Expansion Formula Research Articles

Fourier Expansion, Integral Representation and Explicit Formula at Rational Arguments of the Tangent Polynomials of Higher-Order

In this paper, Fourier series expansion of Tangent polynomials are derived and the integral representation and explicit formula at rational arguments of these polynomials are established.

The hypergeometric function, the confluent hypergeometric function and WKB solutions

Relations between the hypergeometric function with a large parameter and Borel sums of WKB solutions of the hypergeometric differential equation with the large parameter are established. The confluent hypergeometric function is also investigated from the viewpoint of exact WKB analysis. As applications, asymptotic expansion formulas for those classical special functions with respect to parameters are obtained.

Fundamental-ray aberration analysis of off-axial optical systems: analytical formulae of first-order aberrations.

In our previous paper [Appl. Opt.59, 4466 (2020)10.1364/AO.389600], we presented a fundamental-ray aberration analysis that extends ray matrix analysis to the third-order aberration region. The analysis results shown in that paper were applicable to coaxial rotationally symmetric optical systems. This time, we have extended the fundamental-ray aberration analysis so that it can be applied to off-axial optical systems. Here we present new analysis formulae for fundamental-ray aberration analysis of the first-order aberration region. In addition, we newly present first-order aberration expansion formulae for four-element fundamental-ray aberrations and calculation formulae for the fundamental-ray aberration coefficients of the first order, which are necessary for this extension.

New expansion formulas for cluster algebras from surfaces

In 2011, Musiker-Schiffler-Williams proved the positivity for cluster algebras from surfaces by giving the expansion formulas for the cluster variables, in terms of perfect matching, γ-symmetric perfect matching and γ-compatible perfect matching. In this paper, we introduce two lattices L(To,γ(q)) and L(To,γ(p,q)) to give a unified expansion formula for cluster algebras from surfaces.

An Expansion Formula for Decorated Super-Teichmüller Spaces

Motivated by the definition of super-Teichm\"uller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichm\"uller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super $\lambda$-lengths associated to arcs in a bordered surface. In the special case of a disk, we are able to give combinatorial expansion formulas for the super $\lambda$-lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's $T$-path formulas for type $A$ cluster algebras. We further connect our formulas to the super-friezes of Morier-Genoud, Ovsienko, and Tabachnikov, and obtain partial progress towards defining super cluster algebras of type $A_n$. In particular, following Penner-Zeitlin, we are able to get formulas (up to signs) for the $\mu$-invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra.

Intersection numbers on {overline{M}}_{g,n} and BKP hierarchy

In their recent inspiring paper, Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar description for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.

Surface gravity wave interaction with a horizontal flexible floating plate and submerged flexible porous plate

An analytical study associated with surface gravity wave interaction with a horizontal flexible floating and a submerged porous plate is presented under linearized water wave theory and small amplitude structural response. The expansion formulae of the referred problem based on Green's function technique is derived using the two-dimensional Laplace equation and fundamental solution of source potentials in finite and infinite water depths. The velocity potential for the free oscillations in a closed basin is derived and the hydroelastic wave characteristics in floating and submerged flexural modes are analyzed in specific cases. Further, the usefulness of the expansion formula is demonstrated by analyzing a real physical problem of wave interaction with a moored flexible floating plate and a moored submerged flexible porous plate of finite dimensions in finite water depth. The convergence of the present solution is checked and the effect of a moored finite submerged porous plate on the moored finite flexible floating plate with different design parameters are analyzed. It is observed that the present analysis may be helpful for better understanding in designing a floating and submerged flexible porous plate system for application as a breakwater.

Expansion formula for the magnetic field of a periodically deformed circular current loop

A method is derived to obtain an expansion formula for the magnetic field generated by a closed planar wire carrying a steady electric current. The parametric equation of the loop is , with R the radius of the circle, H ∈ [0, R) the radial deformation amplitude, and f(ϕ)∈[−1, 1] a periodic function. The method is based on the replacement of the factor by an infinite series in terms of Gegenbauer polynomials, as well as the use of the Taylor series. This approach makes it feasible to write as the circular loop magnetic field contribution plus a sum of powers of H/R. Analytic formulas for the magnetic field are obtained from truncated finite expansions outside the neighborhood of the wire. These showed to be computationally less expensive than numerical integration in regions where the dipole approximation is not enough to describe the field properly. Illustrative examples of the magnetic field due to circular wires deformed harmonically are developed in the article, obtaining exact expansion coefficients and accurate descriptions. The method can be also applied when the deformation function f(ϕ) is a linear combination of harmonic functions. In that case, we provide the exact formulas for the expansion coefficients of . Error estimates are calculated to identify the regions in where the analytical expansions perform well. Finally, first-order deformation formulas for the magnetic field are studied for generic even deformation functions f(ϕ).

