Arbitrary Multiplicity Research Articles

Parametric stabilization for descriptor linear systems via state-proportional and -derivative feedback

This paper considers eigenstructure assignment problem for descriptor linear systems via state-proportional and -derivative feedback. New complete parametric approaches for the problem are proposed in two cases. General complete parametric expressions in direct closed forms for the closed-loop eigenvectors associated with the finite closed-loop eigenvalues are presented. Based on a recently proposed complete parametric solution to the generalized Sylvester matrix equation AV−EVJ=BWDJ+BWP, very simple complete parametric solutions to the feedback gains are also established. The approach guarantees arbitrary assignment of rank[EB] number finite closed-loop eigenvalues with arbitrary given algebraic and geometric multiplicities. It also guarantees the closed-loop regularity and impulse-freeness when several simple constraints on the design parameters are added. A numerical example shows the effectiveness of the proposed approaches.

More on soft theorems: Trees, loops, and strings

We study soft theorems in a broader context, addressing their fate at loop level and their universality in effective field theories and string theory. We argue that for gauge theories in the planar limit, loop-level soft gluon theorems can be made manifest already at the integrand level. In particular, we show that the planar integrand for N=4 SYM satisfies the tree-level soft theorem to all orders in perturbation theory and provide strong evidence to this effect for integrands in N<4 SYM. We consider soft theorems for non-supersymmetric Yang-Mills theories and gravity, and show the validity of integrand soft theorem, while loop corrections to the integrated soft theorems are intimately tied to the presence of conformal anomalies. We then address the question of universality of the soft theorems for various theories. In effective field theories with F^3 and R^3 interactions, the soft theorems are not modified. However for gravity theories with R^2 phi interactions, the sub-sub-leading order soft graviton theorem, which is beyond what is implied by the extended BMS symmetry, requires modifications at tree level for non-supersymmetric theories, and at loop level for N<5 supergravity due to anomalies. Finally, for superstring amplitudes at finite alpha', via explicit calculation for lower-point examples as well as world-sheet OPE analysis for arbitrary multiplicity, we show that the superstring amplitudes satisfy the same soft theorem as its field-theory counterpart. This is no longer true for bosonic closed strings due to the presence of R^2 phi interactions.

Computing decay rates for new physics theories with FeynRules and MadGraph 5_aMC@NLO

We present new features of the FeynRules and MadGraph 5_aMC@NLO programs for the automatic computation of decay widths that consistently include channels of arbitrary final-state multiplicity. The implementations are generic enough so that they can be used in the framework of any quantum field theory, possibly including higher-dimensional operators. We extend at the same time the conventions of the Universal FeynRules Output (or UFO) format to include decay tables and information on the total widths. We finally provide a set of representative examples of the usage of the new functions of the different codes in the framework of the Standard Model, the Higgs Effective Field Theory, the Strongly Interacting Light Higgs model and the Minimal Supersymmetric Standard Model and compare the results to available literature and programs for validation purposes. Program summaryProgram title: MadWidthCatalogue identifier: AEXY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEXY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1854258No. of bytes in distributed program, including test data, etc.: 21136244Distribution format: tar.gzProgramming language: Mathematica and Python.Computer: Platforms on which Mathematica and Python are available.Operating system: Operating systems on which Mathematica and Python are available.Classification: 11.1, 11.6.External routines: FeynRules 2.0 or higher, MadGraph5_aMC@NLO 2.2 or higher.Nature of problem: The program is a module for the FeynRules and MadGraph5_aMC@NLO packages that allows the computation of tree-level decay widths for arbitrary BSM models. The module consists of two parts: 1.A FeynRules part, which allows one to compute analytically all tree-level two-body decay rates and to output them in the UFO format.2.A MadGraph5_aMC@NLO part, which allows the numerical computation of many-body decay rates.Solution method:1.For the FeynRules part, the analytic expressions for the three-point vertices can be squared to obtain analytic formulas for two-body decay rates.2.For the MadGraph5_aMC@NLO part, MadGraph is used to generate all Feynman diagrams contributing to the decay, and diagrams that correspond to cascade decays are removed.Restrictions: Mathematica version 7 to 9. As the package is a module relying on FeynRules and MadGraph5_aMC@NLO all restrictions of these packages apply.Running time: The computation of the Feynman rules from a Lagrangian, as well as the computation of the decay rates, varies with the complexity of the model, and runs from a few seconds to several minutes. See Section 5 of the present manuscript for more information.

