Form Of Solution Research Articles

Eclipse analysis for small-eccentricity orbits using analytical model

The orbital eclipse is an old but fundamental problem in astrodynamics, and it has been fully studied in many papers and books. However, the properties of the orbital eclipse under the critical situation and the effects of the eccentricity on the eclipse have not been clearly revealed. Applying the first-order approximation to the eccentricity, the simple analytical solution to the conical eclipse of the elliptical orbit with a small eccentricity is developed, which is similar to the circular-orbit solution in form. The proposed model is validated by numerical simulation, and the properties of the critical situation between the cases with and without orbital eclipses and the effects of the eccentricity on the eclipse are both analyzed. The results show that when the eccentricity is less than 0.01, the proposed analytical model agrees well with the numerical method in common situations. When the orbital sun angle is close to the critical value defined by the ratio of the Earth radius to the semimajor axis, the precision of the proposed analytical model would be degraded. In that situation, the penumbra arc length would be no longer a small value, and the statement in previous studies that the penumbra arc was short and neglectable would not always be right. The difference between the eclipse of the small-eccentricity orbit and that of the circular orbit increases as the eccentricity increases. Moreover, when the argument of perigee is in the same or the opposite direction with the projection direction of the Sun in the orbital plane, the variation in the eclipse length caused by the eccentricity reaches the maximum value. When the above two directions are perpendicular to each other, the variation in the eclipse arc center caused by the eccentricity reaches the maximum value.

New type of multiple lump and rogue wave solutions of the (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation

In this paper, a new version of the solution in form of Grammian for the (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation is obtained with the aid of the polynomial function. A family of multiple lump solutions are constructed. All the center points of the lumps form a triangular structure as the auxiliary variable tends to the negative or positive large values. A unified scheme for constructing the high-order rogue wave solutions is proposed. This new type of rogue wave reflects the appearance and disappearance of multiple lump waves in the background of N-kink solitons. The results can well mimic complex waves and explain complex dynamics in fluids.

A Semi-Analytical Solution of Liquid Response in a Two-Dimensional Rectangular Tank with a Vertical Rigid Baffle Under Pitching Excitation

The present work uses a semi-analytic mathematical model to study the dynamic response of the partially liquid-filled rectangular tank equipped with a vertical rigid baffle at the bottom subjected to pitching excitation. The velocity potential consists of the rigid container velocity potential and the disturbance potential, which contains a Stokes–Joukowski potential. The liquid domain is divided into four simple sub-domains. The formal solution of the Stokes–Joukowski potential in each sub-domain is solved using the superposition principle and the separation of variables method. The dynamic response equation is established by using the free liquid surface equation. The ADINA solution is compared with the present method solution, which verifies the correctness of the proposed method. Finally, the effect of baffle parameters on liquid sloshing for a baffled tank subjected to harmonic and seismic pitching excitations is discussed in detail.

Cramer's rule for implicit linear differential equations over a non-Archimedean ring

We consider a linear nonhomogeneous $m$-th order differential equation in a ring of formal power series with coefficients from some field of characteristic zero. This equation has infinite many solutions in this ring -- one for each initial condition of the corresponding Cauchy problem. These solutions can be found using classical methods of differential equation theory. Let us suppose the coefficients of the equation and the coefficients of nonhomogeneity belong to some integral domain $K$. We are looking for a solution in the form of a formal power series with coefficients from this integral domain. The methods of classical theory do not allow us to find out whether there exists an initial condition that corresponds to the solution of the coefficients from $K$ and do not allow find this initial condition. To solve this problem, we use the method proposed by U. Broggi. This method allows to find a formal solution of the linear nonhomogeneous differential equation in the form of some special series. In previous articles, sufficient conditions for the existence and uniqueness of a solution were found for a certain class of rings $K$ with a non-Archimedean valuation. If these conditions hold, the formal power series obtained using the Broggi’s method is considered. Its coefficients are the sums of series that converge in the non-Archimedean topology considered. It is shown that this series is the solution from $K[[x]]$ of our equation. Note that this equation over a ring of formal power series can be considered as an infinite linear system of equations with respect to the coefficients of unknown formal power series. In this article it is proved that this system can be solved by some analogue of Cramer's method, in which the determinants of infinite matrices are found as limits of some finite determinants in the non-Archimedean topology.

