115,629 research outputs found

### Vanishing Viscosity Approach to the Compressible Euler Equations for Transonic Nozzle and Spherically Symmetric Flows

We are concerned with globally defined entropy solutions to the Euler
equations for compressible fluid flows in transonic nozzles with general
cross-sectional areas. Such nozzles include the de Laval nozzles and other more
general nozzles whose cross-sectional area functions are allowed at the nozzle
ends to be either zero (closed ends) or infinity (unbounded ends). To achieve
this, in this paper, we develop a vanishing viscosity method to construct
globally defined approximate solutions and then establish essential uniform
estimates in weighted $L^p$ norms for the whole range of physical adiabatic
exponents $\gamma\in (1, \infty)$, so that the viscosity approximate solutions
satisfy the general $L^p$ compensated compactness framework. The viscosity
method is designed to incorporate artificial viscosity terms with the natural
Dirichlet boundary conditions to ensure the uniform estimates. Then such
estimates lead to both the convergence of the approximate solutions and the
existence theory of globally defined finite-energy entropy solutions to the
Euler equations for transonic flows that may have different end-states in the
class of nozzles with general cross-sectional areas for all $\gamma\in (1,
\infty)$. The approach and techniques developed here apply to other problems
with similar difficulties. In particular, we successfully apply them to
construct globally defined spherically symmetric entropy solutions to the Euler
equations for all $\gamma\in (1, \infty)$.Comment: 32 page

### Global Entropy Solutions to the Gas Flow in General Nozzle

We are concerned with the global existence of entropy solutions for the
compressible Euler equations describing the gas flow in a nozzle with general
cross-sectional area, for both isentropic and isothermal fluids. New
viscosities are delicately designed to obtain the uniform bound of approximate
solutions. The vanishing viscosity method and compensated compactness framework
are used to prove the convergence of approximate solutions. Moreover, the
entropy solutions for both cases are uniformly bounded independent of time. No
smallness condition is assumed on initial data. The techniques developed here
can be applied to compressible Euler equations with general source terms

### A universal approximate cross-validation criterion and its asymptotic distribution

A general framework is that the estimators of a distribution are obtained by
minimizing a function (the estimating function) and they are assessed through
another function (the assessment function). The estimating and assessment
functions generally estimate risks. A classical case is that both functions
estimate an information risk (specifically cross entropy); in that case Akaike
information criterion (AIC) is relevant. In more general cases, the assessment
risk can be estimated by leave-one-out crossvalidation. Since leave-one-out
crossvalidation is computationally very demanding, an approximation formula can
be very useful. A universal approximate crossvalidation criterion (UACV) for
the leave-one-out crossvalidation is given. This criterion can be adapted to
different types of estimators, including penalized likelihood and maximum a
posteriori estimators, and of assessment risk functions, including information
risk functions and continuous rank probability score (CRPS). This formula
reduces to Takeuchi information criterion (TIC) when cross entropy is the risk
for both estimation and assessment. The asymptotic distribution of UACV and of
a difference of UACV is given. UACV can be used for comparing estimators of the
distributions of ordered categorical data derived from threshold models and
models based on continuous approximations. A simulation study and an analysis
of real psychometric data are presented.Comment: 23 pages, 2 figure

### The Thermal Abundance of Semi-Relativistic Relics

Approximate analytical solutions of the Boltzmann equation for particles that
are either extremely relativistic or non-relativistic when they decouple from
the thermal bath are well established. However, no analytical formula for the
relic density of particles that are semi-relativistic at decoupling is yet
known. We propose a new ansatz for the thermal average of the annihilation
cross sections for such particles, and find a semi-analytical treatment for
calculating their relic densities. As examples, we consider Majorana- and
Dirac-type neutrinos. We show that such semi-relativistic relics cannot be good
cold Dark Matter candidates. However, late decays of meta-stable
semi-relativistic relics might have released a large amount of entropy, thereby
diluting the density of other, unwanted relics.Comment: 22 pages, 5 figures. Comments and references adde

### Continuum thermodynamics of chemically reacting fluid mixtures

We consider viscous, heat conducting mixtures of molecularly miscible
chemical species forming a fluid in which the constituents can undergo chemical
reactions. Assuming a common temperature for all components, we derive a closed
system of partial mass and partial momentum balances plus a mixture balance of
internal energy. This is achieved by careful exploitation of the entropy
principle and requires appropriate definitions of absolute temperature and
chemical potentials, based on an adequate definition of thermal energy
excluding diffusive contributions. The resulting interaction forces split into
a thermo-mechanical and a chemical part, where the former turns out to be
symmetric in case of binary interactions. For chemically reacting systems and
as a new result, the chemical interaction force is a contribution being
non-symmetric outside of chemical equilibrium. The theory also provides a
rigorous derivation of the so-called generalized thermodynamic driving forces,
avoiding the use of approximate solutions to the Boltzmann equations. Moreover,
using an appropriately extended version of the entropy principle and
introducing cross-effects already before closure as entropy invariant couplings
between principal dissipative mechanisms, the Onsager symmetry relations become
a strict consequence. With a classification of the factors in the binary
products of the entropy production according to their parity--instead of the
classical partition into so-called fluxes and driving forces--the apparent
anti-symmetry of certain couplings is thereby also revealed. If the diffusion
velocities are small compared to the speed of sound, the Maxwell-Stefan
equations follow in the case without chemistry, thereby neglecting wave
phenomena in the diffusive motion. This results in a reduced model with only
mass being balanced individually. In the reactive case ..

### Uncertainty quantification in mechanistic epidemic models via cross-entropy approximate Bayesian computation

This paper proposes a data-driven approximate Bayesian computation framework
for parameter estimation and uncertainty quantification of epidemic models,
which incorporates two novelties: (i) the identification of the initial
conditions by using plausible dynamic states that are compatible with
observational data; (ii) learning of an informative prior distribution for the
model parameters via the cross-entropy method. The new methodology's
effectiveness is illustrated with the aid of actual data from the COVID-19
epidemic in Rio de Janeiro city in Brazil, employing an ordinary differential
equation-based model with a generalized SEIR mechanistic structure that
includes time-dependent transmission rate, asymptomatics, and hospitalizations.
A minimization problem with two cost terms (number of hospitalizations and
deaths) is formulated, and twelve parameters are identified. The calibrated
model provides a consistent description of the available data, able to
extrapolate forecasts over a few weeks, making the proposed methodology very
appealing for real-time epidemic modeling

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