122683434
Convergence of an iterative neural network learning algorithm for linearly dependent patterns
The authors show that a local iterative learning algorithm of Diederich and Opper (1987) converges for any set of patterns, including linearly dependent sets, and that the resulting coupling constants for a given set of patterns are the same as those given by the pseudoinverse rule for any maximal linearly independent subset of the given set. Thus, the matrix of couplings produced by the algorithm is the desired projection matrix onto the subspace spanned by the training set.
117116150
Exact Renormalization Group for Gauge Theories
Renormalization group ideas have been extremely important to progress in our understanding of gauge field theory. Particularly the idea of asymptotic freedom leads us to hope that nonabelian gauge theories exist in four dimensions and yet are capable of producing the physics we observe — quarks confined in meson and baryon states. For a thorough understanding of the ultraviolet behavior of gauge theories, we need to go beyond the approximation of the theory at some momentum scale by theories with one or a small number of coupling constants. In other words, we need a method of performing exact renormalization group transformations, keeping control of higher order effects, nonlocal effects, and large field effects that are usually ignored.
13103184
An Exchange Identity for Non-linear Fields
We establish a useful identity for intertwining a creation or annihilation operator with the heat kernel of a self-interacting bosonic field theory.
124187143
On the approach to the critical point
The mass perturbation yields a new construction for Even + Linear quantum field models. For rp4 models, the Schwinger functions are differentiable in the parameter 6 = 5y~, in the single phase region, and are continuous ac. For the Isingd model, d >_ 2 classical bounds on the critical exponents, e. g. 1 y 2v, are a consequence of an Ornstein-Zernike type upper bound on the two point function.
116890750
Constructive field theory
117474017
Absolute bounds on vertices and couplings
We prove absolute upper and/or lower bounds on ~ and cp6 dimensionless vertices and physical coupling constants. 1. DEFINITIONS AND ASSUMPTIONS We derive absolute bounds on ~p4 and qJ6 dimensionless coupling constants g. For instance, in a pure ~p4 model we prove ( 1.1 ) 0 g const., in the single phase region (no symmetry breaking). We also obtain bounds on associated vertex functions and on connected parts. Our general methods presumably have other consequences for n-particle amplitudes, n >_ 8, which we do not pursue here. DEFINITIONS. Let (1) Supported in part by the National Science Foundation under Grant MPS 74-13252. (2) Supported in part by the National Science Foundation under Grant MPS 73-05037. Annales de l’Institut Henri Poincaré Section A Vol. XXII, n° 2 1975. 8 98 J. GLIMM AND A. JAFFE denote the n-point (Euclidean) Schwinger function, and let ( 1 ... n )~ denote its connected part. We let denote the Fourier transform of 1 ... We encounter often, and denote it x(p). We use the standard spectral representation where dp(a) is the spectral measure. Furthermore, we let denote the n-point Euclidean vertex function (thus r(n) is the amputated, one particle irreducible part of ST"~(~)). In an even theory, we have for example, where pi + ... + 7?e = 0, where (X, ~ X) is a partition of 0?i,..., p6) into two subsets of three elements, and where Px is the sum of the momenta in X. We let d 4 denote the space-time dimension and let m denote the mass gap. Note that has the dimension of mass to the power d 2). We define the dimensionless amplitude by and the associated coupling constant by (1.9) = For example, where x * x(0). Annales de l ’Institut Henri Poincaré Section A 99 ABSOLUTE BOUNDS ON VERTICES AND COUPLINGS ASSUMPTIONS. We make some or all of the following assumptions. In the case of weakly coupled assumptions (a)-( f) have been proved, while for single phase, even models (h)-(i) have been proved. We expect (a)-( f ) to hold for single phase and models, while (h)-(i) are expected for all single phase even ~ models, d 4. The evidence for the validity of (1.4) in qJ1 comes from perturbation theory. The proof of (i) in even, single phase has been established by Cartier (private communication). (a) The Wightman axioms for a scalar boson field, or the Osterwalder. Schrader axioms for the corresponding Euclidean theory. The two point function satisfies (1.4). In particular we are assuming that is finite. (b) A positive mass gap, m > 0. M~=0. (d) The second Griffiths inequality, where (Ai, A2) is a partition of (1, ..., n). For instance, ( f ) The measure dp(a) contains a delta function at a = m2, of unit strength (This is a renormalization hypothesis). (h) In the case of a cp4 interaction, it is appropriate to assume the Lebowitz inequality ( 1234 )~ ~ 0, or the set of Lebowitz inequalities, where W is an (odd) subset of U. (i) In the case of a pure cp4 model, we assume ( 1.13) ( 123456 ~T >_ 0. 2. BOUNDS ON QUARTIC COUPLINGS THEOREM 2 .1. (i) Assume (a)-(d) above. Then Vol. XXII, no 2 1975. 