The matching law

This article introduces the quantitative analysis of choice behavior by describing a number of equations developed over the years to describe the relation between the allocation of behavior under concurrent schedules of reinforcement and the consequences received for alternative responses. Direct proportionality between rate of responding and rate of reinforcement was observed in early studies, suggesting that behavioral output matched environmental input in a mathematical sense. This relation is termed "strict matching," and the equation that describes it is referred to as "the matching law." Later data showed systematic departures from strict matching, and a generalized version of the matching equation is now used to describe such data. This equation, referred to as "the generalized matching equation," also describes data that follow strict matching. It has become convention to refer to either of these equations as "the matching law." Empirical support for the matching law is briefly summarized, as is the applied and practical significance of matching analyses

TEMPORAL COMPARISON OF HAMSTRING CO-CONTRACTION LATENCY IN ANTERIOR CRUCIATE DEFICIENT KNEES - A PILOT STUDY

Recently, anterior cruciate ligament (ACL) research has been focusing on strengthening the hamstring muscles, as previous studies have found there to be a significant stabilizing effect by the hamstrings in those subjects with anterior cruciate deficiencies. Research has primarily focused on whether differences existed between the strength (Moore & Wade, 1989; Seto, Wino, Morrissey, Medeiros & Mason, 1988), the electromyographical (EMG) output (Tibone & Antich, 1988), or the muscle activation patterns between the quadriceps and hamstrings. (Beard, Kyberd, Fergusson & Dodd, 1993; Jennings Seedhom, 1994). A key factor in hamstring co-contraction during gait is that of timing. Should hamstring cocontraction occur near to heelstrike, where the greatest anterior shear force is applied to the knee during a normal gait, or occur too far behind heelstrike, muscular strength is rendered useless. In such an event, stabilization cannot occur if the joint has already been displaced, or counteracting force has already been exhausted. This pilot study proposed to determine if latency exists between injured and uninjured knees, examining only temporal data relative to heelstrike. Similar studies involving static positions with an applied shear force have been performed and have had contradicting results. (Beard, Kyberd, Fergusson & Dodd, 1993; JeMings & Seedhom, 1994). To date, no temporal measures have been examined in dynamic movement. This may affect results as most joint receptors require motion for proprioception. Data collection procedures involved using a 3-channel EMG monitor to track the time heelstrike, hamstring contraction and quadriceps contraction during a 5-second period of normal walking/jogging on a treadmill. Speeds of 5 km/h and 7 km/h were used on incline, decline and flat surfaces. Data was collected at three randomly selected times for each condition and each leg. Differences found between injured and uninjured legs represent an area of concern for those in the field of rehabilitation. Research into variations in treatment methods to correct for latency in the hamstrings could provide a level of stability not before seen in non-surgical or even surgical patients. The next step to further this research will involve an eight-channel EMG monitor to track bilateral timing. A thorough patient history including Cincinnati and Lysholm scales of subjective functional ability will be included in the patient selection process

Reinforcer magnitude and demand under fixed-ratio schedules with domestic hens

This study compared three methods of normalizing demand functions to allow comparison of demand for different commodities and examined how varying reinforcer magnitudes affected these analyses. Hens responded under fixed-ratio schedules in 40-min sessions with response requirement doubling each session and with 2-s, 8-s, and 12-s access to wheat. Over the smaller fixed ratios overall response rates generally increased and were higher the shorter the magazine duration. The logarithms of the number of reinforcers obtained (consumption) and the fixed ratio (price) were well fitted by curvilinear demand functions (Hursh et al., 1988. Journal of the Experimental Analysis of Behavior 50, 419–440) that were inelastic (b negative) over small fixed-ratios. The fixed ratio with maximal response rate (Pmax) increased, and the rate of change of elasticity (a) and initial consumption (L) decreased with increased magazine duration. Normalizing consumption using measures of preference for various magazine durations (3-s vs. 3-s, 2-s vs. 8-s, and 2-s vs. 12-s), obtained using concurrent schedules, gave useful results as it removed the differences in L. Normalizing consumption and price (Hursh and Winger, 1995. Journal of the Experimental Analysis of Behavior 64, 373–384) unified the data functions as intended by that analysis. The exponential function (Hursh and Silberberg, 2008. Psychological Review, 115, 186–198) gave an essential value that increased (i.e., α decreased significantly) as magazine duration decreased. This was not as predicted, since α should be constant over variations in magazine duration, but is similar to previous findings using a similar procedure with different food qualities (hens) and food quantities (rats)