Repeated eigenstructure assignment for controlled invariant subspaces

This paper is concerned with the computation of basis matrices for the subspaces that lie at the core of the so-called geometric approach to control theory, namely the supremal output-nulling, reachability and stabilisability subspaces. Importantly, we also consider the problem of computing the feedback matrices that render these subspaces invariant with respect to the closed loop, while simultaneously assigning the assignable eigenstructure of the closed loop. Differently from the classical techniques presented in the literature so far on this topic, which are based on the standard pole assignment algorithms and are therefore applicable only in the non-defective case, the method presented in this paper can be applied in the case of closed-loop eigenvalues with arbitrary multiplicity.

FD-метод для задачі на власні значення в гільбертовому просторі у випадку базової задачі з власними значеннями довільної кратності

FD-метод для задачі на власні значення в гільбертовому просторі у випадку базової задачі з власними значеннями довільної кратності

General formulation of the sector-improved residue subtraction

The main theoretical tool to provide precise predictions for scattering cross sections of strongly interacting particles is perturbative QCD. Starting at next-to-leading order (NLO) the calculation suffers from unphysical IR-divergences that cancel in the final result. At NLO there exist general subtraction algorithms to treat these divergences during a calculation. Since the LHC demands for more precise theoretical predictions, general subtraction methods at next- to-next-to-leading order (NNLO) are needed.These proceedings outline the four-dimensional formulation of the sector improved residue subtraction. The subtraction scheme STRIPPER and in particular its extension to arbitrary multiplicities is explained. Therefore, it furnishes a general framework for the calculation of NNLO cross sections in perturbative QCD.

Unitary Equivalence and Similarity to Jordan Models for Weak Contractions of Class C0

AbstractWe obtain results on the unitary equivalence of weak contractions of class C0 to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool in this paper is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product.

From six to four and more: massless and massive maximal super Yang-Mills amplitudes in 6d and 4d and their hidden symmetries

A self-consistent exposition of the theory of tree-level superamplitudes of the 4d N=4 and 6d N=(1,1) maximally supersymmetric Yang-Mills theories is provided. In 4d we work in non-chiral superspace and construct the superconformal and dual superconformal symmetry generators of the N=4 SYM theory using the non-chiral BCFW recursion to prove the latter. In 6d we provide a complete derivation of the standard and hidden symmetries of the tree-level superamplitudes of N=(1,1) SYM theory, again using the BCFW recursion to prove the dual conformal symmetry. Furthermore, we demonstrate that compact analytical formulae for tree-superamplitudes in N=(1,1) SYM can be obtained from a numerical implementation of the supersymmetric BCFW recursion relation. We derive compact manifestly dual conformal representations of the five- and six-point superamplitudes as well as arbitrary multiplicity formulae valid for certain classes of superamplitudes related to ultra-helicity-violating massive amplitudes in 4d. We study massive tree superamplitudes on the Coulomb branch of the N=4 SYM theory from dimensional reduction of the massless superamplitudes of the six-dimensional N=(1,1) SYM theory. We exploit this correspondence to construct the super-Poincare and enhanced dual conformal symmetries of massive tree superamplitudes in N=4 SYM theory which are shown to close into a finite dimensional algebra of Yangian type. Finally, we address the fascinating possibility of uplifting massless 4d superamplitudes to 6d massless superamplitudes proposed by Huang. We confirm the uplift for multiplicities up to eight but show that finding the uplift is highly non-trivial and in fact not of a practical use for multiplicities larger than five.

Acceleration-induced scalar field transitions ofn-particle multiplicity

In this paper we calculate the effect of acceleration on the decay and excitation rates of scalar fields into a final state of arbitrary multiplicity. The analysis is carried out using standard field operators as well as an Unruh-DeWitt detector. Using the equivalence of the two methods, we show how to correctly setup the computation and interpret the results in terms of the particle content of the initial and final state Rindler and Minkowski spacetimes. We find the dominant transition pathway, and thus final state multiplicity, is acceleration dependent. The formalisms developed are then used to analyze the electron and muon system. We compute the transition rates and lifetimes for accelerated electrons and muons as well as the branching fractions for muon decay.

Multiparticle SYM equations of motion and pure spinor BRST blocks

In this paper a multiparticle generalization of linearized ten-dimensional super Yang--Mills superfields is proposed. Their equations of motions are shown to take the same form as in the single-particle case, supplemented by contact terms. A recursive construction of these superfields is inspired by the iterated OPEs among massless vertex operators in the pure spinor formalism. An enlarged set of BRST-covariant pure spinor blocks is then defined in a streamlined fashion and combined to multiparticle vertex operators. The latter can be used to universally describe tree-level subdiagrams in the perturbative open and closed superstring, regardless of the loop order. The inherent symmetries of the multiparticle superfields are reproduced by structure constants of the gauge group, hinting a natural appearance of the BCJ-duality between color and kinematics in the fundamentals of super Yang--Mills theory. We present one-loop applications where known scalar cohomology objects are systematically recomputed and a novel vector cohomology particularly relevant to the closed string is constructed for arbitrary multiplicity.

