On the determination of numbers by their sums of a fixed order

It is shown that the set of all networks of fixed order n form a semigroup that is isomorphic to the semigroup BX of binary relations on a set X of cardinality n. Consequently, BX provides for Green’s L,R,H, and D equivalence classifications of all networks of fixed order n. These classifications reveal that a fixed-order network which evolves within a Green’s equivalence class maintains certain structural invariants during its evolution. The “Green’s symmetry problem” is introduced and is defined as the determination of all symmetries (i.e., transformations) that produce an evolution between an initial and final network within an L or an R class such that each symmetry preserves the required structural invariants. Such symmetries are shown to be solutions to special Boolean equations specific to each class. The satisfiability and computational complexity of the “Green’s symmetry problem” are discussed and it is demonstrated that such symmetries encode information about which node neighborhoods in the initial network can be joined to form node neighborhoods in the final network such that the structural invariants required by the evolution are preserved, i.e., the internal dynamics of the evolution. The notion of “propensity” is also introduced. It is a measure of the tendency of node neighborhoods to join to form new neighborhoods during a network evolution and is used to define “energy”, which quantifies the complexity of the internal dynamics of a network evolution.

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On The Determination of Sets by Sets of Sums of Fixed Order

Canadian Journal of Mathematics ◽

10.4153/cjm-1968-059-6 ◽

1968 ◽

Vol 20 ◽

pp. 596-611 ◽

Cited By ~ 8

Author(s):

John A. Ewell

Keyword(s):

Abelian Group ◽

General Sense ◽

Fixed Order

The present investigation is based on two papers: “On the determination of numbers by their sums of a fixed order,” by J. L. Self ridge and E. G. Straus (4), and “On the determination of sets by the sets of sums of a certain order,” by B. Gordon, A. S. Fraenkel, and E. G. Straus (2).First of all, we explain the terms implicit in the above titles. Throughout these considerations we use the term “set” to mean “a totality having possible multiplicities,” so that two sets will be counted as equal if, and only if, they have the same elements with identical multiplicities. In the most general sense the term “numbers” of (4) can be replaced by “elements of any given torsioniree Abelian group.”

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New approach to hard corrections in precision QCD for LHC and FCC physics

Modern Physics Letters A ◽

10.1142/s0217732316501261 ◽

2016 ◽

Vol 31 (22) ◽

pp. 1650126

Author(s):

B. F. L. Ward

Keyword(s):

Hadron Collider ◽

High Energy ◽

Lepton Pairs ◽

General Applicability ◽

New Approach ◽

Lhc Physics ◽

Fixed Order ◽

Obvious Generalization ◽

Single Formula

We present a new approach to the realization of hard fixed-order corrections in predictions for the processes probed in high energy colliding hadron beam devices, with some emphasis on the large hadron collider (LHC) and the future circular collider (FCC) devices. We show that the usual unphysical divergence of such corrections as one approaches the soft limit is removed in our approach, so that we would render the standard results to be closer to the observed exclusive distributions. We use the single [Formula: see text] production and decay to lepton pairs as our prototypical example, but we stress that the approach has general applicability. In this way, we open another part of the way to rigorous baselines for the determination of the theoretical precision tags for LHC physics, with an obvious generalization to the FCC as well.

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On the Determination of Numbers by Their Sums of Fixed Order

American Mathematical Monthly ◽

10.2307/2315034 ◽

1965 ◽

Vol 72 (8) ◽

pp. 873

Author(s):

Nathan Eljoseph

Keyword(s):

Fixed Order

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Determination of perturbative QCD coupling from ALEPH $$\tau $$ decay data using pinched Borel–Laplace and Finite Energy Sum Rules

The European Physical Journal C ◽

10.1140/epjc/s10052-021-09664-x ◽

2021 ◽

Vol 81 (10) ◽

Author(s):

César Ayala ◽

Gorazd Cvetič ◽

Diego Teca

Keyword(s):

Perturbation Theory ◽

Perturbative Qcd ◽

Sum Rules ◽

Finite Energy ◽

Operator Product ◽

Decay Data ◽

Fixed Order ◽

Principal Value ◽

Adler Function

AbstractWe present a determination of the perturbative QCD (pQCD) coupling using the V+A channel ALEPH $$\tau $$ τ -decay data. The determination involves the double-pinched Borel–Laplace Sum Rules and Finite Energy Sum Rules. The theoretical basis is the Operator Product Expansion (OPE) of the V+A channel Adler function in which the higher order terms of the leading-twist part originate from a model based on the known structure of the leading renormalons of this quantity. The applied evaluation methods are contour-improved perturbation theory (CIPT), fixed-order perturbation theory (FOPT), and Principal Value of the Borel resummation (PV). All the methods involve truncations in the order of the coupling. In contrast to the truncated CIPT method, the truncated FOPT and PV methods account correctly for the suppression of various renormalon contributions of the Adler function in the mentioned sum rules. The extracted value of the $${\overline{\mathrm{MS}}}$$ MS ¯ coupling is $$\alpha _s(m_{\tau }^2) = 0.3116 \pm 0.0073$$ α s ( m τ 2 ) = 0.3116 ± 0.0073 [$$\alpha _s(M_Z^2)=0.1176 \pm 0.0010$$ α s ( M Z 2 ) = 0.1176 ± 0.0010 ] for the average of the FOPT and PV methods, which we regard as our main result. On the other hand, if we include in the average also the CIPT method, the resulting values are significantly higher, $$\alpha _s(m_{\tau }^2) = 0.3194 \pm 0.0167$$ α s ( m τ 2 ) = 0.3194 ± 0.0167 [$$\alpha _s(M_Z^2)=0.1186 \pm 0.0021$$ α s ( M Z 2 ) = 0.1186 ± 0.0021 ].

