Assumption Of Symmetry Research Articles

On the cosmological constant as a quantum operator

We regard the cosmological fluid within an exponentially expanding FLRW spacetime as the probability fluid of a nonrelativistic Schroedinger field. The scalar Schroedinger particle so described has a mass equal to the total (baryonic plus dark) matter content of the Universe. This procedure allows a description of the cosmological fluid by means of the operator formalism of nonrelativistic quantum theory. Under the assumption of radial symmetry, a quantum operator proportional to [Formula: see text] represents the cosmological constant [Formula: see text]. The experimentally measured value of [Formula: see text] is one of the eigenvalues of [Formula: see text]. Next we solve the Poisson equation [Formula: see text] for the gravitational potential [Formula: see text], with the cosmological constant [Formula: see text] playing the role of a source term. It turns out that [Formula: see text] includes, besides the standard Newtonian potential [Formula: see text], a correction term proportional to [Formula: see text] identical to that appearing in theories of modified Newtonian dynamics.

Testing for main fixed effects: The symmetry assumption and monotone incomplete data

We consider the balanced two-way mixed effects design with some empty cells. A test procedure for the hypothesis of no main fixed effects is developed under violation of the assumption of variance homogeneity and symmetry. The asymptotic null distribution of the test statistics is studied under the condition that the number of levels of the random effects tends to infinity as both the number of complete and incomplete observations tend to infinity. An illustrative example is given.

L-DED numerical model for sensor embedding

Sensor integration is one of the drivers in modern industry for obtaining real-time data and enabling transition to Industry 4.0. Sensor integration on production systems and tooling is one of the key points for data acquisition. Although several techniques can be applied for sensor integration, Laser Directed Energy Deposition (L-DED) is becoming one of the most relevant, since the sensor can be placed into the manufactured layer-by-layer structure. However, the thermal nature of the L-DED poses a challenge when heat-sensitive parts, such as thermocouples, are to be embedded. In order to ease parametrization and anticipate the behavior of the L-DED process, modeling is an interesting tool that has attracted the attention of academia in the last years. Nevertheless, most models are highly complex and focused on a very local scale or include symmetry assumptions that restrict their use for real applications. In view of this need, in the present research work a thermal model that considers material addition and determines the clad geometry is developed. The model includes an automatic meshing algorithm that adapts the element size by refining the mesh where required. Besides, the model enables 5 axis L-DED, in-process variation of the machine feed rate, and allows to switch on and off the laser to simulate not only the material deposition, but also the idle movements. The model is validated in two steps: single clad deposition on a flat surface and single clads on a 0.3 mm thick thermocouple sheath. Finally, the validated model is used for defining the maximum laser power for embedding virtually a 3 mm diameter K-type thermocouple with a 0.3 mm thick sheath. The results of the simulation are also corroborated by experimental integration of the same thermocouple, which functionality is tested afterwards. Therefore, the L-DED modeling is proven to be an effective tool for manufacturing complex parts on the first try.

Existence and Uniqueness for Plane Stationary Navier–Stokes Flows with Compactly Supported Force

We study the stationary Navier--Stokes equations in the whole plane with a compactly supported force term and with a prescribed constant spatial limit. Prior to this work, existence of solutions to this problem was only known under special symmetry and smallness assumptions. In the paper we solve the key difficulties in applying Leray's {\it invading domains method} and, as a consequence, prove the existence of $D$-solutions in the whole plane for arbitrary compactly supported force. The boundary condition at infinity are verified in two different scenarios: (I) the~limiting velocity is sufficient large with respect to the external force, (II) both the total integral of force and the~limiting velocity vanish. Hence, our method produces large class of new solutions with prescribed spatial limits. Moreover, we show the uniqueness of $D$-solutions to this problem in a perturbative regime. The main tools here are two new estimates for general Navier-Stokes solutions, which have rather simple forms. They control the difference between mean values of the velocity over two concentric circles in terms of the Dirichlet integral in the annulus between them.

