Separation Of Variables Research Articles

Accurate analysis of functionally graded piezoelectric beams within symplectic framework

Analytical solutions of the functionally graded piezoelectric beams are investigated under symplectic framework. The material parameters vary along the length in an identical exponential form. The method of separation of variables is adopted to transform the original problem into a symplectic eigenproblem. The eigensolutions of particular eigenvalues are presented which represent the basic mechanical and electric properties, and the general eigensolutions are obtained which describe the localized behaviors which usually covered up by Saint-Venant principle. The accurate elasticity solutions are linear combinations of particular and general eigensolutions. Numerical examples are considered to verify the effectiveness and accuracy of the proposed solutions.

Mathematical modelling of couple stress fluid flow around a semi‐permeable sphere enclosing a solid core

AbstractThe focus of this article is to study theoretically the steady‐axisymmetric creeping flow dynamics of a couple stress fluid external to a semi‐permeable sphere containing a solid core. This problem is motivated by emulsion hydrodynamics in chemical engineering where rheological behaviour often arises in addition to porous media effects. The non‐Newtonian Stokes’ couple stress fluid model features couple stresses and body couples that are absent in the classical Navier–Stokes viscous model. It provides a robust framework for simulating emulsions, complex suspensions and other liquids which possess microstructure. The physical regime is delineated into two zones – the interior of the semi‐permeable zone (region II) which engulfs the solid core and the external couple stress fluid zone (region I). The model is formulated using a spherical polar coordinate system in terms of the stream function ψ. The Brinkman‐extended Darcy model is deployed for the porous medium hydrodynamics and isotropic permeability is considered. Analytical expressions are derived for dimensionless pressure, tangential stress and the couple stress components using the method of separation of variables and Gegenbauer functions of the first kind. The integration constants are evaluated with appropriate boundary conditions on the inner and outer boundary of the semi‐permeable zone with the aid of Mathematica symbolic software. Solutions for the drag force exerted by the couple stress fluid on the semi‐permeable sphere and volumetric flow rate are also derived with corresponding expressions for the drag coefficient and non‐dimensional volumetric flow rate. The influence of permeability (k), separation parameter (l), couple stress viscosity coefficient (η), couple stress inverse length dependent parameter (λ = √(μ/η)) and couple stress viscosity ratio (τ = η/η/) on all key variables is studied graphically. Additionally, streamline contours are computed for a range of parameters including inverse permeability parameter (α = 1/k). The computations show that increasing couple stress inverse length dependent parameter (λ) greatly reduces the dimensionless volumetric flow rate in particular at high values of separation parameter (l). Flow rate is, however, markedly enhanced with permeability (k) and also couple stress viscosity parameter (η). In the presence of the solid core, a much greater drag coefficient is observed at all values of couple stress inverse length dependent parameter (λ) relative to the case without a solid core. A significant distortion in streamlines is computed with increasing separation parameter (l) with a dual vortex structure emanating. With greater couple stress viscosity parameter (τ), no tangible modification is observed in the streamline contours. The present work generalizes the earlier study of Krishnan and Shukla to consider a semi‐permeable (porous medium) outer sphere and furthermore presents detailed streamline visualizations of the creeping flow for a range of emerging parameters of relevance to non‐Newtonian chemical engineering processes including emulsion droplet dynamics in porous materials.

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.

Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method

The buckling behavior of rectangular thin plates, which are supported at their four corner points with four edges free, is a matter of great concern in the field of plate and shell mechanics. Nevertheless, the complexities arising from the boundary conditions and governing equations present a formidable obstacle to the attainment of analytical solutions for these problems. Despite the availability of various approximate/numerical methods for addressing these challenges, the literature lacks accurate analytic solutions. In this study, we employ the symplectic superposition method, a recently developed method, to effectively analyze the buckling problem of rectangular thin plates analytically. These plates have four supported corners and four free edges. To achieve this, the problem is divided into two sub-problems and solve them separately using variable separation and symplectic eigen expansion, leading to analytical solutions. Finally, we obtain the resolution to the initial issue by superposing the sub-problems. The current solution method can be regarded as a logical, analytical, and rational approach as it begins with the basic governing equation and is systematically derived without assuming the forms of the solutions. To examine various aspect ratios and in-plane load ratios of rectangular thin plates, which are supported at their four corner points with four edges free, we provide numerical examples that demonstrate the buckling loads and typical buckling mode shapes.

