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The length of the solar activity cycle fluctuates considerably. The temporal evolution of the corresponding cycle phase, that is, the deviation of the epochs of activity minima or maxima from strict periodicity, provides relevant information concerning the physical mechanism underlying the cyclic magnetic activity. An underlying strictly periodic process (akin to a perfect "clock"), with the observer seeing a superposition of the perfect clock and a small random phase perturbation, leads to long-term phase stability in the observations. Such behavior would be expected if cycles were synchronized by tides caused by orbiting planets or by a hypothetical torsional oscillation in the solar radiative interior. Alternatively, in the absence of such synchronization, phase fluctuations accumulate and a random walk of the phase ensues, which is a typical property of randomly perturbed dynamo models. Based on the sunspot record and the reconstruction of solar cycles from cosmogenic C14, we carried out rigorous statistical tests in order to decipher whether there exists phase synchronization or random walk. Synchronization is rejected at significance levels of between 95% (28 cycles from sunspot data) and beyond 99% (84 cycles reconstructed from C14, while the existence of random walk in the phases is consistent with all data sets. This result strongly supports randomly perturbed dynamo models with little inter-cycle memory., Comment: Astronomy & Astrophysics, in press

The twist of the magnetic field above a sunspot is an important quantity in solar physics. For example, magnetic twist plays a role in the initiation of flares and coronal mass ejections (CMEs). Various proxies for the twist above the photosphere have been found using models of uniformly twisted flux tubes, and are routinely computed from single photospheric vector magnetograms. One class of proxies is based on $\alpha_z$, the ratio of the vertical current to the vertical magnetic field. Another class of proxies is based on the so-called twist density, $q$, which depends on the ratio of the azimuthal field to the vertical field. However, the sensitivity of these proxies to temporal fluctuations of the magnetic field has not yet been well characterized. We aim to determine the sensitivity of twist proxies to temporal fluctuations in the magnetic field as estimated from time-series of SDO/HMI vector magnetic field maps. To this end, we introduce a model of a sunspot with a peak vertical field of 2370 Gauss at the photosphere and a uniform twist density $q= -0.024$ Mm$^{-1}$. We add realizations of the temporal fluctuations of the magnetic field that are consistent with SDO/HMI observations, including the spatial correlations. Using a Monte-Carlo approach, we determine the robustness of the different proxies to the temporal fluctuations. The temporal fluctuations of the three components of the magnetic field are correlated for spatial separations up to 1.4 Mm (more than expected from the point spread function alone). The Monte-Carlo approach enables us to demonstrate that several proxies for the twist of the magnetic field are not biased in each of the individual magnetograms. The associated random errors on the proxies have standard deviations in the range between $0.002$ and $0.006$ Mm$^{-1}$, which is smaller by approximately one order of magnitude than the mean value of $q$., Comment: accepted in A&A

Surfaces of the Sun and other cool stars are filled with magnetic fields, which are either seen as dark compact spots or more diffuse bright structures like faculae. Both hamper detection and characterisation of exoplanets, affecting stellar brightness and spectra, as well as transmission spectra. However, the expected facular and spot signals in stellar data are quite different, for instance they have distinct temporal and spectral profiles. Consequently, corrections of stellar data for magnetic activity can greatly benefit from the insight on whether the stellar signal is dominated by spots or faculae. Here, we utilise a surface flux transport model (SFTM) to show that more effective cancellation of diffuse magnetic flux associated with faculae leads to spot area coverages increasing faster with stellar magnetic activity than that by faculae. Our calculations explain the observed dependence between solar spot and facular area coverages and allow its extension to stars more active than the Sun. This extension enables anticipating the properties of stellar signal and its more reliable mitigation, leading to a more accurate characterisation of exoplanets and their atmospheres., Comment: 10 pages, 6 Figures, accepted for publication in ApJL

The solar meridional flow is an important ingredient in Babcock-Leighton type models of the solar dynamo. Global variations of this flow have been suggested to explain the variations in the amplitudes and lengths of the activity cycles. Recently, cycle-related variations in the amplitude of the $P_2^1$ term in the Legendre decomposition of the observed meridional flow have been reported. The result is often interpreted in terms of an overall variation in the flow amplitude during the activity cycle. Using a semi-empirical model based upon the observed distribution of magnetic flux on the solar surface, we show that the reported variations of the $P_2^1$ term can be explained by the observed localized inflows into the active region belts. No variation of the overall meridional flow amplitude is required., Comment: Accepted in ApJ

In this paper the authors evaluate Yeh-Wiener integrals (which apply to functionals of a variable continuous function of two arguments) in terms of multiple Wiener integrals (which apply to functionals of several variable continuous functions of one argument). First somewhat specialized cases are given where the multiplicity of the Wiener integral is finite, and then quite general Yeh-Wiener integrals are evaluated in terms of limits of n-folά Wiener integrals as n->oo.

