Compact Formula Research Articles

The operating profit flow method as a profitability analysis tool for hospitality enterprises

This study introduces the flow-through method, focusing on retaining incremental revenue as an alternative approach. Unlike conventional ratios, this method explains causal relationships between income statement items and aids in future projections. The hospitality industry, with varied revenue streams, faces challenges in applying uniform methods, limiting adoption due to practitioner unawareness. This research analyzes the financials of Hyatt, Marriott, and Hilton from 2021 to 2022 using the flow-through method. The study employs both exploratory and quantitative methods to understand how revenue streams and departmental operations affect profits. It proposes a compact formula for calculating the stream from incremental revenue changes, starting with determining departmental expenses. Results indicate hoteliers achieved remarkable success, demonstrating efficient revenue-profit conversion during post-pandemic recovery. This method provides valuable insights for hospitality managers and potential cross-application in various sectors. Future studies could investigate cross-industry applications, integrate technological tools, and perform longitudinal analyses to enhance and broaden the method's utility.

Theory of bunched beam transverse Schottky signals with stationary rf system and high-order chromaticity

Theory of bunched beam transverse Schottky signals with stationary rf system and high-order chromaticity

Tensor reduction for Feynman integrals with Lorentz and spinor indices

We present an efficient graphical approach to construct projectors for the tensor reduction of multi-loop Feynman integrals with both Lorentz and spinor indices in D dimensions. An ansatz for the projectors is constructed making use of its symmetry properties via an orbit partition formula. The graphical approach allows to identify and enumerate the orbits in each case. For the case without spinor indices we find a 1 to 1 correspondence between orbits and integer partitions describing the cycle structure of certain bi-chord graphs. This leads to compact combinatorial formulae for the projector ansatz. With spinor indices the graph-structure becomes more involved, but the method is equally applicable. Our spinor reduction formulae are based on the antisymmetric basis of γ matrices, and make use of their orthogonality property. We also provide a new compact formula to pass into the antisymmetric basis. We compute projectors for vacuum tensor Feynman integrals with up to 32 Lorentz indices and up to 4 spinor indices. We discuss how to employ the projectors in problems with external momenta.

Mellin amplitudes for AdS3× S3

There are holographic superconformal theories in all dimensions between two and six which allow arbitrary tree-level four-point functions to be fixed by basic consistency conditions. Although Mellin space is usually the most efficient setting for imposing these contraints, four-point functions in two dimensions have thus far been an exception due to their more intricate dependence on the conformal cross-ratios. In this paper, we introduce a simple fix which exploits the relation between a parity-odd conformal block in two dimensions and a parity-even conformal block in four dimensions. We then apply the resulting toolkit to a study of the paradigmatic holographic theory in two dimensions which is the D1-D5 CFT. For correlators involving Kaluza-Klein modes of the tensor multiplet, this analysis reproduces results which were previously obtained using hidden conformal symmetry. With four Kaluza-Klein modes of the graviton multiplet, it yields new results including a compact formula for the correlators of all pairwise identical operators.

Low and moderate x gluon contribution to exclusive Compton scattering processes

We revisit the high energy semi-classical description of the exclusive processes DVCS, TCS, and Double DVCS by explicitly keeping track of the Feynman x dependence in both the hard and the hadronic matrix elements. This is achieved by a modification of the standard shock wave approximation to derive the effective Feynman rules, which leads to a generic expression on which we then perform a partial twist expansion to get rid of quantities suppressed by the proper physical scales. We obtain a compact factorized master formula that can be used to investigate the Bjorken limit at leading twist. In particular, we recover the full one-loop result in the collinear limit for pure gluon exchange with the target. Finally, we discuss the subtleties in taking the simultaneous collinear and small x limit.

High-Accuracy Positivity-Preserving Finite Difference Approximations of the Chemotaxis Model for Tumor Invasion.

