scholarly journals The R-matrix bootstrap for the 2d O(N) bosonic model with a boundary

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Martin Kruczenski ◽  
Harish Murali
Keyword(s):  
Dual Problem ◽  
Analytic Form ◽  
Dual Function ◽  
R Matrix ◽  
Infinite Dimensional ◽  
Reflection Matrix ◽  
S Matrix ◽  
Free Parameters ◽  
Dimensional Section ◽  
Bosonic Model

Abstract The S-matrix bootstrap is extended to a 1+1d theory with O(N) symmetry and a boundary in what we call the R-matrix bootstrap since the quantity of interest is the reflection matrix (R-matrix). Given a bulk S-matrix, the space of allowed R-matrices is an infinite dimensional convex space from which we plot a two dimensional section given by a convex domain on a 2d plane. In certain cases, at the boundary of the domain, we find vertices corresponding to integrable R-matrices with no free parameters. In other cases, when there is a one-parameter family of integrable R-matrices, the whole boundary represents integrable theories. We also consider R-matrices which are analytic in an extended region beyond the physical cuts, thus forbidding poles (resonances) in that region. In certain models, this drastically reduces the allowed space of R-matrices leading to new vertices that again correspond to integrable theories. We also work out the dual problem, in particular in the case of extended analyticity, the dual function has cuts on the physical line whenever unitarity is saturated. For the periodic Yang-Baxter solution that has zero transmission, we computed the R-matrix initially using the bootstrap and then derived its previously unknown analytic form.

Comparison of reaction theories for multilevel interference—the 14C + p reactions

1969 ◽  
Vol 47 (24) ◽  
pp. 2763-2777 ◽  
Author(s):  
C. T. Tindle ◽  
E. Vogt
Keyword(s):  
Compound Nucleus ◽  
Cross Sections ◽  
Curve Fitting ◽  
Low Energy ◽  
Analytic Method ◽  
R Matrix ◽  
Strong Interference ◽  
S Matrix ◽  
Fitting Procedures ◽  
Good Agreement

A comparison is made between the R-matrix and S-matrix theories of low-energy compound nucleus resonances for the particular case of two-level interference. The (p,γ) and (p,n) cross sections of 14C for proton energies between 0.7 and 1.5 MeV are analyzed using both theories. The 15N compound nucleus in this region exhibits strong two-level interference. The two theories provide equally good fits to the data, but the parameters describing the compound-nucleus levels are quite different. A general analytic method of relating the two sets of parameters is derived and shown to give good agreement with the results obtained by curve-fitting procedures. Remarks are made concerning the general behavior of the parameters under strong interference conditions and also on the inclusion of many channels into the analysis.

scholarly journals An Improved Subgradiend Optimization Technique for Solving IPs with Lagrangean Relaxation

2013 ◽  
Vol 61 (2) ◽  
pp. 135-140
Author(s):  
M Babul Hasan ◽  
Md Toha
Keyword(s):  
Dual Problem ◽  
Optimization Problems ◽  
Optimization Technique ◽  
Concave Function ◽  
Optimization Method ◽  
Duality Gap ◽  
Dual Function ◽  
Subgradient Optimization ◽  
Step Size ◽  
Primal Problem

The objective of this paper is to improve the subgradient optimization method which is used to solve non-differentiable optimization problems in the Lagrangian dual problem. One of the main drawbacks of the subgradient method is the tuning process to determine the sequence of step-lengths to update successive iterates. In this paper, we propose a modified subgradient optimization method with various step size rules to compute a tuning- free subgradient step-length that is geometrically motivated and algebraically deduced. It is well known that the dual function is a concave function over its domain (regardless of the structure of the cost and constraints of the primal problem), but not necessarily differentiable. We solve the dual problem whenever it is easier to solve than the primal problem with no duality gap. However, even if there is a duality gap the solution of the dual problem provides a lower bound to the primal optimum that can be useful in combinatorial optimization. Numerical examples are illustrated to demonstrate the method. DOI: http://dx.doi.org/10.3329/dujs.v61i2.17059 Dhaka Univ. J. Sci. 61(2): 135-140, 2013 (July)

Integrable Boundary Conditions Associated with the Zn × Zn Belavin Model and Solutions of Reflection Equation

1997 ◽  
Vol 12 (16) ◽  
pp. 2809-2823 ◽  
Author(s):  
Heng Fan ◽  
Kang-Jie Shi ◽  
Bo-Yu Hou ◽  
Zhong-Xia Yang
Keyword(s):  
Boundary Conditions ◽  
Complete Solution ◽  
Vertex Model ◽  
Solvable Models ◽  
R Matrix ◽  
Reflection Equation ◽  
Reflection Matrix ◽  
The One ◽  
Integrable Boundary Conditions

We construct solvable models with nontrivial boundary from the well known Belavin R-matrix. The reflection equation for the boundary reflection matrix is studied and the complete solution for the one associated with the eight-vertex model is found.

