Recursion relations for
            hp
            ‐FEM Element Matrices on quadrilaterals

ABSTRACT Perturbation theory is an indispensable tool for studying the cosmic large-scale structure, and establishing its limits is therefore of utmost importance. One crucial limitation of perturbation theory is shell-crossing, which is the instance when cold-dark-matter trajectories intersect for the first time. We investigate Lagrangian perturbation theory (LPT) at very high orders in the vicinity of the first shell-crossing for random initial data in a realistic three-dimensional Universe. For this, we have numerically implemented the all-order recursion relations for the matter trajectories, from which the convergence of the LPT series at shell-crossing is established. Convergence studies performed at large orders reveal the nature of the convergence-limiting singularities. These singularities are not the well-known density singularities at shell-crossing but occur at later times when LPT already ceased to provide physically meaningful results.

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New ideas for handling of loop and angular integrals in D-dimensions in QCD

Journal of High Energy Physics ◽

10.1007/jhep06(2021)066 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Valery E. Lyubovitskij ◽

Fabian Wunder ◽

Alexey S. Zhevlakov

Keyword(s):

Computer Algebra ◽

Cross Sections ◽

Orthogonal Basis ◽

Computer Algebra Systems ◽

Covariant Formalism ◽

Linear Combinations ◽

Recursion Relations ◽

New Ideas ◽

Rotational Invariant ◽

Geometric Picture

Abstract We discuss new ideas for consideration of loop diagrams and angular integrals in D-dimensions in QCD. In case of loop diagrams, we propose the covariant formalism of expansion of tensorial loop integrals into the orthogonal basis of linear combinations of external momenta. It gives a very simple representation for the final results and is more convenient for calculations on computer algebra systems. In case of angular integrals we demonstrate how to simplify the integration of differential cross sections over polar angles. Also we derive the recursion relations, which allow to reduce all occurring angular integrals to a short set of basic scalar integrals. All order ε-expansion is given for all angular integrals with up to two denominators based on the expansion of the basic integrals and using recursion relations. A geometric picture for partial fractioning is developed which provides a new rotational invariant algorithm to reduce the number of denominators.

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Open associahedra and scattering forms

Journal of High Energy Physics ◽

10.1007/jhep12(2020)134 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

Aidan Herderschee ◽

Fei Teng

Keyword(s):

Scattering Amplitudes ◽

Canonical Form ◽

Recursion Relations ◽

Fiber Product ◽

New Phenomena

Abstract We continue the study of open associahedra associated with bi-color scattering amplitudes initiated in ref. [1]. We focus on the facet geometries of the open associahedra, uncovering many new phenomena such as fiber-product geometries. We then provide novel recursion procedures for calculating the canonical form of open associahedra, generalizing recursion relations for bounded polytopes to unbounded polytopes.

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Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

Abstract and Applied Analysis ◽

10.1155/2014/890456 ◽

2014 ◽

Vol 2014 ◽

pp. 1-24 ◽

Cited By ~ 2

Author(s):

David W. Pravica ◽

Njinasoa Randriampiry ◽

Michael J. Spurr

Keyword(s):

Fourier Transform ◽

Theta Function ◽

Legendre Polynomial ◽

Legendre Polynomials ◽

Inverse Fourier Transform ◽

Recursion Relations ◽

Jacobi Theta Function ◽

Convergence Results ◽

The Family ◽

Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

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Random crystal field effects on antiferromagnetic spin-1 Blume–Capel model

Modern Physics Letters B ◽

10.1142/s0217984921502869 ◽

2021 ◽

pp. 2150286

Author(s):

Erhan Albayrak

Keyword(s):

Crystal Field ◽

Critical Points ◽

Bethe Lattice ◽

Honeycomb Lattice ◽

Recursion Relations ◽

First Order ◽

Field Effects ◽

Number Formula ◽

First Order Phase ◽

Antiferromagnetic Spin

The outcome of the random crystal field effects on the antiferromagnetic spin-1 Blume–Capel model and external magnetic field are examined on the Bethe Lattice in terms of exact recursion relations. It is assumed that the crystal field is either turned on or off randomly with probability [Formula: see text] and [Formula: see text], respectively. The phase diagrams are constructed from the thermal analysis of the order parameters with the coordination number [Formula: see text] which corresponds to honeycomb lattice. It is explored that the system goes both second- and first-order phase transitions, along with the reentrant behavior and a few critical points. The reentrant behavior is stronger for lower values of [Formula: see text] and disappears as [Formula: see text] gets closer to 1.0. The first-order lines are observed to be either linked to the tricritical points or decomposed. The critical end points and double critical points are also observed.

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On-Shell Recursion Relations for Effective Field Theories

Physical Review Letters ◽

10.1103/physrevlett.116.041601 ◽

2016 ◽

Vol 116 (4) ◽

Cited By ~ 52

Author(s):

Clifford Cheung ◽

Karol Kampf ◽

Jiri Novotny ◽

Chia-Hsien Shen ◽

Jaroslav Trnka

Keyword(s):

Effective Field Theories ◽

Effective Field ◽

Field Theories ◽

Recursion Relations

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Origin shifts in divergent Feynman integrals

Canadian Journal of Physics ◽

10.1139/p69-161 ◽

1969 ◽

Vol 47 (12) ◽

pp. 1263-1269 ◽

Cited By ~ 10

Author(s):

Robert E. Pugh

Keyword(s):

Sum Rules ◽

Tensor Rank ◽

Linear Quadratic ◽

Arbitrary Degree ◽

Recursion Relations ◽

Feynman Integrals ◽

Shift Of Origin

The surface terms arising from a shift of origin in divergent Feynman integrals are considered. Sum rules and recursion relations between these terms are derived for an arbitrary degree of divergence and tensor rank. These relations are explicitly solved for linear, quadratic, cubic, and quartic divergences.

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