Showing papers on "Constant (mathematics) published in 1990"

NOSER: An algorithm for solving the inverse conductivity problem

Abstract: The inverse conductivity problem is the mathematical problem that must be solved in order for electrical impedance tomography systems to be able to make images. Here we show how this inverse conductivity problem is related to a number of other inverse problem. We then explain the workings of an algorithm that we have used to make images from electrical impedance data measured on the boundary of a circle in two dimensions. This algorithm is based on the method of least squares. It takes one step of a Newton's method, using a constant conductivity as an initial guess. Most of the calculations can therefore be done analytically. The resulting code is named NOSER, for Newton's One-Step Error Reconstructor. It provides a reconstruction with 496 degrees of freedom. The code does not reproduce the conductivity accurately (unless it differs very little from a constant), but it yields useful images. This is illustrated by images reconstructed from numerical and experimental data, including data from a human chest.

A new table of constant weight codes

Abstract: A table of binary constant weight codes of length nl28 is presented. Explicit constructions are given for most of the 600 codes in the table; the majority of these codes are new. The known techniques for constructing constant weight codes are surveyed, and a table of (unrestricted) binary codes of length nl28 is given

Unitary-matrix models as exactly solvable string theories.

Abstract: Exact differential equations are presently found for the scaling functions of models of unitary matrices which are solved in a double-scaling limit, using orthogonal polynomials on a circle. For the case of the simplest, k = 1 model, the Painleve II equation with constant 0 is obtained; possible nonperturbative phase transitions exist for these models. Equations are presented for k = 2 and 3, and discussed with a view to asymptotic behavior.

Constant-temperature molecular dynamics

Abstract: A review is given of how molecular dynamics methods have been modified to perform simulations in the constant-temperature condition. One usually considers a system which is thermally connected with a huge external system (a heat reservoir) to describe a canonical ensemble in statistical mechanics. The way in which this situation is reflected is a key factor for simulations under isothermal conditions. The total kinetic energy is kept to a constant value in constraint methods. In stochastic methods, interactions with a heat bath are treated as random collisions with hypothetical atoms or random forces acting on particles. In the extended-system method, a degree of freedom which mimics a heat bath is introduced, and the total energy of a physical system is allowed to fluctuate.

Thermal ink jet printhead with constant operating temperature

Abstract: A thermal ink jet printer is disclosed which has a printhead that is maintained at a substantially constant operating temperature during printing Printing on demand is accomplished by the ejection of ink droplets from the printhead nozzles in response to energy pulses selectively applied to heating elements located in ink channels upstream from the nozzles which pulses vaporize the ink to form temporary bubbles To prevent printhead temperature fluctuations during printing, especially in translatable carriage printers, the heating elements not being used to eject droplets are selectively energized with energy pulses having insufficient magnitude to vaporize the ink

Closed-form solution of pure proportional navigation

Abstract: The closed-form solution of the equations of motion of an ideal missile pursuing a nonmaneuvering target according to the pure proportional navigation law is obtained as a function of the polar coordinates for all real navigation constants N>or=2. The solution is given in the form of a uniformly convergent infinite product which reduces to a product of a finite number of factors if the navigation constant is a rational number. The solution is discussed, and necessary and sufficient conditions are stated for vanishing, bounded, and unbounded missile acceleration in the final phase of pursuit. >

An analytical solution for one‐dimensional transport in heterogeneous porous media

Abstract: An analytical solution for describing the transport of dissolved substances in heterogeneous porous media with a distance-dependent dispersion relationship has been developed. The analytical solution can be used to characterize differences in the transport process relative to the classical convection-dispersion equation which assumes that the hydrodynamic dispersion in the porous medium remains constant. The form of the hydrodynamic dispersion function used in the analytical solution is D(x) = α(x(v¯ + diffusion, where α(x) is the distance-dependent dispersivity and v¯ is the average pore water velocity. It is shown that for models which differ only in how the dispersion function is expressed, erroneous model parameters may result from parameter estimation techniques which assume a constant hydrodynamic dispersion coefficient if the porous medium is more accurately characterized by a distance-dependent dispersion relationship. For such situations, the proposed model could be used to provide an alternate means for obtaining these parameters. The overall features of the solution are illustrated by several examples for constant concentration and flux boundary conditions.