Complete Nagy-Soper subtraction for next-to-leading order calculations in QCD

We extend the Helac-Dipoles package with the implementation of a new subtraction formalism, first introduced by Nagy and Soper in the formulation of an improved parton shower. We discuss a systematic, semi-numerical approach for the evaluation of the integrated subtraction terms for both massless and massive partons, which provides the missing ingredient for a complete implementation. In consequence, the new scheme can now be used as part of a complete NLO QCD calculation for processes with arbitrary parton masses and multiplicities. We assess its overall performance through a detailed comparison with results based on Catani-Seymour subtraction. The importance of random polarization and color sampling of the external partons is also examined.

Converting metamodels to graph grammars: doing without advanced graph grammar features

In this paper, we present a method to convert a metamodel in the form of a UML class diagram into a context-sensitive graph grammar whose language comprises precisely the set of model graphs (UML object diagrams) that conform to the input metamodel. Compared to other approaches that deal with the same problem, we use a graph grammar formalism that does not employ any advanced graph grammar features, such as application conditions, precedence rules, and production schemes. Specifically, we use Rekers and Schurr's Layered Graph Grammars, which may be regarded as a pure generalization of standard context-sensitive string grammars. We show that elementary grammatical features, i.e., grammar labels and context-sensitive graph rewrite rules, suffice to represent metamodels with arbitrary multiplicities and inheritance. Inspired by attribute string grammars, we also propose a graph-grammar-based approach to the semantic analysis of model graphs.

Ramanujan’s Master Theorem for the Hypergeometric Fourier Transform Associated with Root Systems

Ramanujan’s Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the “Master Theorem”. In this paper we prove an analogue of Ramanujan’s Master theorem for the hypergeometric Fourier transform associated with root systems. This theorem generalizes to arbitrary positive multiplicity functions the results previously proven by the same authors for the spherical Fourier transform on semisimple Riemannian symmetric spaces.

On the Topology of Invariant Subspaces of a Shift of Higher Multiplicity

Following Beurling’s theorem R. Douglas and C. Pearcy have studied topology of invariant subspace lattice. R.Yang offered a description of path connected components of invariant subspace lattice for shift of multiplicity one. This paper generalizes the result to arbitrary finite multiplicity. We show that there exists a one to one correspondence between the invariant subspace lattice of shift of arbitrary finite multiplicity and the space of inner functions.

Unified performance analysis of maximal‐ratio combining for spectrum sharing systems with antenna correlation

SUMMARYA comprehensive analytical framework of a spectrum sharing system, in which secondary receiver (SR) is equipped with multiple antennas, is presented over Rayleigh fading channels. The impact of spatial fading correlation on the performance of spectrum sharing system with maximal‐ratio combining at the SR is investigated. In particular, we explicitly derive the CDF and moment generating function of the end‐to‐end signal‐to‐noise ratio by assuming a correlation model with arbitrary eigenvalue multiplicities at the SR. On the basis of these results, we conduct an evaluation of secondary system performance in terms of the outage probability, the average symbol error probability, and the ergodic capacity. All analytical results are verified by Monte Carlo simulations. Furthermore, to obtain an insight of system performance, the asymptotic analysis is studied to obtain the diversity gain and the coding gain in the peak transmit power dominated region. Our results show that the diversity gain is unaffected by antenna correlation whereas the coding gain is degraded. Copyright © 2013 John Wiley &amp; Sons, Ltd.

On the Unilateral Shift as a Hilbert Module over the Disc Algebra

We study the unilateral shift (of arbitrary countable multiplicity) as a Hilbert module over the disc algebra and the associated extension groups. In relation with the problem of determining whether this module is projective, we consider a special class of extensions, which we call polynomial. We show that the subgroup of polynomial extensions of a contractive module by the adjoint of the unilateral shift is trivial. The main tool is a function theoretic decomposition of the Sz.-Nagy–Foias model space for completely non-unitary contractions.

