Abstract: The purpose of this paper is to extend to Hardy spaces in several variables certain well known factorization theorems on the unit disk. The extensions will be carried out for various spaces of holomorphic functions on the unit ball of C" as well as for Hardy spaces defined by the Riesz systems on R". These results together with their proofs yield new characterizations of the space BMO (Bounded Mean Oscillation) and show a close relationship between BMO functions and certain linear operators on various LI and H2 spaces. The main tools are the result of Fefferman and Stein [8] on the duality between HI and BMO and a new characterization of BMO in terms of boundedness on L2 of the commutator of a singular integral operator with a multiplication operator. We begin by illustrating these ideas in the one dimensional case: Let F be holomorphic in {I z I < 1} and satisfy sup, 5 F(rete) I dO ? 1 (i.e., F is in H'(dO)). It is well known that F = GG2 with G1, G2 holomorphic and sup, I G,(rel0) 1' ! 1 (i.e., G, e H2(dO)). Write F = f + if, G, = gj + ig withf, g1, g, real. Thenf = Im(GG2) = sg1 1 + gi. Thusafunction f is an imaginary (or real) part of an HI function if and only if it can be represented as glg2 + g192 for L2 functions g, and g2. Furthermore,

Abstract: A very primitive version of Gotlieb’s timetable problem is shown to be NP-complete, and therefore all the common timetable problems are NP-complete. A polynomial time algorithm, in case all teachers are binary, is shown. The theorem that a meeting function always exists if all teachers and classes have no time constraints is proved. The multicommodity integral flow problem is shown to be NP-complete even if the number of commodities is two. This is true both in the directed and undirected cases.

Abstract: In this paper we establish a theoretical basis for utilizing a penalty-function method to estimate sensitivity information (i.e., the partial derivatives) of a localsolution and its associated Lagrange multipliers of a large class of nonlinear programming problems with respect to a general parametric variation in the problem functions. The local solution is assumed to satisfy the second order sufficient conditions for a strict minimum. Although theoretically valid for higher order derivatives, the analysis concentrates on the estimation of the first order (first partial derivative) sensitivity information, which can be explicitly expressed in terms of the problem functions. For greater clarity, the results are given in terms of the mixed logarithmic-barrier quadratic-loss function. However, the approach is clearly applicable toany algorithm that generates a once differentiable "solution trajectory".

Abstract: Let A, B be $n \times n$ matrices, f a vector-valued function. A and B may both be singular. The differential equation $Ax' + Bx = f$ is studied utilizing the theory of the Drazin inverse. A closed form for all solutions of the differential equation is given when the equation has unique solutions for consistent initial conditions.

Abstract: The effects of output price uncertainty on a competitive firm's supply and factor demands have recently been explored under the assumption that all decisions are made before the price is observed. Agnar Sandmo has shown that a risk-averse, competitive firm with a nonrandom cost function will produce a smaller output if the price is random than it would if the price were known with certainty to equal its mean. Raveendra Batra and Aman Ullah and I have extended the analysis by considering the effects of output price uncertainty on factor demands. Among other things, Batra and Ullah show that the firm will choose its inputs to minimize the cost of producing whatever level of output is chosen. This result, combined with the Sandmo result that the presence of uncertainty reduces output, implies that the effects of uncertainty on factor demands depend on what effect the decreased output due to uncertainty has on the cost minimizing levels of inputs. Except for the rare case of inferior factors, the presence of uncertainty reduces factor demands. Finally, it is clear from these analyses that if the firm is risk neutral, the uncertainty has no effect on supply and factor demands. In this paper I relax the assumption that all inputs are chosen before the output price is observed. My conclusions show that the results noted above are really rather sensitive to that particular assumption. A simple two-input, one-output model of the firm is employed. One of the inputs, which I call capital, is quasi fixed in the sense that it must be chosen before the output price is observed. The other input, which I call labor, is variable since it is not chosen until the output price is observed. Clearly, this implies that the level of output is not determined until its price is known. Although labor may be a poor name for the variable input in view of the recent discussions of its "quasi-fixed" character, it seems apparent that in many situations there are inputs which can be varied on short notice. By allowing a variable input in this sense we considerably alter the situation facing the firm. If, after observing the output price, it turns out that the firm made a "poor" choice regarding the quasi-fixed factor, it is able to partially "adjust" by choosing an appropriate level of the variable input. This ability to make adjustments for what, ex post, appears to be a poor decision is totally lacking if all inputs must be chosen before the uncertainty is resolved. In Section I the basic model is presented. In Section II the analysis for a risk-neutral firm is carried out, and Section III contains an example. In Section IV the analysis is extended to a risk-averse firm. The final section contains some brief concluding comments.