A new continuous-time model for current-mode control with constant frequency, constant on-time, and constant off-time, in CCM and DCM

Abstract: A novel small-signal model for current-mode control of PWM (pulse width modulation) converters is extended to the analysis of converters with constant on-time control, constant off-time control, and discontinuous conduction mode (DCM). Constant on or off-time converters in continuous conduction mode (CCM) have two complex right-half-plane (RHP) zeros in the current loop feedback, but reduced modulator gain eliminates current-loop instability. The modulator gain exhibits a frequency-dependent phase which introduces an extra gain term in the circuit model. The new model for DCM operation does not have RHP zeros in the current feedback loop, but the buck converter exhibits an instability for higher conversion ratios. >

Perfect zero-knowledge in constant rounds

Abstract: Quadratic residuosity and graph isomorphism are classic problems and the canonical examples of zero-knowledge languages. However, despite much research effort, all previous zero-knowledge proofs for them required either unproven complexity assumptions or an unbounded number of rounds of message exchange. For both (and similar) languages, we exhibit zeroknowledge proofs that require 5 rounds and no unproven assumptions. Our solution is essentially optimal, in this setting, due to a recent lower bound argument of Goldreich and Krawczyk. 1 I n t r o d u c t i o n Interactive proofs and especially zero-knowledge ones have found many applications, most notably in the field of secure protocols. In all such proofs, interaction is the crucial resource, as prover and verifier exchange messages in rounds. The fundamental problem here is whether the number of rounds induces a hierarchy. Tha t is, can we prove more languages in zero knowledge given more rounds? In a cryptographic sett ing the answer is no. But little is known for perfect zero-knowledge proofs. These are the ones tha t can be proven to be zero-knowledge without making use of unproven complexity assumptions. For this reason, perfect zero-knowledge is the right context for studying the intrinsic properties of this notion. There is a gap in what we are able to prove about perfect Zero-Knowledge (ZK): Goldreich and Krawczyk [3] * Suppo r t ed in pa r t by NSF g ran t CCR-87-19689. t S u p p o r t e d in pa r t by NSF g ran t DCR-84-13577 a n d A R O g ran t DAALO3-86-K-0171 . I Pa r t of th is work was done at B o s t o n Univers i ty , D e p a r t m e n t of C o m p u t e r Science, par t ia l ly s u p p o r t e d by NSF g ran t DCR-8607492. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. show that a language outside BPP requires more than 3 rounds from any perfect ZK proof. On the other hand, the classic examples for perfect zero-knowledge, the languages of quadratic residuosity and graph isomorphism, required an unbounded number of rounds. In this paper we show T h e o r e m The languages of graph isomorphism and quadratic residuosity have 5 round perfect zero knowledge interactive proofs. More generally, we show that any random self-reducible language [7] has a 5 round perfect zero knowledge interactive proof. Perfect zero-knowledge works by carefully exploiting the structure of the problem at hand. It is important not only in practice (where we do not know which functions are one-way), but also in a theoretical setting. For example, our best way to prove that a language L, for which no efficient algorithm is known, is not NPcomplete involves exhibiting a perfect zero-knowledge proof for it [2]. Let us now see, at a very high level, why achieving perfect ZK in constant rounds is hard. Essentially, in a ZK proof, the confidence that a theorem is true is transferred by discrete amounts , or tokens. Each token consists of an elementary protocol that decreases the probabil i ty of error by a factor of, say, 2. Thus, after k tokens have been exchanged, this probabil i ty will be reduced to 2 -k . Tokens can be exchanged sequentially or in parallel ( that is concurrently). The ZK constraint implies that the messages exchanged for a single token can be simulated in expected polynomial time. Tha t is, a probabilistic, efficient simulator can output a token in expected, say, 2 trials without any intervention of the prover, and knowing nothing about the proof. Thus, if the proof transfers tokens one at a time, it is easy for the simulator to generate the view relative to k of them: in expected two trials the simulator will output a "good" first token; after that , in expected two more trials, will output a second "good" token, and so on. Thus, overall, the entire view of the protocol can be © 1990 ACM 089791-361-2/90/0005/0482 $1.50 482 simulated in expected 2k trials. If, however, the prover transfers tokens "all together," the job of the simulator is much harder: the expected number of trials so that all k of them are simultaneously "good," is 2 k. This phenomenon which will be made more precise in the context of graph isomorphism is what defeated researchers ever since ZK came about. The main result of this paper is a technique for squeezing the number of rounds in a interactive proof, while preserving simulatability. The proof of our main theorem involves several new ideas; in particular, a non-cryptographic committal scheme and a novel method of simulation in modes. In fact, although the protocol only requires 5 rounds and is surprisingly simple, the simulation argument is complex. Moreover, our result is essentially optimal due to the general 4-round lower bound of Goldreich and Krawczyk [3]. In this extended abstract we confine ourselves to a proof that graph isomorphism has a 5 round perfect zero knowledge interactive proof. 2 D e f i n i t i o n s 2 .1 P r o b a b i l i t y S p a c e s a n d A l g o r i t h m s These notations and conventions for probabilistic algorithms are derived from [6] and further extended. We emphasize the number of inputs received by an algorithm as follows. If algorithm A receives only one input we write "A(.)"; if it receives two we write "A(., .)", and so on. If A is a probabilistic algorithm then, for any input i the notation A(i) refers to the probability space which to the string (r assigns the probability that A, on input i, outputs or. If S is a probability space we denote by Ps (e ) the probability that S associates with element e. We denote by IS] the set of elements to which S assigns positive probability. If f ( . ) and g( . , . . . ) are probabilistic algorithms then f ( g ( . , . . . ) ) is the probabilistic algorithm obtained by composing f and g (i.e. running f on g's output) . For any inputs x , y , . . , the associated probability space is denoted f (g (x , y , . . . ) ) . If 5: is a probability space then x *-S denotes the algorithm which assigns to x an element randomly selected according to S (that is, x is assigned the value e with probability P s ( e ) ) (in the case that [,9] consists of 3nly one element e we write x ~ e rather than x +--{e}). For probability spaces S, T . . . . , the notation P ( p ( x , y , . . . ) : x ~--S ; y ~--T ; . . . ) denotes the probability that the predicate p(x, y , . . . ) is true after the (ordered) execution of the algorithms x ~-S, y ~ T, etc. The notation f f ( x , y , . . . ) : x ~-S ; y *--7 " ; . . . } t denotes the probability space which to the string ~r assigns the probability P(cr = f ( x , y , . . . ) : x *-S ; y *-T ; . . . ) , f being some function. If S is a finite set we will identify it with the probability space which assigns to each element of S the uniform probability U(S) = ]-~. (Then x ~-S denotes the operation of selecting an element of 5' uniformly at random, and Ps (e ) = U(S) for any e E S). We let P P T denote the set of probabilistic (expected) polynomial time algorithms. 2 . 2 I n t e r a c t i v e T M s a n d P r o t o c o l s Full descriptions of interactive Turing Machines and protocols appear in [5]; we will only summarize the notation we use. The probability that (A, B) accepts the common input x is denoted P ( ( A , B ) accepts x ) , and the probability space of all conversations between A and B on input x is denoted (A ~ B) (x ) (the probability in both cases is taken over the random tapes of both A and B). Sometimes we want to make the coin tosses of B explicit. For any R E {0, 1}* we write B ( R ) for the (deterministic) machine B with R as its random tape. Then (A ~-~ B (R) ) ( x ) denotes the probability space of conversations between A and B(R) on input x (the probability is over the random tapes of A). We let B(R;x ,oq131 . . . a i l f l i -1 ) denote the next message that B ( R ) sends when the conversation up to this point was cq/31 . . . oli_l~i_ 1. Finally, it is convenient to define the state of the ITM B at any point in its computation as consisting of the contents of its tapes, the positions of the heads on these tapes, and the internal state of the machine. 2 . 3 I n t e r a c t i v e P r o o f S y s t e m s a n d Z e r o K n o w l e d g e Def in i t i on 2.1 An interactive protocol (P, V) is an interactive proof system for the language L if the following conditions hold: • Completeness: For every x E L, P ( ( P , V) accepts x) > 1 2 -I~l .

Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space

Abstract: The Gauss map of a spacelike constant mean curvature hypersurface of Minkowski space is a harmonic map to hyperbolic space. The properties of such hypersurfaces are interpreted in terms of the harmonic mapping. Given an arbitrary closed set in the ideal boundary at infinity of hyperbolic space, there exists a complete entire spacelike constant mean curvature hypersurface whose Gauss map is a diffeomorphism onto the interior of the hyperbolic space convex hull of the set. Identifying ideal infinity with the light cone, this set corresponds to the lightlike directions of the hypersurface. In terms of this extrinsic data we give conditions for the hyperbolicity or parabolicity of this hypersurface. For example, if the set of lightlike directions has nonempty interior in the unit sphere, then this hypersurface can be constructed so as to admit nontrivial bounded harmonic functions. This gives many new examples of harmonic maps of the disk and the complex plane to the hyperbolic plane, which are of full rank. There are relatively few examples of harmonic maps between noncompact manifolds. A class of such maps arises as the Gauss maps of entire spacelike constant mean curvature hypersurfaces of Minkowski space. These are harmonic maps into hyperbolic space by a verson of the RuhVilms theorem. We construct certain constant mean curvature hypersurfaces and analyse theit geometric and function theoretic properties. Thus, we are able to answer a question of Eells and Lemaire [17] by constructing many harmonic maps from R to the hyperbolic plane, which have rank two everywhere. We also construct maps of the hyperbolic plane into itself. A completely elementary example of family of nonconformal harmonic diffeomorphisms of the hyperbolic plane to itself has been constructed using similar methods [14]. The relation between harmonic maps and constant mean curvature surfaces has been studied by T. K. Milnor Received December 7, 1988 and, in revised form, March 8, 1989. The first author was partially supported by National Science Foundation grant DMS8503327, and the second author by The Sloan Foundation and National Science Foundation grant DMS8700783. 776 HYEONG IN CHOI & ANDREJS TREIBERGS [31]. Akutagawa and Nishikawa [1] and O. Kobayashi [25], [26] have used harmonic maps to construct constant mean curvature surfaces. Building on the earlier work of Cheng-Yau [11] and Treibergs [38], we obtain a structure theory of the Gauss map. The entire spacelike hypersurfaces u of constant mean curvature are asymptotically null in certain directions called the lightlike directions, Lu. We prove that after splitting off a trivial factor, the Gauss map is a harmonic diffeomorphism onto the interior of the convex hull of Lu in hyperbolic space. The convex hull is taken in the following sense: the hyperbolic space is canonically identified with the hyperboloid consisting of unit future-pointing timelike vectors and then the lightlike rays are naturally identified with the points at infinity of hyperbolic space. The convex hull of Lu is thus the usual convex hull in hyperbolic space with respect to the hyperbolic metric. The proof is based on the observations that the entire constant mean curvature hypersurfaces are graphs of convex functions u, and that the Gauss map into the Klein model of hyperbolic space is the tangential mapping of u. Conversely, any closed subset of the boundary at infinity is the set of lightlike directions of some entire spacelike constant mean curvature hypersurface. This is a direct interpretation of the solubility of the mean curvature equation with boundary values at infinity [38]. By sharpening the existence theory, we construct constant mean curvature hypersurfaces with given asymptotic behavior along the lightlike directions showing that there are many different surfaces with the same Lu . We can also construct solutions which have good asymptotic estimates in interior directions of Lu . This guarantees that there is an abundant supply of interesting harmonic maps to hyperbolic space. The existence of entire prescribed mean curvature hypersurfaces has also been studied in Minkowski space [8], [36] as well as in more general Lorentzian manifolds (e.g. [6], [7], [16], [18].). To understand the mappings obtained as Gauss maps of constant mean curvature hypersurfaces, we are able to determine the function theory of the hypersurfaces from the lightlike set in some cases. We show that if the lightlike set contains a ball, then the hypersurface is hyperbolic in the sense that it admits a nontrivial bounded harmonic function. In this case the geometry is well controlled in the sector corresponding to the ball. It is quasi-isometric to a sector of the hyperbolic space, and the sectional curvature of the sector is pinched between two negative constants. This implies that there are nonconstant bounded harmonic functions on u. The proof of these facts depends on using the asymptotic behavior to estimate the metric and integral Gauss-Kronecker curvature. Then we prove a local Harnack-like mean value inequality and apply it to the Gauss-Kronecker curvature to conclude the result.

Knotted solutions of the Maxwell equations in vacuum

Abstract: It is shown that there exist electromagnetic knots, which are standard electromagnetic fields in which any pair of magnetic lines whatsoever (or of electric lines) are linked and the magnetic helicity has the value integral A.Bd 3r-na, n being a topological constant of the motion given by a Hopf index and a a certain action constant. A family of n=1 knots is obtained, the projection of spin on three-momentum being equal to a.

