Showing papers on "Product (mathematics) published in 1968"

Minimum Uncertainty Product, Number‐Phase Uncertainty Product, and Coherent States

Abstract: The number‐phase uncertainty products proposed by Carruthers and Nieto are studied to determine whether they are minimized by coherent states. It is found that coherent states do not minimize these products. States that do minimize some of the uncertainty products are constructed. Variational techniques for the study of arbitrary uncertainty products are developed.

A New Algorithm for Inner Product

Abstract: —In this note we describe a new way of computing the inner product of two vectors. This method cuts down the number of multiplications required when we want to perform a large number of inner products on a smaller set of vectors. In particular, we obtain that the product of two n×n matrices can be performed using roughly n3/2 multiplications instead of the n3multiplications which the regular method necessitates.

Chapter 5 – Coverings of Automata

Abstract: Publisher Summary This chapter discusses the coverings of automata. The chapter reviews Moore and Mealy machines. Simplicity of an automaton can be measured, for example, by the number of its states. Homomorphisms of automata, and homomorphism and covering are discussed. The admissible output-consistent decompositions, and reduction of covering of an automaton to covering of a semi-automaton are reviewed. Various theorems are proved in the chapter. Some properties of the notion of covering of semi-automata are derived. Construction of an auxiliary semi-automaton is reviewed. The chapter reviews the direct product of semi-automata, and the cascade product of semi-automata. The construction of a direct product can be, in an obvious way, generalized to any finite number of semi-automata with a common or appropriately translated set of inputs.

The neuron as a Lie group germ and a Lie product

Abstract: A basic Lie group derived earlier that describes the \"constancies\" of visual perception is extended to one more dimension to take into account the nonlinear flow of an afferent volley of nerve impulses through the layers of the visual cortex. One is then led, via the usual determination of the solutions of a Lagrange partial differential equation in terms of an associated Pfaffian system of ordinary differential equations, to a correspondence between neuron cell body, Lie group germ, and critical point of the system of ordinary differential equations governing the orbits. The local phase portraits of the latter bear a marked resemblance to one or the other of the neuron types defined by Sholl. Since \"brains are as different as faces\", the concept of structural stability plays an important role in analyzing the connectivity of the neural network. Finally, Lukasiewicz's theory of parentheses is used to obtain a graph-theoretic representation of the Jacobi identity, which then serves to explain the branching of neuronal processes (dendrites).

Multifunction production monitor

Abstract: A system for monitoring the production of a work station wherein a plurality of process functions must be performed for a satisfactory product. The plurality of functions are sensed in the system to actuate a product count. The functions can be defined by limits on a range as temperature, pressure or curing time. Sequencing of the functions can be critical as where pressure must be imposed prior to heating an article being processed to produce a count of a satisfactory product. Combinations of function limits and sequences of functions can be utilized to indicate an acceptable process cycle. Several processes can be imposed on the product with a grading of the final product dependent upon the grading of intermediate product and process parameters. Several grades of product can be sensed and counted in the system depending upon the values or sequences of functions being performed. For example, a cure time outside the limits for a prime grade product, if within the looser limits of a secondary grade product causes the resultant product, other functions being within limits, to be counted as a secondary grade product.

Representations and Classes in Groups of Finite Order

Abstract: There are two main results in this paper. First, it is shown that we can develop a theory of classes in close analogy to the usual theory of representations. We can introduce concepts, such as reducible and irreducible classes, sum and product of classes, reduction of a class when going from a group to a subgroup, etc. Second, it is shown that it is possible to associate a ``magic square'' to each group. It is related to the numbers of pairs of commuting elements between classes and it can be used immediately to find the structure of the ``tensor operators'' of the group.