Exact solution of time-fractional differential-difference equations: invariant subspace, partially invariant subspace, generalized separation of variables

Exact solution of time-fractional differential-difference equations: invariant subspace, partially invariant subspace, generalized separation of variables

Some Novel Fusion and Fission Phenomena for an Extended (2+1)-Dimensional Shallow Water Wave Equation

An extended (2+1)-dimensional shallow water wave (SWW) model which can describe the evolution of nonlinear shallow water wave propagation in two spatial and temporal coordinates, is systematically studied. The multi-linear variable separation approach is addressed to the extended (2+1)-dimensional SWW equation. The variable separation solution consisting of two arbitrary functions is obtained, by assumption, from a specific ansatz. By selecting these two arbitrary functions as the exponential and trigonometric forms, resonant dromion, lump, and solitoff solutions are derived. Meanwhile, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions, and periodic waves are studied with graphical and analytical methods. The results can be used to enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

Dynamic response of a large-diameter end-bearing pile in permafrost

Vertically dynamic model of a large-diameter pile in frozen soil is established, in which the frozen soil is described to a saturated frozen porous media, and the large diameter end-bearing pile is simplified to a one-dimensional rod considering the influence of the transverse inertia effect. Analytical solutions of the longitudinal coupling vibration between the end-bearing pile and the frozen soil are obtained using Helmholtz decomposition and variable separation methods in the frequency domain. By comparing the dynamic responses of the longitudinal vibration of the large diameter end-bearing pile with the traditionally one-dimensional pile, as well as the impedance factor of the frozen soil layer induced by the pile vibration, these demonstrate the influence of the transverse inertia effect on the high frequency vibration of large diameter pile is significant, and the influence on the pile with a smaller slenderness ratio is larger. The temperature and the Poisson’s ratio also have significant effects on the vertical vibration of large diameter piles in frozen soil, which cannot be ignored in the analysis.

Analytical models for electroosmotic consolidation of layered soil systems considering the boundary effects of horizontal electrodes

AbstractThe pore water can usually flow through the horizontal electrodes, resulting in the top sand layer and the bottom unelectroosmotic layer inevitably affecting the consolidation process of the electroosmotic layer. However, the traditional assumption of the fully drained or undrained electrode boundaries is unable to reflect these boundary effects. In this study, one‐dimensional analytical models for electroosmotic consolidation of the layered soil systems considering the boundary effects of horizontal electrodes are proposed. The solutions for the models are derived using the separate variable method and Laplace transform method. The rationality of the one‐dimensional model is also illustrated by comparing it with the results produced by the two‐dimensional model and the accuracies of the solutions are verified through a series of analytical and numerical validation examples. Parametric studies are conducted to investigate the effects of the sand layer and the unelectroosmotic layer on the consolidation process. Results show that both the sand layer and the unelectroosmotic layer may significantly delay the consolidation progress. The time factor corresponding to 90% average degree of consolidation (Tv_90%U*) can increase by over 1.5 times in comparison with the model without a sand layer when the permeability coefficient ratio of the subsoil to the upper layer reaches 10. The delayed effect is particularly pronounced when the thickness of the unelectroosmotic layer with higher compressibility is relatively larger, potentially resulting in a tenfold or even greater increase in Tv_90%U*.