Lifting the dual immaculate functions

We introduce a reverse variant of the dual immaculate quasisymmetric functions, mirroring the dichotomy between quasisymmetric Schur functions and Young quasisymmetric Schur functions, and establish a lift of this basis to the polynomial ring. We show that taking stable limits of these reverse dual immaculate slide polynomials produces the reverse dual immaculate quasisymmetric functions, and we establish positive formulas for expansions of these polynomials into the fundamental slide and quasi-key bases for polynomials. These formulas mirror connections between dual immaculate quasisymmetric functions, fundamental quasisymmetric functions, and Young quasisymmetric Schur functions, extending these connections from the ring of quasisymmetric functions to the full polynomial ring. We also use this lift to provide a lift of the dual immaculate quasisymmetric functions, and we establish analogous expansion formulas. We moreover use the reverse dual immaculate quasisymmetric functions to establish a connection between the dual immaculate quasisymmetric functions and the Demazure atom basis for polynomials.

Minimization of Binary Decision Diagrams for Systems of Completely Defined Boolean Functions using Algebraic Representations of Cofactors

In design systems for digital VLSI (very large integrated circuits), the BDD is used for VLSI verification, as well as for technologically independent optimization, performed as the first stage in the synthesis of logic circuits in various technological bases. BDD is an acyclic graph defining a Boolean function or a system of Boolean functions. Each vertex of this graph corresponds to the complete or reduced Shannon expansion formula. Having constructed BDD representation for systems of Boolean functions, it is possible to perform additional logical optimization based on the proposed method of searching for algebraic representations of cofactors (subfunctions) of the same BDD level in the form of a disjunction or conjunction of other cofactors of this BDD level. The method allows to reduce the number of literals by replacing the Shannon expansion formulas with simpler formulas that are disjunctions or conjunctions of cofactors, and to reduce the number of literals in specifying a system of Boolean functions. The number of literals in algebraic multilevel representations of systems of fully defined Boolean functions is the main optimization criterion in the synthesis of combinational circuits from library logic gates.

Macroscopic transports in a rotational system with an electromagnetic field

In this work, we explore macroscopic transport phenomena associated with a rotational system in the presence of an external orthogonal electromagnetic field. Simply based on the lowest Landau level approximation, we derive nontrivial expressions for chiral density and various currents consistently by adopting small angular velocity expansion or Kubo formula. While the generation of anomalous electric current is due to the pseudo gauge field effect of the spin-rotation coupling, the chiral density and current can be simply explained with the help of Lorentz boosts. Finally, Lorentz covariant forms can be obtained by unifying our results and the magnetovorticity effect.

Friezes, weak friezes, and T-paths

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T-paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables.This paper will introduce the concepts of weak friezes and T-paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T-path formula.We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued.

2-Variable Fubini-degenerate Apostol-type polynomials

This work deals with the mathematical inspection of a hybrid family of the degenerate polynomials of the Apostol-type. The inclusion of the derivation of few series expansion formulas, explicit representations and difference equations for this hybrid family brings a novelty to the existing literature. Moreover, certain connection formulas and several novel identities for these polynomials are established and investigated. The graphical representations of certain degenerate polynomials are explored and several new interesting pattern of the zeros are observed.

