17 research outputs found

### Stability Estimates for Fractional Hardy-SchrĂ¶dinger Operators

In this chapter, we derive optimal Hardy-Sobolev type improvements of fractional Hardy inequalities, formally written as Lsuâ„wxxÎžu2ââ1, for the fractional SchrĂ¶dinger operator Lsu=âÎsuâkn,sux2s associated with s-th powers of the Laplacian for sâ01, on bounded domains in Rn. Here, kn,s denotes the optimal constant in the fractional Hardy inequality, and 2â=2nâÎžnâ2s, for 0â€Îžâ€2s<n. The optimality refers to the singularity of the logarithmic correction w that has to be involved so that an improvement of this type is possible. It is interesting to note that Hardy inequalities related to two distinct fractional Laplacians on bounded domains admit the same optimal remainder terms of Hardy-Sobolev type. For deriving our results, we also discuss refined trace Hardy inequalities in the upper half space which are rather of independent interest

### Quantum modes in DBI inflation: exact solutions and constraints from vacuum selection

We study a two-parameter family of exactly solvable inflation models with
variable sound speed, and derive a corresponding exact expression for the
spectrum of curvature perturbations. We generalize this expression to the slow
roll case, and derive an approximate expression for the scalar spectral index
valid to second order in slow roll. We apply the result to the case of DBI
inflation, and show that for certain choices of slow roll parameters, the
Bunch-Davies limit (a) does not exist, or (b) is sensitive to stringy physics
in the bulk, which in principle can have observable signatures in the
primordial power spectrum.Comment: 10 pages, LaTeX; V2: version submitted to PRD. References added,
minor error in text correcte

### Layer dynamics for the one dimensional Î”-dependent CahnâHilliard/AllenâCahn equation

The final publication is available at Springer via http://dx.doi.org/10.1007/s00526-021-02085-4We study the dynamics of the one-dimensional Î”-dependent Cahn-Hilliard / Allen-Cahn equation within a neighborhood of an equilibrium of N transition layers, that in general does not conserve mass. Two different settings are considered which differ in that, for the second, we impose a mass-conservation constraint in place of one of the zero-mass flux boundary conditions at
x = 1. Motivated by the study of Carr and Pego on the layered metastable patterns of Allen-Cahn in [10], and by this of Bates and Xun in [5] for the Cahn-Hilliard equation, we implement an N-dimensional, and a mass-conservative Nâ1-dimensional manifold respectively; therein, a metastable state with N transition layers is approximated. We then determine, for both cases, the essential dynamics of the layers (ode systems with the equations of motion), expressed in terms of local coordinates relative to the manifold used. In particular, we estimate the spectrum of the linearized Cahn-Hilliard / Allen-Cahn operator, and specify wide families of Î”-dependent weights ÎŽ(Î”), Â”(Î”), acting at each part of the operator, for which the dynamics are stable and rest exponentially small in Î”. Our analysis enlightens the role of mass conservation in the classification of the general mixed problem into two main categories where the solution has a profile close to Allen-Cahn, or, when the mass is conserved, close to the Cahn-Hilliard solution

### Biomechanic and Hemodynamic Perspectives in Abdominal Aortic Aneurysm Rupture Risk Assessment

Abdominal aortic aneurysms (AAAs) pose a significant source of mortality for the elderly, especially if they go on undetected and ultimately rupture. Therefore, elective repair of these lesions is recommended in order to avoid risk of rupture which is associated with high mortality. Currently, the risk of rupture and thus the indication to intervene is evaluated based on the size of the AAA as determined by its maximum diameter. Since AAAs actually present original geometric configurations and unique hemodynamic and biomechanic conditions, it is expected that other variables may affect rupture risk as well. This is the reason why the maximum diameter criterion has often been proven inaccurate. The biomechanical approach considers rupture as a material failure where the stresses exerted on the wall outweigh its strength. Therefore, rupture depends on the pointwise comparison of the stress and strength for every point of the aneurysmal surface. Moreover, AAAs hemodynamics play an essential role in AAAs natural history, progression and rupture. This chapter summarizes advances in AAAs rupture risk estimation beyond the âone size fits allâ maximum diameter criterion

### Inflationary potentials in DBI models

We study DBI inflation based upon a general model characterized by a
power-law flow parameter $\epsilon(\phi)\propto\phi^{\alpha}$ and speed of
sound $c_s(\phi)\propto\phi^{\beta}$, where $\alpha$ and $\beta$ are constants.
We show that in the slow-roll limit this general model gives rise to distinct
inflationary classes according to the relation between $\alpha$ and $\beta$ and
to the time evolution of the inflaton field, each one corresponding to a
specific potential; in particular, we find that the well-known canonical
polynomial (large- and small-field), hybrid and exponential potentials also
arise in this non-canonical model. We find that these non-canonical classes
have the same physical features as their canonical analogs, except for the fact
that the inflaton field evolves with varying speed of sound; also, we show that
a broad class of canonical and D-brane inflation models are particular cases of
this general non-canonical model. Next, we compare the predictions of
large-field polynomial models with the current observational data, showing that
models with low speed of sound have red-tilted scalar spectrum with low
tensor-to-scalar ratio, in good agreement with the observed values. These
models also show a correlation between large non-gaussianity with low tensor
amplitudes, which is a distinct signature of DBI inflation with large-field
polynomial potentials.Comment: Minor changes, reference added. Version submitted to JCA