On etablit l'existence de l'integrale de Feynmann sequentielle pour des classes plus grandes de fonctionnelles contenant l'algebre de Banach SM On donne des formules pour evaluer les integrales de ces fonctionnelles. On presente un theoreme de Fubini et un theoreme de sommation pour manipuler ces integrales

Presented to the Society April 20, 1957; received by the editors February 15, 1957. 1 This research was supported by the United States Air Force, through the office of Scientific Research of the Air Research and Development Command, under contract No. AF 18(603)-30. Reproduction in whole or in part is permitted for any purpose of the United States Government. 2 The author wishes to thank Mr. James Yeh for carefully checking the manuscript of this paper. I See, for instance, [4].

In a recent paper [2], an operator valued function space integral was defined by the authors as follows.

In our proof of a translation theorem for analytic Feynman integrals [3], a function of two complex variables arose which was analytic in each variable if the other variable was restricted to real values. In this paper we establish some results relating analyticity and separate analyticity for functions of several complex variables. These results differ from the classical ones [1] in that the functions are not defined initially on complete neighborhoods, only on lower dimensional subsets, and also differ from recent results [2], [4, p. 766] in that our proof gives a precise estimate of the domain of analyticity. This estimate is used in later theorems. It is convenient to begin with a lemma treating the case of two complex variables in complete detail.

In a recent paper by the same title [II], hereafter referred to as Part I, R. H. Cameron and the present author investigated the structure of the class of additive functionals of class L2 on the Wiener space C [VI] and obtained certain other results concerning functionals additive and merely measurable on C. Recently Cameron conjectured that every functional additive and measurable on C was of class L2(C), and hence that the characterization and representation theory previously developed applied to all functionals additive and measurable on C. It is the purpose of this paper to establish Cameron's conjecture. Unfortunately the proof is not easy and depends on a number of preliminary results. However, the techniques of proof are novel enough to be of some interest. In addition, the results permit considerable strengthening of our earlier theorems and a curious characterization of the class of continuous linear functionals on the Hilbert space of real-valued functions of class L2 on the unit interval. We begin by introducing a number of preliminary concepts and theorems. DEFINITION 1. Let {I'pk} be a C. 0. N. set (complete orthonormal set in the L2 sense) on [0, 1], each of whose elements is of bounded variation, and let f(t)x E L2 on [0, 1]. Let fjt) be the nth partial sum of the orthogonal development of f(t) in terms of {I ik ; that is, let fn(t) = EL' Ckkp A(t), where Ck = If (t)k(t) dt. Then we define the P. W. Z. (Paley-Wiener-Zygmund) integral f f(t) dx(t) by the equation ff(t) dx(t) = lim ffn(t) dx(t), x E C O ~~~~~n -o 00 whenever the limit on the right exists. It has been shown by Paley, Wiener, and Zygmund [IV] that the limit defining the P. W. Z. integral exists for almost all x E C and that this limit is essentially independent of the particular choice of the C. 0. N. set {Ipk I in the sense that if IOk} and {lk* } are any two C. 0. N. sets all of whose elements are of B. V., then 1 1 (, f f(t) dx(t) = (O*) ] f(t) dx(t) for almost all x E C. It has also been o ~ ~~~~ I shown in [IV] that f fn(t) dx(t) converges in the L2(C) mean to a f(t) ax(t) that the P. W. Z. integral agrees with the ordinary Riemann-Stieltjes integral almost everywhere on C when f(t) is of B. V., and that if G(u1, **, un) is

Introduction. Let C be the space of all real-valued functions x(t) continuous in 0 < t ?1, and vanishing at t = 0. Wiener has defined a measure over the space C and in terms of this measure he has defined an average or integral over C which is intimately related among other things to the theory of the Brownian motion [1, 2](1). The present authors have recently investigated certain aspects of the Wiener integral [3, 4] and have obtained for instance in [4] a result which shows how the integral is transformed under translations. Irn the present paper we determine how the integral transforms under a certain class of linear homogeneous transformations. This result is also combined with the earlier result on translations to yield a transformation formula for the nonhomogeneous transformation-translation plus linear homogeneous transformation. By applying the transformation formula to the special linear transformation