Об особенностях идентификации переменных термомеханических характеристик функционально-градиентного прямоугольника

The inverse thermoelastic problem of identification of the variable properties of a functionally graded rectangle is studied. Unsteady vibrations are excited by applying mechanical and thermal loads to the upper side of the rectangle. To solve the direct problem in Laplace transforms, the method of separation of variables and the shooting method for harmonics are used. Transformants are inverted by expanding the origin in terms of shifted Legendre polynomials. The method proposed for solving the direct problem is verified by comparison with a finite element solution. The influence of the laws of change of variable characteristics on the boundary physical fields is analyzed. The displacement components give additional information on the mechanical loading, and the temperature measured on the upper side of the rectangle over a certain time interval – on the thermal loading. Assuming that the additional information admits expansion in Fourier series, the two-dimensional inverse problem is reduced to one-dimensional problems for various harmonics. The solution of the obtained nonlinear inverse problems is carried out on the basis of an iterative process, at each stage of which, in order to find corrections for thermomechanical characteristics, systems of Fredholm integral equations of the 1st kind are solved. The possibility of simultaneous reconstruction of several characteristics is investigated. The results of computational experiments on the phased reconstruction of thermomechanical characteristics are presented. The influence of the thermomechanical coupling parameter on the results of the thermal stress coefficient reconstruction was clarified.

CREEPING FLOW OF COUPLE STRESS FLUID OVER A SPHERICAL FIELD ON A SATURATED BIPOROUS MEDIUM

This problem emphasizes the dynamic interaction between a biporous medium and a couple stress fluid of laminar flow. The flow around a permeable field engulfed in a couple stress fluid is examined. When examining the motion of an oil droplet in a porous collector that is surrounded by an aqueous medium (oil-in-water emulsion) and is subject to an external pressure drop, this formulation of the problem is typical. A similar issue arises when lymph enters the tissues of humans or animals: the inside permeable spherical field saturated with viscous fluid and outside region saturated with couple stress fluid. The Brinkman equations are utilized to characterize the couple stress fluid flow in a saturated biporous medium. The couple stress tensor and velocity fields are expressed using Gegenbauer polynomials and Macdonald functions. For the axially symmetric motion, both pressure distribution and the stream function solution are explicitly solved. The method of variable separation is used to investigate an analytical resoluteness for the flow field. The drag force on a saturated biporous medium and the drag coefficient <i>D<sub>N</sub></i> are calculated, and the impacts of the permeability κ, the ratio of viscosity (γ<sup>2</sup> = μ<sub>1</sub> /μ<sub>2</sub>), the couple stress viscosity ratio (τ = η'/η), and the parameter of couple stress (λ = √μ/η). The appropriate dependencies are graphically delineated and reviewed, including the permeability κ, couple stress parameter λ, viscosity ratio γ<sup>2</sup>, and couple stress viscosities (η, η'). According to the findings, increasing permeability gradually raises the drag coefficient, which is used to describe a spherical field’s surface with a high level resistance of flow. Limits statements are used to illustrate specific cases that are well-known. The current study is significant primarily in the course through a layer formed by penetrable particles and has very important and compelling applications in both nature and innovation, with a variety of potential outcomes.

Analytical and numerical investigation of viscous fluid-filled spherical slip cavity in a spherical micropolar droplet

<abstract><p>This article presents an analytical and numerical investigation on the quasi-steady, slow flow generated by the movement of a micropolar fluid drop sphere of at a concentrical position within another immiscible viscous fluid inside a spherical slip cavity. Additionally, the effect of a cavity with slip friction along with the change in the micropolarity parameter on the movement of the fluid sphere is introduced. When Reynolds numbers are low, the droplet moves along a diameter that connects their centres. The governing and constitutive differential equations are reduced to a computationally convenient form using appropriate transformations. By using the resulting linear partial differential equations for the stream functions and using the method of separation variables, we can obtain their solutions. General solutions for velocity fields are found using spherical coordinate systems, which are based on the concentric point of the cavity; this allows to obtain solutions to the Navier-Stokes equations internal and external to the spherical droplet. The vorticity-microrotation boundary condition is used in regard to the micropolar droplet case in a viscous fluid. The normalised drag forces acted upon the micropolar drop are illustrated via graphs and tables for diverse values of the viscosity ratio and drop-to-wall radius ratio, with the change of the spin parameter that attaches the microrotation to vorticity. The correction wall factor is shown to increase with an increase in the drop-to-wall radius ratio, when moving from the gas bubble case to the solid sphere case, with an increase in the micropolarity parameter, and with an increase in the slip frictional resistance. This study is relevant due to its potential uses in a variety of biological, natural, and industrial processes, including the creation of raindrops, the investigation of blood flow, fluid-fluid extraction, the forecasting of weather conditions, the rheology of emulsions, and sedimentation phenomena.</p></abstract>

