On the Interpretation of the Balance Function

We construct a simple toy model and explicitly demonstrate that the balance function (BF) can become negative for some values of the rapidity separation and hence cannot have any probabilistic interpretation. In particular, the BF cannot be interpreted as the probability density for the balancing charges to occur separated by the given rapidity interval.

Comment on Modified Fare Ratio in a Two-Class Static Revenue Management Model with Buy-up Behavior

We review the optimal booking limit in the two-class static revenue management model with customers’ buy-up behavior. This is when a deterministic fraction of the low-fare customer class that cannot book early are willing to book the higher fare later. This simple model with dependent demands is difficult to analyze. Some well-known publications, such as Talluri and van Ryzin ( 2004 ) and Phillips ( 2005 ), treat this model incorrectly. In this note, we correct an erroneous formula for the modified fare ratio with the proper probabilistic interpretation. The correction was established previously by Brumelle et al. ( 1990 ). Numerical examples reveal that the corrected modified fare ratio provides a lower optimal booking limit, resulting in a higher expected revenue than those obtained by using the incorrect modified fare ratio.

General helicity formalism for two-hadron production in e+e− annihilation within a TMD approach

We present the complete leading-order results for the azimuthal dependences and polarization observables in e+e−→ h1h2 + X processes, where the two hadrons are produced almost back-to-back, within a transverse momentum dependent (TMD) factorization scheme. We consider spinless (or unpolarized) and spin-1/2 hadron production and give the full set of the corresponding quark and gluon TMD fragmentation functions (TMD-FFs). By adopting the helicity formalism, which allows for a more direct probabilistic interpretation, single- and double-polarization cases are discussed in detail. Simplified expressions, useful for phenomenological analyses, are obtained by assuming a factorized Gaussian-like dependence on intrinsic transverse momenta for the TMD-FFs.

Anti-PT Transformations and Complex PT-Symmetric Superpartners

A quantum mechanical system with unbroken super-and parity-time (PT)-symmetry is derived and analyzed. Here, we propose a new formalism to construct the complex PT-symmetric superpartners by extending the additive shape invariant potentials to the complex domain. The probabilistic interpretation of a PT-symmetric quantum theory is correlated with the calculation of a new linear operator called the C operator, instead of complex conjugation in conventional quantum mechanics. At the present work, we introduce an anti-PT (A PT) conjugation to redefine a new version of the inner product without any additional considerations. This PT-supersymmetric quantum mechanics, satisfies essential requirements such as completeness, orthonormality as well as probabilistic interpretation.

A Review of Matrix SIR Arino Epidemic Models

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”, which may be useful to obtain approximate control policies.