State-Dependent Utilities

Abstract: Several axiom systems for preference among acts lead to the existence of a unique probability and a state-independent utility such that acts are ranked according to their expected utilities. These axioms have been used as a foundation for Bayesian decision theory and the subjective probability calculus. In this paper, we note that the uniqueness of the probability is relative to the choice of what counts as a constant outcome. Although it is sometimes clear what should be considered constant, there are many cases in which there are several possible choices. Each choice can lead to a different “unique” probability and utility. By focusing attention on state-dependent utilities, we determine conditions under which a truly unique probability and utility can be determined from an agent’s expressed preferences among acts. Suppose that an agent’s preference can be represented in terms of a probability P and a utility U. That is, the agent prefers one act to another if and only if the expected utility of the one act is higher than that of the other. There are many other equivalent representations in terms of probabilities Q, which are mutually absolutely continuous with P, and state-dependent utilities V, which differ from U by possibly different positive affine transformations in each state of nature. An example is described in which two different but equivalent state-independent utility representations exist for the same preference structure. What differs between the two representations is which acts count as constants. The acts involve receiving different amounts of one or the other of two currencies and the states are different exchange rates between the currencies. It is easy to see how it would not be possible for constant amounts of both currencies to simultaneously have constant values across the different states. Savage (Foundations of statistics. John Wiley, New York, 1954, sec. 5.5) discovered a situation in which two seemingly equivalent preference structures are represented by different pairs of probability and utility. Savage attributed the phenomenon to the construction of a “small world”. We show that the small world problem is just another example of two different, but equivalent, representations treating different acts as constants. Finally, we prove a theorem (similar to one of Karni, Decision making under uncertainty. Harvard University Press, Cambridge, 1985) that shows how to elicit a unique state-dependent utility and does not assume that there are prizes with constant value. To do this, we define a new hypothetical kind of act in which both the prize to be awarded and the state of nature are determined by an auxiliary experiment.

Transparent layer constancy.

Abstract: Perceived transparency was studied as a constancy problem. In the episcotister (E-) model of scission, luminances are partitioned into layer and background components; four luminances determine values of two layer parameters that specify constancy of a transparent layer on different backgrounds. The E-model was tested in an experiment in which 12 Ss matched 24 pairs of four-luminance patterns by adjusting two luminances of the comparison pattern. Both the standard and the comparison were perceived as a transparent layer on a checkerboard. The E-model predicts matches when layer values are identical in the two patterns. One parameter was constant, constraining the adjustment along the second dimension. Obtained values corresponded well with E-predictions. Alternative models based on local luminance or average contrast ratios accounted for less variability. Results indicate that transparency models should utilize luminance, not reflectance, as the independent variable. Language: en

Competition on a divided and ephemeral resource testing the assumptions ii. association

Abstract: (1) We test the assumptions of the 'aggregation model' of coexistence on a divided and ephemeral resource: (i) that the competing stage is aggregated; (ii) that the distributions of individuals can be represented by a negative binomial distribution; and (iii) that the degree of aggregation can be represented by a constant parameter. The assumption of no association between species is tested in a following paper. (2) Tests are made on 360 datasets representing distributions of Diptera, particularly drosophilids, on a total of 7638 resource patches. Results indicate that for this data: (i) competing stages are strongly aggregated; (ii) the negative binomial is not an exact description but may be adequate; and (iii) aggregation varies with density. (3) We relax the assumption of constant aggregation and develop a model of two-species competition where aggregation varies with density. We show that, with observed estimates of parameters for drosophilids, coexistence between pairs of species with no resource partitioning remains likely despite strong local competition.

Motion of a contact line on a heterogeneous surface

Abstract: Using the lubrication approximation, we calculate the dynamics of a contact line on a periodic heterogeneous plate which is advanced at constant force F or constant velocity u. For a constant force, motion starts when F exceeds a threshold Fc which is simply related to the static advancing (receding) contact angle. If the heterogeneity is smooth, the force increases with velocity as F−Fc∝u2. Alternating patches of constant wettability produce a linear relation. For constant velocity experiments, we identify weak and strong pinning regimes. In the weak pinning regime, the threshold force is zero and the force–velocity relation approaches that of a uniform surface. In the strong pinning regime, the threshold F0 is finite and approaches Fc as the strength of the heterogeneity increases. For smooth heterogeneity, F−F0∝u2/3, while alternating patches produce a linear response. The relevance of these results to experimental surfaces with random heterogeneity is discussed.