HAMSTRING CONTRACTILE TIMING IN ANTERIOR CRUCIATE LIGAMENT DEFICIENT PATIENTS

The purpose of this research is to examine contractile timing of the hamstrings in patients with anterior cruciate ligament (ACL) deficient knees. Ten patients were tested for hamstring contractile timing using Electromyographic (EMG) recordings during gait speeds of 3km/h and 5km/h on both a level and ten degree incline. An ANOVA analysis was used to demonstrate timing onset differences between the deficient and normal leg for each test condition and muscle onset timing was related to the Lysholm scale for perceived knee function. Significant interaction differences were found for legs and speed variations plus a negative correlation between functional scores and hamstring timing onset

Numerical framework for transcritical real-fluid reacting flow simulations using the flamelet progress variable approach

An extension to the classical FPV model is developed for transcritical real-fluid combustion simulations in the context of finite volume, fully compressible, explicit solvers. A double-flux model is developed for transcritical flows to eliminate the spurious pressure oscillations. A hybrid scheme with entropy-stable flux correction is formulated to robustly represent large density ratios. The thermodynamics for ideal-gas values is modeled by a linearized specific heat ratio model. Parameters needed for the cubic EoS are pre-tabulated for the evaluation of departure functions and a quadratic expression is used to recover the attraction parameter. The novelty of the proposed approach lies in the ability to account for pressure and temperature variations from the baseline table. Cryogenic LOX/GH2 mixing and reacting cases are performed to demonstrate the capability of the proposed approach in multidimensional simulations. The proposed combustion model and numerical schemes are directly applicable for LES simulations of real applications under transcritical conditions.Comment: 55th AIAA Aerospace Sciences Meeting, Dallas, T

Analytical approximation for the sphere-sphere Coulomb potential

A simple analytical expression, which closely approximates the Coulomb potential between two uniformly charged spheres, is presented. This expression can be used in the optical potential semiclassical analyses which require that the interaction be analytic on and near the real r-axis.Comment: 4 pages including 3 figures and 1 tabl

Low Temperature Static and Dynamic Behavior of the Two-Dimensional Easy-Axis Heisenberg Model

We apply the self-consistent harmonic approximation (SCHA) to study static and dynamic properties of the two-dimensional classical Heisenberg model with easy-axis anisotropy. The static properties obtained are magnetization and spin wave energy as functions of temperature, and the critical temperature as a function of the easy-axis anisotropy. We also calculate the dynamic correlation functions using the SCHA renormalized spin wave energy. Our analytical results, for both static properties and dynamic correlation functions, are compared to numerical simulation data combining cluster-Monte Carlo algorithms and Spin Dynamics. The comparison allows us to conclude that far below the transition temperature, where the SCHA is valid, spin waves are responsible for all relevant features observed in the numerical simulation data; topological excitations do not seem to contribute appreciably. For temperatures closer to the transition temperature, there are differences between the dynamic correlation functions from SCHA theory and Spin Dynamics; these may be due to the presence of domain walls and solitons.Comment: 12 pages, 14 figure

Real-Gas Effects and Phase Separation in Underexpanded Jets at Engine-Relevant Conditions

A numerical framework implemented in the open-source tool OpenFOAM is presented in this work combining a hybrid, pressure-based solver with a vapor-liquid equilibrium model based on the cubic equation of state. This framework is used in the present work to investigate underexpanded jets at engine-relevant conditions where real-gas effects and mixture induced phase separation are probable to occur. A thorough validation and discussion of the applied vapor-liquid equilibrium model is conducted by means of general thermodynamic relations and measurement data available in the literature. Engine-relevant simulation cases for two different fuels were defined. Analyses of the flow field show that the used fuel has a first order effect on the occurrence of phase separation. In the case of phase separation two different effects could be revealed causing the single-phase instability, namely the strong expansion and the mixing of the fuel with the chamber gas. A comparison of single-phase and two-phase jets disclosed that the phase separation leads to a completely different penetration depth in contrast to single-phase injection and therefore commonly used analytical approaches fail to predict the penetration depth.Comment: Preprint submitted to AIAA Scitech 2018, Kissimmee, Florid