100 J. GLIMM AND A. JAFFE (ii) Assume ( f ) also, then x >_ m 2, so (2.2) const. (iii) Assume (h) also, then (2 . 3) 0 g~4~ const. x 2m ~ const. Remark 1. The constants are pure numbers, independent of all parameters. They can be determined explicitly by the proof. Remark 2. The upper bound on is related to the picture of « critical point dominance » discussed in [7]. According to this picture, for d 4, g~4~ assumes its maximum value at the critical point, i. e. at the onset of phase transitions. According to standard ideas, m = 0 at the critical point. Remark 3. For d 4, the limit ~,o ( = bare cp4 coupling constant) oo with m fixed plays a role in the application of the Callan-Symanzik equations to the study of critical phenomena [2]. We use Theorem 2.1 below, combined with bounds from [3], to show that either this limit exists, or that in this limit g(4) 0 (see Corollary 2 . 4). If g~4~ is monotone in the dimensionless charge À-omg-4, then the alternative g(4) ~ 0 is excluded. The possibility that g~4~ is monotone is discussed in [1], and is a consequence of conventional assumptions made in the study of the Callan-Symanzik equations. Remark 4. Other bounds on quartic couplings have been established by Lukaszuk and Martin [4], see also Healy [5]. Their bounds require different assumptions and methods, and yield different conclusions. In the case of a pure cp4 model we also obtain bounds on g~4~(p) 0. THEOREM 2.2. Assume (a)-(h) above. There exists 5 > 0, such that if 6m, then g~4~(p) is analytic in p and bounded uniformly by Proof of Theorem 2.1. By the positivity of dp (Wightman positivity condition) (2. 5) 12~0, so the right side of (1.11) is positive. By symmetry we then have Annales de l’Institut Henri Poincaré Section A 101 ABSOLUTE BOUNDS ON VERTICES AND COUPLINGS Here each Ai is either or else a permutation of (12~(34~13X24). We bound each Ai using Proposition A2.1, of Appendix 2, An elementary estimate shows that each At is integrable over dxidx2dx3. By homogeneity, and (A2.1), from which we conclude By ( 1. 8), with n = 4, and (1. 6) we have (2 .1 ). The theorem then follows. Proof of Theorem 2 . 2. Assuming (h) and (2 . 6), We bound the power series coefficients of expanding about p = 0. The derivatives become multiplication by ix j in the Fourier transform, and are dominated by the exponential decrease of (2. 8), estimated by (A2.1). Thus the power series in momenta converges for I Pi I m/3, and establishes the analyticity and boundedness of St4)(p), Finally, the analyticity and boundedness of follows from (1.4), for I p I m, yielding Thus g(4)(p) is analytic in p and with the constant independent of p, m, A for This completes the proof. We have also proved. COROLLARY 2.3. Assuming (a)-(h), Si-4)(p) is analytic for I ~ ~m and bounded uniformly by (2. 9). We now let be the n-point Schwinger function for a model satisVol. XXII, nO 2 -1975. 102 J. GLIMM AND A. JAFFE fying (a)-(h) and labelled by the index j (for instance, may be the cp2 model, labelled by the bare coupling and mass m). COROLLARY 2.4. Assume mj is bounded away from zero. Then either there exists a convergent subsequence of Schwinger functions i with satisfying (a)-(h), or else 0. Proof Assume for some p, with sm, there is a subsequence of gr(p) bounded away from zero. Then by (2.4) This bound (uniform in r) on proves a uniform bound for a suitable Schwartz space norm j . 109. Thus by Theorem 1 of [3], a convergent subsequence of Schwinger functions exists, and is a tempered distribution. The properties (a)-(h) follow from the corresponding properties in the approximate theories. Conversely, if 0 for each p with |pi| I 6m, the uniform bound (2.4) ensures that the family of analytic functions gj(p) converges to the analytic function zero. 3. BOUNDS ON In this section we prove preliminary bounds on ST. THEOREM 3.1. (i) Assume (a)-(h). Then there exists 6 > 0 such that if then Si:6)(p) is analytic and bounded by (ii) Assume (a)-(i). Then The constants are pure numbers, independent of the parameters. Annales de l’Institut Henri Poincaré Section A 103 ABSOLUTE BOUNDS ON VERTICES AND COUPLINGS Proof. We consider p = 0. The extension to p # 0 follows the proof of Theorem 2.2. Using (/), we have The definition of the six point connected part is Here Y = (Yi, Y2) ranges over the fifteen partitions of (1, 2, ..., 6) into two and four element subsets, while Z = (Zi, Z2, Z3) ranges over the fifteen partitions of (l, ..., 6) into three pairs. Thus where E’ omits the partition ( 12 )( 3456 ). We use (h) and the definition of( 1234 )r to obtain where E’ omits the three partitions which contain the factory Zi) =(12). By symmetry, where j = 1, 2, ..., 15 labels the two element subsets of (1, ... , 6), and (Z~, Z2’~, Z3’~) one of the twelve partitions of (1, ..., 6) into three pairs, no one of which is j. We now follow the proof of Theorem 2.1 to obtain proving one side of (3 .1 ). We next use (1.12) (see [3]) with U = (3456) to obtain where W ranges over the eight odd subsets of U. Thus by (3.4), Vol. XXII, nO 2 1975. 104 J. GLIMM AND A. JAFFE where E" ranges over the six partitions containing both 1 and 2 in the four element subset. By the definition of ( 1234 )T, By (1.11) we note that -.