Numerical evaluation of virtual corrections to multi-jet production in massless QCD

We present a C++ library for the numerical evaluation of one-loop virtual corrections to multi-jet production in massless QCD. The pure gluon primitive amplitudes are evaluated using NGluon (Badger et al., (2011) [62]). A generalized unitarity reduction algorithm is used to construct arbitrary multiplicity fermion–gluon primitive amplitudes. From these basic building blocks the one-loop contribution to the squared matrix element, summed over colour and helicities, is calculated. No approximation in colour is performed. While the primitive amplitudes are given for arbitrary multiplicities, we provide the squared matrix elements only for up to 7 external partons allowing the evaluation of the five jet cross section at next-to-leading order accuracy. The library has been recently successfully applied to four jet production at next-to-leading order in QCD (Badger et al., 2012 [92]). Program SummaryProgram title:NJetCatalogue identifier: AEPF_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPF_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 250047No. of bytes in distributed program, including test data, etc.: 2138947Distribution format: tar.gzProgramming language: C++, Python.Computer: PC/Workstation.Operating system: No specific requirements — tested on Scientific Linux 5.2. and Mac OS X 10.7.4.Classification: 11.5.External routines: QCDLoop (http://qcdloop.fnal.gov/), qd (http://crd.lbl.gov/dhbailey/mpdist/), both included in the distribution file.Nature of problem:Evaluation of virtual corrections for multi-jet production in massless QCDSolution method:Purely numerical approach based on tree amplitudes obtained via Berends–Giele recursion combined with unitarity method.Restrictions:Full colour and helicity summed corrections only up to 5 final state jets.Running time:Full colour and helicity summed 2→4 channels take around 0.5–8 s per point depending on the number of fermion lines. Realistic times obtained during Monte Carlo integration will be highly dependent on the specific application.

Discrete eigenvalues of periodic operators with distant perturbations

We consider a selfadjoint periodic operator with distant perturbations in a periodic domain and study the behavior of the spectrum of the perturbed operator in the case where the distance between domains where perturbations are localized tends to infinity. We prove that the essential spectrum is stable under perturbations and the spectrum of the perturbed operator converges to the spectrum of the limit operator. We also establish the existence of eigenvalues of the perturbed operator and study their asymptotic behavior; moreover, we represent eigenvalues and eigenfunctions of the perturbed operator in the form of converging series and obtain expressions for their terms. Bibliography :8 titles. In this paper, we consider a periodic operator with finitely many distant perturbations in a multi-dimensional periodic domain. The perturbed operator is selfadjoint and satisfies certain conditions. Perturbations are arbitrary abstract symmetric localized operators. Localization is described by weight functions satisfying certain conditions guaranteeing the smoothness and decay at infinity. Note that no restrictions are imposed on the decay rate. The perturbations considered in this paper were earlier used in [1] for problems of other type, and the behavior of the resolvent of the same perturbed operator was studied in [2]. We study the behavior of the spectrum of the perturbed operator when the distance between the regions of perturbation localizations tends to infinity. We prove that the essential spectrum is invariant under perturbations and the eigenvalues (with multiplicity taken into account) of the perturbed operator converge to the eigenvalues of the limit operator. The main result of this paper consists in complete decompositions of eigenvalues and eigenfunctions of the perturbed operator. We prove the uniform convergence of the constructed series and obtain expressions for their terms. In the case of an arbitrary multiplicity of the limit eigenvalue and two distant (not necessarily symmetric) perturbations, we show that the first correction of the eigenvalues of the perturbed operator are symmetric relative to the origin. The existence and convergence of eigenvalues of the perturbed operator are proved on the basis of the results of [2]–[4] concerning the uniform resolvent convergence. The complete asymptotic expansions for eigenvalues and eigenfunctions of the perturbed operator are obtained (cf.

New irreducible modules for Heisenberg and affine Lie algebras

We study Z-graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine Kac–Moody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function φ:N→{±} we associate a φ-highest weight module over the Heisenberg Lie algebra and a φ-imaginary Verma module over the affine Lie algebra. We show that any φ-imaginary Verma module of nonzero level is irreducible.

Radiation generated by single and multiple volume reflection of ultrarelativistic electrons and positrons in bent crystals

Recently discovered processes of volume reflection and multiple volume reflection of charged particles in bent crystals are accompanied by specific electromagnetic (e.m.) radiation. The mechanism of radiation emission is quite complex due to the coexistence of different regimes for particle dynamics, resulting in e.m. generation with features characteristic of each regime. We present a simulation taking into consideration both the nondipole nature and arbitrary multiplicity of radiation accompanying volume reflection. This approach is worked out for electrons or positrons of any energy and is based on the local straight-crystal approximation and Baier-Katkov formula for radiation, the integration of which is considerably facilitated by the fast Fourier transform method. The radiation generated by multiple volume reflection with vertical and skew planes has been studied, too. A large axial contribution to the hard part of the radiative energy loss spectrum as well as the strengthening of planar radiation, with respect to the single volume reflection case, in the soft part of the spectrum are demonstrated. Both the wide spectrum and high intensity make this type of radiation quite suitable for the conversion of electron and positron beams to hard $\ensuremath{\gamma}$ radiation.