Dimensional reduction of a poromechanical cardiac model for myocardial perfusion studies

In this paper, we adapt a previously developed poromechanical formulation to model the perfusion of myocardium during a cardiac cycle. First, a complete model is derived in 3D. Then, we perform a dimensional reduction under the assumption of spherical symmetry and propose a numerical algorithm that enables us to perform simulations of the myocardial perfusion throughout the cardiac cycle. These simulations illustrate the use of the proposed model to represent various physiological and pathological scenarios, specifically the vasodilation in the coronary network (to reproduce the standard clinical assessment of myocardial perfusion and perfusion reserve), the stenosis of a large coronary artery, an increased vascular resistance in the microcirculation (microvascular disease) and the consequences of inotropic activation (increased myocardial contractility) particularly at the level of the systolic flow impediment. Our results show that the model gives promising qualitative reproductions of complex physiological phenomena. This paves the way for future quantitative studies using clinical or experimental data.

An Approximate MSE Expression for Maximum Likelihood and Other Implicitly Defined Estimators of Non-Random Parameters

An approximate mean square error (MSE) expression for the performance analysis of implicitly defined estimators of non-random parameters is proposed. An implicitly defined estimator (IDE) declares the minimizer/maximizer of a selected cost/reward function as the parameter estimate. The maximum likelihood (ML) and the least squares estimators are among the well known examples of this class. In this paper, an exact MSE expression for implicitly defined estimators with a symmetric and unimodal objective function is given. It is shown that the expression reduces to the Cramer-Rao lower bound (CRLB) and misspecified CRLB in the large sample size regime for ML and misspecified ML estimation, respectively. The expression is shown to yield the Ziv-Zakai bound (without the valley filling function) for the maximum a posteriori (MAP) estimator when it is used in a Bayesian setting, that is, when an a-priori distribution is assigned to the unknown parameter. In addition, extension of the suggested expression to the case of nuisance parameters is studied and some approximations are given to ease the computations for this case. Numerical results indicate that the suggested MSE expression not only predicts the estimator performance in the asymptotic region; but it is also applicable for the threshold region analysis, even for IDEs whose objective functions do not satisfy the symmetry and unimodality assumptions. Advantages of the suggested MSE expression are its conceptual simplicity and its relatively straightforward numerical calculation due to the reduction of the estimation problem to a binary hypothesis testing problem, similar to the usage of Ziv-Zakai bounds in random parameter estimation problems.

On a planar Schrödinger–Poisson system involving a non-symmetric potential

AbstractWe prove the existence of a ground state positive solution of Schrödinger–Poisson systems in the plane of the form \[ -\Delta u + V(x)u + \frac{\gamma}{2\pi} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \quad\text{in}\ \mathbb{R}^2, \]where $p\ge 4$, $\gamma,b>0$ and the potential $V$ is assumed to be positive and unbounded at infinity. On the potential we do not require any symmetry or periodicity assumption, and it is not supposed it has a limit at infinity. We approach the problem by variational methods, using a variant of the mountain pass theorem and the Cerami compactness condition.

A local test for global extrema in the dispersion relation of a periodic graph

We consider a family of periodic tight-binding models (combinatorial graphs) that have the minimal number of links between copies of the fundamental domain. For this family we establish a local condition of second derivative type under which the critical points of the dispersion relation can be recognized as global maxima or minima. Under the additional assumption of time-reversal symmetry, we show that any local extremum of a dispersion band is in fact its global extremum if the dimension of the periodicity group is three or less, or (in any dimension) if the critical point in question is a symmetry point of the Floquet--Bloch family with respect to complex conjugation. We demonstrate that our results are nearly optimal with a number of examples.

New Edgeworth-type expansions with finite sample guarantees

We establish higher-order nonasymptotic expansions for a difference between probability distributions of sums of i.i.d. random vectors in a Euclidean space. The derived bounds are uniform over two classes of sets: the set of all Euclidean balls and the set of all half-spaces. These results allow to account for an impact of higher-order moments or cumulants of the considered distributions; the obtained error terms depend on a sample size and a dimension explicitly. The new inequalities outperform accuracy of the normal approximation in existing Berry-Esseen inequalities under very general conditions. Under some symmetry assumptions on the probability distribution of random summands, the obtained results are optimal in terms of the ratio between the dimension and the sample size. The new technique which we developed for establishing nonasymptotic higher-order expansions can be interesting by itself. Using the new higher-order inequalities, we study accuracy of the nonparametric bootstrap approximation and propose a bootstrap score test under possible model misspecification. The results of the paper also include explicit error bounds for general elliptic confidence regions for an expected value of the random summands, and optimality of the Gaussian anti-concentration inequality over the set of all Euclidean balls.

