Functions Fi Research Articles

An Asymptotic Formula for the Iterates of a Function

Let IK be either IR or ℂ and D an open set of IK containing 0 and starlike with respect to 0 (i.e. an open interval containig 0 in the case IK = IR). If f: D » IK is a continuous function with fixed point 0, then under certain conditions stated below we can prove for the kn- th iterates of f the following asymptotic formula: $$f^{(kn)}\bigg({x \over n}\bigg )=\sum_{i-1}^r{1\over (nk)^i} f_i(kx)+o \bigg({1\over n^r}\bigg),$$ (1) for n » ∞, k, n and r beeing positive integers and x close enough to 0. The functions fi are continuous and uniquely determined by f.

Fonctions hypergéométriques en plusieurs variables et espaces des modules de variétés abéliennes

The monodromy groups of the Appell-Lauricella functions Fi in several variables can be discontinuous, but this does not in general entail that they be arithmetically defined ([DM], [M]). Nonetheless, we associate to them families of abelian varieties for which they play a role similar to that of modular groups. We obtain in this way natural modular embeddings of them into arithmetic groups, giving applications to automorphic functions, transcendence questions and to the classification of the algebraic functions among the functions Fi.

A description of random walks with collision anisotropy and with a nonconstant mean free path

We study various characteristics of a particle's random walk both analytically and with the help of Monte-Carlo simulation techniques. With the analytical approach, we derive the expression[Formula: see text]which relates the mean-square displacement [Formula: see text] to (i) the number of steps N in the walk, (ii) the mean-square displacement [Formula: see text] on each of the steps, and (iii) a coefficient of collision anisotropy A defined as the average value of the cosine of the scattering angle θ. This expression is general in the sense that it holds for any value of N and A. It is, however, restricted to cases where the mean free path is constant throughout the random walk. The results of the simulations allow a further generalization to random walks with a nonconstant mean free path. They also allow the study of the radial distribution f(r) of particles after the walk. We find that a set of six functions fi(r) is necessary to give a satisfactory description of the particles' radial distribution for arbitrary values of N and A.

Genotoxic activity of some mycotoxins using the SOS chromotest

The genotoxic activity of 11 mycotoxins was investigated in Escherichia coli K 12. The induction of the SOS function sfi A whose level of expression is monitored by means of a sfi A::lac Z operon fusion was assayed by measuring the beta-galactosidase activity in the PQ 37 strain. Most of these fungal metabolites did not induce SOS response in this bacterial test. Only aflatoxicol, a reduced metabolite of aflatoxin B1 was well detected as an SOS inducer if metabolic activation was performed. Patulin, penicillic acid and viomellein are only weak inducing agents. The other fungal compounds tested failed to demonstrate a positive SOS inducing activity. Relationship between SOS chromotest, mutagenicity to Salmonella typhimurium and in vivo carcinogenicity was discussed.

Convergence Rates for the Optimal Values of Allocation Processes

The optimal allocation process is of the following type. A sequence of functions (production functions) f1, f2, …, is given, each mapping R+n into [0, ∞). Along with it a vector q > 0 in R+n (vector of resources) is provided. For each integer k we consider the problem [Formula: see text] Here R+n is the nonnegative orthant of the n-dimensional Euclidean space. Inequalities between vectors are taken coordinatewise. In this paper we consider the case where the functions fi are subject to randomness. Specifically, they are generated as independent random drawings from a common distribution. In particular fi = fi(x, ω) with ω being in a probability space, and the solutions xi might depend on u, as well as on k and q. The detailed technical conditions are displayed in §2. We are interested in limit properties, as k → ∞, of the optimal values of the allocation process.

Anharmonic corrections for linear triatomic molecules subject to the Rennet-Teller effect

The effects of vibrational anharmonic terms and of the gK -correction on the energy levels of a triatomic molecule in a degenerate electronic state are considered. The electronic wavefunctions are described using the approach first suggested in the original paper of Renner. Formulae for the anharmonic corrections in a number of different situations are derived. For an electronic Π state the corrections are given in the form where i runs over the various contributions and x 1, x 2, … depend on the anharmonic force constants. The functions Fi can be determined numerically (see equations (4.3) and 4.7)). For the case without spin-orbit interaction the Fi s are given explicitly to first order in ∈ in table 1. Further-more, the same results apply for levels with |K|=v 2+1 even if the spinorbit interaction is not negligible. Explicit results for levels with |K|<v 2 including spin-orbit interaction are given in tables 2 and 3. The cases with larger values for Λ (2, 3, …) are also considered. The energy level formulae for a Δ state differ from those derived earlier by Merer and Travis. In particular, the small separation between the vibronic Φ and Π levels with v 2=1 is now found to be 4gK . The additional terms that arise from end-over-end rotation of the molecule are discussed in § 6.