Analytical analysis of concentration distribution and diffusion-induced stress of finite-length cylindrical electrode under galvanostatic operation

A cylindrical electrode is approximated as a long cylinder in most of existing models in which a generalized plane strain condition/plane strain is used. Based on the theory of elasticity, analytical expressions are derived for concentration distribution and stress component in a finite-length cylindrical electrode under galvanostatic operation. Using the superposition theorem, the Li-concentration is a sum of the concentration due to axial diffusion and the concentration due to lateral diffusion, and the separation of variable method is used to solve diffusion equations. By using the Boussinesq-Papkovich function, the generalized stress component distribution of a linearly combined product of the exponential-type Fourier-Bessel series is derived. The spatiotemporal distribution of concentration and diffusion-induced stresses are calculated in a cylindrical electrode with traction-free condition. The results are compared with the simulation results from a finite element software. For the concentration distribution, the numerical result and simulation result are almost the same. For the stress component, no significant difference exists between the two results, the largest relative difference for radial stress in the center is found to be about 4% and state of charge (SOC) = 17.9%. The radial stress decreases with radial position increasing, and decreases to zero at the surface, which is consistent with the results under the boundary condition. The hoop stress is tensile stress around the center of electrode, and becomes a compressive stress near the surface. Owing to the fact that the tensile hoop stress is attributed to the crack initiation, this implies that when plastic deformation is negligible, cracks first form in the center. The stress components with different length-to-radius ratios are calculated. It is found that the stress caused by lateral diffusion increases with length-to-radius ratio increasing, while the stress induced by axial diffusion decreases with length-to-radius ratio increasing. This is because the lateral diffusion has a greater influence on Li-concentration distribution in a cylinder electrode with length-to-radius ratio increasing.

Adaptive Event-Triggered Fuzzy Control for Stochastic Highly Nonlinear Systems With Time Delay and Nontriangular Structure Interconnections

The problem of adaptive fuzzy control based on the switching event-triggered mechanism for stochastic highly nonlinear system with time-varying delays, actuator faults and non-triangular structure interconnections is investigated in this article. By constructing the corresponding Lyapunov-Krasovskii functional and variable separation method, the complex nonlin- ear functions with time-varying delays and highly nonlinearity can be handled. With the help of the fuzzy logic systems and adaptive compensation technique, the processing difficulty of the unknown nonlinear terms and the parameters of actuator faults is reduced. Simultaneously, the proposed control scheme can greatly decrease the waste of communication resources by a switching event-triggered mechanism. Then, all signals in the system are ensured to be bounded under Lyapunov stability theorem. Ultimately, the simulation example demonstrates the availability of the developed control scheme.

Experimental verification and electromagnetic analysis of axial flux permanent magnet motor using the subdomain method

This study investigates a subdomain method for performing the characteristic analysis of an axial flux permanent magnet motor (AFPM). Initially, the AFPM was simplified for the analytical model and analysis regions were defined to apply the subdomain method. Subsequently, a general solution was derived for the governing equations of each analysis region using the variable separation method. Subsequently, the undetermined coefficients of the general solution were obtained from the matrix using appropriate boundary conditions, and the magnetic flux density in the air-gap region was derived based on the defined magnetic vector potential. Finally, the induced voltage and electromagnetic torque of the analytical model were calculated. The analytical results were verified using the finite element method and the experimental results.

Analysis of sinusoidal post-buckling deformation of horizontal coiled tubing with initial residual bending.