Numerical simulation of the complex evolution process for tumor invasion plays an extremely important role in-depth exploring the bio-taxis phenomena of tumor growth and metastasis. In view of the fact that low-accuracy numerical methods often have large errors and low resolution, very refined grids have to be used if we want to get high-resolution simulating results, which leads to a great deal of computational cost. In this paper, we are committed to developing a class of high-accuracy positivity-preserving finite difference methods to solve the chemotaxis model for tumor invasion. First, two unconditionally stable implicit compact difference schemes for solving the model are proposed; second, the local truncation errors of the new schemes are analyzed, which show that they have second-order accuracy in time and fourth-order accuracy in space; third, based on the proposed schemes, the high-accuracy numerical integration idea of binary functions is employed to structure a linear compact weighting formula that guarantees fourth-order accuracy and nonnegative, and then a positivity-preserving and time-marching algorithm is established; and finally, the accuracy, stability, and positivity-preserving of the proposed methods are verified by several numerical experiments, and the evolution phenomena of tumor invasion over time are numerically simulated and analyzed.

Surfaceology for colored Yukawa theory

Arkani-Hamed and collaborators have recently shown that scattering amplitudes for colored theories can be expressed as integrals over combinatorial objects simply constructed from surfaces decorated by kinematic data. In this paper we extend the curve integral formalism to theories with colored fermionic matter and present a compact formula for the all-loop, all-genus, all-multiplicity amplitude integrand of a colored Yukawa theory. The curve integral formalism makes certain properties of the amplitudes manifest and repackages non-trivial numerators into a single combinatorial object. We also present an efficient formula for L-loop integrated amplitudes in terms of a sum over 2L combinatorial determinants.

Multiple solitons and breathers on periodic backgrounds in the complex modified Korteweg–de Vries equation

This study explores multiple soliton and breather solutions on periodic backgrounds in the complex modified Korteweg–de Vries equation. The compact determinant formulas and their detailed derivation process for these solutions are provided via the bilinear method. We confirm that on periodic backgrounds, soliton amplitudes exhibit regular periodic behaviors, while breather amplitudes display quasi-periodic behaviors, as is expected for a breather with one period propagating over a periodic wave with another period. The asymptotic expressions for the solitons and breathers, which establish the high accuracy of the derived solutions, are provided to reveal the soliton and breather dynamics on the periodic backgrounds.

Analytical QNMs of fields of various spin in the Hayward spacetime

By employing an expansion in terms of the inverse multipole number, we derive analytic expressions for the quasinormal modes (QNMs) of scalar, Dirac and Maxwell perturbations in the Hayward black hole (BH) background. The metric has three interpretations: as a model for a radiating BH, as a quantum-corrected BH owing to the running gravitational coupling in the Asymptotically Safe Gravity, and as a BH solution in the Effective Field Theory. We show that the obtained compact analytical formulas approximate QNMs with remarkable accuracy for ℓ > 0.

Higher-order phase-space moments for off-diagonal rotating Morse oscillators

The present paper proposes an advancement beyond the recent mathematical formulation [Cherroud O and Yahiaoui S-A 2023 Eur. Phys. J. Plus. 138 534], by extending their approach to the evaluation of off-diagonal matrix elements 〈xˆmpˆl〉n,n′J,J′ for the rotation-vibration Morse oscillators. Calculations are conducted within the framework of phase-space quantum mechanics, considering (r → r eq). This methodology facilitates the derivation of analytical expressions for 〈xˆs〉n,n′J,J′ and 〈pˆs〉n,n′J,J′ matrix elements, where s = 1, 2, …. This approach enables the evaluation of both diagonal ( n=n′,J=J′≠0 ) and off-diagonal ( n≠n′,J=J′≠0 ) matrix elements through a more explicit and compact analytic formula, applicable for cases where J≠J′ as well as for scenarios involving non-rotating effects J=J′=0 .

On a finite sum of cosecants appearing in various problems

Finite trigonometric sums is a challenging and often quite difficult object of study. In this paper we investigate the finite sum of cosecants ∑csc⁡(φ+aπl/n), where the summation index l runs through 1 to n−1 and φ and a are arbitrary parameters, as well as several closely related sums, such as similar sums of a series of secants, of tangents and of cotangents. These trigonometric sums appear in various problems in mathematics, physics, and a variety of related disciplines. Their particular cases were fragmentarily considered in previous works, and it was noted that even a simple particular case ∑csc⁡(πl/n) does not have a closed–form, i.e. a compact summation formula (similar sums of sines, of cosines, of tangents, of cotangents and even of secants, if they exist, possess such expressions). In this paper, we derive several alternative representations for the above–mentioned sums, study their properties, relate them to many other finite and infinite sums, obtain their complete asymptotic expansions for large n and provide accurate upper and lower bounds (e.g. the typical relative error for the upper bound is lesser than 2×10−9 for n⩾10 and lesser than 7×10−14 for n⩾50, which is much better than the bounds we could find in previous works). Our researches reveal that these sums are deeply related to several special numbers and functions, especially to the digamma function (furthermore, as a by-product, we obtain several interesting summation formulæ for the digamma function). Asymptotical studies show that these sums may have qualitatively different behaviour depending on the choice of φ and a. In particular, as n increases some of them may become sporadically large and we identify the terms of the asymptotics responsible for such a behaviour. Finally, we also provide several historical remarks related to various sums considered in the paper. We show that some results in the field either were rediscovered several times or can easily be deduced from various known formulæ, including some formulæ dating back to the XVIIIth century.