Limit Theorems for the Multi-urn Ehrenfest Model

Abstract: In the multi-urn Ehrenfest model $N$ balls are distributed among $d + 1 (d \geqq 2)$ urns. If we label the urns $0, 1, \cdots, d,$ then the system is said to be in state $\mathbf{i} = (i_1, i_2, \cdots, i_d)$ when there are $i_j$ balls in urn $j (j = 1, 2, \cdots, d)$ and $N - \mathbf{1}\cdot\mathbf{i}$ balls in urn 0. (The vector $\mathbf{1}$ has all its components equal to 1 and $\mathbf{x} \cdot \mathbf{y}$ is the usual scalar product.) At discrete epochs a ball is chosen at random from one of the $d + 1$ urns; each of the $N$ balls has probability $1/N$ of being selected. The ball chosen is removed from its urn and placed in urn $i (i = 0, 1, \cdots, d)$ with probability $p^i,$ where the $p^i$'s are elements of a given vector, $(p^0, \mathbf{p})$, satisfying $p^i > 0$ and $\sum^d_{i=0} p^i = 1.$ We shall let $\mathbf{X}_N(k)$ denote the state of the system after the $k$th such rearrangement of balls. Our interest in this paper is to obtain limit theorems for the sequence of processes $\{\mathbf{X}_N(k): k = 0, \cdots, N\}$ as $N$ tends to infinity. For the classical Ehrenfest model $(d = 1, p^0 = p^1 = \frac{1}{2})$ Kac [7] showed that the distribution of $(X_N(\lbrack Nt\rbrack) - N/2)/(N/2)^{\frac{1}{2}}$ converges as $N \rightarrow \infty$ to the distribution of the Ornstein-Uhlenbeck process at time $t$ having started at $y_0$ at $t = 0,$ provided $X_N(0) = \lbrack(N/2)^{\frac{1}{2}}y_0 + N/2\rbrack.$ (The symbol $\lbrack x\rbrack$ denotes the integer part of $x$.) Recently, Karlin and McGregor [8] obtained a similar result for the continuous time version of the model with $d = 2$; in this version the random selection of balls is done at the occurrence of events of an independent Poisson process. In addition, they obtained a local limit theorem for the transition function. The proof in [7] depended on the continuity theorem of characteristic functions. On the other hand, the proof in [8] used the properties of the spectral representation of the $n$-step transition probabilities which is available for these processes. These results suggested the direction we shall follow in this paper. A preliminary calculation indicates that the process $\{\mathbf{X}_N(k): k = 0, \cdots, N\}$ is attracted to the pseudo-equilibrium state $N\mathbf{p}$ and that states far from $B\mathbf{p}$ will only occur rarely. Thus it is natural to consider the fluctuations of $\mathbf{X}_N(k)$ about $N\mathbf{p}$ measured in an appropriate scale. For our purposes the appropriate processes to consider are $\{\mathbf{Y}_N(k): k = 0, \cdots, N\}$, where $\mathbf{Y}_N(k) = (\mathbf{X}_N(k) - N\mathbf{P})/N^{\frac{1}{2}}.$ Next we define a sequence of stochastic processes $\{\mathbf{y}_N(t): 0 \leqq t \leqq 1\}$ which are continuous, linear on the intervals $((k - 1)N^{-1}, kN^{-1})$, and satisfy $\mathbf{y}_N(kN^{-1}) = \mathbf{Y}_N(k)$ for $k = 0, 1, \cdots, N.$ In other words we let $\mathbf{y}_N(t) = \mathbf{Y}_N(k) + (Nt - k)(\mathbf{Y}_N(k + 1) - \mathbf{Y}_N(k))$ if $kN^{-1} \leqq t \leqq (k + 1)N^{-1}$. Throughout this paper we shall let $X_N^i(0) = \lbrack N^{\frac{1}{2}}y_0^i + Np^i\rbrack,$ where $y_0 = (y_0^1, \cdots, y_0^d)$ is an arbitrary, but fixed, element of $R^d.$ (It will always be understood that $N$ is sufficiently large so that $0 \leqq X_N^i(0) \leqq N$ for all $i = 1, 2, \cdots, d$, where $X_N^i(\cdot)$ is the $i$th component of the vector $\mathbf{X}_N(\cdot). R^d$ is $d$-dimensional Euclidean space.) Observe that this initial condition implies that $|Y_N^i (0) - y_0^i| \leqq N^{-\frac{1}{2}}.$ With this initial condition and the Markov structure of the model, the processes $\{\mathbf{X}_N(k): k = 0, \cdots, N\}$ for $N = 1, 2, \cdots$ can be defined on a probability triple $(\Omega_N, \mathscr{F}_N, P_N).