In the process of horizontal well construction, the working environment in the well is very complicated. It is difficult to determine the actual deformation and force of the coiled tubing in the wellbore from the load change at the wellhead along. In addition, the coiled tubing wrapped around the reel and entering the wellbore through the injection head may cause an initial residual bending. In this paper, the mathematical model of the coiled tubing under axial load during buckling deformation is established, and the analytical solution with the time term is derived by reasonable simplification of the model. The effect of residual bending on the post-sinusoidal buckling of the coiled tubing in the horizontal well was investigated using the separation variable method. The effect of residual bending variations on the well wall contact forces is analyzed. The research shows that the initial parameter Г is the main factor influencing the post-sinusoidal buckling of the coiled tubing with residual bending. The larger the parameter value Г, the greater the effect of the residual bending on the post-sinusoidal buckling deformation of the coiled tubing, and the earlier the critical point of the mixed sinusoidal-helical buckling of the coiled tubing appears. The compression velocity of the coiled tubing has a significant effect on the well wall contact force. The faster the compression, the greater the contact force. By introducing the time term and the separation variable, this paper provides a new method and theoretical basis for further study of the process of entering sinusoidal-helical buckling of the coiled tubing with initial residual bending.

Systemic inflammation associates with and precedes cord atrophy in progressive multiple sclerosis.

In preclinical models of multiple sclerosis, systemic inflammation has an impact on the compartmentalized inflammatory process within the central nervous system and results in axonal loss. It remains to be shown whether this is the case in humans, specifically whether systemic inflammation contributes to spinal cord or brain atrophy in multiple sclerosis. Hence, an observational longitudinal study was conducted to delineate the relationship between systemic inflammation and atrophy using magnetic resonance imaging: the SIMS (Systemic Inflammation in Multiple Sclerosis) study. Systemic inflammation and progression were assessed in people with progressive multiple sclerosis (n = 50) over two and a half years. Eligibility criteria included: (i) primary or secondary progressive multiple sclerosis; (ii) age ≤ 70; and (iii) Expanded Disability Status Scale ≤ 6.5. First morning urine was collected weekly to quantify systemic inflammation by measuring the urinary neopterin-to-creatinine ratio using a validated ultra-performance liquid chromatography mass spectrometry technique. The urinary neopterin-to-creatinine ratio temporal profile was characterized by short-term responses overlaid on a background level of inflammation, so these two distinct processes were considered as separate variables: background inflammation and inflammatory response. Participants underwent MRI at the start and end of the study, to measure cervical spinal cord and brain atrophy. Brain and cervical cord atrophy occurred on the study, but the most striking change was seen in the cervical spinal cord, in keeping with the corticospinal tract involvement that is typical of progressive disease. Systemic inflammation predicted cervical cord atrophy. An association with brain atrophy was not observed in this cohort. A time lag between systemic inflammation and cord atrophy was evident, suggesting but not proving causation. The association of the inflammatory response with cord atrophy depended on the level of background inflammation, in keeping with experimental data in preclinical models where the effects of a systemic inflammatory challenge on tissue injury depended on prior exposure to inflammation. A higher inflammatory response was associated with accelerated cord atrophy in the presence of background systemic inflammation below the median for the study population. Higher background inflammation, while associated with cervical cord atrophy itself, subdued the association of the inflammatory response with cord atrophy. Findings were robust to sensitivity analyses adjusting for potential confounders and excluding cases with new lesion formation. In conclusion, systemic inflammation associates with, and precedes, multiple sclerosis progression. Further work is needed to prove causation since targeting systemic inflammation may offer novel treatment strategies for slowing neurodegeneration in multiple sclerosis.

Integrating time and knowledge to understand organizational evolution: towards a conceptual framework