### Inflation over the hill

We calculate the power spectrum of curvature perturbations when the inflaton
field is rolling over the top of a local maximum of a potential. We show that
the evolution of the field can be decomposed into a late-time attractor, which
is identified as the slow roll solution, plus a rapidly decaying non-slow roll
solution, corresponding to the field rolling ``up the hill'' to the maximum of
the potential. The exponentially decaying transient solution can map to an
observationally relevant range of scales because the universe is also expanding
exponentially. We consider the two branches separately and we find that they
are related through a simple transformation of the slow roll parameter $\eta$
and they predict identical power spectra. We generalize this approach to the
case where the inflaton field is described by both branches simultaneously and
find that the mode equation can be solved exactly at all times. Even though the
slow roll parameter $\eta$ is evolving rapidly during the transition from the
transient solution to the late-time attractor solution, the resultant power
spectrum is an exact power-law spectrum. Such solutions may be useful for
model-building on the string landscape.Comment: 11 pages, 1 figure (V3: Version accepted by PRD, title changed by
journal

### Tensors, non-Gaussianities, and the future of potential reconstruction

We present projections for reconstruction of the inflationary potential
expected from ESA's upcoming Planck Surveyor CMB mission. We focus on the
effects that tensor perturbations and the presence of non-Gaussianities have on
reconstruction efforts in the context of non-canonical inflation models. We
consider potential constraints for different combinations of
detection/null-detection of tensors and non-Gaussianities. We perform Markov
Chain Monte Carlo and flow analyses on a simulated Planck-precision data set to
obtain constraints. We find that a failure to detect non-Gaussianities
precludes a successful inversion of the primordial power spectrum, greatly
affecting uncertainties, even in the presence of a tensor detection. In the
absence of a tensor detection, while unable to determine the energy scale of
inflation, an observable level of non-Gaussianities provides correlations
between the errors of the potential parameters, suggesting that constraints
might be improved for suitable combinations of parameters. Constraints are
optimized for a positive detection of both tensors and non-Gaussianities.Comment: 12 pages, 5 figures, LaTeX; V2: version submitted to JCA

### Non-canonical generalizations of slow-roll inflation models

We consider non-canonical generalizations of two classes of simple
single-field inflation models. First, we study the non-canonical version of
"ultra-slow roll" inflation, which is a class of inflation models for which
quantum modes do not freeze at horizon crossing, but instead evolve rapidly on
superhorizon scales. Second, we consider the non-canonical generalization of
the simplest "chaotic" inflation scenario, with a potential dominated by a
quartic (mass) term for the inflaton. We find a class of related non-canonical
solutions with polynomial potentials, but with varying speed of sound. These
solutions are characterized by a constant field velocity, and we dub such
models {\it isokinetic} inflation. As in the canonical limit, isokinetic
inflation has a slightly red-tilted power spectrum, consistent with current
data. Unlike the canonical case, however, these models can have an arbitrarily
small tensor/scalar ratio. Of particular interest is that isokinetic inflation
is marked by a correlation between the tensor/scalar ratio and the amplitude of
non-Gaussianity such that parameter regimes with small tensor/scalar ratio have
{\it large} associated non-Gaussianity, which is a distinct observational
signature.Comment: 12 pages, 3 figures, LaTeX; V2: version submitted to JCAP. References
adde

### Classification of Blood Rheological Models through an Idealized Symmetrical Bifurcation

The assumed rheological behavior of blood influences the hemodynamic characteristics of numerical blood flow simulations. Until now, alternative rheological specifications have been utilized, with uncertain implications for the results obtained. This work aims to group sixteen blood rheological models in homogeneous clusters, by exploiting data generated from numerical simulations on an idealized symmetrical arterial bifurcation. Blood flow is assumed to be pulsatile and is simulated using a commercial finite volume solver. An appropriate mesh convergence study is performed, and all results are collected at three different time instants throughout the cardiac cycle: at peak systole, early diastole, and late diastole. Six hemodynamic variables are computed: the time average wall shear stress, oscillatory shear index, relative residence time, global and local non-Newtonian importance factor, and non-Newtonian effect factor. The resulting data are analyzed using hierarchical agglomerative clustering algorithms, which constitute typical unsupervised classification methods. Interestingly, the rheological models can be partitioned into three homogeneous groups, whereas three specifications appear as outliers which do not belong in any partition. Our findings suggest that models which are defined in a similar manner from a mathematical perspective may behave substantially differently in terms of the data they produce. On the other hand, models characterized by different mathematical formulations may belong to the same statistical group (cluster) and can thus be considered interchangeably