Oscillations and potentials of mesons and baryons

In the following, the oscillations and potentials of mesons and baryons are examined and analyzed in detail. The oscillations result from a simple formula that describes the resonance energy at which the corresponding particle can absorb energy and thus appear. The potentials describe three mechanisms that describe the fine splitting of the masses of the elementary particles. These potentials can be read off and derived from the experimentally determined masses of the elementary particles as coefficients. The three mechanisms are internal mass charge binding energy, external mass charge binding energy, and Coulomb interaction.

Accounting Start-up Time of Parallel Processes in Amdahl’s Law

The conventional form of Amdahl’s law states that speedup of calculations in a multiprocessor machine is limited by the definite constant value just due to the existence of some non-parallelizable part in any algorithm. This brief paper considers one more general reason, which prevents a growth of parallel performance: processes that implement distributed task cannot start simultaneously and hence every process adds some start-up time, also reducing by that the gain from a parallel processing. The simple formula, proposed here to extend Amdahl’s law, leads to a less optimistic picture in comparison with classical results: for large amount of processor units the modified law does not approach to constant but vanishes. This is the result of competition between two factors: decreasing of calculation duty and increasing of start-up time when a number of parallel processes grows. The effect may be subdued by means of specific regularity in launching parallel processes.

A New Method to Compare the Interpretability of Rule-Based Algorithms

Interpretability is becoming increasingly important for predictive model analysis. Unfortunately, as remarked by many authors, there is still no consensus regarding this notion. The goal of this paper is to propose the definition of a score that allows for quickly comparing interpretable algorithms. This definition consists of three terms, each one being quantitatively measured with a simple formula: predictivity, stability and simplicity. While predictivity has been extensively studied to measure the accuracy of predictive algorithms, stability is based on the Dice-Sorensen index for comparing two rule sets generated by an algorithm using two independent samples. The simplicity is based on the sum of the lengths of the rules derived from the predictive model. The proposed score is a weighted sum of the three terms mentioned above. We use this score to compare the interpretability of a set of rule-based algorithms and tree-based algorithms for the regression case and for the classification case.

A necessary condition for discrete branching laws for Klein four symmetric pairs

We show a necessary condition for Klein four symmetric pairs [Formula: see text] satisfying the condition (D.D.); that is, there exists at least one infinite-dimensional simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. This work is a continuation of [A criterion for discrete branching laws for Klein four symmetric pairs and its application to [Formula: see text], Int. J. Math. 31(6) (2020) 2050049]. Moreover, we define associated Klein four symmetric pairs, and we may use these tools to compute that a class of Klein four symmetric pairs do not satisfy the condition (D.D.); for example, [Formula: see text].

Longtime behaviour and bursting frequency, via a simple formula, of FitzHugh–Rinzel neurons

The longtime behaviour of the FitzHugh–Rinzel (FHR) neurons and the transition to instability of the FHR steady states, are investigated. Criteria guaranteeing solutions boundedness, absorbing sets, in the energy phase space, existence and steady states instability via oscillatory bifurcations, are obtained. Denoting by $$ \lambda ^{3} + \sum\nolimits_{{k = 1}}^{3} {A_{k} } (R)\lambda ^{{3 - k}} = 0 $$ λ 3 + ∑ k = 1 3 A k ( R ) λ 3 - k = 0 , with R bifurcation parameter, the spectrum equation of a steady state $$m_0$$ m 0 , linearly asymptotically stable at certain value of R, the frequency f of an oscillatory destabilizing bifurcation (neuron bursting frequency), is shown to be $$ f=\displaystyle \frac{\sqrt{A_2(R_\mathrm{H})}}{2\pi } $$ f = A 2 ( R H ) 2 π with $$R_\mathrm{H}$$ R H location of R at which the bifurcation occurs. The instability coefficient power (ICP) (Rionero in Rend Fis Acc Lincei 31:985–997, 2020; Fluids 6(2):57, 2021) for the onset of oscillatory bifurcations, is introduced, proved and applied, in a new version.

CALCULATION OF ELLIPSE CIRCUMFERENCE AS A CYLINDRICAL SECTION

In this article, ellipse is studied as cylindrical section and a simple formula is presented to calculate the circumference of an ellipse approximately. 2000 Mathematics Subject Classification: 51N25

DISCRIMINATION-FREE INSURANCE PRICING

We consider the following question: given information on individual policyholder characteristics, how can we ensure that insurance prices do not discriminate with respect to protected characteristics, such as gender? We address the issues of direct and indirect discrimination, the latter resulting from implicit learning of protected characteristics from nonprotected ones. We provide rigorous mathematical definitions for direct and indirect discrimination, and we introduce a simple formula for discrimination-free pricing, that avoids both direct and indirect discrimination. Our formula works in any statistical model. We demonstrate its application on a health insurance example, using a state-of-the-art generalized linear model and a neural network regression model. An important conclusion is that discrimination-free pricing in general requires collection of policyholders’ discriminatory characteristics, posing potential challenges in relation to policyholder’s privacy concerns.

Invariant Hermitian forms on vertex algebras

We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary [Formula: see text]-algebra. We show that for a minimal simple [Formula: see text]-algebra [Formula: see text] this form can be unitary only when its [Formula: see text]-grading is compatible with parity, unless [Formula: see text] “collapses” to its affine subalgebra.