$ We shall let $C_d\lbrack 0, 1\rbrack$ denote the product space of $d$ copies of $C\lbrack 0, 1\rbrack$, the space of continuous functions on $\lbrack 0, 1\rbrack$ with the topology of uniform convergence, and endow $C_d\lbrack 0, 1\rbrack$ with the product topology. The topological Borel field of $C_d\lbrack 0, 1\rbrack$ will be denoted by $\mathscr{C}_d$. Clearly, the transformation taking the sequence $\{\mathbf{X}_N(k): k = 0, \cdots, N\}$ into $\{\mathbf{y}_N(t): 0 \leqq t \leqq 1\}$ is measurable and induces a probability measure on $\mathscr{C}_d$. We shall denote this induced measure by $\mu_N(\cdot; \mathbf{y}_0)$. The general notion of weak convergence of a sequence of probability measures is defined as follows. Let $S$ be a metric space and $\mathscr{S}$ be the Borel field generated by the open sets of $S$. If $ u_N$ and $ u$ are probability measures on $\mathscr{S}$ and if the $\lim_{N\rightarrow\infty} \int _sf d u_N = \int _sf d u$ for every bounded, continuous function $f$ on $S$, then we say that $ u_N$ converges weakly to $ u$ and write $ u_N\Longrightarrow u$. The principal result of this paper is that $\mu_N(\cdot; \mathbf{y}_0) \Longrightarrow \mu (\cdot; \mathbf{y}_0)$ as $N \rightarrow \infty$, where $\mu(\cdot; y_0)$ is the probability measure on $\mathscr{C}_d$ of a $d$-dimensional diffusion process, $\mathbf{y}(\cdot)$, starting at the point $\mathbf{y}_0$. The limit process $\mathbf{y}(\cdot)$ is a $d$-dimensional analog of the Ornstein-Uhlenbeck process whose distribution at time $t$ is a multi-variate normal with mean vector $e^{-t}\mathbf{y}_0$ and covariance matrix $\Sigma$, where the elements of $\Sigma$ are \begin{equation*}\begin{split}\sigma_{ij} = (1 - e^{-2t})p^i(1 - p^i),\quad i = j, \\ = -(1 - e^{-2t})p^ip^j,\quad i eq j.\end{split}\end{equation*} For applications it is useful to note that $\mu_N(\cdot; \mathbf{y}_0)\Longrightarrow\mu(\cdot;\mathbf{y}_0)$ is equivalent to the statement that $\lim_{N\rightarrow\infty} \mu_N(\{f(\mathbf{y}(\cdot)) \leqq \alpha\}; \mathbf{y}_0) = \mu(\{f(\mathbf{y}(\cdot)) \leqq \alpha\}; \mathbf{y}_0)$ for all functionals $f$ on $C_d\lbrack 0, 1\rbrack$ which are continuous almost everywhere with respect to $\mu(\cdot; \mathbf{y}_0)$. To establish weak convergence two steps are usually required. First, the convergence of the finite-dimensional distributions (fdd) of the approximating processes, $\{\mathbf{y}_N(t): t \geqq 0, N = 1, 2, \cdots\}$ in our case, to the corresponding fdd of the limiting process must be obtained. Second, the probability that the approximating processes can have large fluctuations between points at which these processes are determined by their fdd must be shown to be small. The notion of weak convergence is intimately related to the so-called invariance principles. An invariance principle was first introduced for the case of sums of independent, identically distributed random variables by Erdos and Kac [4] and generalized by Donsker [3]. The method of proof used by Erdos-Kac and Donsker was later modified by Billingsley [1] for dependent random variables. In carrying out the second step outlined above we shall follow Billingsley's argument in Theorems 2.3 and 3.1 of [1]. A program similar in nature to ours was carried out by Lamperti [10] for a particular class of Markov processes. This paper is organized into the following sections: Section 2 is devoted to our analog of the central limit theorem (clt), namely, that the distribution of $\mathbf{y}_N(t)$ converges to the distribution of $\mathbf{y}(t)$ at a single fixed value of $t$. In Section 3 the limit process $\mathbf{y}(t)$ is identified and the properties of the process needed here are discussed. Section 4 completes the proof of the convergence of the fdd of $\{\mathbf{y}_N(t)\}$ to those of $\{\mathbf{y}(t)\}$. The proofs in both Sections 2 and 4 are carried out using the Levy continuity theorem for characteristic functions. Section 5 provides the proof required to show weak convergence. The main tool here, in addition to Billingsley's theorems mentioned above, is the result of Stone (1961) on the weak convergence of random walks. Finally, in Section 6 applications are mentioned along with a suggestion as to how the multi-urn Ehrenfest model might be used to study certain problems in statistical mechanics, networks of queues, and epidemic theory.