Integral relation for the t-dependence of the It = 1 ππ absorptive part

1973 ◽  
Vol 45 (1) ◽  
pp. 48-52 ◽  
Author(s):  
J.L. Basdevant ◽  
C. Schomblond
Keyword(s):  
Absorptive Part ◽  
Integral Relation

scholarly journals A conformal dispersion relation: correlations from absorption

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Dean Carmi ◽  
Simon Caron-Huot
Keyword(s):  
Dispersion Relation ◽  
Perfect Matching ◽  
Elliptic Integral ◽  
Absorptive Part ◽  
Integral Relation ◽  
Integral Function ◽  
Conformal Block ◽  
Tree Level ◽  
Conformal Field ◽  
Free Fields

Abstract We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its “absorptive part”, defined as a double discontinuity, times a theory-independent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the “inverted” conformal block with the ordinary conformal block.

Optimization of Wind Turbines Using Helicoidal Vortex Model

2003 ◽  
Author(s):  
Jean-Jacques Chattot
Keyword(s):  
Minimum Energy ◽  
Energy Condition ◽  
Integral Relation ◽  
Point Of View ◽  
Maximum Output ◽  
Efficient Design ◽  
Profile Data ◽  
Vortex Model ◽  
Viscous Profile ◽  
Optimum Solution

The problem of the design of a wind turbine for maximum output is addressed from an aerodynamical point of view. It is shown that the optimum inviscid design, based on the Goldstein model, satifies the minimum energy condition of Betz only for light loading. The more general equation governing the optimum is derived and an integral relation is obtained, stating that the optimum solution satisfies the minimum energy condition of Betz in the Trefftz plane “in the average”. The discretization of the problem is detailed, including the viscous correction based on the 2-D viscous profile data. A constraint is added to account for the force on the tower. The minimization problem is solved very efficiently by relaxation. Several optimized solutions are calculated and compared with the NREL rotor, using the same profile, but different chord and twist distributions. In all cases, the optimization produces a more efficient design.

Kinematical Theory of Scattering

2020 ◽  
pp. 73-98
Author(s):  
Andrew T. Boothroyd
Keyword(s):  
Correlation Function ◽  
Response Function ◽  
Detailed Balance ◽  
Absorptive Part ◽  
Spectral Weight ◽  
Static Approximation ◽  
Dynamic Correlations ◽  
Fluctuation Dissipation ◽  
Intermediate Scattering Function ◽  
Kinematical Theory

The chapter introduces the kinematical theory of scattering, which is based on the Born approximation. It is shown that the neutron scattering response function can be written as the time Fourier transform of a correlation function, or intermediate scattering function. Several general properties of the correlation function are derived, and the response function is shown to satisfy the Principle of Detailed Balance. The distinction between static and dynamic correlations is explained, and their correspondence to elastic and inelastic scattering is established. The meaning of the static approximation is explained, and the link between the dynamical part of the response function and the absorptive part of the generalized susceptibility via the Fluctuation-Dissipation theorem is established. Some general sum rules are proved, and a spectral-weight function is defined. Response functions are obtained for some simple models.

An integral relation for tensor polynomials

2011 ◽  
Vol 166 (2) ◽  
pp. 186-193 ◽  
Author(s):  
P. A. Vshivtseva ◽  
V. I. Denisov ◽  
I. P. Denisova
Keyword(s):  
Integral Relation

Eigenvalue integral relation and stability condition for a mode

1989 ◽  
Vol 5 (3) ◽  
pp. 278-284
Author(s):  
Zhu Ruzeng ◽  
Wu Hanming
Keyword(s):  
Stability Condition ◽  
Integral Relation

Construction of Higher-Order Accurate Time-Step Integration Algorithms by Equal-Order Polynomial Projection

2005 ◽  
Vol 11 (1) ◽  
pp. 19-49 ◽  
Author(s):  
T. C. Fung
Keyword(s):  
Integral Relation ◽  
Higher Order ◽  
Numerical Results ◽  
Numerical Dissipation ◽  
Time Interval ◽  
Time Step ◽  
Higher Order Terms ◽  
Optimal Polynomial ◽  
Integration Algorithms ◽  
New Framework

This paper presents a new framework to construct higher-order accurate time-step integration algorithms based on weakly enforcing the differential/integral relation. The dependent variable and its time derivatives are assumed to be polynomials of equal order. A differential equation is then transformed into an algebraic equation directly. The main issue is how to approximate the integral of a polynomial by another polynomial of the same degree. Various methods to determine the optimal representation (or projection) are considered. It is shown that to reproduce numerical results equivalent to the Padé or generalized Padé approximations, the coefficients of the optimal polynomial representation are related to the weighting parameters derived previously for time-step integration algorithms with predetermined coefficients. A special feature for the present formulation is that the same procedure can be used to solve first-, second-, and even higher-order non-homogeneous initial value problems in a unified manner. The resultant algorithms are higher-order accurate and unconditionally A-stable with controllable numerical dissipation. It is also shown that for the numerical results to maintain higher-order accuracy at the end of a time interval, the higher-order terms in the excitation have to be projected as polynomials of lower degree within the present framework as well. Numerical examples are given to illustrate the validity of the present formulations.

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