PurposeHow does the interaction between time and knowledge affect the evolution of organizations? Past research in organizational evolution has mostly investigated time and knowledge as two separate variables. In contrast, theoretical perspectives integrating these variables are still seemingly scant. The authors believe that filling this literature gap needs attention. Thus, this study aims to contribute by developing a conceptual framework.Design/methodology/approachThis is a conceptual study. The framework is centred on the concept of “co-evolutionary time”, which the authors explain through a business example from the tourism industry. Supported by a narrative-based style, from a methodological point of view the framework is featured by the attempt to synthesize specific, extant literature into new theoretical development.FindingsAs its main theoretical contribution, the co-evolutionary time suggests how firms can adapt in a way that, from an evolutionary perspective, proves fitting both in terms of contents and methods, thus opening possibilities for new long-term social construction and reconstruction. As its main practical contribution, co-evolutionary time can constitute not only a temporary source of organizational success and competitive advantage but also an agent of enduring change and long-term business survival.Originality/valueAs its main novelty, the framework is developed through merging two literature streams. In particular, the authors first consider the literature about time, with a focus on its objective and subjective dimensions. The authors then consider the literature about organizational evolution, with a focus on the co-evolutionary nature of the firm/environment relationship.

The Effect of Digital Capability, Digital Orientation on Business Resilience with Digital Transformation as a Mediating Variable

In the advent of information technology advancement, digitalization in business becomes pivotal in sustaining competitive advantage, especially for small scale enterprises. Notwithstanding the influx of digitalized business processes, their impact on business resilience is arguably impacted by other variables. This study is aimed at analyzing the influence of digital capabilities, and digital orientation on business resilience with digital transformation as a mediating variable. As a case in point, 380 small business owners in Greater Jakarta area are selected as respondents. The data was collected using a Likert-scale questionnaire that was distributed online. Data was analyzed using Smart PLS. The study finds that digital capabilities, digital orientation, and digital transformation as separate variables are significantly influenced by business resilience with digital capability showing the highest significance value. The digital transformation partially mediates the impact of digital capabilities and digital orientation on business resilience. The findings indicate the limited success of the Indonesian government initiatives in accelerating small businesses’ exposure to digitalization during the pandemic period. The next proposed recommendation would be to initiate efforts to increase digital transformation of small businesses since this variable comes across as the bridge between digitalization and business resilience. Keywords: Digital Capability, Digital Orientation, Business Resilience, Digital Transformation

Application of Genetic Algorithm for Synthesizing Onedimensional Thinned Antenna Arrays to form Twodimensional Thinned MIMO Antenna Arrays

The possibility of using thinned classical linear antenna arrays to synthesize two-dimensional thinned MIMO antenna arrays on their basis is considered. The MIMO antenna array which receiving and transmitting elements are located on two mutually orthogonal lines is regarded. On the example of this array it is shown that the directivity diagram of its virtual aperture has the property of separability of spatial variables. It allows using the arrangement of elements of a thinned classical linear antenna array for the receiving and transmitting elements lines. It is demonstrated that a thinned MIMO antenna array can form a radiation pattern with a lower level of side lobes than a full array while keeping the width of the main lobe unchanged. The dependence allowing to determine the peak level of side lobes of a twodimensional thinned MIMO antenna array at a given thinning factor is given.

Probabilistic Keyphrase Generation From Copy and Generating Spaces.

Keyphrase generation is one of the most fundamental tasks in natural language processing (NLP). Most existing works on keyphrase generation mainly focus on using holistic distribution to optimize the negative log-likelihood loss, but they do not directly manipulate the copy and generating spaces, which may reduce the generability of the decoder. Additionally, existing keyphrase models are either unable to determine the dynamic numbers of keyphrases or produce the number of keyphrases implicitly. In this article, we propose a probabilistic keyphrase generation model from copy and generating spaces. The proposed model is built upon the vanilla variational encoder-decoder (VED) framework. On top of VED, two separate latent variables are adopted to model the distribution of data within the latent copy and generating spaces, respectively. Specifically, we adopt a von Mises-Fisher (vMF) distribution to obtain a condensed variable for modifying the generating probability distribution over the predefined vocabulary. Meanwhile, we utilize a clustering module, which is designed to promote Gaussian Mixture learning and subsequently extract a latent variable for the copy probability distribution. Moreover, we utilize a natural property of the Gaussian mixture network and use the number of filtered components to determine the number of keyphrases. The approach is trained based on latent variable probabilistic modeling, neural variational inference, and self-supervised learning. Experiments on social media and scientific article datasets outperform the state-of-the-art baselines in generating accurate predictions and controllable keyphrase numbers.