Resurgent functions and nonlinear systems of differential and difference equations

The principal aim of this paper is to establish an iteration method on the space of resurgent functions. We discuss endless continuability of iterated convolution products of resurgent functions associated with rooted trees and derive their estimates. We show the resurgence of formal series solutions of nonlinear differential and difference equations by applying the estimates.

Perspective Studies on Perugino’s and Raffaello’s Painted Architecture

The architecture that the Renaissance artists depicted in their works constitutes a vast reservoir of formal solutions that influenced (and were reciprocally influenced by) built architecture. Generally painted according to a rigorous perspective structure, most painted architecture can be restituted and modelled to become part of the Virtual Heritage that develops and extends its knowledge to a wider range of people and scholars. These procedures are here applied to some of the works of Pietro Perugino and Raffaello, a master and his student, in order to define their specific approach to composition, perspective, and architecture. The application of these procedures produced some primary results—the architecture restitutions in plan and elevation or section—and some secondary results concerning the way painted architecture was conceived, represented in perspective, and received as well as it was related to actual architecture.

The Multicomponent Higher-Order Chen–Lee–Liu System: The Riemann–Hilbert Problem and Its N-Soliton Solution

It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. Then, we apply the trace identity to obtain the bi-Hamiltonian structure of mHOCLL system, which certifies that the constructed system is integrable. Considering the spectral problem of the Lax pair, a related Riemann–Hilbert (RH) problem of this integrable system is naturally constructed with zero background, and the symmetry of this spectral problem is given. On the one hand, the explicit expression for the mHOCLL solution is not available when the RH problem is regular. However, according to the formal solution obtained using the Plemelj formula, the long-time asymptotic state of the mHOCLL solution can be obtained. On the other hand, the N-soliton solutions can be explicitly gained when the scattering problem is reflectionless, and its long-time behavior can still be discussed. Finally, the determinant form of the N-soliton solution is given, and one-, two-, and three-soliton solutions as specific examples are shown via the figures.

Follow the Solo: The Formal Evolution of the Concerto in the Eighteenth-Century

AbstractWe examine the evolution of the form of the first movement in the eighteenth-century concerto from the Baroque ritornello plan to the eighteenth-century Classical concerto by means of a statistical corpus study. Our data yields a more nuanced understanding of several key characteristics of this evolution, characteristics that have been observed by both contemporary and present-day theorists. We demonstrate how the rise in both quantity and thematic quality of the soloist’s part enabled the convergence of ritornello practices with sonata-form features. We conclude with analytical case studies of works by C. P. E. Bach and Josef Mysliveček, showing how their formal solutions fit within this evolutionary process.

Exact solutions of (1+2)-dimensional non-linear time-space fractional PDEs

PurposeIn this paper, the author presents a hybrid method along with its error analysis to solve (1+2)-dimensional non-linear time-space fractional partial differential equations (FPDEs).Design/methodology/approachThe proposed method is a combination of Sumudu transform and a semi-analytc technique Daftardar-Gejji and Jafari method (DGJM).FindingsThe author solves various non-trivial examples using the proposed method. Moreover, the author obtained the solutions either in exact form or in a series that converges to a closed-form solution. The proposed method is a very good tool to solve this type of equations.Originality/valueThe present work is original. To the best of the author's knowledge, this work is not done by anyone in the literature.

Novel aspects of integrability for nonlinear sigma models in symmetric spaces

We obtained the formal solution of the auxiliary system of nonlinear sigma models (NLSMs), whose target space is a rank 1 symmetric space based on the indefinite orthogonal group $O(p,q)$, corresponding to an arbitrary solution of the NLSM. This class includes anti-de Sitter, de Sitter, and hyperbolic spaces, which are of interest in view of the $\mathrm{AdS}/\mathrm{CFT}$ correspondence. The formal solution is related to the Pohlmeyer reduction of the NLSM, constituting another link between the NLSM and the reduced theory. Besides deriving the solution, we also review the Pohlmeyer reduction of such models. Finally, we comment on the implications for the monodromy matrix and its eigenvalues.