Spin coating apparatus

Abstract: The present invention relates to a spin coating apparatus for feeding a clean liquid at a fixed rate. In any of the feeding apparatus in the prior art, no constituent other than a filter has the function of eliminating foreign matter, and the operation of feeding a liquid under precise control is not attained. This results in the problem of the mixing of the foreign matter (and air bubbles) in the feed liquid, and the problem of nonuniformity in a feed speed as well as a feed amount. As expedients for solving these problems, the present invention provides the functions of automatically sensing and excluding factors for the appearances of the foreign matters, and devices for automatically and precisely controlling feed control elements such as a pump, thereby feeding the clean liquid in a constant amount and at a constant speed.

Constructing the procedure call multigraph

Abstract: An algorithm for constructing a precise call multigraph for languages that permit procedure parameters, extending the method of B. Ryder (see ibid., vol.5, no.3, p.216-225 (1979)) for handling recursion, is presented. If it is assumed that there is a constant upper bound on the number of procedure parameters to any procedure in the program, then the algorithm is polynomial in the total number of procedures in the program. >

Functions of Modulus 1 on Z n Whose Fourier Transforms Have Constant Modulus, and “CYCLIC n -ROOTS”

Abstract: We consider on one hand elements x in C n as functions on the cyclic group Z n . A function which has a constant modulus and whose Fourier transform also has a constant modulus, is called bi-equimodular. On the other hand, we study the set CR(n) of “cyclic n-roots”, i. e. solutions of a certain system (2.1) of algebraic equations. We prove that there is essentially a one-one correspondence between the bi-equimodular functions and the cyclic n-roots of modulus one. For prime p, we give preliminary results about the structure of the bi-equimodular p-functions, starting from our negative answer in [2] to the question asked by Enflo whether the “classical” ones known already to Gauss are the only ones existing. The structure is related to the subgroups of the multiplicative group on Z p \ {0}, e. g. the group of quadratic residues. For n ≤ 7, CR(n) is completely known. The proofs (in [3]) depend on computer work, using symbolic algebra. Related material can be found in the book [7] by Schroeder and in the paper [6] by McGehee.

The ice/water interface: A molecular dynamics simulation using the simple point charge model

Abstract: The ice/water interface is investigated using the simple point charge (SPC) model for water. Molecular dynamics simulations at constant NVE demonstrate that the model interface is stable on the time scale of at least 300 ps. The density profile and mean square displacement of molecules are measured, and the results are nearly identical to those obtained earlier using the TIP4P model for water. This is additional evidence that the properties measured in the simulation are intrinsic to the ice/water interface and do not depend on the particular model potentials employed.

Optimized Strategies for Scaling Goal-Directed Dynamic Limb Movements

Abstract: A recurring theme of the previous five chapters has been a consideration of control strategies for goal-directed movements involving one of the upper limbs. In each chapter specific classes of tasks were considered. For each task, specific instructions were given to adult subjects who presumably complied as well as possible with the stated task goals. Their movement strategies were a function of a specific neuromusculoskeletal apparatus which had been tuned through years of constant use. Their performance, however, was also a function of many hard-to-define variables, including “mood”, perception of instructions, the amount of practice, the level of mental and/or physical fatigue, and choice of limb. These factors contribute to intra- and intersubject variability. As pointed out in Chapter 34 (Pedotti and Crenna), however, such variability may be related to different strategies as opposed to “noise” about a single strategy.

Use Of Constant Time Delay In Small Group Instruction: A Study Of Observational And Incidental Learning

Abstract: This investigation examined the effectiveness and efficiency of constant time delay in small group instruction. Four secondary-age students with mild and moderate mental retardation were taught to identify local and federal service and government agencies and over-the-counter medications. The amount of information learned when each subject was presented with two target and six observational stimuli (same-task, different-stimuli condition) was compared to when each student in the group was taught the same eight target stimuli (same-task, same-stimuli condition). The subjects' acquisition of incidental information presented in the descriptive praise statement following correct responses to the target stimuli was assessed. A multiple probe design across behaviors was used. The results indicated that (a) constant time delay was effective across all facts, students, and conditions; (b) the same-task, different-stimuli condition produced more efficient learning than the same-task, same-stimuli condition; (c) st...