Minimization of binary decision diagrams for systems of completely defined Boolean functions using Shannon expansions and algebraic representations of cofactors

In the systems of digital VLSI design (Very Large Integrated Circuits), the BDD (Binary Decision Diagram) is used for VLSI verification, as well as for technologically independent optimization as the first stage in the synthesis of logic circuits in various technological bases. The BDD is an acyclic graph defining a Boolean function or a system of Boolean functions. Each vertex of this graph corresponds to the complete or reduced Shannon expansion formula. When BDD representation for systems of Boolean functions is constructed, it is possible to perform additional logical optimization based on the proposed method of searching for algebraic representations of cofactors (subfunctions) of the same BDD level in the form of a disjunction, conjunction either exclusive-or of cofactors of the same level or lower levels of BDD. A directed BDD graph for a system of functions is constructed on the basis of Shannon expansion of all component functions of the system by the same permutation of variables. The method allows to reduce the number of literals by replacing the Shannon expansion formulas with simpler formulas that are disjunctions or conjunctions of cofactors, and to reduce the number of literals in specifying a system of Boolean functions. The number of literals in algebraic multilevel representations of systems of fully defined Boolean functions is the main optimization criterion in the synthesis of combinational circuits from librarian logic elements.

Fractional derivatives and expansion formulae of incomplete $H$ and $\overline{H}$-functions

In this paper, we investigate the fractional derivatives and expansion formulae of incomplete $H$ and $\overline{H}$-functions for one variable. Further, we also obtain results for repeated fractional order derivatives and some special cases are also discussed. Various other analogues results are also established. The results obtained here are very much helpful for the further research and useful in the study of applied problems of sciences, engineering and technology.

Certain expansion formulae of incomplete $I$-functions associated with the Leibniz rule

In this paper, we determine some expansion formulae of the incomplete I-functions in affiliation with the Leibniz rule for the Riemann-Liouville type derivatives. Further, expansion formulae of the incomplete $\overline{I}$-function, incomplete $\overline{H}$-function, and incomplete H-function are conferred as extraordinary instances of our primary outcomes.

Energy and Water Savings during Backwashing of Rapid Filter Plants

This paper describes an analysis of the effects of adjusting the intensity of filter backwash to the water temperature. The consequences of the lack of such adjustment for the life of filter beds, the amount of water used for backwashing, the amount of energy used for backwashing and the quality of the first filtrate are presented. In order to determine the losses and profits resulting from controlling the intensity of backwash water depending on its temperature, an analysis was carried out at a water treatment plant in southern Poland. Laboratory measurements were used to determine the granulation and specific gravity of sand grains filling the filtration beds. On the basis of measurements on a semi-technical scale, the magnitudes of filter bed expansion were determined for average monthly wash water temperatures. They were first calculated from the Richardson–Zaki equation, using different formulae for the value of the exponent of the power in this equation. Due to significant differences in the density and shape of grains covered with a permanent deposit after several years of filter operation, a satisfactory match between the formulae known from the literature and the results of expansion measurements was not obtained. Therefore, an new formula for the bed expansion was developed based on the Richardson–Zaki equation. A good fit of this formula to the experimental results was obtained. Monthly average values of water temperature were compiled, and on this basis the required amount of backwash water and energy was computed. The computations were made for 25% of fluidized bed expansion. Possible energy and water savings were estimated, as well as further gains from keeping the required expansion of the porous bed constant regardless of the wash water temperature.

HANKEL DETERMINANTS OF FACTORIAL FRACTIONS

AbstractBy making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam [‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’, Integral Transforms Spec. Funct.30(7) (2019), 581–593].

Hydroelastic behaviour of a submerged horizontal flexible porous structure in three-dimensions

A hydroelastic model is developed for the problem of linear wave interaction with a submerged horizontal flexible porous structure in three-dimensions. The generalized expansion formulae with orthogonal mode-coupling relations are derived by expanding Fourier cosine series in a channel width. Using eigensolutions, various aspects of hydroelastic wave behaviour in a channel with a porous plate submerged in finite water depth is investigated. Further, the usefulness of the expansion is analysed through a real physical problem of wave diffraction by a submerged flexible porous plate connected with mooting lines in three-dimensions. The accurateness of the three-dimensional solution is inspected and the hydroelastic behaviour of the submerged porous structure is analysed for oscillation modes, mooring stiffness, and porous-effect parameters on numerical assessment of reflection, dissipation coefficients, and plate displacements in different cases. The analysis of current hydroelastic behaviour will be beneficial for future design of a device for wave energy absorption in three-dimensions.

The Parseval identity for q-Sturm\u2013Liouville problems with transmission conditions

In the present study, we investigate the existence of spectral functions and obtain the Parseval identity and expansion formula in eigenfunctions for the singular q-Sturm–Liouville problem with transmission conditions.