Abstract: The solution of pipe flow problems encountered in engineering practice requires an intermediate step of computation of friction factor. The friction factor of commercial pipes is described by a semi-empirical equation developed by Colebrook. This equation that expresses the friction factor as a function of relative roughness and Reynolds number is an implicit one requiring a trial-and-error procedure for its determination. Wood has pointed out obvious disadvantages of an implicit relationship and recognizing the necessity of an explicit equation proposed one to replace the implicit equation for friction factor. It has, however, been possible to develop a simple and more accurate explicit equation for friction factor, based on the Colebrook-White formula, and is presented herein.

Abstract: where y is the investor's wealth, which is a random variable, y is the expected wealth, U(y) is the utility function, f(y) the probability density function, and pi the discrete probability. The calculation can be performed without any difficulties in the discrete case. The expected value of utility is a simple addition of the weighted utility-arguments U(yi)pi. In the continuous case, for some special cases, computing can be done by direct analytic integration. A standard device for computing the value of integral (1) is a Taylor's series expansion around the expected value y (see Paul Samuelson and S. C. Tsiang). Expressing the value of utility U(y) for the argument y, we obtain

Abstract: The probability-density function of the sum of lognormally distributed random variables is studied by a method that involves the calculation of the Fourier transform of the characteristic function; this method is exact. When the number of terms in the sum is large, we employ an asymptotic series in N−1, where N is the number of terms, developed by Cramer. This method is employed in order to show that the permanence of the lognormal probability-density function is a consequence of the fact that the skewness coefficient of the lognormal variables is nonzero. Finally, a simplified proof, by use of the Carleman criterion, is presented to show that the lognormal is not uniquely determined by its moments.

Abstract: Pulse propagation in a random medium is determined by the two-frequency mutual coherence function which satisfies a parabolic equation. In the past, approximate or numerical solutions of this equation have been reported. This paper presents an exact analytical solution for a plane wave case when the random medium is approximated by A(0) − A(ρ) ∝ ρ2. Analytic expressions for two-frequency mutual coherence function, pulse shape, temporal and angular spectra are presented in universal forms, and should be applicable to a large number of practical problems.

Abstract: The total X-ray intensity as a function of h (h is the radial coordinate in reciprocal space), scattered by an isotropic system of particles of equal shapes but of different sizes R, can, under certain conditions, be expressed as an integral over the particle size distribution function D(R), multiplied by a common single-particle function of hR which can be calculated from the assumed particle shape. In the first method D(R) is calculated from this relation by the method of least squares, in which values of D at a limited number of particle sizes are the unknowns. To avoid oscillations in the D curve, constraints are imposed on the D values. The proper weight to be assigned to these constraints must be determined by trial and error. The method has been adapted to suit various assumptions and requirements as to the shape of the particles, the type of distribution function to be calculated, and experimental conditions (slit or pinhole focusing). The second method is essentially the one described by Schmidt, Weil & Brill [X-ray & Electron Methods of Analysis, pp. 86–100. (1968), New York: Plenum], which, however, is adapted to the use of slit-smeared intensities. Both methods may give rise to artefacts in the calculated distribution functions in the range of the smallest particle sizes, which are sensitive to the setting of the various parameters and to experimental errors. However, the position and shape of the main maxima can usually be determined quite well. The agreement between the results obtained by the two methods is satisfactory.

Abstract: The smallest nonempty class ~c4 of subsets of a real analytic space M which is closed under the formation of locally finite unions, intersections, complements, and connected components and which contains Ac~g ~t {0} for any A in ,~ and realvalued function g analytic in a neighborhood of Clos A is the class of semianalytic subsets of M (see [6, w 1] for a list of references, the most basic being [11]). The smallest such class also containing all proper real analytic images of semianalytic sets is the (strictly larger, for dim M > 3 ) class of subanalytic subsets of M ([6, 8, 14, 15]). A continuous function from a subset of real analytic space M into a real analytic space N is called a subanalytic map if its graph is a subanalytic subset of M • N. Here triangulations of subanalytic sets are constructed using only subanalytic maps throughout. A map is proper (respectively, light) if its inverse image preserves compact (respectively, discrete) sets. Our main results are:

Abstract: A subject reproducing a long duration, t, may time out either a single interval of duration t or a succession of n intervals, each of duration t/n. It is shown that a class of timing models obeying Weber’s law predicts the variance of reproductions of t to be a decreasing function of the number of subdivisions, n. In contrast, a second class of proportional variance models, which includes Creelman’s pulse counter model (1962), predicts no change in the variance as a function of n. Data are presented from a duration reproduction experiment in which subjects counted silently at a specified rate up to a given number and then responded. Several statistics involving the variance of the reproduced durations are shown to be predicted significantly better by the Weber’s law class of models than by the proportional variance class of models.