Continuous multi-query optimization for subgraph matching over dynamic graphs

There is a growing need to perform real-time analytics on dynamic graphs in order to deliver the values of big data to users. An important problem from such applications is continuously identifying and monitoring critical patterns when fine-grained updates at a high velocity occur on the graphs. A lot of efforts have been made to develop practical solutions for these problems. Despite the efforts, existing algorithms showed limited running time and scalability in dealing with large and/or many graphs. In this paper, we study the problem of continuous multi-query optimization for subgraph matching over dynamic graph data. (1) We propose annotated query graph, which is obtained by merging the multi-queries into one. (2) Based on the annotated query, we employ a concise auxiliary data structure to represent partial solutions in a compact form. (3) In addition, we propose an efficient maintenance strategy to detect the affected queries for each update and report corresponding matches in one pass. (4) Extensive experiments over real-life and synthetic datasets verify the effectiveness and efficiency of our approach and confirm a two orders of magnitude improvement of the proposed solution.

Best practices of business process improvement: towards a representation on top of the Quintessence kernel

PurposeOrganizational process improvement plays an important role for sustaining business in a competitive environment. Therefore, enterprise leaders are increasingly prone to adopt business process improvement (BPI) practices. However, organizations are unable to implement formal and reusable solutions, representing a gap between academic research and practical use. In addition, companies adopt discipline-dependent solutions, lacking BPI representations of best practices applicable to all organizational divisions. This paper aims to propose some constructs on top of the Quintessence kernel for representing the practice systematic development of the BPI in the BPI lifecycle and we conduct two case studies in a multinational company.Design/methodology/approachA quantitative research design was adopted for recognizing gaps in previous approaches and identifying best BPI practices. Subsequently, characterization of practices and activities are represented based on a unified definition model and the Quintessence kernel. Finally, two case studies are developed for applying the solution.FindingsThe formal representation is applicable to multiple disciplines in organizational environments. Besides, the sub-alphas (abstract level progress health attribute) states and the work products resulting from each activity completion criteria evidence the health and progress accomplished during the practice execution.Originality/valueThe practice representation serves as a formal, graphical, reusable and multidiscipline guide compiling activities and tasks for systematically developing BPI during the radical/incremental improvement lifecycle.

Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables

In this paper we study systems of autonomous algebraic ODEs in several differential indeterminates. We develop a notion of algebraic dimension of such systems by considering them as algebraic systems. Afterwards we apply differential elimination and analyze the behavior of the dimension in the resulting Thomas decomposition. For such systems of algebraic dimension one, we show that all formal Puiseux series solutions can be approximated up to an arbitrary order by convergent solutions. We show that the existence of Puiseux series and algebraic solutions can be decided algorithmically. Moreover, we present a symbolic algorithm to compute all algebraic solutions. The output can either be represented by triangular systems or by their minimal polynomials.

Hierarchical heterogeneous one-dimensional problem in linear viscoelastic media

We address the homogenisation of a linear viscoelastic and hierarchical composite material in a one-dimensional (1D) framework via a three-scale asymptotic homogenisation method. We consider a family of heterogeneous problems with periodic, rapidly-oscillating and piece-wise coefficients that model a structure with two hierarchical levels of organisation. Here, we assume continuity contact conditions at the interfaces among the constituents and set a straightforward geometrical configuration in order to gain a better insight of the multiscale problem. The main goal is to provide a general overview of the procedure and validate the approach by means of a comparison between the solution of the original heterogeneous problem, the homogenised problem and the formal asymptotic solution. In addition, we show that the three-scale approach presents a clear improvement over the recursive two-scale one, and we illustrate the convergence of the solutions towards the solution of the homogenised problem when the asymptotic parameters approach.

A parallel Monte Carlo method for solving electromagnetic scattering in clusters of dielectric objects

In this work, we develop a novel Monte Carlo method for solving the electromagnetic scattering problem for clusters of photonic particles. The method is based on a formal solution of the scattering problem as a modified Born series whose coefficients are found by a conformal transformation. The terms of the Born series are approximated by sampling random elements of its matrix representation, computed by the Method of Moments. Unlike other techniques like the Fast Multiple Method, this Monte Carlo method does not require communication between processors, which makes it suitable for large parallel executions.

Time-weighted average shear wave velocity profiles from surface wave tests through a wavelength-depth transformation

We investigate the possibility of obtaining time-weighted average shear wave velocity profiles through a wavelength-depth transformation of experimental dispersion curves from surface wave tests, without a formal solution of the inverse problem. We evaluate this approach on a wide flat-file database (Polito Surface Wave Database, PSWD) of experimental dispersion curves and related shear wave velocity profiles, both from dispersion curve inversion and invasive tests. The results show that the proposed wavelength-depth transformation can be valuable for seismic site evaluations offering an estimation of time-weighted average shear wave velocity profiles very similar to a state-of-the-art inversion of the experimental dispersion curves and with similar uncertainty with respect to invasive tests. This transformation has the advantage of avoiding time-consuming inversion processes, with related uncertainty sources, and any assumption on layer parameterization and a-priori information. Moreover, in conjunction with an experimental evaluation of the fundamental frequency of the site, from independent surveys, the wavelength-depth transformation can be used to get a direct and fast estimate of the position of the engineering bedrock.