Cosmological simulations of self-interacting Bose-Einstein condensate dark matter

Fully 3D cosmological simulations of scalar field dark matter with self-interactions, also known as Bose-Einstein condensate dark matter, are performed using a set of effective hydrodynamic equations. These are derived from the non-linear Schrödinger equation by performing a smoothing operation over scales larger than the de Broglie wavelength, but smaller than the self-interaction Jeans’ length. The dynamics on the de Broglie scale become an effective thermal energy in the hydrodynamic approximation, which is assumed to be subdominant in the initial conditions, but become important as structures collapse and the fluid is shock-heated. The halos that form have Navarro-Frenk-White envelopes, while the centers are cored due to the fluid pressures (thermal + self-interaction), confirming the features found by Dawoodbhoy et al. (2021, MNRAS, 506, 2418) using 1D simulations under the assumption of spherical symmetry. The core radii are largely determined by the self-interaction Jeans’ length, even though the effective thermal energy eventually dominates over the self-interaction energy everywhere, a result that is insensitive to the initial ratio of thermal energy to interaction energy, provided it is sufficiently small to not affect the linear and weakly non-linear regimes. Scaling relations for the simulated population of halos are compared to Milky Way dwarf spheroidals and nearby galaxies, assuming a Burkert halo profile, and are found to not match, although they conform better with observations compared to fuzzy dark matter-only simulations.

A stable and causal model of magnetohydrodynamics

We formulate the theory of first-order dissipative magnetohydrodynamics in an arbitrary hydrodynamic frame under the assumption of parity-invariance and discrete charge symmetry. We study the mode spectrum of Alfvén and magnetosonic waves as well as the spectrum of gapped excitations and derive constraints on the transport coefficients such that generic equilibrium states with constant magnetic fields are stable and causal under linearised perturbations. We solve these constraints for a specific equation of state and show that there exists a large family of hydrodynamic frames that renders the linear fluctuations stable and causal. This theory does not require introducing new dynamical degrees of freedom and therefore is a promising and simpler alternative to Müller-Israel-Stewart-type theories. Together with a detailed analysis of transport, entropy production and Kubo formulae, the theory presented here is well suited for studying dissipative effects in various contexts ranging from heavy-ion collisions to astrophysics.

Detecting equatorial symmetry breaking with LISA

The equatorial symmetry of the Kerr black hole is generically broken in models of quantum gravity. Nevertheless, most phenomenological models start from the assumption of equatorial symmetry, and little attention has been given to the observability of this smoking gun signature of beyond-GR physics. Extreme mass-ratio inspirals (EMRIs), in particular, are known to sensitively probe supermassive black holes near their horizon; yet estimates for constraints on deviations from Kerr in space-based gravitational wave observations (e.g. with LISA) of such systems are currently based on equatorially symmetric models. We use modified "analytic kludge" waveforms to estimate how accurately LISA will be able to measure or constrain equatorial symmetry breaking, in the form of the lowest-lying odd parity multipole moments $S_2, M_3$. We find that the dimensionless multipole ratios such as $S_2/M^3$ will typically be detectable for LISA EMRIs with a measurement accuracy of $\Delta(S_2/M^3) \sim 1\%$; this will set a strong constraint on the breaking of equatorial symmetry.