The Improvement of Probability Judgements

SUMMARY A subject S assesses the probability for an event A as q. In this paper, it is shown how the assessment can be improved. Improvement requires two functions fi(q) (i = 0, 1); the probability for q when A is true (i = 1) and when A is false (i = 0). It is shown that data on S's probability assessments can be used to update these functions. Specific, normal forms for them are examined. The connection with calibration is explored. 1. THE TRANSFORMATION OF PROBABILITY JUDGEMENTS CONSIDER a subject, S, who has provided his personal probability for an event A. In this paper we address the problem of whether his probability assessment can be improved, not by providing information additional to that which S already has, or even by S remembering something he had temporarily forgotten, a process Brown and Lindley (1980) have termed digging in S's psychological field; but by a better understanding of the assessment mechanism. It is shown that improvement is possible and the method of making it is described. Let the event A under consideration initially have probability y. Because Bayes' theorem in log-odds form is linear, it is convenient to suppose S's statement of probability assessment is that the log-odds for A is q. Our question is: can we transform q to a new value, q* say, which, in some sense, is a better assessment of the log-odds? Two examples may be illuminating. Example 1 S is given an almanac question, with two possible answers, say an upper and a lower one. Data set (a) below provides an example. Before seeing the question he is told that one, and only one, of the answers is correct and that they are equally likely to be correct. On seeing the question he is asked for the probability that the upper answert is correct. Here y =-. The

Solution of a game with a right to the first move and inexact information on the partner's aim

The present paper examines a modification of the game with unopposed interests first stated and solved in [ll. Our statement of the game differs from that in [II in just one respect: it is assumed that the first player does not know precisely the second player’s pay-off function, he only knows its range of variation, i.e. he has information on the functions fi’(x,, x2), fz+(Xl, x2), where f,+ (x1, x,) >/f2(xI, x2) >/f2-(x,, x,); the functions f,, fi’ and f2+ are assumed continuous, while x1 and x, belong to compacta X, and X,. AS in [II, therefore, the first player, counting on his information on the concrete choice of the second player 1c, s X,, chooses his strategy x,(x,> and communicates it to the second player. The final choice of strategy x,(x2) is fully determined by the first player’s attempt to ensure himself under these conditions the best guaranteed result, starting from the pay-off function fi(x,, n,). The second player’s behaviour is fully described by his attempt to increase his pay-off function f2(xl, x,) (not exactly known to the first player), by selecting x, E X,, knowing the function x1(x,).

The calculation of resonances in electron-ion scattering using the distorted wave approximation

In the distorted wave approximation, radial functions Fi for the colliding electron are calculated using a central potential Vi(r). The Kohn variational method is then used to calculate the R matrix; this is the matrix, used in quantum defect theory, which varies slowly as a function of the energy. For closed channels the function Fi are calculated at the eigen-energies for the central-potential problem, and suitably normalized. The resonance structures are calculated using methods of quantum defect theory. Illustrative results are obtained for the 2s2 1S-2s2p3P transition in C III and in O V, and compared with results obtained from exact solutions of coupled integro-differential equations. It is concluded that, for the calculation of rate coefficients for the excitation of more highly ionized positive ions: (i) it is necessary to take resonances into account and; (ii) good results can be obtained using the distorted wave method.

Computer programs for the calculation of electron-atom collision cross sections. I. General formulation

The wavefunction for electron collisions with an N-electron atom is expanded in the form Psi = Sigma i Theta i+ Sigma j Phi jcj where Theta i is an antisymmetrized product of an atomic eigenfunction times an orbital function theta i for the colliding electron; theta i contains a radial function Fi; Phi j has the form of a bound state function for the (N+1)-electron problem. The condition is imposed that (Pgamma mod Fi)=0 if lgamma =li, where Pgamma is an atomic radial function and where lgamma and li are orbital angular momenta associated with Pgamma and Fi; this condition does not imply a restriction on Psi so long as a suitable set of states Phi j is included. The variational principle is discussed and two approximations are described: (i) The distorted wave (DW) approximation is valid when the coupling is not too strong. Particular attention is paid to the normalization of the DW functions Fi. The coefficients cj are determined using the variational principle. (ii) The variational principle is used to obtain a set of coupled integro-differential (ID) equations for the determination of the functions Fi and the coefficients cj.

