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The d-c dynamic braking torque speed curve of a-c motors, dependent upon a magnetizing reactance which varies progressively throughout the speed range, is approximated by a discontinuous curve dependent upon the magnetizing-reactance values at standstill and top speed only. The former reactance corresponds to the full d-c excitation, while the latter is the unsaturated value. By this means, the torque speed curve is expressed by a simple mathematical relationship. The validity of this simplification is investigated by comparison with an accurate method of deducing the curve, published recently by the author, for any degree of magnetic saturation and value of the leakage reactance. Good agreement is obtained at all except the largest values of torque, which occur over a comparatively small speed range. In consequence, the stopping time and total energy loss of the motor, which are dependent upon the integration of the inverse torque throughout the speed range for an inertia load, are readily determined with considerable accuracy by the simplified theory, using derived mathematical relationships. Further, the value of secondary resistance which provides the minimum stopping time and energy loss can be readily deduced. Experimental confirmation of the theory is included.

We have applied a stopping theory of high velocity heavy ions to the problem of nuclear recoil from alpha decay, and have used straightforward ideas to derive the stopping time. The case of 237Np stopping in NpO2 has been worked out in detail, and is discussed in light of recent experiments on the Mossbauer effect following α-decay of 241Am. The computed range of 93 keV 237Np recoil nuclei is 2·36 × 10−6 cm, and the stopping time is approximately 3·9 × 10−13 sec. As these results are relatively insensitive to the particular system studied, we conclude that recoil stopping times following α-decay are extremely short in comparison to lifetimes of nuclear states likely to emit Mossbauer γ-rays. Hence the relaxation processes which occur at the end of the recoil track will set the influential time scale in Mossbauer experiments.

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: А. К. Алешкявичене. Локальные теоремы для времени до первого достижения в случайном блуждании A. Aleškevičienė. Lokalinės teoremos pirmo patekimo laikui atsitiktiniame klaidžiojime

The usual (non-stochastic stopping) control problem is extended to the case of random terminal time. The more general model presented hero should be particularly useful when a system will change while being controlled. Systems which are otherwise deterministic, and systems with additive noise in the dynamic equation are considered. Results concerning the relevant aspects of reliability and Markov process theories are presented. We show that stochastically stopped control optimality conditions are simply extensions of the usual conditions and that the limits of the criteria and optimal control, as the variance of the stopping probability distribution approaches zero, are the corresponding quantities for the non-stochastic problem.

Publisher Summary This chapter focuses on stopping time in sequential samples from multivariate exponential families. The chapter presents a general theorem that proves exponential boundedness of the stopping time N in invariant sequential probability ratio tests if certain conditions are satisfied. The theorem is applied to two examples, the first being a test about the characteristic roots of the covariance matrix in a bivariate normal population with zero mean, and the second being a test about the norm of the mean vector in a multivariate normal population with identity covariance matrix.

Publisher Summary This chapter presents a theorem on exponentially bounded stopping time of invariant sequential probability ratio tests (SPRTs) with applications. It presents an assumption that Z 1 , Z 2 , … are iid random variables with common distribution P and the joint distribution of the Zs is also denoted as P . The chapter presents a description of a model based on this assumption a specified family of distributions ρ = {P θ : θ ∈ Ξ} where Ξ is some index set, also called the parameter space. P may or may not be a member of ρ. The chapter also highlights that a priori there is no reason why Stein's result for Wald's SPRT could not also be valid for invariant SPRT's. However, as a result of the difficulty of the problem, the beginnings of the investigation of N for invariant SPRT's were very modest.

We are motivated by Stein's proof (Stein (1946), Wald (1947), pages 157-158) of the termination of a sequential probability ratio test in the case of independent and identically distributed random variables. Extending his ideas to take certain "dependencies" into account we examine the rank-order sequential probability ratio test based on a Lehmann alternative studied in a paper with the above title by I. R. Savage and the author (1966) (referred to as SS I in the rest of this paper). We prove that this test terminates with probability one and that the stopping time has a finite moment generating function under a very mild condition on the bivariate random variables which resembles the Stein-condition, namely that a certain random variable $V(X_1, Y_1)$, defined in (32), is not identically equal to 0. Finally the asymptotic normality of the logarithm of the likelihood ratio of the rank order is established using the well-known Chernoff-Savage Theorem.

Let $N$ be the stopping variable of a SPRT for testing one composite hypothesis against another, based on i.i.d. observations $Z_1, Z_2, \cdots$ with common distribution $P. P$ need not belong to the model. $N$ is termed exponentially bounded if for every choice of stopping bounds there exists $c < \infty$ and $\rho < 1$ such that $P\{N > n\} < c\rho^n$; if this does not hold $P$ is called obstructive. The main theorem presents sufficient conditions, both on the model and on $P$, for $N$ to be exponentially bounded. Under weaker conditions the theorem proves $P\{N < \infty\} = 1$. Two applications of the theorem are given: 1. In the problem of testing $\sigma = \sigma_1$ against $\sigma = \sigma_0$ in a normal population with unknown mean it is proved that $N$ is exponentially bounded for every $P$ except if $P\{Z_1 = \zeta \pm a\} = \frac{1}{2} (\zeta$ arbitrary and $a^2$ a given function of $\sigma_1$ and $\sigma_2)$ in which case $P$ is obstructive. 2. In the sequential $t$-test it is proved that $N$ is exponentially bounded for every $P$ for which $Z_1^2$ has finite $\operatorname{mgf}$ and is not a member of a certain family of two-point distributions.

Experimental results from a number of small ruby lasers show a linear relationship between the pump power and the inverse of the time delay between pump flash and the onset of laser oscillation. These experimental results are quantitatively predicted by a simple rate-equation analysis, yielding a single theoretical curve with no adjustable parameters and in good agreement with the experiments. Additional evidence is also presented verifying the onset at higher pump powers of additional "bouncing-ball" or "light-pipe" modes of oscillation in ruby rods with polished side walls. These additional modes account, in particular, for the abrupt cessation of laser output after an anomalously short duration of laser action, as is typically observed in such rods. These anomalous modes, which are usually not well coupled to the detection circuitry, apparently suppress the normal mode of oscillation once they begin and limit the available output energy in the normal mode. The anomalous modes are eliminated by roughening the sides of the rod, after which normal duration of laser action is observed. The laser stopping time is also predicted with reasonable accuracy by the rate-equation analysis.