41221098
Planck intermediate results. XII: Diffuse Galactic components in the Gould Belt system
We perform an analysis of the diffuse low-frequency Galactic components in the southern
 part of the Gould Belt system (130° ≤ l ≤ 230° and
 −50° ≤ b ≤ −10°). Strong ultra-violet flux
 coming from the Gould Belt super-association is responsible for bright diffuse foregrounds
 that we observe from our position inside the system and that can help us improve our
 knowledge of the Galactic emission. Free-free emission and anomalous microwave emission
 (AME) are the dominant components at low frequencies
 (ν < 40 GHz), while synchrotron emission is
 very smooth and faint. We separated diffuse free-free emission and AME from synchrotron
 emission and thermal dust emission by using Planck data, complemented by
 ancillary data, using the correlated component analysis (CCA) component-separation method
 and we compared our results with the results of cross-correlation of foreground templates
 with the frequency maps. We estimated the electron temperature
 Te from Hα and free-free emission using two
 methods (temperature-temperature plot and cross-correlation) and obtained
 Te ranging from 3100 to 5200K for an effective fraction of
 absorbing dust along the line of sight of 30% (fd = 0.3). We
 estimated the frequency spectrum of the diffuse AME and recovered a peak frequency (in
 flux density units) of 25.5 ± 1.5 GHz. We verified the reliability of this result with
 realistic simulations that include biases in the spectral model for the AME and in the
 free-free template. By combining physical models for vibrational and rotational dust
 emission and adding the constraints from the thermal dust spectrum from
 Planck and IRAS, we are able to present a good description of the AME
 frequency spectrum for plausible values of the local density and radiation field.
119513048
Observations of galactic and extra-galactic sources From the BOOMERANG and SEST telescopes
We present millimeter-wave observations of three extra-galactic and six Galactic sources in the Southern sky. Observations were made at 90, 150, 240 and 400 GHz with resolutions of 18, 10, 14 and 12 arcmin respectively during the 1998 Antarctic long duration balloon flight of BOOMERANG. Observations were also made with the SEST telescope, at 90 and 150 GHz with resolutions of 57 and 35 arcsec respectively. These observations can be used for calibrations of Cosmic Microwave Background (CMB) experiments as well as an understanding of the physical processes of the sources.
38675087
DETECTION OF GRAVITATIONAL WAVES FROM INFLATION
Recent measurements of temperature fluctuations in the cosmic microwave background (CMB) indicate that the Universe is flat and that large-scale structure grew via gravitational infall from primordial adiabatic perturbations. Both od these observations seem to indicate that we are on the right track with inflation. But what is the new physics responsible for inflation? This question can be answered with observations of the polarization of the CMB. Inflation predicts robustly the existence of a stochastic background of cosmological gravitational waves with an amplitude proportional to the square of the energy scale of inflation. This gravitational-wave background induces a unique signature in the polarization of the CMB. If inflation took place at an energy scale much smaller than that of grand unification, then the signal will be too small to be detectable. However, if inflation had something to do with grand unification or Planck-scale physics, then the signal is conceivably detectable in the optimistic ...
121823176
Searching for Non-Gaussian Signals in the BOOMERANG 2003 CMB Maps
We analyze the BOOMERANG 2003 (B03) 145 GHz temperature map to constrain the amplitude of a non-Gaussian, primordial contribution to CMB fluctuations. We perform a pixel-space analysis restricted to a portion of the map chosen in view of high-sensitivity, very low foreground contamination and tight control of systematic effects. We set up an estimator based on the three Minkowski functionals which relies on high-quality simulated data, including non-Gaussian CMB maps. We find good agreement with the Gaussian hypothesis and derive the first limits based on BOOMERANG data for the nonlinear coupling parameter fNL as -300 < fNL < 650 at 68% CL and -800 < fNL < 1050 at 95% CL.
18690323
Cosmological parameters from the 2003 flight of BOOMERANG
We present the cosmological parameters from the CMB intensity and polarization power spectra of the 2003 Antarctic flight of the BOOMERANG telescope. The BOOMERANG data alone constrain the parameters of the ΛCDM model remarkably well and are consistent with constraints from a multiexperiment combined CMB data set. We add LSS data from the 2dF and SDSS redshift surveys to the combined CMB data set and test several extensions to the standard model including running of the spectral index, curvature, tensor modes, the effect of massive neutrinos, and an effective equation of state for dark energy. We also include an analysis of constraints to a model that allows a CDM isocurvature admixture.