Prompt cusp formation from the gravitational collapse of peaks in the initial cosmological density field

ABSTRACT I present an analytic model for the early post-collapse evolution of a spherical density peak on the coherence scale of the initial fluctuations in a universe filled with collisionless and pressure-free ‘dust’. On a time-scale which is short compared to the peak’s collapse time t0, its inner regions settle into an equilibrium cusp with a power-law density profile, ρ ∝ r−12/7. Within this cusp, the circular orbit period P at each radius is related to the enclosed mass M by P = t0(M/Mc)2/3 where Mc is a suitably defined characteristic mass for the initial peak. The relaxation mechanism which produces this cusp gives insight into those which are active in high-resolution simulations of first halo formation in cold or warm dark matter universes, and, indeed, a simple argument suggests that the same power-law index γ = −12/7 should describe the prompt cusps formed during the collapse of generic peaks, independent of any symmetry assumption. Further work is needed to investigate the additional factors required to explain the slightly flatter exponent, γ ≈ −1.5, found in high-resolution numerical simulations of peak collapse.

Testing the mirror symmetry of Birkeland and ionospheric currents with respect to magnetic latitude, dipole tilt angle, and IMF By

It is often assumed that on average, polar ionospheric electrodynamics in the Northern and Southern Hemispheres are mirror symmetric or antisymmetric with respect to the interplanetary magnetic field By component and the dipole tilt angle ψ. For example, one might assume that the average Birkeland current density j at magnetic latitude λ is equal to the current density at magnetic latitude −λ if the signs of By and ψ are reversed and all other parameters are equal: j(λ, By, ψ, … ) = j(−λ, −By, −ψ, … ). This is a convenient assumption for empirical models, since it effectively doubles the amount of information that a measurement made in one hemisphere contains. In this study we use the Average Magnetic field and Polar current System (AMPS) model to quantify to what extent the assumption holds for Birkeland and ionospheric currents. The AMPS model is an empirical model based on Swarm and CHAMP magnetic field measurements, with no constraints on hemispheric symmetries, and with differences in main magnetic field geometry as well as biases in data point distributions in magnetic coordinates accounted for. We show that when averaged over IMF clock angle orientation, the total ionospheric divergence-free current in each hemisphere largely satisfies the mirror symmetry assumption. The same is true for the total Birkeland current in each hemisphere except during local winter, during which the Northern Hemisphere tends to dominate. We show that this local winter asymmetry is consistent with the average winter hemispheric asymmetry in total precipitating electron current derived from Fast Auroral SnapshoT (FAST) satellite observations. We attribute this and other more subtle deviations from symmetry to differences in sunlight distribution in magnetic coordinates, as well as magnetic field strength and its influence on ionospheric conductivity. Important departures from mirror symmetry also arise for some IMF clock angle orientations, particularly those for which IMF Bz > 0, as suggested by other recent studies.

Instability of closed orbits obtained by minimization* *The author has been supported by Spanish ERDF Project MTM2017-82348-C2-1-P.

We study the dynamics around closed orbits of autonomous Lagrangian systems. When the configuration space is two-dimensional and orientable we show that every closed orbit minimizing the free-period action functional is orbitally unstable. This result applies even when the minimizers are degenerate or nonisolated, but a particularly strong form of instability holds in the isolated case. Under some symmetry assumptions, free-period action minimizers are unstable also in the higher-dimensional case. Applications to geodesics and celestial mechanics are given.

Acoustic black hole analogy to analyze nonlinear acoustic wave dynamics in accelerating flow fields

We present a physical model and a numerical method based on a space- and time-dependent Galilean-type coordinate transformation to simulate acoustic waves in the presence of an accelerating background flow field with sonic transition. Kinematically, the coordinate transformation is designed so as to maintain the well-posedness of the transformed wave equation, which is solved in a fixed computational domain using standard finite differences. Considering an acoustic black hole analogy, we analyze the nonlinear dynamics of acoustic waves in a stationary but non-uniformly accelerating flow field under the assumption of spherical symmetry. The choice of the acoustic black hole analogy is motivated by the fact that the steady-state spherical sonic horizon allows us to parameterize the wave-flow configuration in terms of a Helmholtz number He=c2/(λagh), which is expressed as a function of the speed of sound c, the emitted wavelength λa, and the flow acceleration at the sonic horizon, that is, the acoustic surface gravity gh. The results of the numerical simulations show that He describes geometrically similar sets of wave characteristics for different combinations of gh and λa. However, we also observe nonlinear variations of the wave amplitude along the wave characteristics, which are attributed to nonlinear Doppler modulations. It appears that these amplitude modulations depend on the acceleration of the flow field and can, therefore, differ for geometrically similar characteristics.