Presented to the Society, April 27, 1951; received by the editors August 29, 1952. 1 On the summability of certain orthogonal developments of nonlinear functionals, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 130-145. 2 A simpler proof of this theorem has since been given by G. Maruyama, K6dai Mathematical Seminar Reports, no. 3, 1950, pp. 41-44. 3 Here the double limit is to be interpreted in the sense that N never actually assumes the value + oo and X never actually assumes the value 1.

where A(x I t) is a functional depending on the function x and the number t and satisfying certain smoothness conditions. (The number t ranges over the interval 0 < t < 1, and the function x ranges over a measurable subset r of the space C of continuous functions on 0< t

The function u(t) need not be a member of C, but can be any Borel measurable function defined on 0 _ t _ 1, and may even be permitted to take on infinite values. The Wiener integral of a functional is simply the Lebesgue integral of the functional with respect to Wiener's measure [4]1 in C. This measure is not invariant under translations, but is in other respects a Lebesgue measure based on intervals of the form

The subject treated in this paper forms a part of a more general theory of heat flow involving moving sources of heat. The present problem deals with the temperature distribution around a point source moving at a constant speed along the axis of a cylinder. It is shown that the rise of temperature produced at a given cross section of the cylinder, as a result of the passage of the heat source, tends rapidly to become constant. The factor which governs this trend is a dimensionless product of the following units: (1) The speed at which the source travels; (2) the radius of the cylinder; and (3) the converse of the heat diffusivity.

When one studies periodic transformations such as, for example, rotations, he often encounters transformations which are not periodic but which are, in a very real and non-technical sense, almost periodic. For instance, repeated rotation through an angle which is an irrational part of a revolution will never bring a point set back point-for-point into itself; yet this object may be as nearly attained as we please by repeating the process an appropriate number of times. Moreover, such "appropriate" numbers are relatively denset among the integers. This example suggests the definition of an a.p. (almost periodic) transformation; it being only necessary to make precise the meaning of "as nearly as possible" when applied to bringing the points of a set back into themselves. Consider, for example, a set Z of uniformly continuous transformations which take each point of a complete metric space G into a point of G. Let Z contain the identity and the product of any two of its elements. Then if t is a variable point of ( and 4(a) and T(t) are any two elements of Z, let the smaller of the two quantities, unity and the least upper bound for all t in ( of the distance between the points b(a) and T'(), be called the distance between the transformations 4(a) and T'(), and let it be indicated by I| 4(a), IT (t) I. Then || 4(e), t || represents one way of telling how nearly the points 4i(t) approximate the points t. Moreover a transformation of Z will be called a.p. if to each positive number e there corresponds a positive integer L so great that among each L successive positive integers there is an integer N satisfying || 4 N(t) I | I-e. This is merely an example of a definition of an a.p. transformation. A more general definition will be given in the next section. Transformations will be thought of as points in a new space, and a.p. points will be defined. Moreover, for simplicity in wording and notation, most of the theorems on a.p. transformations will be stated in terms of a.p. points. However, the reader may readily re-phrase them in terms of the more natural and significant a.p. transformations.

Presented to the Society, September 1, 1965; received by the editors August 25, 1965. (1) Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-381-63. (2) Formula (1) given here differs slightly from those in [2] and [3] in that [a, b] is used instead of [0, 1] and the Wiener integrals are normalized on the basis of "unit variance per unit time" instead of "half unit variance per unit time." (3) The integral Jb x'(t) dx(t) in (1) denotes the Riemann-Stieltjes integral obtained by replacing x'(t) by a function of bounded variation which is equal to x'(t) almost everywhere. (4) This restriction on x'(t) may be weakened to x'0(.) eL2 provided that we interpret b x'o(t) dx(t) in the sense of [5].

Existence theorems and evaluation formulas are obtained for the analytic Feynman integral corresponding to functionals of the form given below in (1.1).

The usefulness and limitations of needle aspiration and CT scanning are illustrated in three case reports. In conjunction with the clinical evaluation, these modalities can provide useful information, although no definitive decision should be made on the basis of limited cytopathologic data. The ultimate treatment is better planned when the information provided by CT and needle aspiration are combined with the clinical impression. We advocate the use of fine needle aspiration in conjunction with CT scanning for selected parotid area masses. Readily palpated masses located almost certainly in the substance of the parotid may be appropriately managed without these two modalities. We believe that these tests can greatly help in the planning treatment of less definite lesions.