An alternative two-phase formulation and analytical solution for a Stefan problem in one-dimension accounting for volume change and sensible heat

Stefan problems, or free boundary problems, in the phase change process occur in many engineering problems. This paper develops a compact formula for a two-phase melting process in a one-dimensional semi-infinite slab with a novel approach. Unlike the classic Stefan problem, the novel approach considers volume change (or density difference) and the net-sensible-heat of the two phases. Furthermore, an analytical solution is proposed. To illustrate, the melting process of ice-water in a pipe is investigated. The results have been validated through an exact solution and comparisons with other relevant models. A special case of a one-phase problem is also proposed. The proposed model can be used as an alternative tool for engineers in estimating the position of the melt-front, depending on the dominant effect on the phase change process.

On local geometric properties of a tokamak equilibrium

To separate the complexities in plasma physics and geometric effects, compact formulas for local geometric properties of a tokamak equilibrium are presented in this paper. They are written in a form similar to the Frenet formulas. All of the geometric quantities are expressed in terms of curvature and torsion of the three spatial curves for the moving local frame of reference, i.e., local orthogonal vector basis. In this representation, the local magnetic shear and the normalized parallel current are just the differences between two torsions of the vector basis. All of the geometric properties are coordinate invariants and form a prime set of quantities for describing tokamak plasma equilibrium. This prime set can be evaluated in both flux coordinates with closed flux surfaces and cylindrical coordinates including areas with open field lines, which may allow the extension of some analysis on the open field lines outside the last closed surface. Fundamental differential operators for stability and transport studies can be expressed explicitly in terms of these geometric properties. It can also be used to simplify analytic studies.

Regular decision processes

We introduce and study Regular Decision Processes (RDPs), a new, compact model for domains with non-Markovian dynamics and rewards, in which the dependence on the past is regular, in the language theoretic sense. RDPs are an intermediate model between MDPs and POMDPs. They generalize k-order MDPs and can be viewed as a POMDP in which the hidden state is a regular function of the entire history. In factored RDPs, transition and reward functions are specified using formulas in linear temporal logics over finite traces, or using regular expressions. This allows specifying complex dependence on the past using intuitive and compact formulas, and building models of partially observable domains without specifying an underlying state space.

Computing compact finite difference formulas under radial basis functions with enhanced applicability

In this work, we use the combination of radial basis functions (RBFs) and polynomials to derive compact finite difference (FD) formulas. We use RBFs and integrated RBFs (IRBFs), to obtain two sets of formulas: RBF-Hermite FD (RBF-HFD) and IRBF weight formulas. The analytical coefficients and truncation errors for these formulations are computed for the first derivative, second derivative, and 2D Laplacian operator. Our presented approach expands the usability of the weights by introducing a general framework for deriving weighting formulas. Finally, the computational results underscore the effectiveness of the introduced approach.

Massive ambitwistor-strings; twistorial models

Ambitwistor-strings are chiral strings whose targets are spaces of complex massless particles, and whose correlation functions directly lead to simple, compact formulae for scattering amplitudes and loop integrands for massless gauge and gravity theories. This article extends the framework to massive particles in 4d, obtained via a symmetry reduction of higher dimensional massless models based on twistors. The target space of the resulting models turns out to be the phase space of 4d massive particles in a twistorial representation, and the worldsheet theory agrees with the two-twistor string previously introduced by the authors. The paper has been written so as to be largely self-contained. We discuss two interesting classes of massive theories in detail. For gauge theories, the reduction procedure is explicitly adapted to supersymmetric gauge theories on the Coulomb branch. For supergravity theories, the reduction is adapted to give theories obtained via Cremmer, Scherk & Schwartz (CSS) reduction, with broken supersymmetry and massive multiplets. The reduction procedure gives explicit and systematic rules to obtain amplitudes for all these theories and their amplitudes from two compact master formulae that have their origins in 6d based on the polarized scattering equations; in the CSS case the formulae are new, and in both cases their derivation is systematic. The freedom to include mass allows the definition of a loop insertion operator, thereby extending the formulae to 1-loop. Unlike the massless 4d twistorial models, these all display a perfect double copy structure, here incorporating massive particles in the relationship between gauge theory and CSS supergravity amplitudes.

