis study is that the Δ++Δ- channel consistently dominates Δ+Δ0, which is in contrast with models that unlike the HRM consider a ΔΔ system in the initial state of the interaction. For such models both channels should have the same strength. These results are important in developing a QCD description of the atomic nucleus.

Growth and gas exchange responses of Brazilian pepper (Schinus terebinthifolius) and native South Florida species to salinity

Schinus terebinthifolius Raddi (Schinus) is an invasive exotic species widely found in disturbed and native communities of Florida. This species has been shown to displace native species as well as alter community structure and function. The purpose of this study was to determine if the growth and gas exchange patterns of Schinus, under differing salinity conditions, were different from native species. Two native upland glycophytic species (Rapanea punctata and Randia aculeata) and two native mangrove species (Rhizophora mangle and Laguncularia racemosa) were compared with the exotic. Overall, the exotics morphologic changes and gas exchange patterns were most similar to R. mangle. Across treatments, increasing salinity decreased relative growth rate (RGR), leaf area ratio (LAR) and specific leaf area (SLA) but did not affect root/shoot ratios (R:S). Allocation patterns were however significantly different among species. The largest proportion of Schinus biomass was allocated to stems (47%), resulting in plants that were generally taller than the other species. Schinus also had the highest SLA and largest total leaf area of all species. This meant that the exotic, which was taller and had thinner leaves, was potentially able to maintain photosynthetic area comparable to native species. Schinus response patterns show that this exotic exhibits some physiological tolerance for saline conditions. Coupled with its biomass allocation patterns (more stem biomass and large area of thin leaves), the growth traits of this exotic potentially provide this species an advantage over native plants in terms of light acquisition in a brackish forested ecosystem.

Personality and job performance: The mediating roles of leader -member exchange quality and action control

The present investigation examined the relationships among personality (as conceptualized by the Big Five Factors), leader-member exchange (LMX) quality, action control, organizational citizenship behaviors (OCB), and overall job performance (OJP). Two mediator variables were proposed and tested in this study: LMX and Action Control. Two-hundred and seven currently employed regular elementary school classroom teachers provided data during the 2000–2001 academic school year. Teachers provided personality, LMX quality (member or subordinate perspective), action control, job tenure, and demographic data. Nine school administrators (i.e., Principals, Assistant Principals) were the source for supervisor ratings of OCB, OJP, and LMX quality (leader or supervisor perspective). In eight of the nine total schools, teachers completed questionnaires during an after-school teacher gathering; in the remaining school location questionnaires were dropped off, distributed to teachers, and re-collected two weeks later. Results indicated a significant relationship between the OCB scale and overall supervisory ratings of OJP. The relationship among the big five factors of personality and OJP did not reach statistical significance, nor did the relationships among personality and OCB. The data indicated that none of the teacher tenure variables (i.e., teacher, school, or time worked with principal tenure) moderated the personality-OCB relationship nor the personality-OJP relationship. Finally, a review of the correlations among the variables of interest precluded conducting a mediation between personality-performance by OCB, mediation of personality-OCB by action control, and mediation of personality-OCB by LMX. In conclusion, the data reveal that personality was not significantly correlated with supervisory ratings of OJP or significantly related to supervisory ratings of overall OCB. Moreover, LMX quality and action control did not mediate the relationships between Personality-OJP nor the Personality-OCB relationship. Significant relationships were found between disengagement and overall LMX quality and between Initiative and overall LMX quality (both LMX-Teacher perspectives) as well as between personality variables and both Disengagement and Initiative action control variables. Despite the limitations inherent in this study, these latter findings suggest “lessons” for teachers and school administrators alike. ^

Three essays on the microstructure of exchange traded funds

Exchange traded funds (ETFs) have increased significantly in popularity since they were first introduced in 1993. However, there is still much that is unknown about ETFs in the extant literature. This dissertation attempts to fill gaps in the ETF literature by using three related essays. In these three essays, we compare ETFs to closed ended mutual funds (CEFs) by decomposing the bid-ask spread into its three components; we look at the intraday shape of ETFs and compare it to the intraday shape of equities as well as examine the co-integration factor between ETFs on the London Stock Exchange and the New York Stock Exchange; we also examine the differences between leveraged ETFs and unleveraged ETFs by analyzing the impact of liquidity and volatility. These three essays are presented in Chapters 1, 2, and 3, respectively. ^ Chapter one uses the Huang and Stoll (1997) model to decompose the bid-ask spread in CEFs and ETFs for two distinct periods—a normal and a volatile period. We show a higher adverse selection component for CEFs than for ETFs without regard to volatility. However, both ETFs and CEFs increased in magnitude of the adverse selection component in the period of high volatility. Chapter two uses a mix of the Werner and Kleidon (1993) and the Hupperets and Menkveld (2002) methods to get the intraday shape of ETFs and analyze co-integration between London and New York trading. We find two different shapes for New York and London ETFs. There also appears to be evidence of co-integration in the overlapping two-hour trading period but not over the entire trading day for the two locations. The third chapter discusses the new class of ETFs called leveraged ETFs. We examine the liquidity and depth differences between unleveraged and leveraged ETFs at the aggregate level and when the leveraged ETFs are classified by the leveraged multiples of -3, -2, -1, 2, and 3, both for a normal and a volatile period. We find distinct differences between leveraged and unleveraged ETFs at the aggregate level, with leveraged ETFs having larger spreads than unleveraged ETFs. Furthermore, while both leveraged and unleveraged ETFs have larger spreads in high volatility, for the leveraged ETFs the change in magnitude is significantly larger than for the unleveraged ETFs. Among the multiples, the -2 leveraged ETF is the most pronounced in its liquidity characteristics, more so in volatile times. ^