On the method of generalized stochastic gradients and quasi-F�jer sequences

The results described in this paper can be extended to the case of Hilbert spaces, with small changes due mostly to the replacement of the word “vector” by the word “element.” Of greater interest from the practical point of view is the generalization of the method (9)–(10) to extremal problems with conditions of the form $$F^i (x_1 , \ldots ,x_n ) \leqslant 0,i = 1, \ldots ,m,$$ for which projection for some reason cannot be made (for example, if the functions Fi(x) are not defined analytically and only statistical estimates of their values can be obtained). A stochastic variant of the Arrow-Hurwitz method was proposed in [7] for the solution of such problems.

Phonon Energy Spectrum in a Ferromagnet

The magnetoelastic interaction provides a mechanism whereby, in a ferromagnet, a phonon may couple to two magnons. The propagation of phonons in the crystal may thus be modified by emission and reabsorption of pairs of magnons. Since interactions between the magnons may significantly influence the intermediate state, we have included the dominant low-temperature contributions via the two magnon t-matrix and calculated the phonon self-energy. We find for a phonon with wave-vector q, polarization λ, and energy Ωqλ, Σqλ(Ωqλ)=(|gqλ|2/J) {fB(qλ)+Σ[fi(qλ)/(Ωqλ−ωs)]},where gqλ is the magnon-phonon coupling constant. The functions fB(qλ) and fi(qλ) give, respectively, the contributions from the two spin-wave band and the bound states of energy ωi. In addition to the poles at the true bound states, we find resonances in Σ from fB, corresponding to damped bound (resonant) states of two spin-waves within the band. Near these resonances ImΣ (the line-width) becomes anomalously large, just as does the corresponding function for a single magnon. An interesting result is that, due to the transformational properties of the bound states for q along the [111] direction, longitudinal phonons couple only to the singly degenerate (s-wave) bound state and the transverse phonons couple only to the doubly degenerate (d-wave) bound state.

A note on the deductive completeness of m-valued propositional calculi

In this note we present a simplification of the axiom schemes which were given in our earlier paper Axiom schemes for m-valued propositional calculi. Except for new concepts that are needed for our present purposes, the terminology used will be the same as that introduced in the earlier paper. Hence, we can say that in addition to listing our simplified specifications for axiom schemes, we shall indicate how our new axiom schemes may be used to establish deductive completeness.The simplification of our axiom schemes is achieved by using only the operators ⊃ and Jk( ) with the resulting elimination of ∨ and ·. This does not constitute an actual reduction in functional complexity since the latter operators can be denned in terms of the former. For example,However, by dispensing with ∨ and · one can greatly reduce the number of axioms without an appreciable increase in the difficulties of proving deductive completeness.Our simplified specifications for axiom schemes will retain conditions (1) and (3) of our earlier set but (2) and (4) will be changed to (2*) and (4*) which we will now state.(2*) Assume that an implication function (P ⊃ Q) can be defined exclusively in terms of the basic functions Fi(P1, …:, Pαi).The statement of what follows is simplified by introducing a chain symbol defined recursively as follows:The variable i of a chain symbol which takes values from u to v will be called the chain variable. When it is clear that i is the chain variable we will use the notation in place of . Hence, denotes (P5 ⊃ (P4 ⊃ (P3 ⊃ Q))).

Abel’s theorem and a generalization of one-parameter groups

1. Let a be a given point on a Riemann surface S of positive genus p and big * I b, p+, an arbitrary set of p + 1 points of S. By Abel's theorem, there exists on S a set of p points, c1, * * *, cp, dependent on the b, such that, given any integral of the first kind on S, with a for lower limit, the sum of the p+ 1 values of the integral with the b as upper limits equals the sum of the p values with the c as upper limits. The set of c may be regarded as a type of product of the p + 1 points b. As the reduction of any number q > p of integrals to p integrals leads generally to a unique result, the product operation has an associative quality. For p = 1, the operation makes S a one-parameter Lie group in which b1 and b2 have cl for product. We study a question of associative analytic combinations which is exemplified by the case of a surface of arbitrary postitive genus('). We use p symmetric functions fi (xi, * * , xP+i), i-1, * , p of p+1 variables, analytic when the variables are all zero. Let Pi (ui, . . , u,) represent, for i =1, * * * , p, the elementary symmetric function (e. s. f.) of degree i of ul, * * , up. We assume that when an xj vanishes, fi reduces to Pi of the x other than xJ. Now let there be given p relations