Hexagonalization of Fishnet integrals. Part II. Overlaps and multi-point correlators

This work presents the building-blocks of an integrability-based representation for multi-point Fishnet Feynman integrals with any number of loops. Such representation relies on the quantum separation of variables (SoV) of a non-compact spin-chain with symmetry SO(1, 5) explained in the first paper of this series. The building-blocks of the SoV representation are overlaps of the wave-functions of the spin-chain excitations inserted along the edges of a triangular tile of Fishnet lattice. The zoology of overlaps is analyzed along with various worked out instances in order to achieve compact formulae for the generic triangular tile. The procedure of assembling the tiles into a Fishnet integral is presented exhaustively. The present analysis describes multi-point correlators with disk topology in the bi-scalar limit of planar γ-deformed N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM theory, and it verifies some conjectural formulae for hexagonalisation of Fishnets CFTs present in the literature. The findings of this work are suitable of generalization to a wider class of Feynman diagrams.

Clar Covers and Zhang-Zhang Polynomials of Zigzag and Armchair Carbon Nanotubes

We derive here compact formulas for the Zhang-Zhang (ZZ) polynomials of two classes of finite open-ended carbon nanotubes: zigzag nanotubes (n, 0) of length d and armchair nanotubes (n, n) of length d. For zigzag nanotubes, the underlying Clar cover theory is trivial; in contrast, for armchair nanotubes, the Clar theory is complex and abundant in results. The ZZ polynomial formulas have been obtained using the interface theory of benzenoids and the transfer matrix methodology.

Asteroseismic g-mode period spacings in strongly magnetic rotating stars

ABSTRACT Strong magnetic fields are expected to significantly modify the pulsation frequencies of waves propagating in the cores of red giants or in the radiative envelopes of intermediate- and high-mass main-sequence stars. We calculate the g-mode frequencies of stars with magnetic dipole fields which are aligned with their rotational axes, treating both the Lorentz and Coriolis forces non-perturbatively. We provide a compact asymptotic formula for the g-mode period spacing and universally find that strong magnetism decreases this period spacing substantially more than is predicted by perturbation theory. These results are validated with explicit numerical mode calculations for realistic stellar models. The approach we present is highly versatile: once the eigenvalues λ of a certain differential operator are pre-computed as a function of the magnetogravity and rotational frequencies (in units of the mode frequency), the non-perturbative impact of the Coriolis and Lorentz forces is understood under a broad domain of validity and is readily incorporated into asteroseismic modelling.

Control of Cascaded/Brushless Doubly-Fed Induction Motors With Real-Time Torque Optimization

Brushless doubly-fed induction machines offer a potential alternative to regular doubly-fed induction generators for wind turbines, due to higher reliability and lower maintenance costs. As motors and generators, the brushless machines could also be part of new technological solutions for hybrid-electric propulsion, including aircraft propulsion. A main advantage is that most of the power would be transferred from generators to motors without electronic conversion. The paper proposes a new speed control algorithm that maximizes the available torque within the limits of the control winding currents and ensures anti-windup protection. The maximum torque and the control variables are obtained through simple analytical computations. The derivations of the paper are based on a complex-variable representation that simplifies the mathematical model and the control algorithm. Compact formulas specify the controller parameters as functions of the machine parameters and of the desired closed-loop behavior. The results are tested on a laboratory testbed using a cascaded doubly-fed induction motor and demonstrate fast responses reaching torque limits that are computed in real-time by the algorithm. While control is achieved through the converter interfaced with the control windings, the data shows that most of the active and reactive power is exchanged directly through the power windings. Similar results are obtained in simulations of a larger brushless doubly-fed induction motor.