Abstract: In the first part of this paper, we summarize and complete earlier results of Delfour-Mitter [3] on the optimal control problem for linear hereditary differential systems (HDS) with a linear-quadratic cost function The properties of the operator Π(t) which characterizes the feedback gains and the reference functionr(t) are fully detailed In a second part, we construct an approximation to the linear HDS in state form and to the linear adjoint state equation and prove convergence In a third part we construct and solve the approximate optimal control problem following the method of J C Nedelec In the last part we construct the approximation to Π(t) andr(t) and prove convergence Finally we give a number of typical examples to illustrate the main features of the kernel of Π(t)

Abstract: Experience indicates that in the identification of the impulse response function of a linear hydrologic system the results are extremely sensitive to minor errors in the input-output data. In particular, low-amplitude random errors in these data tend to cause severe oscillations in the response function, thereby making it often impossible to obtain a physically realizable solution by conventional methods. Artificial filtering of the input-output records may help, but since the extent of noise is seldom known a priori, one cannot be sure about the proper choice of a cutoff frequency. Such filtering also causes a loss of data at the end points of the record and is therefore undesirable when the number of data points is small. Filtering the response function itself is only effective in eliminating high-frequency oscillations, and it is far less effective when the frequency of the oscillations is relatively low. Clearly, the ultimate goal of identification is to determine a solution which optimizes the predictive capabilities of the linear model. To achieve this goal, it is not sufficient that an observed output be correctly reproduced from a given input; an equally important criterion of optimality is that the shape of the response function be physically plausible. It is shown that one way to obtain a stable and physically realizable response function from a relatively short input-output record is to use parametric linear programing. According to this approach, the problem is formulated as a multicriterion decision process under uncertainty in a manner analogous to that previously described by one of the authors in connection with the inverse problem of groundwater hydrology. Parametric programing serves as a means of generating a continuous set of alternative solutions to the identification problem together with a bicriterion function representing these alternatives. The shape of this bicriterion curve is then used as a guide by the hydrologist in selecting a particular solution when he is relying on his own value judgment. If none of the alternative solutions appears to be physically plausible at this stage, the hydrologist has a further option of imposing modality constraints to eliminate undesirable low-frequency oscillations from the response function. The method is illustrated by two examples, and the results are compared with those obtained by another approach developed previously by one of the authors.

Abstract: The dielectric constant of seven 3 He- 4 He mixtures has been measured in the saturated liquid phase at temperatures between 0.55 and 1.86 K. The Clausius-Mosotti function (ɛ−1)/(ɛ+2) is found to be nearly linear in concentration, especially at low temperatures. A method is described for determining the concentration from a measured dielectric constant. Measurements in the two-phase region are used to calculate the phase-separation curve. Molar volumes are calculated using polarizabilities consistent with those reported by Kerr and Sherman for the pure components.

Abstract: The problem of estimating coefficients and initial values in a system of linear differential equations from observations on linear combinations of the system's responses is addressed. Using the Gauss-Newton algorithm, the reqllired function values are obtained by expressing the system's solution in terms of the eigenvalues and eigenvectors of its coefficient matrix aud its initial values. Differentiating this solution gives expressious for the required function derivatives in terms of these same eigenvalues and eigenvectors. The advantage of this approach is that it, uses exact analytic expressions for the required function values and derivatives rather than resorting to numerical integration or secants. An application to compartment, analysis is considered aud results are compared with those obtained by using the SAAM program of Berman and Weiss.

Abstract: : Three principal areas of investigation are as follows; (1) Kernel functions and related areas, Results have been obtained on polynomial density in Ber's Spaces, Berman Spaces over multiply-connected domains, Total Positively and reproducing kernels, Szego kernels and the Riesz projection theorem and Metric on Annuli; (2) BVP (Boundary Value Problems), and IVP (Initial Value Problems), Study has been undertaken of transforming BVP into IVP. In particular, a method whereby a well-posed elliptic boundary-value problem of the Dirichlet type is transformed into a first-order non-linear equation governing the Green's function of an embedded problem is studied; and (3) Singularities, The study of smoothings of analytic singularities is discussed. In particular, generalized complete intersections and their spaces of deformations are analyzed.

Abstract: Nonlinear resistive networks can be characterized by the equation f(x)= y where f(x) is a continuous piecewise-linear mapping of Rn into itself. The n-dimensional Euclidean space is divided into a finite number of regions, and, in each region say region Rm, we can express f by J(m)x + w(m) where J(m) is a constant n × n Jacobian matrix and w(m) is a constant n-vector. In this paper we obtain the following results: If all the Jacobian determinants in the unbounded regions have the same sign, the equation f(x)= y has at least one solution and an algorithm is developed, which obtains one or more solutions in a finite number of steps. The work represents a generalization of early work by Fujisawa, Kuh and Ohtsuki and relaxes the condition imposed on the function. For example, in the bounded regions, the Jacobian matrices can be singular and the sign of Jacobian determinants can be arbitrary.

Abstract: In this paper we demonstrate how the definition of the one‐particle Green’s function, or electron propagator, can be extended to include averages over nonexact reference states without affecting the exact nature of the poles of this function. We also make connections between this Green’s function and an average value of the superoperator resolvent.