Formal Gevrey solutions: in analytic germs—for higher order holomorphic PDEs

We consider a family of holomorphic PDEs whose singular locus is given by the zero set of an analytic map P with P(0)=0. Our goal is to establish conditions for the existence and uniqueness of formal power series solutions and to determine their divergence rate. In fact, we prove that the solution is Gevrey in P, giving new information on divergency while compared to the classical Gevrey classes. If P is not singular at 0, we also provide Poincaré conditions to recover convergent solutions. Our strategy is to extend the dimension and lift the given PDE to a problem where results of singular PDEs can be applied. Finally, examples where the Gevrey class in P is optimal are included.

On the influence of the pipe outer coating nonuniformity on the contact stresses arising under rigid bushes

The work is devoted to the formulation and construction of a solution in analytical form for the mechanical problem of contact interaction between a finite system of rigid cylindrical bushes and a pipe with an external coating, which has longitudinal nonuniformity. In industry and mechanical engineering, the presence of such additional coatings is required, for example, for protection the pipe main layer from external aggressive media, for its electrical or thermal insulation, etc. The nonuniformity of such a layer may be due to various additional requirements, as well as the obtaining process of this layer. When analyzing the stress-strain state of the pipe interacting with the bushes, it is necessary to take into account all the features. The paper shows that the mathematical model of the above-described interaction problem is a system of mixed integral equations containing integral operators of different types. Moreover, the resulting system includes functions describing the variable properties of the outer layer in the area of interaction with the bushings. The special approach described by the main theses in this paper makes it possible to obtain a solution in a form in which all complex functions are allocated as separate multipliers. This makes it possible to perform calculations even in cases where the coating is highly heterogeneous or consists of different materials.

Application of the Qualitative Research Methodology Grounded Theory in Studying of the Needs of Tourist Destination Stakeholders and Developing Digital Industry Platform

The development of digital economy implies not only deep transformation processes for individual enterprises, but also the need for adaptations to new economic conjucture of those stakeholders, who are not ready for full-scale changes in their management and production models. Thus, managing digital transformation of tourist destination require detailed and systematic study of the needs in digitalization and the search for tools for stakeholder's adaptation to new socio-economic conditions. The qualitative research methodology “Grounded theory” is able to provide tools to resolve this problem, which approbation is the main subject of the study. This method allows to form inductively a logical concept that would describe the phenomenon and directly confirm theoretical conclusions in observations. The study resulted in key categories of needs, their interconnections and corresponding digital solutions in form of different modules of digital platform, comined by the central key category - “need to increase the involvement of all stakeholders through a collaborative digital service”. The pilot study conducted according to the proposed methodology confirmed its effectiveness in studying the needs of stakeholders of tourist market and in developing the architecture of a digital industry platform. At the same time, the subsequent prototyping of the proposed digital solutions should be carried out in accordance with the strategy of implementing the industry platform and its integration into the digital ecosystem of the Krasnodar Territory.

Further results about the exact solutions of conformable space–time fractional Boussinesq equation (FBE) and breaking soliton (Calogero) equation

Seeking for the exact solutions of fractional nonlinear partial differential equations (FNPDE) has penetrated into almost every discipline of the natural, engineering, mathematics, and social sciences. In this article, with the aid of Maple software, using the (G′G2)-expansion method and the unified F-expansion method, we successfully investigate the exact solutions about conformable space–time fractional Boussinesq equation (FBE) and breaking soliton (Calogero) equation. Consequently, three types traveling wave solutions are found such as Jacobian double periodic elliptic, simply periodic and the rational function solutions. The (G′G2)-expansion and unified F-expansion methods are relatively complex, but we can find more solutions that have not been seen before. We give the some specific examples solutions such as double-periodic Jacobian elliptic functions which can be molted into single-periodic functions. Also we point the bridge linking among the Weierstrass elliptic, hyperbolic, trigonometric functions and Jacobian elliptic functions. The connection and distinction between these three formal solutions are also indicated in our paper.