Resonance Widths and Unitarity

1972 ◽  
Vol 50 (2) ◽  
pp. 84-92 ◽  
Author(s):  
C. T. Tindle
Keyword(s):  
Energy Dependence ◽  
Cross Section ◽  
Matrix Theory ◽  
Neutron Cross Section ◽  
Matrix Formalism ◽  
R Matrix ◽  
Level Width ◽  
S Matrix ◽  
The Difference ◽  
Energy Dependent

The low energy neutron cross section of 135Xe is analyzed using both the R-matrix theory of Wigner and Eisenbud and the S-matrix theory of Humblet and Rosenfeld. Particular attention is given to the role played by the total resonance level width for it is well known that the R-matrix widths are energy dependent but the S-matrix widths are not. This different energy dependence leads to different analytic forms for the cross section and the n + 135Xe reaction offers what may be the simplest and best physical example for comparing these two forms. To the accuracy of the present data the difference is not detectable. The different energy dependence of the resonance widths is shown to be related to unitarity. A general proof that the R-matrix formalism is always unitary is given. The difficulty of satisfying unitarity in the S-matrix formalism is discussed and it is shown for the n + 135Xe reactions that this can lead to physically unacceptable solutions. This "lack of unitarity" does not, however, lead to any difficulties in fitting the present experimental data.

On threshold resonances

1970 ◽  
Vol 48 (15) ◽  
pp. 1747-1758 ◽  
Author(s):  
C. T. Tindle
Keyword(s):  
Cross Sections ◽  
Matrix Theory ◽  
Threshold Level ◽  
R Matrix ◽  
Neutral Particles ◽  
S Matrix ◽  
Energy Neutron ◽  
Square Well ◽  
Neutron Cross Sections ◽  
Threshold Resonances

A comparison is made of the way R-matrix and S-matrix theories analyze the threshold resonances which occur in the scattering of neutral particles by a square well. Both approaches are found to provide very good approximate formulas. However, modifications of the usual S-matrix expansions must first be made. The behavior of the energy of the threshold level is quite different in the two alternatives. By comparing the two theories in their interpretation of the low-energy neutron cross sections of 1H, 16O, and 208Pb it is concluded that.R-matrix theory provides a better interpretation for unbound levels and S-matrix theory is preferable when the threshold level is bound.

scholarly journals A note on the S-matrix bootstrap for the 2d O(N) bosonic model

2018 ◽  
Vol 2018 (11) ◽  
Author(s):  
Yifei He ◽  
Andrew Irrgang ◽  
Martin Kruczenski
Keyword(s):  
S Matrix ◽  
Bosonic Model

scholarly journals S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Yifei He ◽  
Martin Kruczenski
Keyword(s):  
Scalar Fields ◽  
Upper Bounds ◽  
Linear Functionals ◽  
Convex Minimization Problem ◽  
Primal Problem ◽  
Infinite Dimensional ◽  
Partial Waves ◽  
Symmetry Properties ◽  
S Matrix ◽  
Convex Problem

Abstract The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the 2 → 2 scattering matrix S2→2 such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider 3 + 1 dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves kℓ(s) that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves fℓ(s), for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.

The S-Matrix and Its Analysis

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Acoustic Waves ◽  
Bound States ◽  
Scattering Matrix ◽  
Infinite Dimensional ◽  
Levinson’S Theorem ◽  
S Matrix ◽  
Radial Schrödinger Equation ◽  
Square Well ◽  
Watson Transform ◽  
Remote Past

This chapter focuses on the scattering matrix, or S-matrix, an infinite-dimensional matrix or operator that expresses the state of a scattering system consisting of waves or particles or both in the far future in terms of its state in the remote past. In the case of electromagnetic (or acoustic) waves, the S-matrix connects the intensity, phase, and polarization of the outgoing waves in the far field at various angles to the direction and polarization of the beam pointed toward an obstacle. The chapter first considers the problem of scattering by a square well, located symmetrically with respect to the origin, before discussing bound states and a heuristic derivation of the Breit-Wigner formula. It als describes the Watson transform and Regge poles before concluding with an analysis of the time-independent radial Schrödinger equation and Levinson's theorem.

Compensating the Distributed Effect of a Wave PDE in the Actuation or Sensing Path of Multi-Input and Multi-Output LTI Systems

2010 ◽  
Author(s):  
Nikolaos Bekiaris-Liberis ◽  
Miroslav Krstic
Keyword(s):  
Dual Problem ◽  
Second Order ◽  
Order System ◽  
Backstepping Method ◽  
Complex Type ◽  
Infinite Dimensional ◽  
Lti System ◽  
Single Input ◽  
Pure Delay ◽  
Explicit Feedback

The problem of compensation of infinite-dimensional actuator or sensor dynamics of more complex type than pure delay was solved recently using the backstepping method for PDEs. In this paper we construct an explicit feedback law for a multi-input LTI system which compensates the wave PDE dynamics and stabilizes the overall system. Our design is based on a novel infinite-dimensional backstepping-forwarding transformation. We illustrate the effectiveness of our design with a simulation example of a single-input second order system, in which the wave input enters the system through two different channels, each one located at a different point in the domain of the wave PDE. Finally, we consider a dual problem where we design an exponentially convergent observer that compensates the distributed effect of the wave sensor dynamics.

Sign in / Sign up

Export Citation Format

Share Document