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A first determination of parton distributions with theoretical uncertainties

The European Physical Journal C ◽

10.1140/epjc/s10052-019-7364-5 ◽

2019 ◽

Vol 79 (10) ◽

Cited By ~ 5

Author(s):

Rabah Abdul Khalek ◽

Richard D. Ball ◽

Stefano Carrazza ◽

Stefano Forte ◽

Tommaso Giani ◽

...

Keyword(s):

Covariance Matrix ◽

Hadron Collider ◽

Distribution Functions ◽

Leading Order ◽

Parton Distribution Functions ◽

Parton Distributions ◽

Central Values ◽

Sources Of Uncertainty ◽

Fixed Order

Abstract The parton distribution functions (PDFs) which characterize the structure of the proton are currently one of the dominant sources of uncertainty in the predictions for most processes measured at the Large Hadron Collider (LHC). Here we present the first extraction of the proton PDFs that accounts for the missing higher order uncertainty (MHOU) in the fixed-order QCD calculations used in PDF determinations. We demonstrate that the MHOU can be included as a contribution to the covariance matrix used for the PDF fit, and then introduce prescriptions for the computation of this covariance matrix using scale variations. We validate our results at next-to-leading order (NLO) by comparison to the known next order (NNLO) corrections. We then construct variants of the NNPDF3.1 NLO PDF set that include the effect of the MHOU, and assess their impact on the central values and uncertainties of the resulting PDFs.

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NNLO virtual and real leptonic corrections to muon-electron scattering

Journal of High Energy Physics ◽

10.1007/jhep11(2021)098 ◽

2021 ◽

Vol 2021 (11) ◽

Author(s):

Ettore Budassi ◽

Carlo M. Carloni Calame ◽

Mauro Chiesa ◽

Clara Lavinia Del Pio ◽

Syed Mehedi Hasan ◽

...

Keyword(s):

Electron Scattering ◽

Monte Carlo Code ◽

Radiative Corrections ◽

Leading Order ◽

Muon Anomalous Magnetic Moment ◽

Small Momentum Transfer ◽

Hadronic Contribution ◽

Typical Event ◽

Fixed Order

Abstract The recently proposed MUonE experiment at CERN aims at providing a novel determination of the leading order hadronic contribution to the muon anomalous magnetic moment through the study of elastic muon-electron scattering at relatively small momentum transfer. The anticipated accuracy of the order of 10ppm demands for high-precision predictions, including all the relevant radiative corrections. The fixed-order NNLO radiative corrections due to the emission of virtual and real leptonic pairs are described and their numerical impact is discussed for typical event selections of the MUonE experiment, by means of the upgraded Monte Carlo code Mesmer.

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Galactic orbits of stars

Symposium - International Astronomical Union ◽

10.1017/s0074180900105340 ◽

1966 ◽

Vol 25 ◽

pp. 93-97

Author(s):

Richard Woolley

Keyword(s):

Globular Cluster ◽

Potential Field ◽

High Velocity ◽

Velocity Vector ◽

Variable Stars ◽

The Sun ◽

Proper Motions ◽

Rr Lyrae ◽

The Galaxy

It is now possible to determine proper motions of high-velocity objects in such a way as to obtain with some accuracy the velocity vector relevant to the Sun. If a potential field of the Galaxy is assumed, one can compute an actual orbit. A determination of the velocity of the globular clusterωCentauri has recently been completed at Greenwich, and it is found that the orbit is strongly retrograde in the Galaxy. Similar calculations may be made, though with less certainty, in the case of RR Lyrae variable stars.

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Distance to SN 1987A and the LMC

Symposium - International Astronomical Union ◽

10.1017/s0074180900118959 ◽

1999 ◽

Vol 190 ◽

pp. 549-554

Author(s):

Nino Panagia

Keyword(s):

Angular Size ◽

Large Magellanic Cloud ◽

Magellanic Cloud ◽

Light Curves ◽

Distance Modulus ◽

Absolute Size ◽

Sn 1987A

Using the new reductions of the IUE light curves by Sonneborn et al. (1997) and an extensive set of HST images of SN 1987A we have repeated and improved Panagia et al. (1991) analysis to obtain a better determination of the distance to the supernova. In this way we have derived an absolute size of the ringRabs= (6.23 ± 0.08) x 1017cm and an angular sizeR″ = 808 ± 17 mas, which give a distance to the supernovad(SN1987A) = 51.4 ± 1.2 kpc and a distance modulusm–M(SN1987A) = 18.55 ± 0.05. Allowing for a displacement of SN 1987A position relative to the LMC center, the distance to the barycenter of the Large Magellanic Cloud is also estimated to bed(LMC) = 52.0±1.3 kpc, which corresponds to a distance modulus ofm–M(LMC) = 18.58±0.05.

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International determination of the total motion of the pole

Symposium - International Astronomical Union ◽

10.1017/s0074180900177824 ◽

1961 ◽

Vol 13 ◽

pp. 29-41

Author(s):

Wm. Markowitz

Keyword(s):

Secular Motion ◽

The Future

A symposium on the future of the International Latitude Service (I. L. S.) is to be held in Helsinki in July 1960. My report for the symposium consists of two parts. Part I, denoded (Mk I) was published [1] earlier in 1960 under the title “Latitude and Longitude, and the Secular Motion of the Pole”. Part II is the present paper, denoded (Mk II).

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