Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures,and the solutions admit the concentration of mass.It is found that under the requirement of satisfying the over-compressing entropy condition:(i)there is a unique delta shock solution,corresponding to the case that has two strong classical Lax shocks;(ii)for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave,or two shocks with one being weak,there are infinitely many solutions,each consists of a delta shock and a rarefaction wave;(iii)there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves.These solutions are self-similar.Furthermore,for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data,there always exists a unique delta shock for at least a short time.It could be prolonged to a global solution.Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass(particle).Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified.This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases,that is strictly hyperbolic,and whose characteristics are both genuinely nonlinear.We also discuss possible physical interpretations and applications of these new solutions.

Evaluation of IoT Measurement Solutions from a Metrology Perspective

To professionally plan and manage the development and evolution of the Internet of Things(IoT),researchers have proposed several IoT performance measurement solutions.IoT performance measurement solutions can be very valuable for managing the development and evolution of IoT systems,as they provide insights into performance issues,resource optimization,predictive maintenance,security,reliability,and user experience.However,there are several issues that can impact the accuracy and reliability of IoT performance measurements,including lack of standardization,complexity of IoT systems,scalability,data privacy,and security.While previous studies proposed several IoT measurement solutions in the literature,they did not evaluate any individual one to figure out their respective measurement strengths and weaknesses.This study provides a novel scheme for the evaluation of proposed IoT measurement solutions using a metrology-coverage evaluation based on evaluation theory,metrology principles,and software measurement best practices.This evaluation approach was employed for 12 IoT measure categories and 158 IoT measurement solutions identified in a Systematic Literature Review(SLR)from 2010 to 2021.The metrology coverage of these IoT measurement solutions was analyzed from four perspectives:across IoT categories,within each study,improvement over time,and implications for IoT practitioners and researchers.The criteria in this metrology-coverage evaluation allowed for the identification of strengths and weaknesses in the theoretical and empirical definitions of the proposed IoT measurement solutions.We found that the metrological coverage varies significantly across IoT measurement solution categories and did not show improvement over the 2010–2021 timeframe.Detailed findings can help practitioners understand the limitations of the proposed measurement solutions and choose those with stronger designs.These evaluation results can also be used by researchers to improve current IoT measurement solution designs and suggest new solutions with a stronger metrology base.

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

A Unified Measurement Solution of Software Trustworthiness Based on Social-to-Software Framework

As trust becomes increasingly important in software domain, software trustworthiness--as a complex high- composite concept, has developed into a big challenge people have to face, especially in the current open, dynamic and ever-changing Internet environment. Furthermore, how to recognize and define trust problem from its nature and how to measure software trustworthiness correctly and effectively play a key role in improving users＇ trust in choosing software. Based on trust theory in the field of humanities and sociology, this paper proposes a measurable S2S （Social-to-Software） software trustworthiness framework, introduces a generalized indicator loss to unify three parts of trustworthiness result, and presents a whole metric solution for software trustworthiness, including the advanced J-M model based on power function and time-loss rate for ability trustworthiness measurement, the fuzzy comprehensive evaluation advanced-model considering effect of multiple short boards for basic standard trustworthiness, and the identity trustworthiness measurement method based on the code homology detecting tools. Finally, it provides a case study to verify that the solution is applicable and effective.

THE VANISHING PRESSURE LIMIT OF SOLUTIONS TO THE SIMPLIFIED EULER EQUATIONS FOR ISENTROPIC FLUIDS

In this paper,the Riemann problem of the 1-D reduced model for the 2-D Euler equations is considered and the Riemann solutions are obtained.It is proved that,as the pressure vanishes,they converge to two kinds of Riemann solutions to the 1D reduced model for the 2-D transport equations:one contains δ-shocks,the other contains vacuum.

Steady Compressible Euler Equations of Concentration Layers for Hypersonic-limit Flows Passing Three-dimensional Bodies and Generalized Newton-Busemann Pressure Law

For stationary hypersonic-limit Euler flows passing a solid body in three-dimensional space,the shock-front coincides with the upwind surface of the body,hence there is an infinite-thin layer of concentrated mass,in which all particles hitting the body move along its upwind surface.By proposing a concept of Radon measure solutions of boundary value problems of the multi-dimensional compressible Euler equations,which incorporates the large-scale of three-dimensional distributions of upcoming hypersonic flows and the small-scale of particles moving on two-dimensional surfaces,the authors derive the compressible Euler equations for flows in concentration layers,which is a stationary pressureless compressible Euler system with source terms and independent variables on curved surface.As a by-product,they obtain a formula for pressure distribution on surfaces of general obstacles in hypersonic flows,which is a generalization of the classical Newton-Busemann law for drag/lift in hypersonic aerodynamics.

REMARKS OF DIFFERENTIAL EQUATIONS ON CLOSED SUBSETS IN A BANACH SPACE

We give some extensions of Monch-Harton inequalities with respect to measures of noncompactness. As an example of the application, we obtain two existencetheorems of solutions for Cauchy problems of differential equations on closed setsunder weaker compactness conditions.

Forward-backward doubly stochastic differential equations and related stochastic partial differential equations

The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations （FBDSDEs）. It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBDSDEs. Finally, the probabilistie interpretation for the solutions to a class of quasilinear stochastic partial differential equations （SPDEs） combined with algebra equations is given. One distinctive character of this result is that the forward component of the FBDSDEs is coupled with the backvzard variable.

Optimal Transportation for Generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L（x, u, t）, and considers the following cost function： c（x, y） = inf x（0）=x x（1）=y u∈U∫0^1 L（x（s）, u（x（s）, s）, s）ds, where U is a control set, and x satisfies the ordinary equation x（s） = f（x（s）, u（x（s）, s））.It is proved that under the condition that the initial measure μ0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation：Vt（t, x） ＋ sup u∈U = 0,V（0, x） = Φ0（x）.