π = r arctan x where r and x are rational and |x | < 1 is small. Such an identity would require only one evaluation of the arctangent function and this evaluation would converge quickly. However, identities of this form do not exist and this fact is not mentioned in the literature alongside lists of such multiple-angle identities. The present article gives a very natural proof of this fact usin...

We use data on the deep inelastic structure function F2 in order to constrain the cross-section for scattering a colour dipole off a proton. The data seem to prefer parameterisations which include saturation effects. That is they indicate that the strong rise with energy of the dipole cross-section, which holds for small dipoles, pertains only for r < rs(x) where rs(x) decreases monotonically a...

We analyse power corrections to longitudinal and transverse fragmentation processes in ee annihilation, based on the assumption of ultraviolet dominance of power corrections. Under this assumption, we determine the dependence of power corrections on the scaling variable x from the infrared renormalon asymptotics of leading power coefficient functions. Our results suggest that the longitudinal a...

The objective of this section is to introduce notation and theoretical tools regarding structural models and bipartite graphs that will be used in proofs of results in coming section. The structural representation of a set of equations M with unknown variables X is a bipartite graph with variables and equations as node sets. There is an edge in the graph between a node representing an equation ...

We observe that the saturation model of deep inelastic scattering, which successfully describes inclusive and diffractive data at small x, predicts a geometric scaling of the total γ∗p cross section in the region of small Bjorken variable x. The geometric scaling in this case means that the cross section is a function of only one dimensionless variable τ = Q2R2 0(x), where the function R0(x) (c...

1 Mirror operator ∇ψ * In this section, we discuss properties of distance generating functions and their subdif-ferentials. Let ψ be a proper, closed, convex function, and suppose that X is the effective domain of ψ (i.e. X = {x ∈ R n : ψ(x) < ∞}). The subdifferential of ψ at x ∈ X is ∂ψ(x) = {z ∈ E * : ψ(y) − ψ(x) − z, y − x ≥ 0 ∀y ∈ X }.

Corrigendum: Propagation phasor approach for holographic image reconstruction Wei Luo, Yibo Zhang, Zoltán Göröcs, Alborz Feizi & Aydogan Ozcan Scientific Reports 6:22738; doi: 10.1038/srep22738; published online 11 March 2016; updated 19 October 2016 Here we correct and clarify some of the mathematical expressions in this Article. These do not affect the conclusions and the reported results of ...

For last fifteen years, numerous authors have studied the birational geometry of projective irreducible holomorphic symplectic varieties X, seeking to relate extremal contractions X → X ′ to properties of the Hodge structures on H(X,Z) and H2(X,Z), regarded as lattices under the Beauville-Bogomolov form. Significant contributions have been made by Huybrechts, Markman, O’Grady, Verbitsky, and ma...

The (p,1)-total number λp (G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least p. In this paper we consider the list version. Let L(x) be a list of ...

Optimization approach to define geometry (parameters) of radial matching section in RFQ accelerator is suggested. This approach gives wider opportunities to design of radial matching section because it does not have certain prescribed laws of variation of focusing strength along the section. INTRODUCTION Acceptance at the entrance of RFQ accelerator, as it is known, depends on time and rotates ...