A Continuum-mechanics solution of laminate free edges based on asymmetric stress and strain tensors

A meaningful attempt is made to investigate the impact of using asymmetric stress and strain formulations in the prediction of free-edge response of laminated orthotropic materials. The paper contributes to the process of making the singular free-edge stresses and strains in the literature more proximate to the bounded (realistic) solutions. Related to the singularity issue, it was previously identified that the symmetry assumption for stress and strain tensors could lead to a contradiction with the predicted free-edge stress functions. It is, therefore, reasonable to assert that the classical continuum mechanics based on tensor symmetries is unfaithful for the modelling of free edges. The present study demonstrates that for the free-edge region, stress and strain tensors naturally become asymmetric when they are enabled to be, to reach the minimum strain energy. By comparing with strain predictions generated by Digital Image Correlation technique, the new framework is shown to be able to address the notable discrepancy between the classical theoretical result and experimental result, which suggested the higher fidelity of the new framework. Moreover, by resolving the contradiction, the model is also made less sensitive to mesh refinement which sets an important basis for reaching a bounded solution in the future.

Bayesian Analysis of Masked Competing Risks Data Based on Proportional Subdistribution Hazards Model

Masked issues can emerge when dealing with competing risk data. Such issues are exemplified by the cause of a particular failure not being directly exhibited for all units to observe but only proven to be a subset of possible causes of failure. For assessing the impact of explanatory variables (covariates) on the cumulative incidence function (CIF), a process of Bayesian analysis is discussed in this paper. The symmetry assumption is not imposed on the masking probabilities and independent Dirichlet priors assigned to them. The Markov Chain Monte Carlo (MCMC) technique is utilized to implement the Bayesian analysis. The effectiveness of the developed model is tested via numerical studies, including simulated and real data sets.

Numerical analysis of non-local calculus on finite weighted graphs, with application to reduced-order modeling of dynamical systems

We present an approach to reduced-order modeling that builds off recent graph-theoretic work for representation, exploration, and analysis of computed states of physical systems (Banerjee et al., Comp. Meth. App. Mech. Eng., 351, 501–530, 2019). We extend a non-local calculus on finite weighted graphs to build such models by exploiting polynomial expansions and Taylor series. In the general framework for non-local calculus on graphs, the graph edge weights are intricately linked to the embedding of the graph, and consequently to the definition of the derivatives. In a previous communication (Duschenes and Garikipati, arXiv:2105.01740 [math.NA]), we have shown that radially symmetric, continuous edge weights derived from, for example Gaussian functions, yield inconsistent results in the resulting non-local derivatives when compared against the corresponding local, differential derivative definitions. Taking inspiration from finite difference methods, we algorithmically compute edge weights, considering the embedding of the local neighborhood of each graph vertex. Given this procedure, we ensure the consistency of the non-local derivatives in this setting, a crucial requirement for numerical applications. We show that we can achieve any desired orders of accuracy of derivatives, in a chosen number of dimensions without symmetry assumptions in the underlying data. Finally, we present two example applications of extracting reduced-order models using this non-local calculus, in the form of ordinary differential equations from parabolic partial differential equations of progressively greater complexity.

Cavitation in a soft porous material.

We study the collapse and expansion of a cavitation bubble in a deformable porous medium. We develop a continuum-scale model that couples compressible fluid flow in the pore network with the elastic response of a solid skeleton. Under the assumption of spherical symmetry, our model can be reduced to an ordinary differential equationthat extends the Rayleigh-Plesset equationto bubbles in soft porous media. The extended Rayleigh-Plesset equationreveals that finite-size effects lead to the breakdown of the universal scaling relation between bubble radius and time that holds in the infinite-size limit. Our data indicate that the deformability of the porous medium slows down the collapse and expansion processes, a result with important consequences for wide-ranging phenomena, from drug delivery to spore dispersion.